==== TABLE OF CONTENTS ====
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//main.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/01-introduction.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/02-macrograph.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/03-outer-codes.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/04-outer-programs.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/05-outer-representation.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/06-outer-composition.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/07-inner-algebra.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/08-inner-binary.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/09-inner-gadget.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/10-geometry.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/11-weak-nash.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/12-extensions.tex
The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//references.bib
===========================

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//main.tex ====
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\title{The PCP-for-PPAD conjecture:\\a quasilinear reduction}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove the quasilinear-size PCP-for-PPAD conjecture of Babichenko,
Papadimitriou, and Rubinstein. There are fixed positive rational constants
$\eps$ and $\delta$ and a deterministic polynomial-time reduction that
transforms an End-of-Line instance of binary length $N$ into a generalized
circuit of total binary length $N(\log N)^{O(1)}$. From any rational
assignment of polynomial encoding length that $\eps$-satisfies all but a
$\delta$ fraction of the gates, a solution to the original End-of-Line
instance can be recovered in polynomial time, regardless of which gates
fail. Such assignments always exist, with one fixed polynomial bound on
their encoding length.
\end{abstract}
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-macrograph}
\input{sections/03-outer-codes}
\input{sections/04-outer-programs}
\input{sections/05-outer-representation}
\input{sections/06-outer-composition}
\input{sections/07-inner-algebra}
\input{sections/08-inner-binary}
\input{sections/09-inner-gadget}
\input{sections/10-geometry}
\input{sections/11-weak-nash}
\appendix
\input{sections/12-extensions}
\phantomsection
\addcontentsline{toc}{section}{References}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//main.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/01-introduction.tex ====
\section{Introduction and the exact search relation}\label{sec:introduction}

The class PPAD organizes total search problems whose existence proofs are
based on the parity of endpoints in a directed graph~\cite{Papadimitriou1994}. Its canonical problem,
End-of-Line, asks for an endpoint other than a specified source, when the
graph is described by predecessor and successor circuits. Generalized
circuits provide another useful language for this class: a solution assigns
real values in $[0,1]$ to wires and approximately satisfies local arithmetic,
comparison, and Boolean constraints. The generalized-circuit intermediary
was developed by Chen, Deng, and Teng~\cite[Section~4.2]{CDT2007Preprint};
arithmetic, comparison, and Boolean gadgets also underlie the equilibrium
reductions of Daskalakis, Goldberg, and
Papadimitriou~\cite[Section~5.1]{DGP2009}. Rubinstein established
constant-error hardness with every gate required to be
satisfied~\cite[Theorem~2]{Rubinstein2018}. These results motivate the
circuit language used here, with the precise normalization and thresholds
specified below.

Babichenko, Papadimitriou, and Rubinstein formulated the PCP-for-PPAD
conjecture as a quasilinear reduction from End-of-Line to generalized
circuits in which both the allowed numerical error and the allowed fraction
of violated gates are positive constants~\cite[Conjecture~2]{BPR2016}.
The failure fraction is the essential robustness requirement: the decoder
must succeed even if an adversary chooses which constraints to violate.
The quasilinear size requirement is a separate restriction on the reduction.
Ordinary polynomial-time PPAD-hardness, even with constant numerical error,
does not provide it. For example, hardness for every numerical error below $1/10$ is known
for the standard generalized-circuit problem with every gate required to
be satisfied~\cite[Theorem~4.1]{DFHM2024}. Subsequent uses of PCP-for-PPAD also
consider the weaker conjecture asserting ordinary PPAD-hardness with a
positive failure fraction; the distinction from the original quasilinear
formulation is explicit in~\cite[Section~2.2.2]{Fisher2026}.

Rubinstein's 2016 work combined encoded-path geometry, a locally
evolving version of End-of-Line, and a holographic proof system whose
certificates can be constructed and decoded through restricted
access~\cite[Sections~3, 5, and~6]{Rubinstein2016Settling}. It established
quasipolynomial hardness for constant-error two-player Nash under the
exponential-time hypothesis for PPAD, while leaving the PCP-for-PPAD
conjecture open and identifying composition as a remaining
question~\cite[Theorem~1.2 and Section~1.3]{Rubinstein2016Settling}.
More recently, Golowich derives sharper two-player Nash lower bounds
for vanishing approximation error under PCP for PPAD together with
the exponential-time hypothesis for PPAD~\cite[Theorem~1.1]{Golowich2026}.
These time lower bounds retain their additional exponential-time
hypothesis; the result here concerns the conjectured reduction itself.

We give a positive resolution of the quasilinear PCP-for-PPAD conjecture.
Our size bound counts the complete finite output, including connection
indices, every auxiliary gate, and the bits of all rational parameters.
The gate conventions below are the standard bounded-rational conventions
of~\cite{DFHM2024,Fisher2026}; the totality and witness-size statement is
proved explicitly. When restricted to the nearly linear predecessor and
successor circuits in the original formulation, the theorem gives the
conjectured quasilinear reduction there as well.\footnote{The original
Definition~1 of~\cite{BPR2016} allows $x=0^n$ in its first solution
disjunct $P(S(x))\ne x$. If $P(S(0^n))\ne0^n$, this is an immediately
computable solution. Otherwise the instance satisfies the additional
source promise used below. Thus this normalization requires only circuit
evaluation and a trivial branch of the reduction: output a fixed one-gate
Constant circuit and have the decoder return $0^n$.}

\subsection{The source and target problems}

We fix conventional explicit binary encodings of circuit gate lists,
connection indices, input and output lists, and rational parameters.
Integers are encoded in binary with delimiters. All size bounds concern
these complete encodings. Changing between the usual such explicit
encodings changes only constant or logarithmic factors.

\begin{definition}[End-of-Line]\label{def:eol}
An instance $I$ consists of two explicitly listed, acyclic, bounded-fanin
Boolean circuits
\[
 S,P:\{0,1\}^n\longrightarrow\{0,1\}^n,\qquad n\geq1,
\]
using AND, OR, and NOT gates, with unrestricted fan-out, and satisfying
\[
 P(0^n)=0^n,\qquad S(0^n)\ne0^n,\qquad P(S(0^n))=0^n.
\]
A solution is a string $v\ne0^n$ such that
\begin{equation}\label{eq:end-of-line-solution}
 P(S(v))\ne v\quad\text{or}\quad S(P(v))\ne v.
\end{equation}
The input length $N\geq2$ includes the complete descriptions of both
circuits and their input and output lists.
\end{definition}

\begin{definition}[Generalized circuit]\label{def:gcircuit}
A generalized circuit $G=(V,T)$ has an explicitly listed finite set of
nodes $V$ and a nonempty set of gates $T$. Each gate has one output and at
most two inputs. A node is the output of at most one gate. Directed cycles
and unrestricted fan-out are permitted. Every gate parameter $\alpha$ is
a binary-encoded rational in $[0,1]$.

For an assignment $x:V\to[0,1]$, write $x_a,x_b$ for input values and
$x_z$ for the output. The five arithmetic gate types are
$\eps$-satisfied when the corresponding condition holds:
\begin{align*}
 \mathrm{Constant}(\alpha;z)&:&\quad \abs{x_z-\alpha}&\leq\eps,\\
 \mathrm{Scale}(\alpha;a;z)&:& \abs{x_z-\alpha x_a}&\leq\eps,\\
 \mathrm{Copy}(a;z)&:& \abs{x_z-x_a}&\leq\eps,\\
 \mathrm{Add}(a,b;z)&:& \abs{x_z-\min\{x_a+x_b,1\}}&\leq\eps,\\
 \mathrm{Subtract}(a,b;z)&:& \abs{x_z-\max\{x_a-x_b,0\}}&\leq\eps.
\end{align*}
The comparison gate $\mathrm{Less}(a,b;z)$ has the two implications
\[
 x_a<x_b-\eps\ \Longrightarrow\ x_z\geq1-\eps,
 \qquad
 x_a>x_b+\eps\ \Longrightarrow\ x_z\leq\eps.
\]
The Boolean gates have the following implications:
\begin{align*}
 \mathrm{Or}(a,b;z):\qquad
 (x_a\geq1-\eps\ \text{or}\ x_b\geq1-\eps)&\Longrightarrow x_z\geq1-\eps,\\
 (x_a\leq\eps\ \text{and}\ x_b\leq\eps)&\Longrightarrow x_z\leq\eps;\\[2pt]
 \mathrm{And}(a,b;z):\qquad
 (x_a\geq1-\eps\ \text{and}\ x_b\geq1-\eps)&\Longrightarrow x_z\geq1-\eps,\\
 (x_a\leq\eps\ \text{or}\ x_b\leq\eps)&\Longrightarrow x_z\leq\eps;\\[2pt]
 \mathrm{Not}(a;z):\qquad
 x_a\leq\eps&\Longrightarrow x_z\geq1-\eps,\\
 x_a\geq1-\eps&\Longrightarrow x_z\leq\eps.
\end{align*}
For these four gate types, every applicable implication must hold. If no
antecedent holds, any output in $[0,1]$ is permitted. In particular, the
comparison thresholds are strict and the Boolean thresholds are weak.

Let $M$ be the complete binary encoding length of $G$. For fixed
$\eps,\delta>0$ and a fixed polynomial $p$, an acceptable assignment is
rational, has total binary encoding length at most $p(M)$, and
$\eps$-satisfies at least $(1-\delta)|T|$ gates. Each gate has equal weight;
at most $\lfloor\delta|T|\rfloor$ gates may fail. Failed gates may violate
their relations arbitrarily, but all their nodes still have one globally
consistent value in $[0,1]$.
\end{definition}

The Boolean predicates make the error parameter nonmonotone: increasing
$\eps$ can activate an antecedent that was previously vacuous. This issue
is analyzed by Schuldenzucker and Seuken~\cite[Proposition~1]{SchuldenzuckerSeuken2019}.
Our gate arguments use one actual tolerance throughout. Later upper bounds
on that tolerance ensure explicit margins; they do not reclassify a
satisfied Boolean gate at a larger tolerance.

\subsection{Main theorem}

\begin{samepage}
\begin{theorem}[Quasilinear PCP for PPAD]\label{thm:main}
There are fixed rational constants
\[
 0<\eps<\tfrac1{10},\qquad 0<\delta<1,
\]
a fixed positive integer $A$, a fixed nonnegative integer $b$, a fixed
polynomial $p$ with nonnegative integer coefficients, and deterministic
algorithms $R,D$ with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every generalized circuit in Definition~\ref{def:gcircuit} has an
acceptable rational assignment of encoding length at most $p(M)$.
\item For every End-of-Line instance $I$ of length $N$, the output $R(I)$
is a generalized circuit whose complete encoding length satisfies
\[
 M\leq A N\bigl\lceil\log_2(N+2)\bigr\rceil^b.
\]
\item The construction $R$ runs in time polynomial in $N$, with one fixed
polynomial bound.
\item For every acceptable assignment $x$ of $R(I)$, the algorithm $D(I,x)$
returns a solution of $I$ in time polynomial in $N$ plus the encoding
length of $x$, with one fixed polynomial bound.
\end{enumerate}
All constants, polynomials, and algorithms are chosen once. They use no
advice depending on the input length and no randomized construction.
\end{theorem}
\end{samepage}

The conclusion about $D$ is universal over the locations and values of
the failed gates. In fact, the soundness argument applies to arbitrary
real assignments meeting the gate-failure budget; rationality is needed
for the finite search relation and its computational decoder. The totality
proof supplies an assignment satisfying every gate on a fixed rational
mesh. Neither tolerance tends to zero with $N$.

\subsection{Proof strategy: encoded paths and robust local computation}

We begin with the geometric target of the reduction. Replace the
End-of-Line graph by a locally evolving directed graph with a known
source, no isolated vertices, and the same endpoint-decoding property.
Represent each vertex $w$ by a separated binary word $E_w\in\{0,1\}^m$.
In the box $[-1,2]^{4m}$, viewed as four blocks of $m$ coordinates,
each edge $u\to v$ traces
a polygon that first changes a copy of $E_u$ to $E_v$, changes a repeated
control bit, updates the other copy, and resets the control bit. Rounded
corners join these polygons into paths and cycles. An additional segment
enters the known source, and a separate field prescription handles its
far end. Every other path endpoint encodes an End-of-Line answer.

Section~\ref{sec:geometry} constructs a bounded displacement field $v$
on this box. Close to a path it uses the directed tangent and the inward
normal; farther away it passes to a fixed direction. For a sufficiently
small fixed step $\beta>0$, the clipped move
$\clip(X+\beta v(X))-X$ has norm at least a fixed positive multiple of
$\beta$ outside a fixed neighborhood of the desired endpoints. Here
clipping keeps each coordinate in the box, and the norm is normalized
by the word length. Thus an accurate enough approximate fixed point
identifies an endpoint. The proof must also provide local numerical
access to the field: it cannot inspect the entire encoded graph when
computing one coordinate of $v$.

This use of encoded paths and restricted access has a close predecessor
in Rubinstein's holographic proof system~\cite[Sections~3, 5, and~6]{Rubinstein2016Settling}.
The present reduction needs two further properties simultaneously:
quasilinear total storage and robustness to an arbitrary fixed fraction
of failed gates. The two scales of the construction address these
requirements as follows.

\paragraph{A quasilinear outer word.}
Sections~\ref{sec:macrograph}--\ref{sec:outer-representation} construct
canonical representations
\[
 R_w=(s_k)_{k\in\Gamma},\qquad
 s_k\in\{0,1\}^L,\qquad
 |\Gamma|=N(\log N)^{O(1)},\qquad L=(\log N)^{O(1)}.
\]
A bounded number of symbol positions suffices to test local validity,
propose a neighboring symbol, or perform one correction step. Each such
operation has Boolean complexity polynomial in $L$, with an exponent
fixed before the later correction accuracy. Distinct valid words have
constant symbol distance, and a word within a fixed symbol-error radius
can be decoded globally in polynomial time.

Begin with the sparse transcript graph of Section~\ref{sec:macrograph},
whose vertices record partial computations of the predecessor and successor
circuits and whose edges change one tape position and its occupation bit.
Transcript validity is expressed by bounded-degree polynomial equations in
the tape entries. Sampling an individual equation could miss one false
equation. Sections~\ref{sec:outer-codes} and~\ref{sec:outer-programs}
instead encode polynomial remainders recording the equation residuals,
and retain certificates for the intermediate products and linear
computations that produce them. On exact codewords, a nonzero remainder
differs from zero at a constant fraction of positions. The code's commuting
row and column constraints also permit whole-row or whole-column comparisons
that give a constant gap for an incorrect certificate identity. All stored
components are tested together on one fixed list of neighboring positions,
so more certificates enlarge a symbol without diluting the gap.

One tape edit can change all these encoded arrays. Before making an allowed
edit, Section~\ref{sec:outer-representation} prepares each retained array's
finite difference: its value on the edited tape minus its value on the old
tape. Certificates introduced by this preparation need their own increments
as well. Each new increment has lower assigned polynomial degree in the
logical tape entries, so this recursion has bounded depth. Once preparation
is complete, adding the increments updates the whole retained tuple to its
canonical value on the new tape; the preparation can then be reversed.

The remaining storage issue is the long linear computation needed to prepare
an increment. It is carried out one small group of Fourier frequencies at a
time: each such orbit is preserved by the encoder and has only
polylogarithmically many frequencies. A clock records progress and scratch
space is reused. Thus the preparation's elapsed time lengthens the
represented path rather than the data stored at a vertex. The resulting
words are the locally evolving $R_w$ above.

\paragraph{One fixed binary port.}
A polynomial-size robust construction applied directly to an $N$-bit
vertex would not give the required size. Instead we apply it only to
the $L$-bit outer symbols. The inner construction supplies one injective
encoding
\[
 \mathcal D:\{0,1\}^L\longrightarrow\{0,1\}^{S_0},
 \qquad S_0=\operatorname{poly}(L),
 \qquad E_w=(\mathcal D(s_k))_{k\in\Gamma}.
\]
Its encoding and positive correction radius are chosen before the
number of symbols inspected by a local function. From that fixed radius,
it supports every prescribed fixed average accuracy for requests with
bounded address density. More outer correction rounds may enlarge the number
of symbols inspected, and hence the number of input ports in the resulting
local computation. More inner correction rounds improve its requested
bit accuracy. These choices may enlarge constant factors, but they do
not change $\mathcal D$ or its radius. All polynomial factors in $L$
remain fixed powers of $\log N$.

Sections~\ref{sec:inner-algebra}--\ref{sec:inner-gadget} construct this
port and its robust decision gadget independently of the outer
reduction. For a fixed tuple $s=(s_1,\ldots,s_k)$ of $L$-bit inputs
to a Boolean function $f$, the inner graph is one computation path with
blank work at its source and output $f(s)$ at its sink. Polynomial charts encode its tapes and
certificates. Their dependencies are kept separate for each read-only
input, including throughout the finite-difference closure. Consequently
the input-dependent parts of the source and its first successor have
the same formulas in every later invocation. A one-input template at
these two labels defines $\mathcal D(s_i)$ for one input $s_i$; its bits then supply the
source data for an actual multi-input computation.

The binary representation in Section~\ref{sec:inner-binary} adds
point tables, line tables, and tables for the required products.
Correction keeps both addresses in each product request and preserves
the required bound on their joint distribution. Its error bound is averaged over requested coordinates;
arbitrarily accurate correction at every individual coordinate is
neither asserted nor needed. At the inner scale, the geometric
construction forces a feedback state near the unique sink, where a
positive fraction of encoded coordinates reveal $f(s)$. These become the
many designated outputs of the decision gadget.

\paragraph{From local decisions to an endpoint under gate failures.}
Section~\ref{sec:outer-composition} uses the outer tester and correction
routines to supply the field's local information. Two rounded blocks
serve as possible vertex encodings. On good validation decisions, each
accepted seed has a unique reference vertex; corrected calls supply its incident
flags and the requested bits of its own and neighboring encodings.
The geometric estimator reserves a bounded list of possible pieces and
estimates their statistics before selecting a numerical winner.
This order makes average bit accuracy sufficient even when the selected
piece depends on the output coordinate.

The generalized circuit has many physical replicas of each geometric
coordinate. Write $X_{ir}$ for replica $r$ of component $i$, and
$\bar X_i$ for their mean. A fixed-degree expander average $A$ couples
the replicas. Each output is produced by a circuit that realizes the
approximate relation
\[
 X_{ir}\approx\clip\bigl((AX)_{ir}
                       +\beta\widehat v_{ir}(\bar X)\bigr),
\]
where $\widehat v$ is the bounded local field estimator. First, boundedness
of the update and the expander gap force the replicas close to their
means, without assuming that any decision is correct. This synchronization
then bounds the number of reads far from their means. Randomized
threshold offsets control ambiguous comparisons, and the gate-failure
budget controls the circuits containing a failed gate.
The randomness describes the sampling analysis: the printed circuit assigns
replicas a deterministically enumerated list of fixed tapes of size polynomial
in $L$.

These charges apply to the complete wired computation. All hypothetical
answer branches are allocated before routing, and each physical read
occurrence is routed separately with bounded load. Expensive Boolean
decision gadgets are shared by groups of output replicas. Their many
designated outputs ensure that failures concentrated on extraction
wires still spoil only a controlled fraction of the group. On reliable
outputs the update approximates the true field; on unreliable outputs
whose gates are satisfied, its bounded size still gives an error
proportional to $\beta$, apart from the numerical gate tolerance. An estimate that keeps clipping inside the
replica contraction argument then makes the mean an approximate fixed
point. The displacement lower bound forces $\bar X$ near an endpoint.
Rounding its first block and applying the global binary and outer
decoders yields the required End-of-Line answer.

The same synchronization, error charging, and clipped-mean argument is
used inside the decision gadget. At that scale every output has its own
bounded estimator circuit; the additional sharing of decision gadgets
occurs only in the outer composition. In both uses, a fixed hypothetical
answer branch is analyzed under the original random-tape law, rather
than by conditioning on that branch being executed.

\paragraph{Organization and imported inputs.}
Section~\ref{sec:outer-composition} assembles the main theorem from
the outer representation, the port interface stated there, and the
independent geometric interface. It proves universal decoding, the
complete binary size bound, and rational-witness totality.
Sections~\ref{sec:inner-algebra}--\ref{sec:inner-gadget} prove the port
interface, and Section~\ref{sec:geometry} proves both the displacement
bound and finite numerical access. Section~\ref{sec:weak-nash} gives
the consequences described below. Appendix~\ref{sec:extensions}
collects the separate full-macrograph and unrestricted-product variants;
these are not prerequisites for the fixed-port composition.

The outer code uses ternary Goldbach, fixed-generator prime-quotient
expansion, and an effective least-prime bound
\cite{Helfgott2013,BG2008,Xylouris2011}. Its simultaneous systematic
encoders, matrix twists, and joint correction guarantees are proved
here; a general locally testable code theorem such as~\cite{DEL+2022}
does not by itself supply this combination. The inner binary construction
uses the corrected scalar line test and two group-theoretic inputs
\cite{FS1995,EJZ2010,Shalom1999}; its vector replacement, affine
expansion, and average correction are proved here. The all-modulus
Margulis--Gabber--Galil bound supplies the local and replica walks
\cite{MGGAFP}. The inner fields and groups are introduced afresh and
are distinct from those of the outer construction.

\subsection{Consequences for games and markets}

Section~\ref{sec:weak-nash} applies the theorem to the conditional
reductions for weak approximate Nash and relative bimatrix Nash
\cite{BPR2016}, course allocation~\cite{BPR2016}, markets with
approximately optimal bundles~\cite{Fisher2026}, stationary equilibria
in discounted games~\cite{DGZ2023,PZO2025,FKPP2026}, and approximate
Hylland--Zeckhauser allocation~\cite{HZ2026}. For each target we state
the precise approximation and encoding conventions before its
consequence. The cited reductions provide the transfers to these
problems. We supply the needed gate-basis and fan-out conversions, track
their fixed error and failure losses, and verify the rational witness
and decoder bounds. These consequences use fixed-gap circuit hardness;
the quasilinear complete-output bound is the additional assertion of
Theorem~\ref{thm:main}.
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/01-introduction.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/02-macrograph.tex ====
\section{A sparse macrograph for End-of-Line}\label{sec:macrograph}

We first replace the circuit transitions by a graph in which one edge
changes at most one tape bit and its occupation bit.  The graph is
specified by short labels and local transition rules; its vertices are
not enumerated.  The use of a clock to expose sparse circuit updates
has a close precedent in Rubinstein's LocalEndOfALine
construction~\cite[Sections~5--5.1]{Rubinstein2016Settling}, where a
counter schedules activation and deactivation of individual wires.
We give the transcript, reciprocal proposals, and bounded-incidence
validity description needed here explicitly.

Throughout the outer construction, let $n_0$ be the
number of input bits of the End-of-Line instance and let $N$ be its
complete binary encoding length.  We use the promises
\begin{equation}\label{macro:promises}
 P(0^{n_0})=0^{n_0},\qquad S(0^{n_0})\ne0^{n_0},\qquad
 P(S(0^{n_0}))=0^{n_0}.
\end{equation}
Thus a nonzero string $y$ is an answer if
\begin{equation}\label{macro:answer}
 P(S(y))\ne y\quad\hbox{or}\quad S(P(y))\ne y.
\end{equation}

\subsection{Sparse transcripts and their two orders}\label{macro:transcripts}

Normalize both circuits to gates of fan-in at most two.  In particular,
replace any larger AND or OR by a binary tree.  Give every input and
every gate a distinct tape position.  For each listed output, introduce
a separate terminal copy gate, even if several listed outputs originally
name the same wire.  Replace the outgoing uses of a wire by a binary
tree of copy gates, including its uses as feeds to terminal copies.
A wire with $d$ uses needs $O(d)$ such gates.  The total number of
positions is therefore bounded by a constant times the number of gates,
connections, and input and output list entries in the original instance,
and hence is $O(N)$.

These changes preserve $S$ and $P$.  They also make every tape position
an operand of only a bounded number of internal gate rules.  Choose a
topological order of the nonterminal gates of each circuit, followed by
all its terminal output copies.  Such an order exists because those
copies have no internal consumers: any branching needed for other uses
was placed before them.  Copy gates here are auxiliary transcript rules
and need not belong to the source circuit's original basis.

Pad the tape with dummy positions to a power of two $h=2^\ell$.
An arbitrary fixed lower cutoff on $h$ may be imposed, and still
$h=O(N)$.  Write $[h]=\{0,\ldots,h-1\}$.  On these same positions define
the two permutations
\begin{align*}
 \pi_{\mathrm f}:\;&S\text{ inputs},\ S\text{ gates},\
 P\text{ inputs},\ P\text{ gates},\ \text{dummies},\\
 \pi_{\mathrm r}:\;&P\text{ inputs},\ P\text{ gates},\
 S\text{ inputs},\ S\text{ gates},\ \text{dummies},
\end{align*}
using the chosen gate order inside each block.  For a content array
$D\in\{0,1\}^{[h]}$, let $\operatorname{pref}_{\pi,k}(D)$ agree with
$D$ on $\pi(0),\ldots,\pi(k-1)$ and be zero elsewhere.  Its occupation
array is the indicator of these $k$ positions.  Empty and full tapes
have occupation arrays $\mathbf 0$ and $\mathbf 1$, respectively.
The content and occupation arrays are separate: an occupied position
may contain zero.

For a full content array $D$, the notation
$\operatorname{in}_S(D)$, $\operatorname{out}_S(D)$,
$\operatorname{in}_P(D)$, and $\operatorname{out}_P(D)$ denotes the
vectors at the designated input or terminal output positions.  A
\emph{valid full transcript} $Q_x$ computes $S$ on input $x$, copies
its output to the inputs of $P$, computes $P$, and satisfies
\begin{equation}\label{macro:full-transcript}
 \operatorname{in}_S(Q_x)=x,\qquad
 \operatorname{in}_P(Q_x)=\operatorname{out}_S(Q_x),\qquad
 \operatorname{out}_P(Q_x)=x.
\end{equation}
Its dummy positions are zero.  Consequently $Q_x$ exists precisely
when $P(S(x))=x$, and is unique when it exists.

For every $y\in\{0,1\}^{n_0}$, define two complete computation templates,
without imposing any terminal comparisons.  The template $U_y$ first
computes $S(y)$ and then $P(S(y))$; the template $V_y$ first computes
$P(y)$ and then $S(P(y))$.  Both occupy the same tape coordinates and
have zero dummies.  In $V_y$, the $P$ inputs are $y$ and the $S$ inputs
are $P(y)$.  Thus $U_y$ is a valid full transcript exactly when
$P(S(y))=y$, in which case it is $Q_y$; and $V_y$ is a valid full
transcript exactly when $S(P(y))=y$, in which case it is $Q_{P(y)}$.

\begin{definition}[Computed prefixes]\label{macro:prefixes}
A \emph{forward prefix of length $k$ with parameter $y$} has the first
$k$ positions in $\pi_{\mathrm f}$ occupied.  At occupied positions it
obeys these rules: $S$ inputs copy the corresponding bits of $y$;
$P$ inputs copy the corresponding designated $S$ outputs; every gate
obeys its Boolean or copy rule; and dummies are zero.  In addition,
each occupied designated $P$ output equals the corresponding bit of
$y$.  Unoccupied contents are zero.

A \emph{reconstruction prefix of length $k$ from $y$} uses
$\pi_{\mathrm r}$.  Its occupied $P$ inputs copy $y$, its occupied
$S$ inputs copy the designated $P$ outputs, and its occupied gates
and dummies obey the same rules as above.  Each occupied designated
$S$ output must additionally equal the corresponding bit of $y$.
Unoccupied contents are again zero.
\end{definition}

Every operand in a prefix rule is either an earlier occupied position
or a specified bit of $y$.  In particular, a designated comparison is
an extra test on the newly computed terminal copy, not a substitute
for its computation rule.

\begin{lemma}[Unique prefixes and their first obstruction]
\label{macro:prefix-uniqueness}
For a fixed $y$, the valid forward prefixes exist uniquely for an
initial interval of lengths $0\le k\le k_{\mathrm f}(y)$ and equal
$\operatorname{pref}_{\pi_{\mathrm f},k}(U_y)$.  The valid
reconstruction prefixes likewise exist uniquely for
$0\le k\le k_{\mathrm r}(y)$ and equal
$\operatorname{pref}_{\pi_{\mathrm r},k}(V_y)$.  Moreover,
\begin{align}
 k_{\mathrm f}(y)=h&\quad\Longleftrightarrow\quad P(S(y))=y,
 \label{macro:forward-obstruction}\\
 k_{\mathrm r}(y)=h&\quad\Longleftrightarrow\quad S(P(y))=y.
 \label{macro:reverse-obstruction}
\end{align}
If a maximal length is less than $h$, the next position is a designated
output at which the additional comparison fails.
\end{lemma}

\begin{proof}
Induct along the relevant order.  Input copies, gates, and dummies
uniquely determine the next content bit from data already determined.
These are exactly the bits of $U_y$ or $V_y$.  The only possible
obstruction is an additional terminal comparison.  Shortening a valid
prefix removes requirements without changing retained bits, so the
valid lengths form an initial interval.  At length $h$, all terminal
comparisons have been imposed, giving the two equivalences.
\end{proof}

\subsection{The cyclic schedule and reciprocal proposals}
\label{macro:schedule}

A logical tape vector is
\[
 b=(A,E^A,B,E^B)\in\{0,1\}^{4h},
\]
where $A,B$ are contents and $E^A,E^B$ are occupations.  The schedule
uses four phases.  To see their effect when all computations succeed,
suppose $Q_x$ and $Q_y$ exist and $y=S(x)$.  Starting with $A=Q_x$
and $B$ empty, it fills $B$ with $Q_y$, erases $A$ by reversing
reconstruction from $y$, copies $B$ into $A$, and erases $B$.
The resulting tapes are $A=Q_y$ and $B$ empty.  The identity
$P(y)=x$, which follows from validity of $Q_x$, is what makes the
reconstruction recover the old tape.

For intermediate prefixes, including those at a failed terminal
comparison, validity is defined as follows.  The stage $s$ lies in
$\mathbb Z/(4h)\mathbb Z$.  Write $\mathcal V_s$ for the following
set of valid logical vectors; the endpoint descriptions in adjacent
cases are identified.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item $0\le s\le h$: $A=Q_x$ is full for some $x$, and $B$ is a
 forward prefix of length $s$ with parameter
 $y=\operatorname{out}_S(A)=S(x)$.
 \item $h\le s\le2h$: $B=Q_y$ is full for some $y$, and $A$ is a
 reconstruction prefix of length $2h-s$ from
 $y=\operatorname{in}_S(B)$.
 \item $2h\le s\le3h$: $B$ is a valid full transcript and
 $A=\operatorname{pref}_{\pi_{\mathrm f},s-2h}(B)$.
 This copied prefix has no additional computation requirements.
 \item $3h\le s\le4h$: $A$ is a valid full transcript and
 $B=\operatorname{pref}_{\pi_{\mathrm f},4h-s}(A)$.
\end{enumerate}
Occupations in every case are the stated full or prefix indicators.
Stage $4h$ is stage $0$, at which $A$ is full and $B$ is empty.

\begin{lemma}[Boundary agreement]\label{macro:boundaries}
The two validity descriptions agree at each of $h,2h,3h$, and $4h=0$.
\end{lemma}

\begin{proof}
At stage $h$, the first description gives
\[
 A=Q_x,\qquad B=Q_y,\qquad y=S(x),\qquad P(y)=x.
\]
The reconstruction template from $y$ computes $P(y)=x$ and then
$S(x)=y$, so it agrees with $A$ at every position and satisfies all
its comparisons.  This gives the second description.  Conversely,
the second description at $h$ gives $B=Q_y$ and a full reconstruction
$A=V_y$ with $S(P(y))=y$.  Put $x=P(y)$.  Then
$P(S(x))=P(y)=x$, so $A=Q_x$ and $\operatorname{out}_S(A)=y$.
Since $B=Q_y$, it is the full forward prefix required in the first
description.  These identities include all copy gates and dummies.
At $2h$, both descriptions have $A$ empty and $B$ a valid full
transcript.  At $3h$, both have $A=B$ a valid full transcript.
At $4h=0$, both have $A$ full and $B$ empty.
\end{proof}

For each \emph{start frame} $c\in\{0,\ldots,4h-1\}$, define a
forward proposal $F_c$ from stage $c$ to stage $c+1$, and a reverse
proposal $R_c$ from stage $c+1$ to stage $c$, with stages reduced
modulo $4h$.  Thus at current stage $s$, the successor uses $F_s$
and the predecessor uses $R_{s-1}$, where the latter subscript is
reduced modulo $4h$.  The following table specifies their content changes.
Every fill or copy sets the corresponding occupation to one, and
every erasure sets it to zero.  All other bits remain unchanged.
\begin{center}
\begin{tabular}{@{}llll@{}}
\toprule
Start frame & Position $j$ & $F_c$ & $R_c$ \\
\midrule
$0\le c<h$ & $\pi_{\mathrm f}(c)$ & fill $B_j$ forward & erase $B_j$ \\
$h\le c<2h$ & $\pi_{\mathrm r}(2h-c-1)$ & erase $A_j$ & fill $A_j$ by reconstruction \\
$2h\le c<3h$ & $\pi_{\mathrm f}(c-2h)$ & copy $B_j$ to $A_j$ & erase $A_j$ \\
$3h\le c<4h$ & $\pi_{\mathrm f}(4h-c-1)$ & erase $B_j$ & copy $A_j$ to $B_j$ \\
\bottomrule
\end{tabular}
\end{center}
A computed fill always writes its Boolean rule value, even if the
additional terminal comparison fails.  Thus the proposal itself is
defined separately from the test of whether its result is valid.

The \emph{macrograph} has vertex set
\[
 \mathcal V=\{(s,b):s\in\mathbb Z/(4h)\mathbb Z, b\in\mathcal V_s\}.
\]
It has an edge $(c,b)\longrightarrow(c+1,b')$ precisely when
$b'=F_c(b)$ and both endpoints are valid.

\begin{lemma}[Reciprocity and local existence tests]
\label{macro:local-edges}
Whenever $b\in\mathcal V_c$ and $b'\in\mathcal V_{c+1}$,
\begin{equation}\label{macro:reciprocity}
 b'=F_c(b)\quad\Longleftrightarrow\quad b=R_c(b').
\end{equation}
For a valid input to either proposal, validity of its output is
determined by a bounded number of bits at public, frame-dependent
addresses.  The forward proposal can fail only when $0\le c<h$ and
the new forward terminal comparison fails.  The reverse proposal can
fail only when $h\le c<2h$ and the new reconstruction terminal
comparison fails.  The graph has indegree and outdegree at most one,
and no self-edges.
\end{lemma}

\begin{proof}
In the first range, erasing the newly filled position recovers its
previous blank content and occupation.  Conversely, a valid longer
forward prefix has the uniquely prescribed value at its last
position, so filling its shortened prefix recovers it.  In the second
range, erasure shortens a reconstruction prefix.  Its erased value
is uniquely determined by earlier retained positions and the
unchanged full tape $B$, so reverse reconstruction recovers it.
The converse follows from the same uniqueness.  In the last two
ranges, the unchanged full tape determines the copied value,
and erasure recovers the blank position.  Boundary agreement applies
to each use of these statements at a boundary.  In particular,
at $c=4h-1$, the reverse proposal from stage zero copies
$A_{\pi_{\mathrm f}(0)}$ into the first forward position of the empty
$B$ tape, exactly reversing the wraparound erasure.

Shortening a valid computed prefix is always valid, as are extending
or shortening a copied prefix.  When extending a computed prefix,
all earlier conditions remain true.  Its next value is obtained
from at most two earlier operands or a specified full-tape input
bit, so the only new condition that can fail is the stated terminal
comparison.  That comparison needs one additional specified bit.
Thus the relevant operands, old bits, and comparison bits suffice
to decide both proposals and the corresponding existence flags.
Equation~\eqref{macro:reciprocity} gives at most one predecessor,
and the definition gives at most one successor.  Every edge changes
the stage by one modulo $4h$, and $4h\ge4$, so no edge is a self-edge.
\end{proof}

\subsection{Endpoints, sparse edits, and uniform size}
\label{macro:interface}

The following statement is the interface used by the evolving
representation.

\begin{lemma}[Macrograph interface]\label{lem:macrograph}
From every promised End-of-Line instance of explicit length $N$, one
can deterministically construct, in time polynomial in $N$, the
above tape specification with $h=2^\ell=O(N)$, the stage conditions
$\mathcal V_c$, and the following data and algorithms.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item The macrograph has indegree and outdegree at most one, no
 self-edges, and no isolated vertices.  It has a distinguished source
 $v_*$ whose label and actual successor are computable in polynomial
 time.  Every endpoint other than $v_*$ supplies a nonzero
 End-of-Line answer by reading a designated vector on a full tape.
 \item Forward and reverse proposals at the same start frame are
 reciprocal on valid pairs.  Their edits and existence flags use
 only a fixed number of logical bits at public addresses, with a
 fixed finite set of Boolean decision rules.
 \item There is a fixed finite alphabet
 $\Sigma=\{-1,0,1\}^4$ and public edit vectors $h_\sigma(c)$ such
 that each chosen proposal has the form
 $b\mapsto b+h_\sigma(c)$.  One public address is allotted to each
 logical array.  On actual edges only the active tape's content
 position and its occupation position can change.  The same fixed
 edit formulas are defined over any field, whether or not their
 arguments represent valid Boolean tapes.
 \item For each fixed phase, the local tape requirements have
 bounded arity and bounded operand multiplicity, uniformly in $N$.
 The full tape specification and all frame-dependent rule and
 address tables have $O(h)$ records of $O(\log(N+2))$ bits each.
 A vertex label has $4h+\lceil\log_2(4h)\rceil=O(N)$ bits;
 validity and endpoint decoding take polynomial time.  These
 bounds concern descriptions and individual labels and do not
 require enumeration of the macrograph.
\end{enumerate}
\end{lemma}

\begin{proof}
We first identify all endpoints.  A missing successor can occur
only in the first range of the schedule.  Write its full tape as
$A=Q_x$ and put $y=\operatorname{out}_S(A)=S(x)$.  The current
$B$ tape is its longest valid forward prefix, so
Lemma~\ref{macro:prefix-uniqueness} gives
\begin{equation}\label{macro:sink-identity}
 P(S(y))\ne y,\qquad P(y)=x.
\end{equation}
If $y=0^{n_0}$, then $x=P(0^{n_0})=0^{n_0}$ and
$y=S(0^{n_0})\ne0^{n_0}$, a contradiction.  Thus this designated
vector is a nonzero End-of-Line answer.  Such a vertex has a
predecessor: shorten its forward prefix when its length is positive;
at stage zero use the reverse wraparound copy from the preceding
frame.

A missing predecessor can occur only at a stage $s=c+1$ with
$h\le c<2h$.  Its full tape is $B=Q_y$, where
$y=\operatorname{in}_S(B)$, and its reconstruction prefix has its
maximal length $k_{\mathrm r}(y)<h$.  Hence
\begin{equation}\label{macro:source-identity}
 s=2h-k_{\mathrm r}(y),\qquad S(P(y))\ne y.
\end{equation}
For $y\ne0^{n_0}$ this is again a permitted answer.  The vertex
has a successor obtained by shortening the reconstruction prefix;
if the prefix is empty, the next copying step supplies a successor.
Every proposal in the two copying ranges succeeds in the relevant
direction, except that an incoming reconstruction proposal at stage
$2h$ is covered by the case just considered.  Together with
Lemma~\ref{macro:local-edges}, this exhausts the possible missing
neighbors and proves that no vertex is isolated.

It remains to specify the zero-labelled exception.  The last promise
in \eqref{macro:promises} supplies the unique full transcript
$Q_{0^{n_0}}$, whereas the first two promises give
\[
 S(P(0^{n_0}))=S(0^{n_0})\ne0^{n_0}.
\]
Let $k_*=k_{\mathrm r}(0^{n_0})<h$.  Then $v_*$ is exactly the
vertex at stage $2h-k_*$ whose $B$ tape is $Q_{0^{n_0}}$ and whose
$A$ tape is
$\operatorname{pref}_{\pi_{\mathrm r},k_*}(V_{0^{n_0}})$,
with their prescribed occupations.  Prefix uniqueness proves that
this is the only missing-predecessor vertex with designated vector
zero.  It has no predecessor and has the successor just described.
Evaluate $U_{0^{n_0}}$ and $V_{0^{n_0}}$, locate the first failed
reconstruction terminal comparison, and take the preceding prefix.
This computes $v_*$ and its successor in polynomial time.  This
proves the endpoint claims, including the existence of the public
source.  Other directed paths and directed cycles may remain in the
graph; nothing here requires one connected component.

We next make the local edit interface explicit.  At start frame $c$,
let $j(c)$ be the position in the transition table.  Allot this
position to the content and occupation arrays of the active tape,
and allot the fixed default position $0$ to the two inactive arrays.
Denote these four addresses by $j_D(c)$, for
$D\in\{A,E^A,B,E^B\}$.  If $\mathbf e_{D,j}$ is the unit vector
at position $j$ of array $D$, define, over a field $K$,
\begin{equation}\label{macro:finite-edits}
 h_\sigma(c)=\sum_{D\in\{A,E^A,B,E^B\}}
       \sigma_D\mathbf e_{D,j_D(c)}\in K^{4h},
 \qquad \sigma\in\{-1,0,1\}^4,
\end{equation}
where the integers $-1,0,1$ are mapped into $K$.  For a proposed
Boolean write, choose its content coefficient as the desired bit
minus the old bit and its occupation coefficient as the desired
occupation minus the old occupation.  Set both inactive coefficients
to zero.  Each coefficient lies in $\{-1,0,1\}$, and the resulting
additive edit is precisely the proposal, in either direction.
The vector in \eqref{macro:finite-edits} is public once $c$ and
$\sigma$ are fixed; addition by it is defined on all of $K^{4h}$.
The assertion about valid Boolean proposals does not require a
validity interpretation for other field-valued inputs.

The necessary public data at a frame are its phase, active tape,
write position, rule type, at most two operand addresses, the old
content and occupation addresses, and, when applicable, the extra
comparison address.  A forward fill uses its forward rule; a reverse
fill uses its reconstruction rule.  Lemma~\ref{macro:local-edges}
shows that the existence flag is the new comparison result in the
only direction where failure is possible, and is one otherwise.
These are fixed Boolean decisions once the finitely many addressed
bits and public rule data are supplied.  Define the decisions on
arbitrary Boolean reads by the same formulas, using a fixed default
rule and address for unused roles.  Their agreement with graph
neighbors is asserted on valid input vertices.  No global validity
check or full circuit evaluation is hidden in an existence flag.

For completeness, the bounded-multiplicity assertion includes more
than internal gate edges.  Each internal value has boundedly many
uses after copy-tree normalization.  Each designated output is used
once by the other circuit's input-copy rule and participates in only
a bounded number of terminal or external-vector comparisons.  Each
external-vector bit is used once as an input bit and once for its
corresponding terminal comparison.  Copied-prefix equations use each
full-tape position once.  Occupation indicators condition only the
rules at their positions; prefix monotonicity, if expressed by
adjacent-position tests, reads a position at most twice in each
order.  Thus all local content, bitness, occupation, gate, copy,
comparison, and dummy conditions use bounded-arity records, and each
logical position appears as an operand only boundedly often across
the fixed list of phases.  The requirement that the occupied prefix
have its specified length is separate: one may either inspect the
prescribed occupation pattern directly, or use prefix monotonicity
together with the occupation sum.  The latter is a single global
count condition for each tape and will be represented separately in
the algebraic construction.

All normalization and ordering operations are deterministic.  Their
records and the two permutations occupy $O(h\log(N+2))$ bits.
Tabulate the transition data for the $4h$ frames in the same space
bound, including both directions and defaults.  Scanning the arrays
and these records tests membership in $\mathcal V_s$ in polynomial
time.  The designated endpoint vectors and the circuit inequalities
\eqref{macro:sink-identity} and \eqref{macro:source-identity} give a
polynomial-time endpoint decoder.  At every fixed stage a valid
vertex is determined by the input of its full transcript, since
its other tape is a unique computed or copied prefix.  Consequently
the graph has at most $4h\,2^{n_0}$ vertices, even though an individual
label has only $O(N)$ bits.  The construction outputs the normalized
records, public tables, and algorithms, never this potentially
exponential vertex list.  All constants in these bounds depend only
on the fixed Boolean basis and padding cutoff.  This proves the
remaining uniformity and size assertions.
\end{proof}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/02-macrograph.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/03-outer-codes.tex ====
\section{Outer index groups and systematic square codes}\label{sec:outer-codes}

The outer code has two roles.  Its systematic coordinates carry the data
on which the representation computes, and its redundant coordinates make
local consistency imply proximity in the metric that counts an entire
stored symbol as one position.  We construct the code, including its
matrix twists and simultaneous systematic encoders.  All vector-valued
statements below use joint support: a coordinate is erroneous when any
component is erroneous.  Their constants do not depend on the number of
components or stored slots.  The notation in this section is local to the
outer construction; in particular, the group constructed here is not the
affine group used later for the inner construction.

The index group must provide commuting left and right moves, constant
expansion, and only polylogarithmically many frequencies in each relevant
orbit. The first two properties support local consistency; the orbit
bound makes the local filling in Lemma~\ref{lem:outer-fill} possible.
The code and correction guarantees are summarized in
Theorem~\ref{thm:outer-codes} and Lemma~\ref{lem:outer-uncertainty}.

\subsection{A group of quasilinear order}

Let $h=2^\ell=O(N)$ be the padded tape length supplied by
Lemma~\ref{lem:macrograph}.  Fix a sufficiently large absolute integer
$C$, and choose an even $r$ with
\begin{equation}\label{oc:outer-size}
 \ell+C\le r\le\ell+C+2,
 \qquad M_o=2^r,\qquad G_0=\F_2^r.
\end{equation}
We increase $C$ whenever a fixed lower cutoff is needed.  Thus $4h\le M_o$,
$M_o=O(N)$, and $r=O(\log(N+2))$.  The bits of $x\in G_0$ are ordered
from least to most significant; $\underline{x}\in[0,M_o)$ denotes the
integer they represent.

Helfgott's ternary Goldbach theorem says that every odd integer greater
than five is a sum of three primes \cite[Main Theorem]{Helfgott2013}.
For our even $r$ we can therefore find primes
$p_1,p_2,p_3$ with $r-3=p_1+p_2+p_3$, by exhaustive search in $[r]$.
They need not be distinct or comparable in size.  Partition the $r$ bit
positions into blocks of lengths $p_a+1$, and let
\begin{equation}\label{oc:group}
 H=\prod_{a=1}^3\operatorname{PGL}_2(\F_{p_a}),
 \qquad \Gamma=G_0\rtimes H.
\end{equation}
Each factor acts on its projective line, hence permutes its block of bit
positions.  Since
$|\operatorname{PGL}_2(\F_q)|=q(q^2-1)$, we have
\begin{equation}\label{oc:group-order}
 |H|\le r^9,\qquad |\Gamma|=M_o|H|\le M_or^9.
\end{equation}
A generator of $\F_{p_a^2}^{\times}$, acting on its one-dimensional
$\F_{p_a}$-subspaces, induces a cycle $R_a$ of length $p_a+1$:
the scalar subgroup has order $p_a-1$ and its quotient acts freely and
transitively on these subspaces.  Label each block in this cycle order,
and denote its first unit vector by $e_a$.  The three finite fields,
permutation groups, cycles, and their tables have size polynomial in $r$
and can be found by exhaustive finite-field computations of that size.

The product is $(g,u)(w,v)=(g+uw,uv)$.  It is useful to write a group
element $k=(g,u)$ instead as $(x,u)$, where $x=u^{-1}g$.
Left and right multiplication by $(w,v)$ then have the formulas
\begin{equation}\label{oc:actions}
 (x,u)\longmapsto (x+u^{-1}v^{-1}w,vu),
 \qquad
 (x,u)\longmapsto (v^{-1}(x+w),uv),
\end{equation}
respectively.  These formulas also give polynomial-in-$r$ Boolean
algorithms for multiplication and inversion, using the small tables for
$H$ rather than a table for $\Gamma$.

\begin{lemma}[Uniform expansion]\label{oc:expansion}
For every fixed $0<\lambda<1$, the groups $\Gamma$ in
\eqref{oc:group} admit symmetric labelled Cayley multisets of a fixed
degree whose averaging operator has norm at most $\lambda$ on mean-zero
functions.  The same assertion holds for $H$.  Their labels can be
constructed deterministically in time polynomial in $N$ and represented
with $\operatorname{polylog}(N)$ bits of public data.  Multiplicities and identity
labels are permitted.
\end{lemma}

\begin{proof}
First consider a factor of $H$.  The reductions of the upper and lower
unit elementary matrices, with their inverses, have a uniform Cayley
gap in $\operatorname{SL}_2(\F_p)$.  This is the elementary-generator
case of the congruence gap discussed in \cite[p.~626]{BG2008}, and also
follows directly from \cite[Theorem~1]{BG2008}, since the corresponding
fixed subgroup of $\operatorname{SL}_2(\mathbb Z)$ is non-elementary.
Passing to the quotient by the center preserves the upper gap.  A
half-lazy walk has a uniform absolute gap.  The finitely many small
primes are absorbed by decreasing this fixed gap.

For odd $p$, the subgroup $\operatorname{PSL}_2(\F_p)$ is normal of
index two in $\operatorname{PGL}_2(\F_p)$.  Add a representative of the
nonsquare determinant class and its inverse; in the index-one case pad
with identities.  To check a uniform gap, work in an arbitrary unitary
representation.  The preceding subgroup gap bounds the distance of a
vector to its subgroup-invariant projection by a fixed multiple of its
maximum displacement under the subgroup labels.  Normality makes that
invariant subspace invariant under the extra label, which controls its
quotient of order at most two.  Thus distance to the full invariant
subspace is bounded by a fixed multiple of the maximum displacement
under all these labels.  The three factor projections commute; applying
this argument to the three factors gives the same statement for $H$,
with a fixed-size generating list.  The regular-representation gap used
here applies to every representation of the finite group, by its
decomposition into irreducible representations.

We next give a bounded list of translations which, together with these
$H$ labels, expands $\Gamma$.  Consider a block of $n'$ bits with cycle
$R$.  For a nonzero frequency $y\in\F_2^{n'}$, let $d'$ be the dimension
of the span of its cyclic shifts.  Identifying vectors with polynomials
modulo $Z^{n'}-1$, this span is a cyclic ideal.  Every such ideal of
dimension $d'$ is specified by a complementary monic divisor of degree
$d'$.  There are at most $2^{d'}$ such divisors and at most $2^{d'}$
vectors in each ideal.  In particular, there are at most $2^{2d'}$
frequencies with this span dimension.  This argument uses only the
ideal structure of the polynomial quotient and remains valid when
$Z^{n'}-1$ has repeated factors in characteristic two.

For a uniform seed $a\in\F_2^{n'}$, the correlation vector
$(y\cdot R^ta)_{t=0}^{n'-1}$ is uniform on a shift-invariant linear
space of dimension $d'$.  Take $j_0$ independent seeds.  In the product
of these spaces, the subset with total weight less than $j_0n'/4$ is
invariant under cyclic shifts and permutation of the seeds.  If this
subset is nonempty, its uniform distribution has the same marginal at
every coordinate, with probability of one at most $1/4$.  An information
set of $j_0d'$ coordinates determines a vector in the product space.
Entropy subadditivity consequently bounds the size of this subset by
$2^{j_0d'\mathcal H(1/4)}$, where $\mathcal H$ is binary entropy.
The subset of weight greater than $3j_0n'/4$ has the same bound, by the
marginal entropy estimate with probabilities at least $3/4$; it is
unnecessary for the space to be closed under complementation.
Thus a fixed sufficiently large $j_0$ makes the probability of any
failure at any nonzero frequency at most
\begin{equation}\label{oc:seed-bound}
 2\sum_{d'\ge1}
   2^{-\{j_0(1-\mathcal H(1/4))-2\}d'}<1.
\end{equation}
There exist $j_0$ seeds whose cyclic translates have absolute Walsh
bias at most $1/2$ at every nonzero frequency.  Enumerating all seed
tuples and checking their biases constructs them in time $2^{O(r)}$,
which is polynomial in $N$.

Take the Cartesian product of the three block seed lists.  Averaging
its translations under $\langle R_1\rangle\times\langle R_2\rangle
\times\langle R_3\rangle$ has absolute Walsh bias at most $1/2$ off
zero: the three factors multiply, and a nonzero frequency has a nonzero
block.  Averaging further over cosets in $H$ preserves this bound.
Denote by $T$ the average of the resulting $H$-conjugates of seed
translations.  On the orthogonal complement of $G_0$-invariants, $T$
has all eigenvalues in $[-1/2,1/2]$, by Fourier decomposition for $G_0$.

Let $v$ be a unit vector in a representation of $\Gamma$, and let
$\eta$ be its maximum displacement under the fixed $H$ labels and the
fixed product-seed translations.  The $H$-invariant projection $v_H$
satisfies $\|v-v_H\|\le K\eta$ for an absolute $K$.  Its displacement
under any seed translation is at most $(1+2K)\eta$.  Since $v_H$ is
$H$-invariant, the same holds for every $H$-conjugate of a seed.
Writing $P_{G_0}$ for projection to $G_0$-invariants gives
\[
 \|v_H-P_{G_0}v_H\|^2
 \le 2\langle v_H,(I-T)v_H\rangle
 \le (1+2K)^2\eta^2.
\]
Normality of $G_0$ implies that $P_{G_0}v_H$ is also $H$-invariant.
Distance to the $\Gamma$-invariant subspace is therefore at most a
fixed multiple of $\eta$.  The symmetric half-lazy average on our
fixed generating list has a uniform positive gap: its Dirichlet form
is the average of squared label displacements divided by a fixed
constant.  Its spectrum is nonnegative.  A fixed power of this walk
has mean-zero norm at most the requested $\lambda$.  Reversing words
shows that the powered list remains symmetric.  Its length is fixed,
and the same conclusion holds for left and right regular actions.
\end{proof}

\subsection{Common scalar local codes}

The distance and agreement estimates in this subsection hold over an
arbitrary field $J$, conditional on the scalar check matrices satisfying
the stated minor conditions.  We will choose the special outer field
and those matrices after fixing the constants in these estimates.

The local agreement constant must be fixed before the outer Cayley
degree is chosen.  To obtain this order, we make the label positions
the edges of a separate expander.  For every integer $s_b\ge1$, consider
the Margulis--Gabber--Galil multigraph~\cite{Margulis1973,GabberGalil1981}
on $\mathbb Z_{s_b}^2$ with the
eight labelled moves
\[
 (x\mathbin\pm2y,y),\quad (x\mathbin\pm(2y+1),y),\quad
 (x,y\mathbin\pm2x),\quad (x,y\mathbin\pm(2x+1)).
\]
Its normalized mean-zero norm is at most $5\sqrt2/8$ for every positive
integer modulus, including composite moduli.  The primary formal
development \cite[\texttt{mgg\_graph\_step},
\texttt{mgg\_numerical\_radius}]{MGGAFP} proves exactly this labelled,
symmetric, all-modulus statement.  Add eight identity labels for
half-laziness and take a sufficiently large fixed power.  Viewing the
result as a bipartite graph between two copies of $\mathbb Z_{s_b}^2$
gives a graph $B_0$ with fixed degree $d_0$, permutation labels
$1,\ldots,d_0$, and mean-zero cross-part norm
\begin{equation}\label{oc:base-parameters}
 \lambda_0<10^{-4},\qquad
 c_0=\lfloor d_0/16\rfloor,\qquad \delta_0=1/16.
\end{equation}
Take the fixed power large enough that $c_0\ge1$.
Let $n_b=s_b^2$ and $d=d_0n_b$.  Each of the $d$ edge coordinates
is named by its left base vertex and permutation label.  Both ends
of an outer edge use these same coordinate names.

For each of the four types $C_A^0,C_A^1,C_B^0,C_B^1$, choose two
$c_0\times d_0$ scalar matrices, one for each base side.  Impose the
corresponding checks at every base vertex, with incident edges in
permutation-label order.  There are eight check matrices in total;
their entries are shared by all stars of the same type.  We require
every $c_0$-column minor to be nonzero.  Section~\ref{oc:encoders-subsec}
constructs matrices meeting these conditions and the global encoder
conditions simultaneously.  Until then the following conclusions hold
for every choice satisfying the stated minors.  For a finite-dimensional
$J$-space $U$, notation such as $C_A^s(U)$ means that the scalar
constraints are applied componentwise to a word in $U^d$.

\begin{lemma}[Local distance and tensor agreement]\label{oc:local-agreement}
Set
\begin{equation}\label{oc:delta-one}
 \delta_1=\delta_0(\delta_0-\lambda_0)>0.
\end{equation}
Every nonzero word of any of the four local codes, over any
finite-dimensional $J$-space $U$, has at least $\delta_1d$ nonzero
coordinates.  There is a constant $C_0<\infty$, depending only on
$d_0,\delta_0,\lambda_0$, with the following property for every $s_b$,
every field and check matrices as above, every $U$, and every $s,t$.
If $F$ is a $d\times d$ table with rows in $C_B^t(U)$ and $D$ is such
a table with columns in $C_A^s(U)$, and their joint cell-disagreement
fraction is $z$, there is a table $W$ whose rows and columns obey both
codes and such that
\begin{equation}\label{oc:tensor-agreement}
 \Pr_i[W(i,\cdot)\ne F(i,\cdot)]
 +\Pr_j[W(\cdot,j)\ne D(\cdot,j)]\le C_0z.
\end{equation}
In particular, $C_0$ is independent of $d$, $|J|$, and $\dim_JU$.
\end{lemma}

\begin{proof}
The maximal-minor condition gives star distance at least
$c_0+1\ge\delta_0d_0$: a nonzero word of smaller support would give a
linear dependence among at most $c_0$ columns, which could be enlarged
to a dependence among $c_0$ columns.  This also proves the assertion
for vector entries, either by applying a nonzero scalar component or
by solving the independent-column equations componentwise.

For a nonzero whole local word, let $l,u$ be the fractions of left and
right base vertices incident to its support, and let $a=\max(l,u)$.
Its edge-support fraction $e$ obeys, by star distance and mixing,
\[
 \delta_0a\le e\le lu+\lambda_0\sqrt{lu}
 \le a(a+\lambda_0).
\]
Hence $a\ge\delta_0-\lambda_0$ and $e\ge\delta_1$.  Mixing here and
below follows by subtracting the two constant means and applying the
cross-part operator norm to the indicator functions.

For agreement, apply the A-base checks to the columns of $F$.  Mark
every base star at which any check syndrome is nonzero.  Each nonzero
syndrome, as a row indexed by the $d$ columns, is a scalar linear
combination of B-code rows, and is therefore a nonzero B-codeword over
$U$.  Its support has size at least $\delta_1d$.  In each such column
there must be a discrepancy with $D$ on an edge incident to the marked
star, since $D$ satisfies that star's check.  A discrepancy cell is
incident to at most two base stars.  If $w_0$ is the total marked-star
count divided by $n_b$ (counting both base sides), then
\begin{equation}\label{oc:marked-size}
 w_0\le \frac{2zd^2}{\delta_1dn_b}
       =\frac{2d_0z}{\delta_1}.
\end{equation}

We now confine every low-discrepancy difference column to edges inside
a small set of base vertices.  Deleting those row indices, together
with the columns having large discrepancy, will leave a rectangle on
which $F$ and $D$ agree.

Close the marked set by adding a vertex whenever at least
$(\delta_0/2)d_0$ incident edges lead to the set.  If the initial set
is empty it stays empty.  Otherwise suppose its normalized size first
reaches $w\ge2w_0$.  A single addition is at most $1/n_b\le w_0$, so
$w\le3w_0$.  Edges made internal by additions give internal edge
fraction at least $(\delta_0/2)(w-w_0)$.  Mixing gives the upper bound
$w^2/4+\lambda_0w/2$.  Since $w\ge2w_0$, these inequalities imply
$\delta_0\le w+2\lambda_0$.  They are incompatible when
$3w_0<\delta_0-2\lambda_0$.  For sufficiently small $z$, therefore,
the final closure $T$ satisfies $|T|/n_b\le2w_0$, with strict
inequality when $w_0>0$.
Every star outside $T$ still satisfies all checks on $F$, and fewer
than $(\delta_0/2)d_0$ of its edges enter $T$.

Fix a constant
\[
 0<z_0<\delta_0(\delta_0/2-\lambda_0).
\]
Consider a column whose discrepancy fraction is less than $z_0$.
The difference between its $F$ and $D$ columns satisfies all checks
outside $T$.  An outside vertex incident to its support consequently
has at least $\delta_0d_0$ support edges in total and more than
$(\delta_0/2)d_0$ support edges to other outside vertices.  The left
and right fractions of such outside vertices are each at most
$z_0/\delta_0$, by counting their total support edges.  If $a$ is
their maximum fraction, mixing on the outside support would give
\[
 (\delta_0/2)a\le a(a+\lambda_0),
 \qquad a\le z_0/\delta_0<\delta_0/2-\lambda_0.
\]
Thus $a=0$.  The difference column vanishes on every edge having an
endpoint outside $T$.

Delete the row indices corresponding to edges internal to $T$, and
delete columns with discrepancy fraction at least $z_0$.  The former
fraction is at most $|T|/(2n_b)\le w_0$, since each internal edge consumes
two incidences in $T$, and the latter fraction is at most $z/z_0$.
The two tables agree on the complementary rectangle.  Choose a fixed
$z_*>0$, depending only on the base parameters, so small that for
$z<z_*$ the closure inequality holds and each deleted fraction is
less than $\delta_1$.  Puncturing either scalar local code to its
retained coordinates is then injective by distance.  Its inverse
on the image is a scalar linear map.  Lift the common rectangle in
the B direction and then in the A direction.  Applying scalar linear
maps in one direction preserves the constraints in the other; hence
the resulting table belongs to the full tensor code.  By injectivity
it agrees with every retained whole row of $F$ and every retained
whole column of $D$.  This gives \eqref{oc:tensor-agreement} with
$2d_0/\delta_1+1/z_0$ for $z<z_*$.  For $z\ge z_*$, use the zero
tensor table and the bound $2\le(2/z_*)z$.  The maximum of these two
constants is a suitable $C_0$.  All lifting maps and all support
counts act on joint vector entries, so no dimension factor occurs.
\end{proof}

We now fix, once and for all,
\begin{equation}\label{oc:lambda-choice}
 0<\lambda\le\min\{(8C_0)^{-1},\delta_1/16\}.
\end{equation}
Apply Lemma~\ref{oc:expansion} to obtain an outer Cayley list of degree
$D$.  Choose $s_b$ to be a positive multiple of $D$.  Then
$d=d_0s_b^2$ is divisible by $D$, so label the $d$ local coordinates
by a balanced repetition of the outer list.  This repetition changes
neither averaging operator nor its norm.  Use such lists
$(a_i)_{i=1}^d$ and $(b_j)_{j=1}^d$ for the A and B directions.
All of $d_0,c_0,\lambda_0,\delta_1,C_0,\lambda,s_b,d$ are now fixed
constants.  We now specialize $J$; the field choice will not change
these constants.

Choose an odd characteristic independently of the primes $p_a$
and independently of the characteristic-two translation group.  Put
$m=M_o$.  Linnik's theorem, for example the effective bound in
\cite[Theorem~1.1]{Xylouris2011}, supplies a prime
\begin{equation}\label{oc:prime}
 p\equiv1\pmod m,\qquad m<p\le C_Lm^{L}
\end{equation}
for absolute constants $C_L,L$.  Searching integers and testing
primality, even by trial division, takes polynomial time in $m$.
Search the prime field for an element $\zeta$ of exact order $m$;
such an element exists because its multiplicative group is cyclic.
Since $m$ is a power of two, testing $\zeta^m=1$ and
$\zeta^{m/2}\ne1$ suffices.  We will use
\begin{equation}\label{oc:field}
 J=\F_{p^4}.
\end{equation}
The proof below verifies that degree four supplies enough room for
every simultaneous outer-code choice.  One can find an irreducible
quartic by enumeration and represent $J$ as its polynomial quotient.
Field operations and the root $\zeta$ then use $O(\log N)$ bits and
polynomial-in-$\log N$ Boolean computations.  Our fixed lower cutoff
also ensures $p>m>|H|$, so both $m$ and $|H|$ are invertible in $J$.

\subsection{Twists and simultaneous systematic extension}
\label{oc:encoders-subsec}

There are four outer parts
\[
 V=\Gamma\times\{0,1\}\times\{0,1\}.
\]
At $(k,s,t)$ a full slot stores a $d\times d$ matrix $W_k^{st}$ over
a finite-dimensional $J$-space $U$, with columns in $C_A^s(U)$ and
rows in $C_B^t(U)$.  The A and B edges impose, respectively,
\begin{equation}\label{oc:edge-equations}
 W_{a_ik}^{1t}(i,j)=\alpha_i W_k^{0t}(i,j),
 \qquad
 W_{kb_j}^{s1}(i,j)=\beta_j W_k^{s0}(i,j).
\end{equation}
The maps $\alpha_i,\beta_j$ are invertible $J$-linear maps on $U$,
independent of $k,s,t$, and satisfy
$\alpha_i\beta_j=\beta_j\alpha_i$ for every $i,j$.  An A edge of
label $i$ transports the entire B-coded row $i$, while a B edge of
label $j$ transports the entire A-coded column $j$.  A reverse edge
uses the inverse map.  At their intersection the two transports give
\[
\begin{array}{ccc}
 W_k^{00}(i,j)&\xrightarrow{\ \alpha_i\ }&W_{a_i k}^{10}(i,j)\\
 {\scriptstyle\beta_j}\big\downarrow&&
       \big\downarrow{\scriptstyle\beta_j}\\
 W_{k b_j}^{01}(i,j)&\xrightarrow{\ \alpha_i\ }&W_{a_i k b_j}^{11}(i,j).
\end{array}
\]
Left and right multiplication on $\Gamma$ commute, so the two routes
reach the same index; commuting entry maps give the same entry value.

The scalar type has $U=J$ and identity twists.  The programs in
Section~\ref{sec:outer-programs} will also need to multiply scalar
data $v:\Gamma\to J$ by $(-1)^{e\cdot x}$ for a distinguished bit
$e=e_a$.  The matrix types below supply edge equations compatible
with this operation.  The programs will use them to retain one axis
of code constraints while changing the other.
For each of the three distinguished bits $e=e_a$, define a
representation on the basis
$({\bf e}_{t'})_{t'\in H}$ by
\begin{equation}\label{oc:rho}
 \rho_e(g,u){\bf e}_{t'}
   =(-1)^{(ut'e)\cdot g}{\bf e}_{ut'}.
\end{equation}
Its entries are signed permutation entries in the prime field.
Indeed, for $k=(g,u)$ and $k'=(g',u')$, the exponent in
$\rho_e(k)\rho_e(k'){\bf e}_{t'}$ is
\[
 (u't'e)\cdot g'+(uu't'e)\cdot g
  =(uu't'e)\cdot(g+ug'),
\]
which proves multiplicativity.  In a matrix type let
$U=\operatorname{Mat}_{|H|}(J)$ and independently choose each axis
to be untwisted or to use
\begin{equation}\label{oc:matrix-twists}
 \alpha_i D=\rho_e(a_i)D,
 \qquad \beta_j D=D\rho_e(b_j).
\end{equation}
Left and right matrix multiplication commute.  Right multiplication
in \eqref{oc:matrix-twists} specifies edge maps; no convention about
its order as a group representation is needed.  This gives at most
$1+3\cdot4=13$ full types.  Finite tuples, with arbitrary repetitions
of these types, use direct sums of their entry spaces and block
diagonal twists.

On either local-label graph, let the parity set consist of all edges
with permutation number in $[4c_0]$.  Its size and complement are
\begin{equation}\label{oc:parity-set}
 q_{\rm par}=4c_0s_b^2,\qquad
 I_A=[d]\setminus\text{A-parity set},\qquad
 I_B=[d]\setminus\text{B-parity set}.
\end{equation}
Both systematic sets are nonempty; in fact they contain at least
$3d/4$ labels.

\begin{lemma}[Simultaneous systematic encoders]\label{oc:systematic}
One can choose the eight scalar check matrices over $J$ so that all
their maximal minors are nonzero and every A and B one-axis system
of the thirteen types above is systematically invertible.  Precisely,
arbitrary data at the systematic labels of part zero have a unique
extension to both parts satisfying that axis's local checks and edge
equations.  Consequently arbitrary data on
$\Gamma\times I_A\times I_B$ in part $(0,0)$ have a unique full
extension satisfying both axes, and the two orders of axis extension
give the same answer.  The matrices and the full encoders are
deterministically constructible in polynomial time in $N$.  These
assertions continue to hold for arbitrary finite direct sums of the
listed types, without any further field or check-matrix choices.
\end{lemma}

\begin{proof}
Consider one A-axis system.  Eliminate part-one variables with the
A-edge equations.  At each $k$, the two types of base-star checks in
part zero contribute $2c_0s_b^2$ equations; those in part one contribute
the same number after substitution.  There are exactly
$q_{\rm par}=4c_0s_b^2$ parity-label variables at part zero for each
$k$, so the resulting parity system is square, also after scalarizing
$U$.  Its entries are linear polynomials in the entries of the four
A check matrices.

Its determinant polynomial is nonzero in the prime characteristic.
For this purpose partition the $4c_0$ parity permutation numbers into
four disjoint sets of $c_0$ numbers, one for each pair consisting of an
outer part and a base side.  Specialize each associated check matrix
to selectors of its set.  For a fixed permutation number, every
label-index edge occurs exactly once at the specified base side.
At outer part zero its selected coordinate is unshifted; at part one
it is pulled back through an invertible group shift and an invertible
entry twist.  Thus every scalarized parity coordinate is selected
exactly once, up to an invertible block and a permutation.  The
specialized parity matrix is invertible.  The same proof works for
the B axis and for every allowed twist type.  It does not require
this selector specialization itself to have the desired maximal
minors: it separately witnesses that each determinant is a nonzero
polynomial.  Each maximal-minor polynomial is also nonzero.  Their
product is nonzero because a polynomial ring over a field is an
integral domain.

Here is a degree bound that justifies this field choice without
reference to any future number of slots.  There are exactly
\begin{equation}\label{oc:number-variables}
 8c_0d_0
\end{equation}
scalar indeterminates.  Scalarizing a one-axis parity matrix gives
dimension at most
\begin{equation}\label{oc:parity-dimension}
 q_{\rm par}|\Gamma|\dim_JU
 \le q_{\rm par}m|H|^3\le q_{\rm par}mr^{27}.
\end{equation}
Its determinant has degree at most this number.  At most
$2(1+3\cdot4)=26$ axis systems cover every required type, with
redundancies harmless.  The product of the maximal minors of the
eight check matrices has degree $8c_0\binom{d_0}{c_0}$.  Thus the
total degree is at most
\begin{equation}\label{oc:determinant-degree}
 26q_{\rm par}mr^{27}+8c_0\binom{d_0}{c_0}<m^3.
\end{equation}
For the last inequality increase the fixed cutoff so that
$r^{27}\le m$, $m\ge2\cdot26q_{\rm par}$, and
$m^2\ge2\cdot8c_0\binom{d_0}{c_0}$.  The first term is then at
most $m^3/2$ and the second at most $m^2/2<m^3/2$.
The exponent three is independent of $s_b$; only the cutoff depends
on the already fixed local degree.

A nonzero polynomial of total degree less than a finite field's
cardinality cannot vanish on every tuple over that field.  For
completeness, view it as a polynomial in the last variable; by
induction choose the other variables so that a nonzero leading
coefficient remains nonzero, and then use the univariate root bound.
Since $|J|=p^4>m^3$, such a tuple exists in $J$.  Enumerate all tuples
of the fixed number \eqref{oc:number-variables} of field elements,
testing maximal minors and parity ranks, until one succeeds.  The
field cardinality and every matrix dimension are polynomial in $N$,
so this is a deterministic polynomial-time algorithm with a fixed
exponent.  A direct sum of already invertible parity systems is
invertible with the direct sum of their inverses.  It introduces no
new polynomial condition, even when the number of slots grows with
$N$.

It remains to justify two-axis extension.  Regard an A parity matrix
as a matrix of operators on $U^\Gamma$: each entry is a scalar
combination of left translations and A twists.  B entries are scalar
combinations of right translations and B twists.  They commute
entrywise across axes.  An invertible A parity matrix $L_A$ therefore
intertwines the diagonal actions of any B-entry operator $T$ on its
domain and codomain.  From $L_AT=TL_A$ one obtains
$L_A^{-1}T=TL_A^{-1}$, with the same indexed domain and codomain
interpretation.  Thus the operators making up either axis encoder
commute with every entry operator of the other axis encoder, including
its inverse parity solve.

Start with the systematic rectangle at part $(0,0)$.  Extend along A
for every systematic B column and then along B for every resulting
A row.  The second extension preserves all first-axis constraints
by this commutation.  Both encoders leave their systematic entries
unchanged.  Reversing their order gives the same map.  Finally, one-axis
uniqueness applied in both directions shows that a full word is
uniquely determined by its systematic rectangle.  Gaussian elimination
computes these extensions in polynomial time in $N$, or in polynomial
time in the total explicit input size for an arbitrary direct sum.
\end{proof}

For $v:\Gamma\to U$, give every systematic cell in part $(0,0)$ the
same value $v(k)$.  Its full extension is denoted
\begin{equation}\label{oc:encoder}
 E_{\alpha,\beta}v,
\end{equation}
or $Ev$ when its type is specified.  We call this the
\emph{duplicated-payload} encoder.  It is linear and injective, and
its rate in field coordinates is exactly $1/(4d^2)$: its domain has
$|\Gamma|\dim_JU$ coordinates and its stored word has
$4d^2|\Gamma|\dim_JU$ coordinates.  In the largest individual type
the entry dimension is $|H|^2\le r^{18}$.  Therefore one full slot
occupies $\operatorname{polylog}(N)$ bits at each of $|\Gamma|=O(N\operatorname{polylog} N)$
sites.  Any fixed power of $\log(N+2)$ slots preserves this size form.

\begin{remark}[Operator invariance]\label{oc:orbit-invariance}
The preceding inverse argument also justifies restriction to frequency
orbits, as required by the local filling algorithms.  Expand a function
of $(x,u)$ in the Walsh characters $(-1)^{\nu\cdot x}$, using $m^{-1}$
for the inverse transform over $J$.  By \eqref{oc:actions}, a left pull
preserves each $x$-frequency, with a phase depending on $u$, whereas
a right pull permutes frequencies by $H$.  Constant entry twists do
not change the frequency.  For any $H$-orbit $O\subseteq G_0$, both
the $O$-frequency subspace and its complementary frequency subspace
are invariant under each parity operator.  Its domain and codomain
have equal dimensions on this subspace: the parity-label and
check-label counts are both $q_{\rm par}$.  Global invertibility
therefore implies invertibility of the restriction.  Its scalar
dimension is at most $q_{\rm par}|O||H|\dim_JU\le
q_{\rm par}|H|^4$, polynomial in $r$, so the restriction and its
inverse can be computed by polynomial-in-$\log N$ field operations.
This is an algebraic consequence of the explicit parity matrices;
it does not require reading unspecified global values of a word.
\end{remark}

\subsection{Proximity and one-axis gaps}

Write $\dist_V$ for relative Hamming distance on the four-part vertex
set $V$.  When all four parts at the same $k$ are packed into one
symbol, write $\dist_\Gamma$.  For any two words,
\begin{equation}\label{oc:packing-metric}
 \dist_V\le\dist_\Gamma\le4\dist_V.
\end{equation}
An \emph{A-only word} retains the A local constraints and A agreements
at all four parts; any other label indices and any collection of slots
are regarded as coordinates of a single row symbol.  No B validity
or B agreement is asserted.  Define B-only words symmetrically.

For a tuple of full slots a local check rejects when any component
fails its scalar local checks, and an edge is conflicting when any
slot fails its agreement on the whole row or column carried by that
edge.  In particular, a growing tuple does not enlarge the set of
address maps or cause separate per-slot error charges.

\begin{lemma}[Full-code proximity]\label{oc:full-proximity}
There are fixed $\theta_*>0$ and $K_*<\infty$ such that the following
holds for every finite tuple of allowed full types and every received
word.  Check local validity and all A and B agreements at a uniformly
random packed center, rejecting if any check there fails.  If the
rejection fraction is $\theta\le\theta_*$, the word is at packed
distance at most $K_*\theta$ from a word satisfying every local and
edge constraint.  The constants are independent of the tuple length,
the field size, and the entry dimensions.  Duplicated-payload
constraints are addressed separately below.
\end{lemma}

The proof uses the decreasing-disagreement method of
\cite[Section~4.2, Algorithm~1 and Propositions~4.7--4.8]{DEL+2022}.
We prove the version for joint tuples with commuting twists needed here.

\begin{proof}
First replace every locally invalid whole tuple at a vertex by zero.
There are at most $\theta|V|$ such vertices.  Initially at most
$d\theta|V|$ undirected edges conflict, and this replacement creates
at most $2d\theta|V|$ further conflicts.  Starting with this locally
valid word, repeatedly replace a whole vertex by a locally valid tuple
whenever that strictly decreases its number of incident conflicting
edges.  The total integer conflict count strictly decreases, so at
most $3d\theta|V|$ replacements occur.  This argument establishes
existence and termination; it is not asserted to be an efficient
search through the alphabet.  At most $(1+3d)\theta|V|$ vertices have
been touched, and the final conflict count is no larger than the
count before the replacements.

At a stable vertex $v$, let $p_A(v),p_B(v)$ be the fractions of its
A and B edges that conflict, and put
$p(v)=p_A(v)+p_B(v)$ and $\mu=\E_Vp$.  Neighbors across A edges
suggest rows, transported by the appropriate twist or inverse twist.
They are valid B-code rows.  Neighbors across B edges similarly
suggest valid A-code columns.  At cell $(i,j)$ these suggestions
agree unless at least one of the two opposite square edges conflicts:
the common far corner and commuting twists supply the equality when
both opposite edges agree.  If $P_A,P_B$ average over the respective
neighbor maps, their cell-disagreement fraction is at most
$P_Ap_B(v)+P_Bp_A(v)$.  Lemma~\ref{oc:local-agreement} gives a local
replacement whose incident conflict fraction is at most $C_0$ times
this number.  Stability therefore gives pointwise
\begin{equation}\label{oc:stability}
 p\le C_0(P_Ap_B+P_Bp_A)\le C_0(P_Ap+P_Bp).
\end{equation}

All norms and inner products in this proof use uniform normalized
counting measure on $V$.  Let $\bar p$ be the function equal to the
mean of $p$ on each of the four parts.  Nonnegativity gives
$\|\bar p\|_2\le2\mu$.  Each neighbor operator permutes the part
means and has norm at most $\lambda$ on their orthogonal complement.
Consequently \eqref{oc:stability} implies
\[
 \|p\|_2\le2C_0(2\mu+\lambda\|p\|_2),
 \qquad \|p\|_2\le C_1\mu,\quad C_1=8C_0,
\]
by \eqref{oc:lambda-choice}.  This bound has no dependence on $d$.

Fix an A label $i$, let $S_i$ be the vertices whose incident A edge
of that label conflicts, and let $p_i=\Pr_V(S_i)$.  At such an edge
the difference of its two transported rows is a nonzero B-codeword,
so it has at least $\delta_1d$ nonzero cells.  In each corresponding
B direction, either a B edge conflicts at one A endpoint or an
$i$-label A edge conflicts at the opposite B neighbor.  The
involution exchanging the two A endpoints preserves $S_i$.  Averaging
this implication, and then subtracting the four part means in the
mixing term, yields
\begin{align}
 \delta_1p_i
 &\le 2\E_V[1_{S_i}p_B]
      +\langle1_{S_i},P_B1_{S_i}\rangle \notag\\
 &\le 2\E_V[1_{S_i}p_B]+4p_i^2+\lambda p_i.
 \label{oc:conflict-propagation}
\end{align}
Indeed, the squared norm of the four part means of $1_{S_i}$ is at
most $4p_i^2$, and the squared norm of its remaining part is at most
$p_i$.  Also $p_i\le d\mu$.  For sufficiently small $\mu$, depending
on the now fixed $d$, we have $4d\mu+\lambda<\delta_1/2$.
Average \eqref{oc:conflict-propagation} over $i$ and repeat the
argument with A and B exchanged.  It follows that
\begin{equation}\label{oc:stable-zero}
 (\delta_1/2)\mu
 \le4\E_V[p_Ap_B]\le4\E_V[p^2]\le4C_1^2\mu^2.
\end{equation}
If $0<\mu<\delta_1/(8C_1^2)$ this is impossible.  Since the initial
zeroing left at most $3d\theta|V|$ conflicts, stability has
$\mu\le6\theta$.  For example, choose $\theta_*$ so that
$6\theta_*<\min\{\delta_1/(32d),\delta_1/(16C_1^2)\}$.
Then $\mu=0$.  The final word is exact, and
\eqref{oc:packing-metric} gives $K_*=4(1+3d)$ as a permissible
proximity constant for this normalization.  Common scalar local
checks, block diagonal twists, and joint conflict events justify
every step for an arbitrary tuple without a slot-count factor.
\end{proof}

\begin{lemma}[One-axis distance and exact comparisons]\label{oc:one-axis}
Let $\Delta_A=(\delta_1-\lambda)/4>0$.  Every nonzero A-only or
B-only word with the specified local constraints and invertible twists
has relative vertex support at least $\Delta_A$ on $V$.  The same
holds for differences of two words with the same axis constraints,
and for arbitrary finite joint tuples.  It also holds for full words.

Among exact full words, the duplicated-payload condition has a fixed
gap under the following checks: at $s=0$, compare all systematic A
rows as whole rows, at both values of $t$; at $t=0$, compare all
systematic B columns as whole columns, at both values of $s$.
If any payload is not duplicated, the packed-center rejection
fraction is at least $\Delta_A$.  The tests accept every word $Ev$.

More generally, suppose an expression $X$ and a full target $Y=Ev$
both obey the same A constraints, and the systematic rectangle of
$X$ is duplicated with payload $w$.  Equality of their whole
systematic B columns at $t=0$, including both $s$ parts, holds if
and only if $v=w$.  If $v\ne w$, the comparison rejects at least a
$\Delta_A$ fraction of packed centers.  The symmetric assertion
holds for a shared B axis and comparisons of whole systematic A
rows at $s=0$.  No constraint on the other axis of $X$ is needed.
\end{lemma}

\begin{proof}
For one A slice with fixed $t$, a nonzero vertex has at least
$\delta_1d$ nonzero row symbols by local distance.  Each is carried
to a nonzero symbol at its opposite endpoint, because twists are
invertible.  If $l,u$ are the two vertex-support fractions and
$a=\max(l,u)$, the fraction $e$ of nonzero carried edges obeys
\[
 \delta_1a\le e\le lu+\lambda\sqrt{lu}\le a(a+\lambda).
\]
Thus $a\ge\delta_1-\lambda$.  Support on these two parts is at
least $(\delta_1-\lambda)/2$ in their own normalization and at least
$\Delta_A$ in the four-part normalization.  At least one slice is
nonzero.  The argument for B, differences, and joint entries is
identical.

For two systematic A row labels, their difference at $s=0$, considered
over both $t$ parts and all $k$, is a B-only word: the scalar B checks
and B twists are the same for both rows.  If nonzero it has the
distance just proved.  Differences between systematic B columns are
A-only words.  A nonduplicated systematic rectangle contains two
unequal entries connected either by a change of its A label or a
change of its B label, so one of these comparisons is nonzero.
Conversely, for a duplicated payload, B-axis uniqueness extends the
same systematic row data identically at every systematic A label,
and A-axis uniqueness does the analogous thing for columns.

For the last assertion, if $v=w$, A-axis uniqueness identifies the
two whole columns at every systematic B label.  If $v\ne w$, a
systematic payload cell differs somewhere, so one of the compared
column differences is a nonzero A-only word and has support at least
$\Delta_A$.  The B-sharing assertion is proved in the same way.
These comparisons use whole one-axis codewords, which is why an
arbitrary localized payload disagreement has a constant exact gap.
\end{proof}

The next estimate will control a growing tuple of A-only scratch
slots against one exact reference.  It uses the tuple's joint edge
conflicts, with no factor for its number of components.

\begin{lemma}[One-axis near-code estimate]\label{oc:near-code}
Let $W$ and $V_0$ be locally A-valid joint tuples on $V$, and suppose
$V_0$ has exact A agreement with the same twists as $W$.  Let
$p=\dist_V(W,V_0)$ and let $z_A$ be the A-edge conflict fraction of
$W$, normalized as the probability that a uniform vertex and a
uniform A label give a conflict.  Then
\begin{equation}\label{oc:near-code-inequality}
 \delta_1p\le z_A+4p^2+\lambda p.
\end{equation}
In particular, if $p\le\delta_1/16$, then
$p\le2z_A/\delta_1$.  Both statements hold for every finite tuple
of A-only and full slots, uniformly in its size.  They also hold
with A and B exchanged.
\end{lemma}

\begin{proof}
Let $S$ be the joint vertex-disagreement set.  At each vertex in $S$,
local distance gives at least $\delta_1d$ nonzero difference rows.
For each corresponding label, either the other endpoint also lies
in $S$, or agreement of $V_0$ forces an actual conflict of $W$.
Consequently
\[
 \delta_1p\le z_A+\langle1_S,P_A1_S\rangle
 \le z_A+4p^2+\lambda p,
\]
by the four-part mixing bound used above.  For
$p\le\delta_1/16$ and \eqref{oc:lambda-choice},
$4p+\lambda<\delta_1/2$, giving the stated rearrangement.
The proof concerns one fixed received word and one joint reference.
It can be applied to successively larger prefixes of a tuple without
modifying the received word or accumulating a number-of-prefixes
loss.
\end{proof}

We also allow a clock or control tuple repeated identically at every
$k$.  For any received tuple, let its distinct-value class proportions
be $u_j$.  Equality across an expanding symmetric A list has probability
at most
\begin{equation}\label{oc:clock-gap}
 \sum_j u_j^2+\lambda\left(1-\sum_j u_j^2\right).
\end{equation}
This follows by applying the mean-zero bound to each class indicator
and summing.  If the equality rejection is $\eta$, then
$1-\max_j u_j\le1-\sum_j u_j^2\le\eta/(1-\lambda)$.
Thus a small rejection fraction implies proximity to one constant
tuple, irrespective of its bit length.

Use unique bit representations of field elements and canonical zero
padding.  At a packed center check parsing, ranges, and padding.
Replace a malformed symbol by a fixed parsable default when reasoning
from acceptance.  If all checks read from one fixed finite list
$\Pi$ of permutation addresses, modifying a set of $u|\Gamma|$
symbols can change their center outcomes on at most
$|\Pi|u|\Gamma|$ centers, by taking the union of inverse images.
This applies even when many slots are checked together or branches
use only subsets of the list.  Combining this observation with
Lemma~\ref{oc:full-proximity}, \eqref{oc:clock-gap}, and the
duplicated-payload gap of Lemma~\ref{oc:one-axis} gives proximity
to exact duplicated full encodings and a repeated clock whenever
the total packed rejection is sufficiently small.  Indeed, the
one-time repair changes only $O(\theta)$ centers and hence leaves
duplication rejection $O(\theta)$; choose its threshold below the
fixed exact gap.  Any required public range for a corrected clock
is likewise enforced, since a constant illegal value fails its
local range test everywhere.  These constants may depend on the
fixed address-list size, but not on the slot count.

\begin{theorem}[Outer-code interface]\label{thm:outer-codes}
There exist fixed constants $d$, $\delta_1>0$, $C_0<\infty$,
$\lambda>0$, $\theta_{\mathrm{out}}>0$, and $K_{\mathrm{out}}<\infty$, with
$\lambda\le\min\{(8C_0)^{-1},\delta_1/16\}$, having the following
uniform properties for every input length $N$ and its padded tape
length $h=O(N)$, after the fixed padding in \eqref{oc:outer-size}.
\begin{enumerate}[label=(\roman*)]
\item The group $\Gamma=\F_2^r\rtimes H$ has
$|\Gamma|\le M_or^9=O(N\operatorname{polylog} N)$, where $M_o=2^r=O(N)$.
Its A and B label lists are symmetric, of degree $d$, and have
mean-zero norm at most $\lambda$.  There is a field
$J=\F_{p^4}$ with $p>M_o>|H|$, $p\equiv1\pmod{M_o}$, and a
primitive $M_o$-th root $\zeta\in\F_p$.  All public group and field
data and the common scalar check matrices are deterministically
constructible in polynomial time in $N$, and the data needed at an
index have $\operatorname{polylog}(N)$ size and Boolean computation cost.
\item Each of the scalar and three matrix-representation families,
with either axis independently twisted or untwisted as specified
in \eqref{oc:matrix-twists}, has the simultaneous systematic
encoder \eqref{oc:encoder}.  Arbitrary systematic rectangle data
extend uniquely; the A and B extensions commute.  Duplicated payload
encoding is injective, linear, and has rate $1/(4d^2)$ in field
coordinates.  A full slot has $\operatorname{polylog}(N)$ bits per packed site.
Arbitrary finite tuples reuse the same checks and inverses.
\item The local distance and tensor-agreement statement of
Lemma~\ref{oc:local-agreement} holds uniformly in field size and
joint vector dimension.  Local validity, edge agreement, and
duplicated-payload comparisons have perfect completeness.  For
every finite tuple of full slots, if their joint packed-center
rejection fraction is $\theta\le\theta_{\mathrm{out}}$, their packed
distance to a tuple of exact duplicated encodings is at most
$K_{\mathrm{out}}\theta$.  Canonical parsing and a repeated control
tuple can be included with the same form of conclusion, changing
only the fixed constants.
\item The one-axis distance and shared-axis comparison gaps of
Lemma~\ref{oc:one-axis} hold with $\Delta_A=(\delta_1-\lambda)/4$.
For every actual and reference joint tuple satisfying the hypotheses
of Lemma~\ref{oc:near-code},
$\delta_1p\le z_A+4p^2+\lambda p$ and, when
$p\le\delta_1/16$, $p\le2z_A/\delta_1$.  A-only slots require
no B constraints for these conclusions.  The symmetric B assertions
hold as well.
\end{enumerate}
All constants are independent of $N$ and of the number of later
program slots.  The field search has only $8c_0d_0$ indeterminates,
at most $26$ parity determinant factors, and total degree less than
$M_o^3$; in particular it causes no feedback into $C_0$, $\lambda$,
or $d$.
\end{theorem}

\begin{proof}
Parts (i) and (ii) follow from Lemmas~\ref{oc:expansion} and
\ref{oc:systematic}, the field construction, and the coordinate
count after \eqref{oc:encoder}.  Part (iii) is
Lemma~\ref{oc:local-agreement}, Lemma~\ref{oc:full-proximity}, and
the parsing and exact-gap argument following \eqref{oc:clock-gap}.
Decrease $\theta_{\mathrm{out}}$ and increase $K_{\mathrm{out}}$ once to
cover those fixed costs.  Part (iv) is
Lemmas~\ref{oc:one-axis} and \ref{oc:near-code}.  The degree and
dependency assertions are \eqref{oc:number-variables}--
\eqref{oc:determinant-degree} and the parameter order fixed in
\eqref{oc:lambda-choice}.  No external local-testability theorem is
used for these specialized conclusions.
\end{proof}

\subsection{Correction with value-independent uncertainty supports}

The next guarantee is stronger than correction of a single fixed
word.  It tracks a set on which values are wholly unspecified, and
therefore applies to separate bounded-cone simulations with different
choices on that set.

\begin{lemma}[Outer uncertainty shrinkage]\label{lem:outer-uncertainty}
Fix the data in Theorem~\ref{thm:outer-codes}, and any finite tuple of
full or A-only slots with their allowed A twists.  Optionally include
a control tuple constant over $\Gamma$ and canonical padding.
There is a deterministic synchronous local correction map $\mathcal C$
with the following property.  Let $V_0$ be any reference word satisfying
all A local constraints and A agreements, with constant control and
canonical parsing.  Let $S\subseteq\Gamma$ and let $W$ be any input
equal to $V_0$ outside $S$, with arbitrary values or malformed symbols
on $S$.  Then $\mathcal C W=V_0$ outside
\begin{equation}\label{oc:uncertainty-set}
 S'=
 \left\{k:\frac1d\#\{i:a_ik\in S\}\ge\delta_1/2\right\}
 \ \cup\
 \left\{k:\frac1d\#\{i:a_i^{-1}k\in S\}\ge\delta_1/2\right\}.
\end{equation}
In particular, $S'$ depends on $S$ alone.  If $u=|S|/|\Gamma|<\delta_1/4$,
then
\begin{equation}\label{oc:uncertainty-shrink}
 \frac{|S'|}{|\Gamma|}
 \le\frac{2\lambda^2u}{(\delta_1/2-u)^2}\le u/2.
\end{equation}
The local map uses a fixed number of permutation addresses and time
polynomial in the symbol bit length and the public local-parameter
length, with total default behavior on arbitrary inputs.

For every fixed integer $q\ge0$, the $q$-round map has a bounded
input cone consisting of fixed permutations of its target; the bound
may depend on $q$ but not on $N$.  For an arbitrary initial $S$, the
recursively defined supports contain the errors simultaneously for
all completions on $S$.  If $|S|/|\Gamma|<\delta_1/4$, their densities
are at most $2^{-q}|S|/|\Gamma|$ after $q$ rounds.  Different target-cone
simulations may use different such completions, provided each simulation
assigns one consistent initial value to every address repeated within
that cone.  Under this small-initial-support hypothesis,
$O(\log|\Gamma|)$ rounds correct the whole word exactly in time
polynomial in its explicit size and $N$.
\end{lemma}

\begin{proof}
At a target part $(k,0,t)$, take the row of label $i$ from
$(a_ik,1,t)$ and apply $\alpha_i^{-1}$.  At $(k,1,t)$, take it from
$(a_i^{-1}k,0,t)$ and apply $\alpha_i$.  Combine all other indices
and all slots into one row-symbol vector.  Decode these $d$ row
suggestions in $C_A^s$ up to strictly fewer than $\delta_1d/2$
joint row errors.  There is at most one such local word by the
distance in Lemma~\ref{oc:local-agreement}.  A total deterministic
implementation enumerates all subsets $T\subseteq[d]$ with
$|T|>d-\delta_1d/2$, solves the local scalar constraints together
with the suggested values on $T$ componentwise, and verifies that
the candidate has fewer than $\delta_1d/2$ joint row errors.
Such a puncturing is injective by local distance, so each consistent
system gives at most one candidate.  If the distance promise holds,
the set of all correct suggestions is one enumerated $T$ and gives
the required word.  Return zero if no candidate passes.  Malformed
neighbor symbols are first replaced by the fixed parsable default.
Restore each control bit by majority on the forward A-neighbor list,
with fixed tie behavior, and write canonical padding.

If $k\notin S'$, each relevant list contains fewer than
$\delta_1d/2$ positions of $S$.  Every other row suggestion equals
the corresponding row of $V_0$, by its exact A agreement.  Thus the
local decoder recovers all four reference parts and all slots
jointly.  The same good-neighbor count is strictly below $d/2$, so
control majorities are also correct.  This proves the support
containment for every assignment of values on $S$.

For either forward or inverse averaging operator $P$ on $\Gamma$,
the mean-zero bound gives
\[
 \E_k(P1_S(k)-u)^2
 \le\lambda^2u(1-u)\le\lambda^2u.
\]
When $u<\delta_1/4$, Chebyshev's inequality at threshold
$\delta_1/2-u>0$ and a union bound for the two lists prove the first
inequality in \eqref{oc:uncertainty-shrink}.  Its right side is then
at most $32\lambda^2u/\delta_1^2\le u/2$ by
\eqref{oc:lambda-choice}.

Iterate the deterministic operation on supports in
\eqref{oc:uncertainty-set}.  The preceding reasoning holds for every
completion at each step, so the same iterated supports contain all
possible errors.  A depth-$q$ dependency cone is obtained by
composing at most $q$ forward or inverse A moves, with the finite
part and slot bookkeeping.  For fixed $q$ this is a fixed finite
list of permutations.  A consistent assignment to repeated initial
addresses in one cone extends to a global initial word agreeing
with $V_0$ outside $S$.  Applying the global support statement to
that extension proves the claimed result for that cone.  This
argument can use a different global extension for each target.

Assume now $|S|/|\Gamma|<\delta_1/4$.  Every iterated support stays
in this regime, so its integer cardinality decreases by a factor of
at least two while nonempty.  More than $\log_2|\Gamma|$ rounds make
it zero.  Running these rounds synchronously on the complete array
uses $O(|\Gamma|\log|\Gamma|)$ local calls, each of polynomial
bit complexity as established above.  This proves polynomial-time
global correction without using the existential greedy repair in
Lemma~\ref{oc:full-proximity} as an algorithm.
\end{proof}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/03-outer-codes.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/04-outer-programs.tex ====
\section{Certified programs and persistent tape equations}\label{sec:outer-programs}

We turn local code constraints into certificates for computations on the
tape. This section supplies local reversible filling
(Lemma~\ref{lem:outer-fill}) and a persistent tuple of tape equations
(Lemma~\ref{lem:outer-wiring}); Section~\ref{sec:outer-representation}
will assemble them into the evolving representation.

We retain the outer notation of Sections~\ref{sec:macrograph} and
\ref{sec:outer-codes}: $h=2^\ell$, $M_o=2^r$, the odd-characteristic
field $J$, the group $\Gamma=\mathbb F_2^r\rtimes H$, and coordinates
$k=(g,u)=(ux,u)$.  In particular $r=O(\log N)$,
$|H|\le r^9$, and a field element has $O(\log N)$ bits.
All constants and polynomial exponents in this section are fixed
independently of $N$.  The local star degree $d$ is a constant.
We pack the four parts $(s,t)\in\{0,1\}^2$ at one $k$ into one symbol;
``part distance'' instead normalizes over the four parts separately.
These distances differ by at most a factor of four.

There are two kinds of storage.  A \emph{full slot} has both axis
constraints and duplicated systematic payload, and thus equals
$E_{\alpha,\beta}v$ for a unique payload $v:\Gamma\to U$.
An \emph{A-only scratch slot} has the same local A constraints and
A-edge agreements, but its other labels and entry coordinates are
merely vector coordinates: no B condition is imposed.  Keeping all
intermediate certificates, rather than just the values being computed,
will let us certify a computation at every point of a reversible schedule.

\subsection{Full operations and a fixed query list}

\begin{definition}[Certified full operation]\label{op:full-operation}
A full operation specifies parent full slots, a target encoder
$E_{\mathrm{out}}$, and a local expression $X^{st}_k(i,j)$ in the parents.
For arbitrary exact duplicated full parents, $X$ must have duplicated
systematic payload $v$ and satisfy an allowed A-axis constraint system.
In addition, either its A system is that of the target, or $X$ satisfies
the target's B system.  The required target is $E_{\mathrm{out}}v$.
Scalar coefficients in linear combinations are identical at every
index.  They may depend on an identical clock and on previously
certified, completed public address data.  The specified covariant
primitive $W(k)\mapsto W(k)\rho_e(k)$ below may additionally use its
explicit index-dependent public matrix; it must satisfy the same
axis-constraint requirements on $X$.
\end{definition}

In the first case compare $X$ and the target on all systematic B columns
at $t=0$, retaining every entry in each such column at both A parts.
In the second case compare all systematic A rows at $s=0$, retaining
their entire rows at both B parts.  The one-axis uniqueness and distance
in Theorem~\ref{thm:outer-codes} give a constant $\gamma_{\mathrm{op}}>0$
with the following alternative: the required payload identity is true
and all comparisons hold, or these comparisons fail on at least
$\gamma_{\mathrm{op}}$ of the centers.  Indeed, equality of payloads
fixes each compared column or row by the shared one-axis extension.
If a payload differs, at least one compared column or row is a nonzero
codeword difference for that axis, so its vertex support has constant
density.  The number of labels and parts is fixed.  A comparison of
isolated systematic cells would not have this implication.

We use the following full operations, splitting sums into bounded-arity
stages whenever necessary.
\begin{enumerate}[label=(\roman*)]
\item Scalar linear combinations and right pulls
$R_bF(k)=F(kb)$ share the scalar A system.  Elementary pulls are the
identity, the three first-bit translations $(e_a,1)$, and one-step
Singer-cycle rotations of each block and their inverses.
\item A matrix payload can be reencoded while changing only one axis
type, using the other as the shared axis.  Constant linear maps and
contractions of fully untwisted entry alphabets also give full operations.
\item To multiply a payload by a one-bit Walsh character, we must also
certify the resulting full encoding. An index-dependent scalar
multiplier need not preserve the scalar edge equations. The matrix
representation supplies a shared-axis route through the twisted types.
Starting from an untwisted scalar slot $W$, form
\[
             X^{st}_k(i,j)=W^{st}_k(i,j)\rho_e(k),
\]
where $e$ is a distinguished first bit of a block and $\rho_e$ is the
representation of Section~\ref{sec:outer-codes}.  Multiplicativity gives
\[
 X^{1t}_{a_i k}(i,j)=\rho_e(a_i)X^{0t}_k(i,j),\qquad
 X^{s1}_{k b_j}(i,j)=X^{s0}_k(i,j)\rho_e(b_j).
\]
Scalar local checks continue to hold, since the same entry matrix
multiplies every local scalar.  Reencode first with twisted A and
untwisted B, sharing A, and then with both axes untwisted, sharing B.
The column of $\rho_e(g,u)$ indexed by $1_H$ has just one nonzero entry,
in row $u$, equal to
$(-1)^{(ue)\cdot g}=(-1)^{e\cdot x}$.
Summing this entry-alphabet column therefore produces the full scalar
encoding of $(-1)^{e\cdot x}v_W(k)$.  The contraction is on entry
coordinates, not on outer star labels.
\end{enumerate}
Every intermediate matrix slot in this construction has a specified
homogeneous linear payload recipe.  An arbitrary multiplier depending
on one bit, with values $a_0,a_1\in J$, is
$\frac12(a_0+a_1)I+\frac12(a_0-a_1)M_e$, where
$M_ef(x,u)=(-1)^{e\cdot x}f(x,u)$ is the corresponding modulation.
The divisions by two are valid because $J$ has odd characteristic.

Right rotations move these operations to any bit in a block.  For a
block rotation $v$ the pull convention gives
\[
 R_{(0,v)}R_{(e,1)}R_{(0,v^{-1})}=R_{(ve,1)}.
\]
Conjugating the modulation in the same way gives the modulation of bit
$ve$.  A power of a rotation is implemented as a sequence of single
rotations, returning to the original layout after the operation.
Thus an arbitrary bit operation costs $O(r)$ elementary full stages,
without introducing new query maps.

Fix once and for all a permutation list $\Pi$ containing identity,
the required forward and inverse A and B neighbor maps, left pulls by
a fixed generating list of $H$, the elementary right pulls above,
and right pulls by a fixed generating list of $H$ and its inverses.
Include any bounded compositions used in a local expression.
Its cardinality $q=|\Pi|$ is independent of $N$.
All constraints of all slots are checked together at a center using
the symbols at $\Pi(k)$.  A growing number of slots consequently
increases local computation and symbol size, but not the number of
queried positions.  In particular a single false exact full operation
retains the gap $\gamma_{\mathrm{op}}$, rather than having its gap
divided by the program length.

\subsection{Scalar transforms with certified stages}

\begin{lemma}[Scalar program library]\label{op:scalar-programs}
On scalar tables indexed by $x\in\mathbb F_2^r$, the full operations
above give homogeneous linear programs of $\poly(r)$ stages for:
bit-conjunction and interval masks; controlled bit flips and bit
permutations; integer cyclic shifts; discrete Fourier transforms and
their inverses of lengths $M_o$ and $h$; the Walsh transform; and cyclic
coefficient multiplication by any systematic Lagrange polynomial
$L_j$, $0\le j<h$.  The last operation is uniform also when $j$ is
supplied by already identical address bits.  All intermediate stages
are full slots with fixed encoder types and query maps in $\Pi$.
\end{lemma}

\begin{proof}
A bit-conjunction mask is a product of one-bit masks, applied in
sequence.  For the integer interval $[0,a)$ with $0\le a\le M_o$,
represent $a$ by $r+1$ public or already identical control bits.
Use the zero operator for $a=0$ and the identity for $a=M_o$.
For $0<a<M_o$, use the disjoint prefix rectangles obtained by
specifying the most significant differing bit: at a position where
$a$ has bit one, match its more significant bits and require the
current bit to be zero.  There are at most $r$ rectangles.
More explicitly, writing $a_r$ for the endpoint bit, one fixed
padded program applies the mask
\[
 a_r+(1-a_r)\sum_{i=0}^{r-1}a_i\,
        1[x_i=0]\prod_{j=i+1}^{r-1}1[x_j=a_j].
\]
All endpoint decisions and rectangle coefficients are identical at
every index; the program keeps all candidate stages and uses scalar
coefficient selection, so its topology and polynomial size do not
depend on $a$.  Differences of two such masks give arbitrary intervals.

Let $m$ be the indicator of a conjunction not involving bit $i$.
The pullback of the permutation that flips bit $i$ precisely when
$m=1$ is
\begin{equation}\label{op:controlled-flip}
                    f-mf+R_{(e_i,1)}(mf).
\end{equation}
To verify this, if $m(x)=0$ then $m(x+e_i)=0$, and the result is $f(x)$;
if $m(x)=1$ then $m(x+e_i)=1$, and the result is $f(x+e_i)$.
The independence of $m$ from the flipped bit is essential here.
Three controlled XOR operations exchange two index bits, so they
implement bit reversal and any prescribed bit permutation.

For an integer subtraction of $2^j$ modulo $2^r$, flip bit $i>j$
when the original bits $j,\ldots,i-1$ are all zero, and flip bit $j$
unconditionally.  Process the higher bits in descending order;
then none of the bits used by a condition has yet changed.
The addition rule uses ones instead of zeros.  Pullbacks reverse the
chronological composition of index permutations: thus the table
stages apply these borrow gates in ascending bit order, the reverse
of their descending index-permutation order.  This implements the
desired cyclic table shift.  A conjunction uses $O(r)$ bit
operations, so this procedure is polynomial in $r$.

For the Fourier transform we realize the radix-two Cooley--Tukey
decomposition~\cite[pp.~299--300]{CooleyTukey1965} by the certified full
operations above. Let $D=2^v\in\{h,M_o\}$ and let
$\xi=\zeta^{M_o/D}$ have order $D$.
The transform acts on the low $v$ integer bits, independently within
each block of fixed higher bits, and is
\begin{equation}\label{op:dft}
       \mathcal F_D(f)(y)=\sum_{z=0}^{D-1}\xi^{yz}f(z).
\end{equation}
First reverse these $v$ bits.  At level $j$, pair positions separated
by XOR of bit $j-1$ in each aligned window of length $2^j$.
Writing $a<2^{j-1}$ for the offset and $U,V$ for the lower and upper
entries, replace the pair by
\begin{equation}\label{op:butterfly}
          U+\xi^{aD/2^j}V,\qquad U-\xi^{aD/2^j}V.
\end{equation}
The factor is a product of one-bit factors, since if
$a=\sum_{i<j-1}2^ia_i$, then
$\xi^{aD/2^j}=\prod_{i<j-1}(\xi^{2^iD/2^j})^{a_i}$.
Half-window masks, the bit pull, these factors, and addition implement
the displayed rule.  The identity
\[
 \sum_{z=0}^{D-1}\xi^{yz}f(z)
 =\sum_{z=0}^{D/2-1}(\xi^2)^{yz}f(2z)
  +\xi^y\sum_{z=0}^{D/2-1}(\xi^2)^{yz}f(2z+1)
\]
proves the algorithm by induction, with bit reversal placing the
recursive even and odd subproblems in the required order.  More
explicitly, if $\operatorname{rev}_a$ reverses $a$ bits, after level
$j$ the entry at $b2^j+y$, $0\le y<2^j$, is
\[
 \sum_{z=0}^{2^j-1}(\xi^{D/2^j})^{yz}
 f\bigl(\operatorname{rev}_{v-j}(b)+2^{v-j}z\bigr).
\]
The level-$j$ butterfly combines the even and odd values of $z$ in
the level-$(j-1)$ formula, proving this invariant from the bit-reversed
input at level zero; at level $v$ it is \eqref{op:dft}.
Using $\xi^{-1}$ and multiplying by $D^{-1}$ gives the inverse:
$\sum_{y=0}^{D-1}\xi^{y(z-z')}=D\,1[z=z']$.
Without initial bit reversal, apply the sum/difference butterflies
directly to the input table, one coordinate bit at a time, and use
no twiddle factors.  This gives
\begin{equation}\label{op:walsh}
                   \mathcal W(f)(x)=\sum_y(-1)^{x\cdot y}f(y).
\end{equation}
They can be applied on all $r$ bits or just a specified block.
The Walsh inverse is $M_o^{-1}\mathcal W$ when all bits are used.

Finally set $\omega=\zeta^{M_o/h}$ and
\begin{equation}\label{op:lagrange}
 L_j(Z)=h^{-1}\sum_{a=0}^{h-1}\omega^{-ja}Z^a.
\end{equation}
The root sum just used gives $L_j(\omega^i)=1[i=j]$.
For a length-$M_o$ coefficient table $F$, cyclic multiplication by
$L_j$ has coefficient at $w$ equal to
\begin{equation}\label{op:lagrange-convolution}
 h^{-1}\omega^{-jw}\sum_{a=0}^{h-1}F_{w-a}\omega^{j(w-a)},
\end{equation}
with subscripts modulo $M_o$.  The powers are well defined under
cyclic indexing because $h\mid M_o$.  Each of the two diagonal
factors splits over the bits of $w$ and has coefficients computable
from $j$.  If $T_bf(w)=f(w-b\bmod M_o)$, the middle window sum is
\[
                    \prod_{i=0}^{\ell-1}(I+T_{2^i}).
\]
Every integer in $[0,h)$ has a unique binary expansion, proving this
identity.  The previously constructed shifts implement it.
When $\deg(F L_j)<M_o$, cyclic and ordinary multiplication coincide;
we impose this degree bound at every ordinary-product use.

For clarity, crude stage bounds suffice: a bit operation uses $O(r)$
elementary stages, a conjunction $O(r^2)$, a carry shift $O(r^3)$,
and the window product $O(r^4)$.  The Fourier, Walsh, and range-mask
constructions also fit a fixed polynomial bound.  Coefficient
calculation, layout descriptions and every elementary local operation
use polynomially many bit operations in $r$.  Padding unused steps
with zero or identity operations makes all lengths independent of
the particular control or address values.
\end{proof}

\subsection{Triangular A-only certificates}

An A-only program starts from A-coded public constants or a local
expression $X$ from Definition~\ref{op:full-operation}.  It forms
identical-coefficient linear combinations and right pulls of earlier
slots having the same A twist.  Right pulls commute with left
translations and constant entry twists, and therefore preserve A
constraints.  A fixed generating list of $H$ suffices to compute all
desired right-$H$ pulls through stored paths in its Cayley graph.
Each path has length at most $|H|-1$ after removing repetitions.
Arbitrary bit pulls use the rotation sequences already described.
The number of stored stages is polynomial in $r$, while all recurrence
queries still belong to $\Pi$.

At a fixed clock value, the initialized slots form triangular prefixes
and satisfy their recurrences; every other scratch slot is zero.
Filling one next stage evaluates its recurrence locally.  Erasing in
reverse order replaces that stage by zero while its parents are still
available.  Each step changes one designated slot or fixed family of
slots, with the same definition of every retained slot.

\begin{lemma}[Simultaneous scratch restoration]\label{op:scratch-restoration}
Suppose all scratch recurrences at a legal repeated clock are
triangular and their references, on arbitrary exact duplicated full
parents, satisfy their assigned A constraints.  Public address-focus
recurrences precede every recurrence using the resulting coefficients.
There exist constants $C_{\mathrm{sc}}$ and $\theta_{\mathrm{sc}}>0$,
independent of the number of stages, slots and entry coordinates,
with the following property.  If the packed parsing, clock, full-code,
duplication, scratch local-code, scratch-edge, recurrence and inactive-zero
tests reject on a fraction $\theta<\theta_{\mathrm{sc}}$ of centers,
then after one preprocessing of the parsing, clock and full slots,
all scratch can be replaced simultaneously by its exact triangular
reference at total symbol cost at most $C_{\mathrm{sc}}\theta$.
The references are taken on the corrected full parents; those parents
need not yet satisfy their advertised payload-operation identities.
\end{lemma}

\begin{proof}
Theorem~\ref{thm:outer-codes} first repairs parsing, repetition and
full codes at $O(\theta)$ symbol cost.  A repeated illegal clock is
excluded by its range test.  The full-code duplication gap then
excludes any failure of duplicated payload, for small enough $\theta$.
Replace all locally A-invalid scratch parts by zero.  This is one
further packed $O(\theta)$ change.  Call the resulting word $W$, and
\emph{keep this word fixed throughout the following induction}.

If preprocessing changed at most $C_{\mathrm{pre}}\theta$ packed
sites, any test using $\Pi$ can change its outcome only on their
$q$ inverse images.  After harmless fixed conversions of edge and
part normalizations there are constants $C_z,C_r$ such that
\begin{equation}\label{op:fixed-word-invariant}
 z_A(W)\le C_z\theta,\qquad
 \Pr[\text{some scratch recurrence or inactive-zero test fails on }W]
       \le C_r\theta.
\end{equation}
One can use constant multiples of $1+qC_{\mathrm{pre}}$ for both
constants.  These bounds concern the \emph{entire} packed scratch
tuple, so they contain no slot-count factor.

Order all initialized families topologically, putting public focus
first, and include the inactive-zero families with their zero
references.  Let $V_j$ be the exact reference of the first $j$ families
on the corrected full parents and legal clock, and let $p_j$ be the
joint part distance from the corresponding prefix of $W$ to $V_j$.
The empty prefix has $p_0=0$.  The full parents may have incorrect
advertised identities, but are A-coded; every allowed initial $X$
is A-coded on such parents.  Right pulls and identical coefficients
preserve this property.  Public focus begins with hardwired correct
constant encodings and the identical clock.  Its completed reference
supplies globally identical coefficients to subsequent recurrences.
Consequently every $V_j$ is exactly A-coded.

An error in the new family occurs only at a recurrence-failure center
or at a center reading an erroneous earlier part.  A joint part-error
set of density $p_j$ occupies at most $4p_j$ of the packed sites.
Taking a union over the $q$ maps, and retaining the old prefix, gives
\begin{equation}\label{op:prefix-provisional}
                 p_{j+1}\le C_r\theta+(1+4q)p_j.
\end{equation}
This remains true when a recurrence reads all earlier slots or types
at one of those sites: the error event is already joint.
Put $K=2C_z/\delta_1$ and choose $\theta_{\mathrm{sc}}$ so small that
\begin{equation}\label{op:scratch-smallness}
          (C_r+(1+4q)K)\theta_{\mathrm{sc}}<\delta_1/16.
\end{equation}
If $p_j\le K\theta$, the provisional estimate is in the small-error
range of the A near-code inequality of Theorem~\ref{thm:outer-codes}.
Apply that inequality to the \emph{same fixed word} $W$, restricted
to the enlarged tuple, and to $V_{j+1}$:
\[
 \delta_1p_{j+1}
 \le z_A(W|_{j+1})+4p_{j+1}^2+\lambda p_{j+1}.
\]
The conflict count of a prefix is at most that of the whole tuple.
Since $\lambda\le\delta_1/8$ and $p_{j+1}<\delta_1/16$, this implies
\[
                 p_{j+1}\le 2z_A(W)/\delta_1\le K\theta.
\]
Induction proves this bound for the complete scratch tuple.
It resets an \emph{estimate}, without making successive word repairs.
Only now replace the whole scratch tuple by its reference, at symbol
cost at most $4K\theta$.  The common scalar local checks act on joint
vectors, and the finitely many twist types act by a block-diagonal
invertible map; the near-code estimate is therefore valid in exactly
this joint metric.  Together with preprocessing this proves the lemma.
\end{proof}

\subsection{Mode-serial filling of a full slot}

The target must remain a full codeword while it is being filled, even
though its final value can depend on the parents at every index. We use
a sequence of frequency cutoffs that commute with the encoder. Each
increment is supported on one $H$-orbit, so its encoded contribution
can be computed from a polylogarithmic list of coefficients gathered
locally. A completed certificate for the old cutoff is retained until
the next cutoff and its increment have both been prepared.

Retaining the completed target while reversing its auxiliary computation
follows the reversible-cleanup principle of Bennett~\cite[pp.~525--526]{Bennett1973}.
The two-buffer schedule, local transitions, and certification guarantees
needed here are established in the following proof.

\begin{lemma}[Local reversible filling]\label{lem:outer-fill}
For every full operation in Definition~\ref{op:full-operation}, there
is a canonical reversible schedule from zero target and zero temporary
scratch to its required full target and zero temporary scratch.
Parents and all coefficient data remain fixed throughout.  The schedule
has $O(M_o\poly(r))$ steps, a $\poly(r)$-bit local symbol and local
algorithm, and an $O(\log M_o+\log\poly(r))$-bit microclock.
Its checks and both local transition directions use only $\Pi$.
At each clock location the target is a specified full encoding and
all scratch consists of specified initialized prefixes.  After
simultaneous scratch restoration, failure of the required target
identity has a constant center gap.  Each reference scratch slot
satisfies its A constraints, even when its full parents have not yet
been shown to satisfy their advertised payload identities.
\end{lemma}

\begin{proof}
Write the operation's local expression as $X$ and its duplicated
systematic payload as $v$.  For each label, part and entry coordinate
expand in the $x$ variable:
\[
 X(x,u)=\sum_{\nu\in\mathbb F_2^r}\widehat X_\nu(u)(-1)^{x\cdot\nu},
 \qquad
 \widehat X_\nu(u)=M_o^{-1}\sum_xX(x,u)(-1)^{x\cdot\nu}.
\]
For $0\le\tau\le M_o$ define a Fourier multiplier $P_\tau$ by
\begin{equation}\label{op:cutoff}
 w_\tau(\nu)=|H|^{-1}\sum_{a\in H}
                    1[\underline{a^{-1}\nu}<\tau].
\end{equation}
Thus $P_0=0$ and $P_{M_o}=I$.  The inverses of $|H|$ and $M_o$
exist in $J$.  The multipliers are constant on frequency orbits of
$H$; they are not asserted to be projections.

Here is the required commutation in detail.  A left pull by $(w,a)$
sends $(x,u)$ to $(x+u^{-1}a^{-1}w,au)$, so it preserves each
frequency, changing its coefficient by a known sign and an $H$-index
permutation.  A right pull sends $(x,u)$ to
$(a^{-1}(x+w),ua)$, carrying a frequency $\nu$ to $a\nu$, with
sign $(-1)^{w\cdot a\nu}$.  Hence $P_\tau$ commutes with both pulls.
The constant entry twists act on a different coordinate and also
commute.  Every parity operator and every systematic injection thus
intertwines $P_\tau$ on its domain and codomain.  If an invertible
parity operator $T$ satisfies $TP_\tau=P_\tau T$, then its inverse
does too, by multiplication by $T^{-1}$ on both sides.  Applying the
two one-axis extensions gives
\begin{equation}\label{op:cutoff-extension}
              P_\tau E_{\mathrm{out}}v
                      =E_{\mathrm{out}}(P_\tau v).
\end{equation}
No commutation with a preliminary index-dependent operation such as
$W(k)\mapsto W(k)\rho_e(k)$ is needed: $P_\tau$ is applied to its
already formed expression $X$.

The mode-bit selector
\begin{equation}\label{op:mode-selector}
                   \frac{I+(-1)^bR_{(e_i,1)}}2
\end{equation}
retains precisely those frequencies with $\nu_i=b$.
For each $a\in H$, decompose
$\underline{a^{-1}\nu}<\tau$ into at most $r$ disjoint prefix
rectangles and multiply their selectors.  Sum over rectangles and
average over $a$.  This gives an A-only program computing $P_\tau X$
with $\poly(r)$ stages.  Include all candidate rectangles and use
coefficient bits from the microclock, so topology and length are fixed.
The case $\tau=M_o$ uses the identity with the same padding.
All selectors use the elementary pull list via stored bit alignments.

The partial target at cutoff $\tau$ is
$Y_\tau=E_{\mathrm{out}}(P_\tau v)$.
Compare it with the completed cutoff program using the shared-axis
test of Definition~\ref{op:full-operation}.  In the shared-B case,
intermediate scratch is only A-coded.  The test's justification is
made \emph{after} Lemma~\ref{op:scratch-restoration}: the completed
reference is then exactly $P_\tau X$, which satisfies the target's
B constraints by the commutation above.  One may now use B distance
on its whole systematic A rows.  This argument neither assumes nor
tests B validity of unfinished scratch.  The shared-A case uses the
analogous columns.  Thus equality of the required payload suffices,
and its failure has the same constant gap at every cutoff.

It remains to compute a cutoff increment locally.  Let $a_\tau$ be
the bit vector for the integer $\tau<M_o$, and set $O=Ha_\tau$.
The increment multiplier is zero outside $O$ and, for $\nu\in O$, is
\begin{equation}\label{op:orbit-weight}
 (w_{\tau+1}-w_\tau)(\nu)
 =\frac{\#\{a\in H:aa_\tau=\nu\}}{|H|}
 =\frac1{|O|}.
\end{equation}
For each $\nu$ in the fixed-length list $(aa_\tau)_{a\in H}$,
apply all $r$ mode selectors to $X$.  Repetitions are harmless.
The resulting pure-mode slot is
$X_\nu(x,u)=(-1)^{x\cdot\nu}\widehat X_\nu(u)$.
Also store each right pull $R_{(0,b)}X_\nu$, $b\in H$, through
generator paths and their intermediate stages.  At the current
$k=(x,u)$ its systematic payload is
\begin{equation}\label{op:orbit-focus}
             (-1)^{(b^{-1}x)\cdot\nu}\widehat v_\nu(ub).
\end{equation}
For any desired $u'\in H$, take $b=u^{-1}u'$ and remove the displayed
known sign.  One symbol now contains every coefficient
$\widehat v_\nu(u')$ for $\nu\in O$ and every $u'\in H$.
These are read from locally stored pull slots, not from remote global
payload positions.

We can apply $E_{\mathrm{out}}$ to this coefficient list locally.
Remark~\ref{oc:orbit-invariance} supplies the inverse of each one-axis
parity system on the $O$-frequency subspace: both this subspace and its
complement are invariant, and the restricted domain and codomain have
equal dimensions. Here its Fourier coordinates are indexed by frequency,
$H$, labels, parts and entry coordinates, so the restricted dimension is
at most
\begin{equation}\label{op:orbit-dimension}
         C|O|\,|H|\,\dim_J U\le C|H|^4=\poly(r),
\end{equation}
where $C$ depends only on the fixed labels, and
$\dim_JU\le |H|^2$.  Form its entries using the explicit pulls and
Walsh signs, solve the two successive axis systems by Gaussian
elimination, and sum the output modes at $x$.  Together with
\eqref{op:orbit-weight}, this computes every local entry of
\[
                 E_{\mathrm{out}}((P_{\tau+1}-P_\tau)v).
\]
Matrix assembly, elimination and field arithmetic take $\poly(r)$
bit operations.  Canonical parsing and fixed defaults on malformed
values, illegal clocks or failed pivots make the local algorithm total;
on the promised data the prescribed matrices are invertible.

Use two cutoff-program buffers and one orbit-focus buffer.
Initially the target and all three buffers are zero.
At iteration $\tau$ keep the completed buffer for cutoff $\tau$
when $\tau>0$, and compare the target with it; at $\tau=0$ compare
the entire target with zero.  In the other, empty buffer fill the
cutoff-$\tau+1$ program in dependency order, and fill the orbit focus
in dependency order.  With both complete, perform one pivot adding
the local encoded increment to the target.  Its required comparison
now uses the completed cutoff-$\tau+1$ buffer.  Erase focus in reverse
order and erase the old cutoff buffer in reverse order when present.
Swap the buffer roles, determined by the iteration parity.
At cutoff $M_o$, compare the target directly with $X$ while erasing
the final cutoff buffer.  This finishes with the full desired target
and zero temporary scratch.

At every clock value enforce the full-code and duplication constraints
of the target, all initialized-prefix recurrences, inactive zeros,
and the stated comparison.  While a new proof is being prepared, a
completed old proof remains available.  At a pivot both the next
proof and the increment focus are complete.  In reverse traversal,
erased prefixes are first reconstructed in forward dependency order,
so the reverse pivot subtracts the same locally available increment.
Thus both directions are local, mutually inverse on canonical data,
and never use a partial certificate as a completed one.

For the size bound, each cutoff program uses at most
$O(|H|r^3)$ elementary stages up to fixed padding factors.
The orbit selectors and all right-$H$ paths use at most
$O(|H|r^2+|H|^3)$ such stages, with a polynomial allowance for
metadata.  Even a bound $O(r^{30})$ for both programs and their
descriptions suffices.  Each entry alphabet has at most $r^{18}$
field coordinates, each of $O(r)$ bits.  The two reused buffers
therefore have $\poly(r)$ bits per index.  There are $M_o$ iterations
of a common polynomial sublength, with a special initial omission
and final erasure; their location is decoded by arithmetic on a
logarithmic microclock.  No local step loops over all $M_o$ modes.
The different cutoffs reuse the same zeroed buffers and do not
contribute a factor $M_o$ to the word length.  Separate buffers for
the finitely many required A types keep every slot's type fixed.
These observations prove all bounds and claims of the lemma.
\end{proof}

\subsection{From sparse tape wiring to row systems}

We next construct a persistent tuple $P(b)$ independent of the clock
frame, where $b$ comprises the four tape-content and occupation arrays.
Its canonical formulas are defined for every $b\in J^{4h}$; Boolean
and phase-validity tests are additional conditions.
Testing a uniformly chosen tape row would not detect one false row with
constant probability. Instead we interpolate the row data, certify the
required products, and encode the remainder of each row-equation
polynomial modulo $Z^h-1$. A single false row makes that remainder
nonzero; comparing its full encoding with zero then has the code's
constant gap. The tuple retains the intermediate linear computations
needed to perform these checks at every frame.

\begin{lemma}[Persistent wiring and certification]\label{lem:outer-wiring}
There are $\poly(r)$ persistent full slots $P(b)$, with polynomial
canonical payload formulas in $b\in J^{4h}$, and packed local tests
using $\Pi$, with the following properties.
\begin{enumerate}[label=(\roman*)]
\item Structural tests uniquely determine $P(b)$ from its four logical
arrays.  Each variable factor array is a fixed permutation of one
logical array; a one-address logical edit changes that factor at at
most one publicly determined address.
\item Each primary has assigned logical degree at most a fixed $d_2$.
Every other persistent slot has a homogeneous linear recipe on the
primaries, including all intermediate matrix reencodings and all
phase-specific computations.  Its assigned logical degree can be
the maximum degree of its primary inputs.
\item There are persistent full encodings of
\begin{equation}\label{op:address-transform}
       U_D(x)=\sum_{j=0}^{h-1}(-1)^{x\cdot j}D_j
\end{equation}
for every needed logical singleton $D$, where $j$ denotes its padded
$r$-bit vector.  Its Walsh coefficient at $j<h$ is exactly $D_j$.
\item Conditional on exact duplicated full codes and a legal repeated
clock, either all structural formulas hold and $b$ is a valid
Boolean macrovertex at that clock, or one packed test rejects on
at least a fixed fraction $\gamma_{\mathrm{per}}>0$ of centers.
The same alternative without the macro-validity conclusion holds
if only structural tests are imposed.
\end{enumerate}
The canonical tuple and public data are deterministically constructible
in $\poly(N)$ time.  The number of field coordinates, public local
parameters and bit operations at one index is $\poly(r)$.
\end{lemma}

\begin{proof}
\paragraph{Sparse row equations.}
Use the normalization in Lemma~\ref{lem:macrograph}.
The Boolean tests for the four arrays are $z(z-1)=0$ at each row,
and content must vanish where occupation is zero, by $a(1-e)=0$.
For occupation in a prescribed order $\pi$, impose
\[
        e_{\pi(i+1)}(1-e_{\pi(i)})=0\quad(0\le i<h-1).
\]
Together with Booleanity, these equations say that occupied positions
form a prefix.  Its desired length is imposed separately by its
integer count.  Full occupation may instead be imposed row by row.
Fixed gate truth tables have multilinear representing polynomials
over $J$: for a Boolean input $a\in\{0,1\}^m$ its indicator is
$\prod_{i:a_i=1}z_i\prod_{i:a_i=0}(1-z_i)$, so summing the indicators
weighted by the truth-table values gives the desired polynomial.
The arity is fixed.  Gate, input-copy, designated-comparison, dummy,
and tape-to-tape-copy equations are conditioned by the occupation
of their destination where appropriate.  This expresses exactly
the phase rules in Section~\ref{sec:macrograph}; the phase prescription
ensures that an occupied computation only depends on earlier occupied
positions or the fixed full tape.  In a full $Q_x$, its $S$ inputs
are free, with the terminal and other-input relations imposed as in
its definition.  At shared phase boundaries fix either equivalent
description.

For each of the four phases and each unconditional family, organize
these equations into a fixed number of length-$h$ lists, padding with
zero equations.  Every operand map from list rows to an original
logical array has bounded multiplicity: this follows from bounded
fan-out for internal feeds, separate designated copies, and partial
permutations for adjacency and same-position copies.
An operand map of multiplicity at most $B$ splits into $B$ partial
injections by coloring its preimages separately at each target.
Each partial injection extends to a permutation of $[h]$, by any
bijection between the unused domain and range.  Separate logical
array types and intersect the finitely many operand-color and
active-row cases.  Because arity and $B$ are constants, this causes
only a constant number of cases.  Expand the representing polynomials
into monomials.  Each row list is now a sum of a fixed number of
bounded-length monomials, whose factors are permuted logical arrays
or public coefficient and mask arrays.  Empty monomials represent
constant terms.  Counting coefficient factors in the length still
gives a fixed bound on the degree.

\paragraph{Routing arbitrary permutations.}
For completeness a permutation of $2^\ell$ locations has a binary
switch realization with $2\ell-1$ layers, each layer switching pairs
along one fixed XOR bit, with a public Boolean mask constant on each
pair. This is the binary specialization of the Bene\v{s}
rearrangeability construction~\cite[Lemma~3 and Theorems~2--3]{Benes1964Permutation}.
Pair sources along the top
bit and do the same for targets.  The prescribed item routes form a
bipartite multigraph on these pairs with degree two at every vertex.
Every component is an even cycle, including a pair of parallel edges;
alternate its edges with two colors.  An initial top-bit switch and
a final top-bit switch route each item through the half corresponding
to its color.  Each half now has a permutation of $2^{\ell-1}$ items.
Recurse with the same layer schedule in the two halves.  The base case
$\ell=1$ has one layer and $\ell=0$ none; the recurrence for depth is
$L(\ell)=L(\ell-1)+2$.  Cycle coloring and recursive routing take
polynomial time in $h$.

Introduce a variable factor type for every routed intermediate, and
for each layer introduce the XOR-pulled counterpart $D^*$ of its input
$D$.  The layer output $D'$ satisfies the row equation
\begin{equation}\label{op:switch-row}
                      D'-D-m_{\mathrm{sw}}(D^*-D)=0.
\end{equation}
The mask is pair-constant, so this is exactly either the identity or
a transposition on each pair.  The counterpart identity itself is
checked by a scalar program below.  These equations are always
imposed, including in inactive phases.  Induction through the network
makes each variable factor a known permutation of an original logical
array, for arbitrary field values of that array.
There are only $O(r)$ layers among a fixed number of routing networks,
hence $\poly(r)$ factor and equation types with bounded arity per row.
For an edit of an original logical array at $j$, every factor derived
from it changes only at the image of $j$ under its fixed permutation.
All such addresses for one frame occupy $\poly(r)$ bits.
Their entire frame-indexed tables need not have a succinct random-access
formula; they will be stored as public encoded tables and focused
before use in Section~\ref{sec:outer-representation}.

\paragraph{Primaries and assigned degree.}
For every factor array $D\in J^h$, define its unique degree-$<h$
interpolant on the subgroup grid by
\begin{equation}\label{op:factor-interpolant}
                  F_D(Z)=\sum_{j=0}^{h-1}D_jL_j(Z).
\end{equation}
For each factor-occurrence list of each monomial, include primaries
for every subset $S$ of its occurrences, with payload the
length-$M_o$ coefficient table, padded by zeros, of
\begin{equation}\label{op:subset-primary}
                         F_S(Z)=\prod_{a\in S}F_{D_a}(Z).
\end{equation}
The index set here consists of occurrences: repeated uses of one
factor remain distinct.  The empty subset gives $1$.
Identify shared singleton primaries or impose direct full-word
equalities on copies, and include independent singletons when needed.
The assigned logical degree $k_S$ counts variable occurrences only;
public factors have degree zero.  Let $d_2$ bound the length of every
factor-occurrence list after all switch and row equations have been
included.  It is a fixed constant, and $k_S\le d_2$.
The honest ordinary polynomial degree is at most $d_2(h-1)$.
Choose the earlier size constant so that
\begin{equation}\label{op:degree-slack}
                         M_o>(8d_2+8)h.
\end{equation}
All primary formulas exist for arbitrary $b\in J^{4h}$.

\paragraph{Index independence and support.}
Scalar primaries are intended to depend only on $x$, not on $u$.
Test invariance of their payloads under left $(0,a)$ pulls for the
fixed generating list of $H$.  Such a pull fixes $x$ and changes
$u$ to $au$, and shares B with the original untwisted scalar slot.
Compare whole systematic A rows as above, using the pulled local
expression without introducing a free encoder witness.
The shared-B exact gap forces invariance under each generator;
since these generate $H$, this is precisely independence from $u$.
No independent invariance test is needed on a derived program stage
whose exact recipe already determines it, and matrix intermediates
need not be $u$-independent.

Use range masks from Lemma~\ref{op:scalar-programs} to require
singleton coefficient support in $[0,h)$ and product-primary support
in $[0,d_2h+1)$, with sharper public bounds when appropriate.
Encode the masked complement as a full output and compare that
entire word with zero.  A nonzero output has full-code distance.
Compare each public singleton and each empty product with its
precomputed canonical full encoding.  These are structural conditions.

\paragraph{Product certification.}
Compute the length-$M_o$ DFT of each relevant primary coefficient
table by certified full stages.  Its payload at integer $y$ is the
polynomial's value at $\zeta^y$.  For every nontrivial subset choose
one occurrence $a\in S$ and check, at one fixed systematic payload
cell at $s=t=0$,
\begin{equation}\label{op:primary-product-test}
                   F_S(\zeta^y)
                  =F_{S\setminus\{a\}}(\zeta^y)F_{D_a}(\zeta^y).
\end{equation}
These are pointwise nonlinear tests; the stored transform stages
themselves have linear recipes.  Once support, independence and
transform identities hold, a false equality is the nonzero polynomial
$F_S-F_{S\setminus\{a\}}F_{D_a}$, of degree less than
$(d_2+1)h+1<M_o/2$.  A nonzero polynomial of degree $D$ over a field
has at most $D$ roots, as follows by successively dividing by $Z-z$
at distinct roots.  Thus the false equality fails at more than half
the $M_o$ distinct points $\zeta^y$.  Uniform $k$ has uniform $x$,
and independence from $u$ prevents the extra index from diluting this
gap.  Induction on subset size forces exactly all products
\eqref{op:subset-primary} once these tests hold.

\paragraph{Counterparts and the subgroup annihilator.}
Apply the length-$h$ Fourier transform to the low $\ell$ bits of each
singleton's padded coefficient table.  Its first block is the value
table $D$, and every higher block is zero by coefficient support.
For each XOR counterpart use the certified bit pull on this padded
table and compare its full output with the counterpart's separately
computed padded value table.  Their full-operation gap forces the
exact counterpart identity needed in \eqref{op:switch-row}.

For a row-equation list let $R(Z)$ be the indicated linear combination
of its product primaries.  Its value at $\omega^j$ is exactly the
row expression at $j$.  Its remainder modulo $Z^h-1$ has coefficients
\begin{equation}\label{op:remainder-fold}
       [Z^i](R\bmod(Z^h-1))
          =\sum_{z=0}^{M_o/h-1}[Z^{i+zh}]R,
                \quad 0\le i<h,
\end{equation}
and zero padding elsewhere.  This formula follows from
$Z^{i+zh}\equiv Z^i\pmod{Z^h-1}$, since the coefficient table is
supported below $M_o$.  It is a short linear program: apply
$\prod_{i=\ell}^{r-1}(I+R_{(e_i,1)})$ to sum all high-bit blocks,
then mask to high bits zero.  Require its entire full word to be
zero when the equation is imposed.

The polynomial $Z^h-1$ is exactly the annihilator of the subgroup grid:
it has the $h$ distinct roots $\omega^j$, because $h$ is invertible
in $J$, and its degree is $h$.  The remainder has degree below $h$.
Therefore it vanishes as a polynomial if and only if $R$ vanishes at
all grid points.  In one direction divisibility suffices; in the
other direction a degree-$<h$ polynomial with $h$ distinct roots
is zero.  This proves that the folded full-word checks impose the
row equations exactly, without any random sampling loss over rows.
All structural switch equations and their folded computations are
always present.  All phase-specific folded computations are also
stored persistently; the repeated clock selects only which final
zero conditions are imposed.

\paragraph{Counts and address transforms.}
The constant coefficient of \eqref{op:factor-interpolant} is
\begin{equation}\label{op:occupation-count}
                       [Z^0]F_e=h^{-1}\sum_{j=0}^{h-1}e_j,
\end{equation}
since every $L_j$ has constant coefficient $h^{-1}$.
Mask the occupation coefficient table to index zero, using all bit
masks.  Compare its full word with the public full encoding of
$1[\underline x=0]$ times $c h^{-1}$, where the expected count
$c\in[0,h]$ is computed from the legal repeated stage and phase.
This scalar is used in the comparison; it is not part of a changing
persistent value.  On Boolean occupation data, characteristic
$p>M_o\ge h$ implies that two counts in $[0,h]$ are equal in $J$
only when they are equal as integers.  Together with monotonicity
this enforces the prescribed prefix exactly.

For address access, take the padded length-$h$ value table obtained
above and apply the full $r$-bit Walsh transform.  Its result is
\eqref{op:address-transform}.  Walsh orthogonality gives
\[
 M_o^{-1}\sum_x U_D(x)(-1)^{x\cdot j}
 =\sum_{i<h}D_i
        \left(M_o^{-1}\sum_x(-1)^{x\cdot(i+j)}\right)
 =D_j \qquad(j<h).
\]
In particular the high bits of the zero-padded value table cause
$U_D$ to repeat across the corresponding high output blocks;
they do not introduce an additional normalization factor.

\paragraph{Exact alternatives and complexity.}
Every stage of every mask, transform, counterpart comparison and
remainder computation is stored as a full slot with its specified
linear recipe, including mixed and untwisted matrix intermediates.
Give a derived stage the maximum assigned degree of its primary
ancestors and retain its ancestor subprogram or shared directed
acyclic recipe.  Public indexed arrays are degree-zero inputs with
precomputed encoded constants.

On exact duplicated full codewords, a false independence, support,
public-constant, full-operation, counterpart, folded-zero or count
identity has a constant gap by the shared-axis or full-code distance.
If none is false, the polynomial root argument forces all primary
products.  Singleton support makes the interpolant unique from its
subgroup values.  The counterpart and switch equations then determine
all factors as their prescribed permutations of the four original
logical arrays, and the public factors are fixed.  Thus the structural
tests force precisely $P(b)$.  The remaining Boolean, prefix, count
and selected-phase row equations are exactly the macrovertex rules.
If any is false, its folded or count check has the same exact gap.
All tests are packed at the center; choosing one false relation in
this argument entails no division by their number.  The minimum of
the fixed code gaps and the product gap gives
$\gamma_{\mathrm{per}}>0$ and proves both alternatives.

There are $O(r)$ routed factor and row families before the scalar
program expansions, a constant number of occurrence subsets per
monomial, and polynomially many stages per transform.  Thus the
number of full slots, recipes and local metadata is $\poly(r)$.
The entry dimension and field bit length give a further fixed
polynomial factor.  At one center the checker scans these lists,
performs field operations and evaluates fixed-length control
decisions, all in $\poly(r)$ bit operations.  It reads public encoded
constants only at the publicly specified positions in $\Pi(k)$;
arbitrary hidden-frame table lookup is never assumed.
All public permutations, masks, coefficients, encodings and complete
canonical arrays are constructible globally by polynomial-time
combinatorial and linear-algebra algorithms.  Since
$|\Gamma|=M_o|H|=O(N\operatorname{polylog} N)$ and every scalarized encoding system
has polynomial dimension in $N$, this takes $\poly(N)$ time.
The persistent word length is $O(N\operatorname{polylog} N)$ field-bit storage,
with a fixed exponent in the polylogarithmic factor.  This completes
the construction and proof.
\end{proof}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/04-outer-programs.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/05-outer-representation.tex ====
\section{A locally evolving outer representation}\label{sec:outer-representation}

We now combine the macrograph, the outer codes, and the certified programs
into one representation. A long preparation is part of the represented
path: its current position is recorded by a binary clock, while its scratch
space is reused. Thus preparation time does not multiply the length of a
represented vertex. The main additional requirement is that every retained
certificate, including certificates introduced to verify other
certificates, can change locally when a logical tape entry changes.

All notation in this section belongs to the outer construction. In
particular, $J$, $H$, and $\Gamma=\F_2^r\rtimes H$ are those of
Section~\ref{sec:outer-codes}. We retain $M_o=2^r$, the tape length $h$,
the four logical arrays $b$, and the persistent tuple $P(b)$ of
Lemma~\ref{lem:outer-wiring}. The assigned degree $k_S$ of a primary
$F_S$ counts its variable factor occurrences, and $d_2$ is the fixed upper
bound on the number of factor occurrences. In particular,
\[
 r=O(\log N),\qquad M_o=O(N),\qquad |H|\leq r^9,
 \qquad M_o>(8d_2+8)h.
\]
Every persistent program stage has a specified linear payload recipe in
the primaries. This includes support masks, transforms, routing checks,
remainder computations for inactive phases, and every matrix-valued
reencoding intermediate. Its assigned degree is at most the maximum
assigned degree of the primary inputs to that recipe.

\subsection{Finite differences of the complete certificate tuple}

Fix a start frame $c\in\{0,\ldots,4h-1\}$. The finite edit alphabet
$\Sigma$ of Lemma~\ref{lem:macrograph} describes additive changes
$h_\sigma(c)$ of the logical arrays. Enlarge this fixed alphabet by inverse
edits and a zero edit if necessary. All addresses depend only on
$(c,\sigma)$ and public routing data. The choice of the edit can depend on
the critical logical bits; a recipe for a specified edit cannot.
For a field-valued function $F$ define
\[
 \Delta_\sigma F(b)=F(b+h_\sigma(c))-F(b).
\]
This definition is made on arbitrary field arrays $b$, independently of
whether they satisfy the macrograph conditions.

For a variable singleton occurrence $a$, routing gives either no change or
a change $\epsilon_{a,\sigma}L_{j_{a,\sigma}}(Z)$, where
$\epsilon_{a,\sigma}\in\{-1,0,1\}$ and $j_{a,\sigma}<h$ are public at
this frame. A default address is used when $\epsilon_{a,\sigma}=0$.
Public factors have zero increment. If $S_{\rm var}$ is the set of
variable \emph{occurrences} in $S$, distributivity gives
\begin{equation}\label{oe:primary-difference}
 \Delta_\sigma F_S
 =\sum_{\varnothing\ne A\subseteq S_{\rm var}}
     F_{S\setminus A}(b)
     \prod_{a\in A}\epsilon_{a,\sigma}L_{j_{a,\sigma}}.
\end{equation}
Repeated occurrences are retained separately in this formula. Every
primary on its right has assigned degree at most $k_S-1$. Multiplication
by a specified Lagrange polynomial is the linear coefficient program of
Lemma~\ref{op:scalar-programs}. The number of subsets in
\eqref{oe:primary-difference} is bounded in terms of $d_2$ alone.
Every ordinary polynomial product appearing here has degree at most
$d_2(h-1)<M_o$; consequently its implementation by cyclic coefficient
convolution has no wraparound error. All possible addresses and zero
coefficients use a common padded program topology.

\begin{lemma}[Closure under finite differences]\label{oe:derivative-closure}
For every positive-degree persistent full slot $D$ and every
$\sigma\in\Sigma$, one can allocate a finite certified full program whose
output, in the same encoder type as $D$, is the increment of $D$ under
$b\mapsto b+h_\sigma(c)$. One can do this simultaneously for every
positive-degree full intermediate introduced by these programs, including
all their verification intermediates. The resulting list has
$\operatorname{poly}(r)$ slots and description bits. Every new slot has a
specified linear recipe in persistent primaries of strictly smaller
assigned degree than its parent target. All identities hold on arbitrary
field arrays $b$.
\end{lemma}

\begin{proof}
For a primary use \eqref{oe:primary-difference}. If $D$ is another
persistent stage with linear payload recipe $\mathcal L_D$, apply
$\mathcal L_D$ to the corresponding primary increments, replacing
degree-zero primary increments by zero. At this fixed frame its
coefficients, pulls, and encoder types are fixed. Hence
\[
 \mathcal L_D\bigl((F_S(b+h_\sigma(c)))_S\bigr)
 -\mathcal L_D\bigl((F_S(b))_S\bigr)
 =\mathcal L_D\bigl((\Delta_\sigma F_S(b))_S\bigr).
\]
The operations used in Section~\ref{sec:outer-programs} implement this
identity at the payload level. In particular, multiplying a scalar slot
by the public matrix $\rho_e(k)$ is linear in that scalar; changing one
encoder axis or contracting matrix entries is also linear. These
operations introduce no additional logical degree.

Allocate full slots for every stage of each increment program, with new
slots for its mixed and untwisted verification stages. An identity or zero
stage may be inserted to give an output its required type. Order stages
topologically, so that Lemma~\ref{lem:outer-fill} can initialize one full
target using already completed parents. For every newly allocated stage,
record the ancestor directed acyclic graph that expresses its payload in
the persistent primaries. Its roots have degrees at most one less than
those of the target whose increment is being computed. If its assigned
degree is positive, apply the same construction to this new stage for
every edit. This includes a stage introduced solely for verification: it
will be retained at full preparation and therefore needs its own
increment.

Inlining here means substitution in a directed acyclic graph, with
sharing retained within each target computation. If a primary is used
many times, substitute its increment program once and fan out its result.
Separate target computations may use fresh copies. Along any chain of
newly generated targets the assigned degree drops by at least one.
Therefore the construction stops after at most $d_2$ derivative tiers.
At each tier the number of target graphs, their sizes, their ancestor
descriptions, and their substitutions are bounded by a fixed polynomial
in the preceding total description size and $r$. A fixed number of such
polynomial expansions remains polynomial in $r$.

The programs verify the computed increments by their own stage
relations, starting from the lower-degree primaries. They do not test a
finite-difference assertion about an otherwise unspecified witness.
Neither their construction nor their identities use the final validity
equations. This also ensures that increment recipes remain defined on
field data produced by an edit that is not a legal macrograph move.
\end{proof}

\subsection{Public addresses, logical reads, and a reversible preparation}

Public address tables need not admit succinct evaluation at a hidden
frame $c$. We therefore represent the needed access explicitly. For each
field entry $a(c)$ of the public data table, form the scalar payload
\begin{equation}\label{oe:public-walsh-table}
 V_a(x,u)=\sum_{0\leq c'<4h}a(c')(-1)^{x\cdot c'}.
\end{equation}
Integers in this display are identified with their $r$-bit vectors, and
the coefficient table is zero outside $[0,4h)$. The payload is independent
of $u$. Hardwire its full encoding at every publicly requested symbol
position. The table includes operand addresses, rule types,
terminal-comparison addresses, and choice-rule data for both directions,
together with the
permuted edit addresses of every factor type used by
Lemma~\ref{oe:derivative-closure}. Addresses for inactive phases are
included too. Strings and indices are split into field entries with fixed
parsing conventions. There are only $\operatorname{poly}(r)$ entries per
frame.

The first preparation steps are an A-only public focus. For the bits of
the repeated frame $c$, apply the mode selectors
\[
 \frac{I+(-1)^{c_i}R_{(e_i,1)}}2
\]
successively, implementing a nonfirst-bit pull by the fixed generators
as in Section~\ref{sec:outer-programs}. Applied to
\eqref{oe:public-walsh-table}, their completed output is
$a(c)(-1)^{x\cdot c}$ on the systematic payload. Removing that known sign
recovers $a(c)$ identically at every center. No recurrence using a
table-dependent coefficient is active before the public focus is
complete. Retain this focus until preparation has been reversed. On
malformed data use fixed default field values, indices, and instructions;
thus every local algorithm is total.

Next, in the topological order of Lemma~\ref{oe:derivative-closure}, fill
every transient full computation stage by Lemma~\ref{lem:outer-fill}.
Previously completed targets remain full and their temporary fill scratch
is empty. At every intermediate location check their direct full-operation
relations; apply the two-buffer fill checks to the current target; and
check zero in all still empty full targets and inactive scratch slots.
The persistent tuple $P(b)$ is checked throughout. All these conditions
are tested together at the same center.

Finally prepare the critical logical reads. The persistent slot $U_D$
provided by Lemma~\ref{lem:outer-wiring} has payload
\[
 U_D(x)=\sum_{j<h}(-1)^{x\cdot j}D_j.
\]
Its Walsh coefficient at address $j$ is $D_j$. Using the completed public
address focus, build A-only selector stacks for the fixed list of
critical roles needed by either the forward or reverse macro proposal
and its existence flag. Unused roles use default addresses. Include
enough roles for each logical array type; their number is constant.
Build the identical focus stacks on the full increment outputs for
$U_D$ for every $\sigma$. The recipe for $U_D$ uses just one degree-one
singleton, so each increment input has assigned degree zero. At a fixed
frame its entire focused stack is independent of $b$ as well.

\begin{lemma}[Preparation and local evolution]\label{oe:preparation}
There is an integer $J_{\max}\geq1$ and a canonical preparation indexed by
$j=0,\ldots,J_{\max}$ with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item At $j=0$ all frame-dependent full and A-only transient slots are
zero. At $j=J_{\max}$ all full increment programs and critical focuses are
complete, and every full-target fill buffer and orbit-focus buffer is
zero.
\item For every field array $b$ and frame $c$, every snapshot is uniquely
specified. Its transition to the next or previous snapshot uses a fixed
finite list of permutation-neighbors of each requested center.
\item The number of steps is $J_{\max}=O(M_o\operatorname{poly}(r))$.
The number of stored slots, the schedule description, and the work to
interpret one microclock or perform one local transition are
$\operatorname{poly}(r)$, including entry dimensions and bit lengths.
\item At full preparation, applying an edit $h_\sigma(c)$ changes the
entire retained tuple to its canonical value on $b+h_\sigma(c)$ by local
addition of the stored increment outputs. The public focus is unchanged.
\end{enumerate}
\end{lemma}

\begin{proof}
Concatenate the public focus, all full-target filling schedules, and the
critical-read focuses. The stage lists for selectors and cutoffs have
fixed padded lengths independent of their address values. The
per-target mode schedule consists of $M_o$ iterations of fixed
polylogarithmic sublength, together with its initial and final buffer
steps. The exceptional first iteration and final erasure are specified
separately. Thus a binary microclock can be interpreted by scanning the
polynomial-in-$r$ list of program boundaries and performing arithmetic
on $O(\log M_o+\log\operatorname{poly}(r))$ bits. The interpreter selects
the current iteration, buffer, initialized prefix, and transition; it
does not execute all preceding iterations. Distinct programs may use
distinct scratch indices to keep their shapes and twists fixed.
An idle preparation step supplies $J_{\max}\geq1$ if needed.

For the last assertion, write $D_\sigma(b)$ for the stored full increment
of a positive-degree full slot $D$. At the top perform, simultaneously,
\begin{equation}\label{oe:simultaneous-edit}
 D(b)\longmapsto D(b)+D_\sigma(b)
             =D(b+h_\sigma(c))
\end{equation}
for every persistent and transient full computation slot. Equation
\eqref{oe:simultaneous-edit} is valid in its own encoder because that
encoder is linear. The increment slots themselves are included in this
update whenever their assigned degree is positive; their required
increments exist by Lemma~\ref{oe:derivative-closure}. Degree-zero slots
do not change. All additions read the old tuple before writing the new
one.

The full-target temporary buffers are already empty. Public focus
depends only on $c$ and remains fixed. Each main critical-read stack is a
linear selector-and-pull program at fixed public addresses. Adding the
matching focused increment stack therefore updates all its initialized
stages to their new canonical values. Those increment stacks have degree
zero and stay fixed. This verifies the assertion also for the retained
A-only data. The preparation can now be erased in the reverse order,
using exactly the same recipes on the new $b$ and unchanged frame.
\end{proof}

\subsection{The extended graph and its endpoint payload}

Define an extended valid label to be
\begin{equation}\label{oe:extended-label}
 v=(c,b,\eta,j),\qquad
 0\leq c<4h,\quad \eta\in\{0,1\},\quad 0\leq j\leq J_{\max},
\end{equation}
where $b$ is a valid macrograph state at stage $c$ if $\eta=0$, and at
stage $c+1\pmod{4h}$ if $\eta=1$. We call these modes pre and post,
respectively. Its raw word $R_v$ contains, at every $k\in\Gamma$, all four
parts of the persistent tuple $P(b)$, all full and A-only transient data
in the canonical snapshot $(c,b,j)$, and a canonically encoded repeated
copy of $(c,\eta,j)$ and its range information. Padding conventions will
be fixed below.

In pre mode increase $j$ until the top. At the top use the prepared
critical reads to compute the macrograph's forward choice and existence
flag. If the edge exists, perform \eqref{oe:simultaneous-edit}, leave
$c,j$ fixed, and change the mode to post. In post mode decrease $j$ until
zero, then perform
\begin{equation}\label{oe:frame-advance}
 (c,b,1,0)\longmapsto(c+1\bmod4h,b,0,0).
\end{equation}
The reverse rules use the reverse preparation transition, the reverse
macro choice at the same start frame, and the inverse of
\eqref{oe:frame-advance}. Both directions read their choice and flag
before the top update. At a missing edge the corresponding proposed word
is defined to be the current raw word and its flag is zero. This is a
total proposal convention, not a graph self-edge. At a malformed input
the local proposal and flag likewise have a fixed total default.

For each valid macro vertex at stage $s$, the extended graph contains the
directed path
\begin{equation}\label{oe:replacement-path}
 \begin{split}
 (s-1,b,1,J_{\max})&\longrightarrow\cdots
 \longrightarrow(s-1,b,1,0)\\
 &\longrightarrow(s,b,0,0)
 \longrightarrow\cdots\longrightarrow(s,b,0,J_{\max}),
 \end{split}
\end{equation}
with frames taken modulo $4h$. Macro edges join the corresponding ends
of these paths. Every internal edge changes the clock or mode, and every
top edge changes mode, so none is a self-edge. The graph has indegree and
outdegree at most one. Every label belongs to a path of positive length;
there are no isolated vertices.

Let $(s_*,b_*)$ be the public exceptional macro source of
Lemma~\ref{lem:macrograph}. The unique exceptional extended source and
its actual successor are
\begin{equation}\label{oe:public-source}
 v_*=(s_*-1,b_*,1,J_{\max}),\qquad
 v_*^+=(s_*-1,b_*,1,J_{\max}-1).
\end{equation}
Their entire raw words are deterministically computable in
$\operatorname{poly}(N)$ time. Indeed the macro source is public, all
full encoders and program formulas are computable by global array
algorithms, and even explicitly following this one preparation takes
only polynomially many steps.

An extended endpoint can occur only at a top label of
\eqref{oe:replacement-path}. Map it to its macro vertex by retaining $b$
and taking stage $c+\eta\pmod{4h}$. If it is not $v_*$, the endpoint
decoder of Lemma~\ref{lem:macrograph} returns a nonzero string $y$ with
$P(S(y))\ne y$ or $S(P(y))\ne y$. Thus the endpoint payload consists of
the logical tapes and the repeated label; it does not require finding a
history of the transient computation.

\subsection{A packed tester for valid represented vertices}

Let $\Pi$ contain identity, all forward and inverse A- and B-neighbor
maps needed by the codes, the left $H$ generators used for payload
independence, the right first-bit shifts and single-cycle rotations and
their inverses, and the right $H$ generators used by scratch programs.
Add the finitely many bounded compositions required by a primitive local
expression. Its cardinality $q=|\Pi|$ is fixed independently of $N$.
Arbitrary powers of a rotation or paths through $H$ are represented by
successive stages; they do not add new query maps to this list. At every
center the tester reads $W|_{\Pi(k)}$ and rejects if \emph{any} applicable
condition below fails:
\begin{enumerate}[label=\textup{(\roman*)}]
\item canonical parsing, legal repeated clock, range checks, and zero
padding;
\item all full-slot local codes, A/B agreements, and systematic-payload
duplication, including targets that are partial or inactive;
\item all A-only local codes and A agreements, including inactive
scratch;
\item all structural persistent relations and the macro-validity
conditions selected by $(c,\eta)$;
\item every required current-snapshot relation: completed full-operation
checks, the current partial-target comparison, initialized scratch-prefix
recurrences in both buffers and all focuses, and inactive-zero checks.
\end{enumerate}
In particular, the tester does not choose one relation uniformly from a
growing list. All relations share this fixed set of center permutations.
Write $\operatorname{rej}(W)$ for its rejection probability at uniform
$k\in\Gamma$, and use normalized whole-symbol distance
\[
 \dist(W,W')=\Pr_{k\in\Gamma}[W(k)\ne W'(k)].
\]

\begin{lemma}[Soundness of the valid-word tester]\label{oe:valid-testing}
There are fixed constants $C_{\rm rep}<\infty$ and $g_{\rm rep}>0$
such that, if $\operatorname{rej}(W)<g_{\rm rep}$, some valid extended
label $v$ satisfies
\[
 \dist(W,R_v)\leq C_{\rm rep}\operatorname{rej}(W).
\]
The tester has perfect completeness. Conversely,
$\operatorname{rej}(W)\leq q\dist(W,R_v)$ for every valid label $v$.
Distinct valid labels have raw distance at least a fixed
$\delta_{\rm raw}>0$.
\end{lemma}

\begin{proof}
Put $\theta=\operatorname{rej}(W)$. The parsing, repetition, and full-code
proximity conclusions of Theorem~\ref{thm:outer-codes} first change
$O(\theta)$ centers to give legal constant clock data and exact full
codewords. A corrected repeated clock of illegal range would cause
rejection everywhere, so small enough $\theta$ rules it out. Exact
one-axis distance then forces systematic-payload duplication in every
full slot. Otherwise that one failed duplication relation already has a
fixed positive rejection probability. Zeroing locally A-invalid scratch
parts costs another $O(\theta)$ centers. A change on $z$ centers changes
any simultaneous test on at most $qz$ centers; this estimate uses the
common permutation list and does not depend on the number of slots.

We next restore all scratch relative to these corrected full codewords
and the corrected clock, before using any advertised full-operation
identity. The reference for the public focus is the actual hardwired
public word at the corrected frame. Treat its recurrences first. A later
table-dependent recurrence is active only when this reference focus is
complete and hence gives identical correct coefficients at every center.
Each initial expression $X$ in a mode-fill program is A-coded from its
exact full parents, even if those parents do not yet satisfy their
advertised payload recipes. The critical and increment focuses also
start from exact A-coded full words. Thus every required reference
prefix is exactly A-coded.

These are precisely the reference and ordering hypotheses of
Lemma~\ref{op:scratch-restoration}. The preprocessing above changed only
$O(\theta)$ sites, so the common query list bounds both the joint
A-conflict fraction and the simultaneous recurrence-failure fraction of
this one preprocessed word by $O(\theta)$. Apply the lemma to its entire
scratch tuple, including inactive zeros. Its fixed-word induction keeps
every prefix estimate $O(\theta)$ and makes only one final replacement.
All scratch is therefore restored at cost $O(\theta)$, independently of
the number of stages. No full-operation identity has been used to obtain
this conclusion.

Call the resulting tuple $\widetilde W$. Its remaining tests still fail
with probability at most $C'\theta$ for a fixed $C'$. By
Lemma~\ref{lem:outer-wiring}, either its persistent tuple is exactly
$P(b)$ for logical bits $b$ valid at the selected macro stage, or a
persistent relation fails with fixed positive probability. Choose
$\theta$ below that gap divided by $C'$ to obtain the former alternative.
Now follow the initialization order of the transient full targets.
Completed targets with canonical parents are forced to their exact
outputs by the shared-axis gap following Definition~\ref{op:full-operation}.
For the current partial target, its required cutoff comparison has the
same gap by Lemma~\ref{lem:outer-fill}. The cutoff scratch has already
been restored: its completed value is precisely $P_{t'}X$. In the case
that the target shares only the B axis with $X$, orbit invariance makes
this completed reference B-coded. Only at this point is B-axis distance
applied. No intermediate A-only scratch is assumed to be B-valid.
Inactive full targets are forced to zero by full-word distance.

There are only finitely many kinds of these gaps, uniformly in their
stage indices, program lengths, and vector dimensions. Their minimum is
positive. If any relation failed exactly on $\widetilde W$, it alone
would give at least that gap in the packed test. Decrease $\theta$ once
more to rule out every such failure. All full slots and restored scratch
then have the unique canonical values of $(c,b,\eta,j)$. The total
restoration cost is $C_{\rm rep}\theta$ for a fixed $C_{\rm rep}$.

Canonical snapshots satisfy every test by their construction, proving
perfect completeness. If a neighborhood avoids the disagreement set
between $W$ and $R_v$, its test accepts; a union bound over $\Pi$ proves
the converse. Finally, two labels with different repeated data differ
at every center. If only their logical arrays differ, one logical
singleton interpolant differs. Its full encoded difference is nonzero
and has the fixed distance of Theorem~\ref{thm:outer-codes}, including the
constant conversion between four-part and whole-symbol metrics. This
gives $\delta_{\rm raw}>0$.
\end{proof}

\subsection{Full representation cost and a fixed local Boolean exponent}

We record the size argument before choosing any correction accuracy or
any number of Boolean ports. This order is needed later. All constants
in the following count depend only on the fixed gate basis, $d_2$, and
the already chosen outer code parameters.

One-bit alignment takes $O(r)$ elementary rotations. A conjunction of
bit conditions takes $O(r^2)$ primitive stages, and a carry or borrow
shift takes $O(r^3)$. The Lagrange window product takes $O(r^4)$ stages.
The other transforms and masks in Section~\ref{sec:outer-programs} fit
within this last bound. There are $O(r)$ initial factor and row families:
a fixed number of row lists, with $O(r)$ routing layers for each needed
permutation and a bounded subset list for each monomial. Hence there are
$O(r^5)$ persistent primitive stages. Storing their ancestor graphs,
parent indices, types, and coefficient instructions is bounded, with
ample slack, by $Cr^{20}$ description bits. Public indexed field data
are counted separately as local constant words below; descriptors refer
to their entries by indices, rather than copying an entire length-$h$
array into a descriptor.

Here is an explicit finite-depth bound for derivative expansion. Let $T$
bound the complete current list of graph descriptions and subgraph
descriptions, with each slot, parent occurrence, and instruction counted
at least once. A target graph has at most $T$ nodes and at most $T$
primary roots. Substitution at one root uses a bounded number of the
$O(r^4)$ Lagrange programs. There are at most $T$ target graphs. Recording
an ancestor subgraph for every newly introduced stage, retaining sharing
within it and allowing fresh copies for separate targets, costs at most
the square of the size of its target computation. Public address
instructions and index encodings contribute fixed powers of $r$ and
logarithms of these list lengths. The following deliberately loose bound
therefore covers all these operations at one tier:
\begin{equation}\label{oe:recipe-size-recurrence}
 T_{a+1}\leq C r^{20}(T_a+1)^6,\qquad T_0\leq Cr^{20}.
\end{equation}
For example, the six factors allow a separate factor for choosing a
target, traversing its graph, substituting its roots, and copying the
ancestor descriptions, with two additional factors for indexing and
duplication; the actual substitution keeps more sharing. If
\begin{equation}\label{oe:recipe-exponent}
 E_0=20,\qquad E_{a+1}=6E_a+20,\qquad E=E_{d_2},
\end{equation}
then all descriptions are $O(r^E)$. This recurrence is composed only
$d_2$ times, by Lemma~\ref{oe:derivative-closure}.

For each full target, a cutoff program uses at most $|H|$ group elements,
$r$ prefix rectangles, $r$ selectors per rectangle, and $O(r)$ primitive
alignment stages per selector. This is $O(r^{12})$ stages. For the orbit
focus, there are at most $|H|$ frequencies and $|H|$ requested group
indices. Even finding and storing a separate generator path of length
$|H|$ for every such pull gives $|H|^3$ stages, in addition to the mode
selectors. A bound $O(r^{30})$ covers both focus and cutoff work and the
two buffers. An entry alphabet has dimension at most $|H|^2\leq r^{18}$,
and a field entry uses $O(r)$ bits. Therefore
\begin{equation}\label{oe:raw-size-bound}
 \text{raw data, local public constants, and schedule metadata at one
 center}\ \leq C r^{E+60}.
\end{equation}
This includes public-address focus, critical focus, all slot types and
all four parts. The long sequence of $M_o$ cutoffs is represented by a
clock and reusable buffers, not by $M_o$ copies of these data.

Put $\ell_N=\lceil\log_2(N+2)\rceil$. Fix once an integer $C_L>E+100$
and pad each raw symbol to
\begin{equation}\label{oe:raw-length}
 L=(\ell_N+C_{\rm base})^{C_L}.
\end{equation}
A fixed lower bound on $C_{\rm base}\geq1$ makes
\eqref{oe:raw-size-bound} fit for every $N$. The unpadded construction is
still parameterized by $r$ and the same slot schema. Increasing
$C_{\rm base}$ later adds zero padding only. The raw word and all
center-specific public descriptions have total length
\begin{equation}\label{oe:total-raw-size}
 |\Gamma|\operatorname{poly}(\ell_N)
 =M_o|H|\operatorname{poly}(\ell_N)
 =N\operatorname{poly}(\ell_N).
\end{equation}
This is a bound on the listed bits, including index and field
representations, rather than just the number of slots.

Every local test and proposal can now be implemented by a fixed
polynomial-time bit algorithm on a bounded number of $L$-bit symbols.
Parsing, loops over slots and the explicit group $H$, arithmetic in $J$,
program interpretation, and multiplexing within a supplied
polylogarithmic array use bounded nests of polynomially bounded loops.
The largest local computation is the orbit-restricted parity solve of
Lemma~\ref{lem:outer-fill}. Its dimension is at most
\[
 C|O|\,|H|\,\dim_J(U)\leq C|H|^4\leq CL^4.
\]
Its explicitly stored matrix has $O(L^9)$ bits if each field entry is
bounded by $L$ bits. Gaussian elimination uses $O(L^{12})$ field
operations, and elementary arithmetic and inversion have a schoolbook
bound $O(L^4)$ in this generous bit-length allowance. Even accessing
each entry by a linear scan through the matrix, and scanning the full
slot and metadata lists, adds only fixed polynomial factors. All other
displayed local operations have the same type of finite loop bound.
No routine iterates through the $M_o$ cutoffs to answer one query.
Choose an integer $a$ exceeding the sum of these fixed loop and
arithmetic exponents. A standard direct time-unrolling of a deterministic
bit computation of length $O(L^a)$ gives an acyclic bounded-fanin Boolean
circuit of size $O(L^{2a+2})$: represent each configuration by its used
tape cells and head indicators, and implement each next configuration
by Boolean selection. Thus one fixed exponent $c_{\rm raw}$ bounds all
raw local circuits. Its precise numerical value is immaterial; the
displayed construction fixes it before correction rounds or port arities
are selected.

All lists and constants are uniformly constructible. Global algorithms
enumerate the finite fields, groups, public routing tables, and full
constant encodings, using the deterministic searches and linear algebra
of Sections~\ref{sec:outer-codes} and~\ref{sec:outer-programs}. Their
time is $\operatorname{poly}(N)$, though it need not be quasilinear.
Input-dependent data without succinct evaluation at $c$ are accessed
through \eqref{oe:public-walsh-table}; other public constants are
hardwired only at the bounded, publicly specified neighbors of the
requested center. There is no hidden access to an unrepresented
length-$N$ array.

\subsection{Correction and the complete outer interface}

All full and scratch slots of $R_v$ satisfy their A constraints. Apply
the joint A-axis corrector of Lemma~\ref{lem:outer-uncertainty} to all
of them, together with majority correction of the repeated clock and
canonical padding. The local decoder enumerates at most $2^d$ label
subsets and solves the common scalar constraints componentwise. Since
$d$ is fixed, this again has a fixed polynomial bound in $L$, independent
of the number or dimensions of the packed slots. Enlarge $c_{\rm raw}$
once to cover this one-round routine.

Let $Q\geq2$ bound its neighborhood size and the sizes of the raw
neighborhoods. For any fixed nonnegative integer $t$, unrolling $t$
correction rounds followed by a raw local routine uses at most
\begin{equation}\label{oe:fixed-interface-composition}
 C(1+Q+\cdots+Q^t)=O(Q^t)
\end{equation}
copies of these same $L$-bit circuits. Their intermediate outputs still
have length $L$. It follows that, for a constant $C_t$ depending on $t$,
the composite circuit has size at most $C_tL^{c_{\rm raw}}$. Its ports
are the finitely many compositions of the original permutations to depth
at most $t$, followed by a raw neighborhood. The exponent is independent
of $t$. A later increase of $C_{\rm base}$ changes only the linear work
of checking and writing padding; it changes neither this exponent nor
the number of ports. In particular, fixed multiplicative constants may
be absorbed into a subsequent exponent allowance with slack by increasing
$C_{\rm base}$ after all those constants have been fixed.

\begin{theorem}[Outer representation interface]\label{thm:outer-representation}
There are fixed positive constants $\delta_{\rm raw},\rho_{\rm raw}$,
fixed integers $C_L,c_{\rm raw}$, and uniform deterministic algorithms
with the following properties for every End-of-Line input of length $N$.
They construct a directed graph of valid extended labels, represented by
canonical words
\[
 R_v\in(\{0,1\}^L)^\Gamma,
 \qquad |\Gamma|=N\operatorname{poly}(\ell_N),
 \qquad L=(\ell_N+C_{\rm base})^{C_L},
 \qquad \ell_N=\lceil\log_2(N+2)\rceil,
\]
where $C_{\rm base}$ may be any sufficiently large fixed integer.
\begin{enumerate}[label=\textup{(\roman*)}]
\item \textbf{Graph and endpoint data.} The graph has indegree and
outdegree at most one, no self-edges, and no isolated vertices. It has the
public distinguished source $v_*$ and actual successor $v_*^+$ in
\eqref{oe:public-source}; their entire words are computable in
$\operatorname{poly}(N)$ time. Every other endpoint is mapped in
$\operatorname{poly}(N)$ time to a nonzero solution of the input
End-of-Line instance. Every canonical word is computable from its valid
label in polynomial time.
\item \textbf{Distance and valid-word testing.} Distinct canonical words
have normalized symbol distance at least $\delta_{\rm raw}$. A uniform
center test has perfect completeness and uses only a fixed finite list
$\Pi$ of permutation-neighbors, of cardinality $q$ independent of $N$.
For every fixed $\alpha>0$ there is a fixed $\theta_\alpha>0$ such that
\[
 \operatorname{rej}(W)<\theta_\alpha
 \quad\Longrightarrow\quad
 \dist(W,R_v)<\alpha\quad\hbox{for some valid }v.
\]
Also $\operatorname{rej}(W)\leq q\dist(W,R_v)$ for every valid $v$.
\item \textbf{Local proposals and flags.} At each requested center,
total deterministic routines on $\Pi(k)$ compute a next or previous
proposed symbol and an existence bit. On $R_v$ the bit is independent
of the center and equals the actual edge-existence flag. If it is one,
the proposed symbols form exactly the canonical neighboring word; if it
is zero, the default proposal is $R_v$ itself.
\item \textbf{Value-independent uncertainty correction.} There is a
total local one-round corrector with a bounded permutation neighborhood.
If $|U|/|\Gamma|\leq\rho_{\rm raw}$, it maps every word agreeing with
$R_v$ outside $U$ to a word agreeing with $R_v$ outside a set $U^+$
determined only by $U$, with $|U^+|\leq|U|/2$. Consequently $t$ rounds
leave a prescribed uncertainty support of density at most
$2^{-t}|U|/|\Gamma|$. The guarantee holds for arbitrary values on $U$.
Different requested-center simulations may use different values there,
provided repeated occurrences of a position are consistent within each
individual cone.
\item \textbf{Global recovery.} From any word within
$\rho_{\rm raw}$ symbol distance of a valid word, a deterministic
polynomial-time algorithm recovers its exact extended label and raw
word. It then recovers the logical bits by systematic coefficient
extraction and evaluation on the subgroup. In particular it extracts an
End-of-Line answer whenever the recovered label is a nonexceptional
endpoint.
\item \textbf{Uniform cost and order of choices.} The total raw data and
listed public descriptions have $N\operatorname{poly}(\ell_N)$ bits
and are constructible in $\operatorname{poly}(N)$ time. Each raw local
test, proposed symbol, flag, and one-round corrector has a Boolean
circuit of size at most $CL^{c_{\rm raw}}$ on a fixed number of symbols.
For each fixed $t$, correction for $t$ rounds followed by any such raw
routine has size at most $C_tL^{c_{\rm raw}}$ and at most $O(Q^t)$
symbol ports, with uniform construction for that $t$. The exponents
$C_L,c_{\rm raw}$ are fixed before $t$ and any later port arity. Further
fixed additive padding in $C_{\rm base}$ has only linear processing
cost and does not change these exponents or arities.
\end{enumerate}
\end{theorem}

\begin{proof}
The graph and local evolution were established in
Lemma~\ref{oe:preparation} and \eqref{oe:replacement-path}. The tester
and distance are Lemma~\ref{oe:valid-testing}. For example, its constants
allow
\[
 \theta_\alpha=
 \min\{g_{\rm rep}/2,\alpha/(2C_{\rm rep})\}.
\]
All local proposals use the same primitive permutation list as the
tester, enlarging it by a fixed number of maps if necessary.

For the correction claim, let $d$ and $\delta_1$ be the outer A-code
degree and distance. Choose $\rho_{\rm raw}$ smaller than both
$\delta_{\rm raw}/3$ and $\delta_1/4$, and smaller than the fixed
uncertainty radius in Lemma~\ref{lem:outer-uncertainty}. An admissible
choice of $U^+$ is the set of centers having at least $(\delta_1/2)d$
neighbors in $U$ in a forward or inverse A list. Off this set all row
suggestions have fewer than half the local minimum distance in errors,
so the joint decoder recovers the reference exactly; the clock majority
and padding agree there too. If $u=|U|/|\Gamma|$, the spectral estimate
from that lemma gives
\[
 \frac{|U^+|}{|\Gamma|}
 \leq\frac{2\lambda^2u}{(\delta_1/2-u)^2}
 \leq\frac u2.
\]
Neither this support nor the proof refers to the values inside $U$.
For a requested-center cone with consistent repeated positions, extend
its initially supplied values arbitrarily to the whole word, setting
the reference outside $U$. The same support argument applies. This
proves the assertion even when separate cones choose different
extensions.

After $O(\log|\Gamma|)$ synchronous global rounds the uncertainty set
has cardinality less than one, giving exact recovery in polynomial
time. From a recovered logical singleton slot, read its coefficient
payload and evaluate the degree-$<h$ interpolant at $\omega^j$,
$0\leq j<h$, to obtain the corresponding logical bits. The repeated
clock gives $(c,\eta,j)$. The distance choice makes the promised
reference unique. On inputs outside the recovery promise the algorithm
has a total default; it may check the decoded label and its canonical
word by the global polynomial-time algorithms. Finally,
\eqref{oe:recipe-size-recurrence}--\eqref{oe:total-raw-size} and
\eqref{oe:fixed-interface-composition} give all asserted cost and
parameter-order statements.
\end{proof}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/05-outer-representation.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/06-outer-composition.tex ====
\section{Binary composition and the robust outer circuit}\label{sec:outer-composition}

We now convert the symbol representation into a generalized circuit. Two
features of the conversion require separate arguments. A local Boolean
function may inspect polynomially many bits of a bounded number of symbols;
its implementation must tolerate failures among its own gates. Moreover,
sharing such implementations must not allow one corrupted physical value
to spoil an unbounded number of decisions. We state the first interface,
proved in Section~\ref{sec:inner-gadget}, and then give the construction and
all the counting and synchronization arguments for the second.
The same sequence is used inside a decision gadget in
Section~\ref{sec:inner-gadget}: first obtain replica synchronization without
trusting decisions, then bound unstable comparisons and failed computations,
and finally use the clipped mean displacement to locate an endpoint.
At the outer scale, a completed decision is shared by a group of output
replicas, called a cohort; its internal gates and external read occurrences are charged separately from
each output's own remaining gates.

Throughout this section, a distance between words is relative Hamming
distance, unless a norm is displayed. A constant may depend on previously
fixed accuracies, but never on the End-of-Line input or its length. A
``fixed branch'' of a randomized answer tree means that the earlier
hypothetical answers are fixed when the randomized algorithm is described.
It does not mean conditioning the probability space on receiving those
answers. This distinction applies also to branches never executed.

\subsection{A fixed port and a robust Boolean decision}

The port family will encode every raw symbol in the same binary format.
Its tester lets us recognize symbols close enough to use, and its recovery
radius is fixed before we know how many symbols a corrected local
computation will inspect. Once that arity is fixed, the robust-decision part of
the interface realizes one Boolean output of such a computation. It
provides many designated copies of the answer, so the guarantee remains
useful even when failed gates concentrate on designated outputs.

\begin{theorem}[Standardized ports and robust decisions]\label{thm:port-gadget}
For every fixed positive integer exponent allowance $c_{\rm allow}$, there
are a cutoff $L_{\rm port}$ and a uniform family of injective binary
encodings, for $L\geq L_{\rm port}$,
\[
 \mathcal D:\{0,1\}^{L}\longrightarrow\{0,1\}^{S_0},
 \qquad S_0=\poly(L),
\]
with fixed positive relative distance $d_B$, satisfying \textup{(i)}--
\textup{(ii)} below. The radius $\nu_B$ in \textup{(iii)} is fixed with
this port family; the actual-gadget assertions in that part have the
separate later cutoff specified there.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every coordinate of $\mathcal D(s)$ is a multilinear polynomial in
the bits of $s$ over $\F_2$, of degree at most $10$. Its coefficient list
is computable in time polynomial in $L$. The degree bound is independent
of $c_{\rm allow}$. Encoding and exact recovery from a fixed positive
relative-error radius are deterministic polynomial-time algorithms.
\item A perfect-completeness tester makes a bounded number of bit queries
and performs bounded Boolean answer processing. Its public randomness has
an enumerable polynomial-size rational support and polynomial bit
complexity. For every fixed $a>0$, there are a fixed $\theta_B(a)>0$
and a cutoff $L_B(a)\geq L_{\rm port}$, allowed to depend on $a$, such
that for every $L\geq L_B(a)$ rejection below $\theta_B(a)$ implies
distance less than $a$ from $\mathcal D(\{0,1\}^L)$.
A word at distance $a$ from an encoding has
rejection at most $C_Ba$, for a fixed $C_B$. Each fixed-branch query has a
bounded-density marginal relative to the uniform distribution on the
block. These assertions include arbitrary words and all hypothetical
branches.
\item There is a constant $0<\nu_B<d_B/4$ determined with this encoding,
before any arity is specified. Subsequently fix an arity bound $k$, a
size factor $C_f$, and a desired output loss $\upsilon>0$. There is a
further cutoff $L_g$, allowed to depend on these fixed choices, such that
all actual-gadget assertions below hold for every
$L\geq\max\{L_{\rm port},L_g\}$. Consider any single-bit Boolean
function of at most $k$ length-$L$ strings, described
by an acyclic fan-in-at-most-two circuit over a fixed Boolean basis with
public constants, of size at most $C_f L^{c_{\rm allow}}$. Some inputs
may be declared public $0^L$.

There is a uniform generalized subcircuit for this function, with a
common count scale $G_0=\poly(L)$ for all padded functions under these
bounds, at most $C_gG_0$ gates, and a set of $W$ designated output wires
produced by its gates, where
\begin{equation}\label{out:designated-count}
 c_gG_0\leq W\leq C_gG_0.
\end{equation}
Here $c_g,C_g>0$ are fixed. Each nonpublic external input is a block of
physical reads $X=3x-1$, with $x\in[0,1]$, compared to a public common
rational threshold $\xi\in[2/5,3/5]$. Every external comparison occurrence
in the complete unfolded subcircuit can be routed separately to an
external replica.
For each external port position, the total number of these occurrences is
at most
\begin{equation}\label{out:port-occurrence}
 C_{\rm occ}G_0/S_0,
\end{equation}
uniformly over $\xi$, the function, and its public parameters. This counts
all hypothetical branches, including unsuccessful and default branches.

There are fixed $\delta_g,\gamma_g,\eps_g>0$ with the following universal
soundness property. Fix logical strings $s_1,\ldots,s_k$ and ideal binary
port words $y_j$ satisfying
$\dist(y_j,\mathcal D(s_j))\leq\nu_B$ for each nonpublic port.
For any subsequently chosen fixed physical margin $\omega>0$, use
\[
 0<\eps\leq\eps_g,\qquad \eps<\omega/1000.
\]
Suppose at most $\gamma_gG_0$ external comparison occurrences fail to
have a physical value on the side of $\xi$ specified by the corresponding
bit of $y_j$, at gap greater than $\omega$. Repeated occurrences need not
have consistent physical values. Then every assignment of the internal
nodes in $[0,1]$ with at most $\delta_gG_0$ failed gates makes all but
$\upsilon W$ designated wires correct near-bits for $f(s_1,\ldots,s_k)$:
at most $\eps$ for zero and at least $1-\eps$ for one.
\end{enumerate}
The port construction and $\nu_B$ are independent of $k,C_f,\upsilon$.
The later constants, the gate-count bounds, and the topology bounds may
depend on these fixed choices, but not on $L,\xi$, or the later margin
$\omega$. The threshold may appear as a rational gate parameter.
Construction and other parameter computations take polynomial time in
$L$ and the supplied description lengths, including the bit length of
$\xi$; all parameter lengths are bounded by a polynomial in these
quantities. In particular an arbitrarily long externally supplied
threshold is charged for its bits.
\end{theorem}

Sections~\ref{sec:inner-algebra} and~\ref{sec:inner-binary} construct the
algebraic and binary ingredients. Proposition~\ref{prop:dummy-port}
proves the fixed encoding, tester, and recovery assertions. The remaining
construction in Section~\ref{sec:inner-gadget} proves the physical occurrence
bound~\eqref{ig:external-load}, gate count~\eqref{ig:gate-count}, and robust
output guarantee~\eqref{ig:extraction-loss}, using the independent geometric
procedures proved in Section~\ref{sec:geometry}. None of these constructions
uses the outer circuit constructed here.

\subsection{Port substitution, validation, and average correction}

Apply Theorem~\ref{thm:outer-representation}, writing its raw encoding as
$R_w=(s_k)_{k\in\Gamma}$, with
\[
 \ell_N=\lceil\log_2(N+2)\rceil,\qquad
 L=(\ell_N+C_{\rm base})^{C_L},\qquad
 |\Gamma|=O(N\ell_N^{O(1)}).
\]
Choose $c_{\rm allow}>c_{\rm raw}+12$, with any additional fixed local
polynomial overhead absorbed in this inequality, and construct
$\mathcal D$ before choosing correction rounds. Put
\begin{equation}\label{out:binary-encoding}
 E_w=(\mathcal D(s_k))_{k\in\Gamma},\qquad m=|\Gamma|S_0.
\end{equation}
The product of the two relative distances is a positive Hamming-distance
bound; write $d_*>0$ for its square root, a lower bound on normalized
Euclidean separation. The distinguished source and its actual successor are publicly computable in polynomial time.

There is a fixed radius $\rho_{\rm dec}>0$ from which these words can be
decoded globally. Indeed, let $r_B>0$ be a port recovery radius and
$r_{\rm raw}>0$ a raw recovery radius. If the whole binary distance is
less than $r_Br_{\rm raw}/2$, at most an $r_{\rm raw}/2$ fraction of port
blocks can be at distance at least $r_B$ from their true block. Decode
all ports, using a fixed default on an invalid or unsuccessful decoder
output, and then use the raw decoder. An incorrect decoding at a bad
site need not be detectable; the raw error bound already charges all
such sites. Fix a smaller radius $\rho_{\rm dec}$ if necessary.

The graph in Theorem~\ref{thm:outer-representation} has indegree and
outdegree at most one, no self-edges, and no isolated vertices. Write
$w_*$ for its distinguished source, which has an actual successor, and
call every other missing-neighbor vertex a desired endpoint. The
geometric construction applies to this encoded graph. For an edge
$u\to v$, its four-block polygon successively changes the second, third,
first, and third blocks:
\[
 \begin{aligned}
 (E_u,E_u,0,0)&\longrightarrow(E_u,E_v,0,0)
 \longrightarrow(E_u,E_v,1,0)\\
 &\longrightarrow(E_v,E_v,1,0)
 \longrightarrow(E_v,E_v,0,0).
 \end{aligned}
\]
Here a scalar in a block denotes its constant repetition. The corners
are rounded, and a segment in the fourth block enters the distinguished
source from $(E_{w_*},E_{w_*},0,2)$.
Theorem~\ref{thm:geometry} constructs a displacement field on
\[
 \mathcal B=[-1,2]^{4m},\qquad
 \norm{U}_{\rm b}^2=\frac1m\sum_i U_i^2,
\]
where all distances to subsets of $\mathcal B$ use this norm. Each point
of its endpoint set $\mathcal E$ has first block exactly $E_w$ for a desired
endpoint $w$; the top of the extra segment is excluded. For any sufficiently
small fixed rational $\tau>0$, chosen to meet the validation and
global-recovery bounds in \eqref{out:tau}, there are constants
$B_1,c_*,\beta_0>0$ and a field
$v$ such that, after increasing $B_1$ to an integer,
\begin{equation}\label{out:geometry-displacement}
 \norm{v(U)}_\infty\leq B_1-1,\qquad
 \norm{\clip_{\mathcal B}(U+\beta v(U))-U}_{\rm b}\geq c_*\beta
\end{equation}
whenever $\dist(U,\mathcal E)>10\tau$ and $0<\beta\leq\beta_0$.
These constants precede estimator accuracy and the step $\beta$.
The circuit will keep many replicas of each physical coordinate and
force their mean vector close to $\mathcal E$; rounding the first block
will then permit global recovery.

To approximate the field locally, the numerical procedure of
Lemma~\ref{lem:geometric-procedures} rounds the first two blocks at one
common threshold $\xi\in[2/5,3/5]$. Each rounded word is a seed for
validation. On good decisions an accepted seed must identify one fixed
reference vertex, independent of the random acceptance decision and of
the requested confidence or numerical accuracy; a seed in the intermediate
region may be accepted or rejected. If one of the first two physical
blocks is within $5\tau$ of a valid word, its seed must be accepted with that
reference on good decisions for every allowed threshold. The procedure
then asks for incident-edge flags, bits of each reference and its actual neighbors,
and bits of the distinguished source and its successor. Each word-bit
request uses either the coordinate matching the field component being
estimated or a preallocated uniform coordinate for a bounded scalar
statistic. Each reference and its neighbors determine a bounded list of
incident path pieces; a fixed mesh and estimates of their scalar data
approximate the field near those pieces. Scalar reads use fixed-width
quantization with a public offset. We next supply these calls. Their
number is fixed once the geometric mesh, scalar sample counts and
numerical tolerances have been chosen, before the smaller bit-error
budgets are assigned.

\begin{lemma}[Protecting functional inputs]\label{out:flags}
There are constants $0<a_1<a_2<\nu_B$, fixed before the arity of any
functional invocation, with the following property. A bounded Boolean
flag routine on an arbitrary binary port block can have arbitrarily
small fixed failure probability, unflagging every block within $a_1$ of
an encoding and flagging every block at distance at least $a_2$ from all
encodings. In a functional invocation replace flagged ports by public
$0^L$. Except on the union of the completed flag routines' failure
events, every nonpublic input of every selected invocation is within
$\nu_B$ of its intended encoding.
\end{lemma}
\begin{proof}
Choose $a_2<\nu_B$ and then $a_1$ so small that
$C_Ba_1<\theta_B(a_2)/4$. Estimate port-tester rejection by a fixed
number of independent trials and compare to $\theta_B(a_2)/2$.
Bounded independent variables give any fixed confidence by increasing
the fixed sample count. In the interval between the two promises either
Boolean result is permitted. An unflagged block on the good event is
within $a_2$ of an encoding, unique because $a_2<d_B/4$; take its logical
string as the intended input. A flagged input is explicitly a public
zero, so it requires no external near-port premise. A union bound is
applied to the finite list of completed flag promises, after that list
has been fixed. Their internal tester answer trees are not counted as
new logical promises.
\end{proof}

Fix a physical vector $U$ and initially fix its rounding threshold and
scalar offset. A rounded seed $y=(y_k)_{k\in\Gamma}$ is now a fixed
binary word; use bit one at equality with the threshold. For analysis, define a global nearest-symbol convention
$s^*(k)$ at every site within $a_2$ of a port encoding, and choose a
fixed arbitrary string elsewhere. An ideal functional call computes
its Boolean function on these symbols at unflagged inputs and on
$0^L$ at flagged inputs. Within one correction cone, deduplicate repeated
raw sites and share their flag and initial logical value. Different
cones may make different choices at uncertain sites.

\begin{lemma}[Binary validation]\label{out:validation}
There are fixed $0<\rho_{\rm comp}<\rho_{\rm acc}$, with
$\rho_{\rm acc}$ smaller than the uniqueness radius of
\eqref{out:binary-encoding}, and a finite ideal Boolean validation
procedure such that, with arbitrarily high fixed probability, acceptance
implies distance less than $\rho_{\rm acc}$ from a unique valid $E_w$,
and distance less than $\rho_{\rm comp}$ implies acceptance with the
true reference. The procedure uses port flags, port tests, and bounded
raw-checker functional calls. It remains valid with averaged stationary
expander-walk sampling as described below.
\end{lemma}
\begin{proof}
Choose $\rho_{\rm acc}$ small enough that
$\rho_{\rm acc}/a_1$ is within the raw uncertainty-correction radius.
Choose $\alpha<\rho_{\rm acc}/4$ and take the raw gap
$0<\theta_\alpha\leq1$ from Theorem~\ref{thm:outer-representation}.
Writing $q=|\Pi|$ for its bounded checker-neighborhood size, choose
$0<a<a_1$ and $b_0>0$ satisfying
\begin{equation}\label{out:validation-constants}
 \alpha+a+b_0<\rho_{\rm acc},\qquad
 qb_0<\theta_\alpha/16,
 \qquad p_{\max}=b_0\theta_B(a)>0.
\end{equation}
Decrease $p_{\max}$ if necessary. Let $p$ be the average port-tester
rejection over all sites. Estimate $p$, and independently simulate the
raw checker at sampled centers using the flagged/substituted inputs.
Accept when the first estimate is at most $p_{\max}/2$ and the simulated
raw rejection estimate is at most $\theta_\alpha/2$.

On statistics of error less than $p_{\max}/4$, acceptance gives
$p<p_{\max}$. Consequently the fraction of sites not within $a$ of any
port encoding is less than $b_0$. At a center whose entire neighborhood
avoids these sites, good flag promises leave all inputs unflagged and
make the simulated checker exactly the checker on the global word
$s^*$. Exceptional centers occupy at most $qb_0$. In the analysis,
also estimate the true checker result on $s^*$ to accuracy
$\theta_\alpha/8$ and the exceptional-center fraction to accuracy
$\theta_\alpha/16$. These are fixed bounded statistics; the algorithm
need not compute the hypothetical true statistic. On their good events
and the completed flags' good events,
\[
 \Pr[\text{raw checker rejects }s^*]
 \leq\theta_\alpha/2+\theta_\alpha/8+qb_0+\theta_\alpha/16
 <\theta_\alpha.
\]
Raw soundness makes $s^*$ $\alpha$-close to a valid $R_w$. Charging all
raw-disagreement sites and all non-$a$-close sites at cost one yields
\[
 \dist(y,E_w)<\alpha+a+b_0<\rho_{\rm acc}.
\]
The uniqueness radius makes this reference independent of subsequent
random acceptance decisions for the fixed $y$.

Conversely, if $\dist(y,E_w)<\rho_{\rm comp}$, then
$p\leq C_B\rho_{\rm comp}$. By Markov's inequality at most
$\rho_{\rm comp}/a$ of the ports are at distance at least $a$ from their
true block. Outside their $\Pi$-neighborhoods the simulated checker is
the perfectly accepting raw checker. Choose $\rho_{\rm comp}$ so small
that both means are strictly below one quarter of their acceptance
thresholds; choose the sampling errors with the remaining slack. Flag
error probabilities are set after the sample counts. This proves both
assertions. The proof requires concentration of the sampled exceptional
fraction when $p$ is small. It never requires an arbitrarily long batch
to avoid every exceptional site.
\end{proof}

Choose the geometric radius so that
\begin{equation}\label{out:tau}
 4(10\tau)^2<\rho_{\rm dec},\qquad
 \frac{(5\tau)^2}{(2/5)^2}<\rho_{\rm comp}.
\end{equation}
The second inequality supplies physical validation completeness:
every incorrectly rounded coordinate of a binary word costs at least
$(2/5)^2$ in squared distance for every allowed threshold. Fix the
resulting $B_1,c_*,\beta_0$ and fix $\kappa=1/100$, the replica gap used
below. Set the unscaled estimator error target with slack, smaller than
$c_*\kappa/100$ in normalized root-mean-square norm.

\begin{lemma}[Average correction of every relevant word role]
\label{out:average-correction}
After fixing the high-level mesh and statistical call counts, boundedly
many raw correction rounds suffice to supply every required incident
flag and every requested encoding bit, with any prescribed fixed average
error. The average is over a uniform output component or its
uniform-marginal statistical requests. The conclusion holds
unconditionally for every true reference that can be accepted on good
validation decisions, before selecting pieces or numerical winners.
All resulting single-bit functions fit the fixed exponent allowance of
Theorem~\ref{thm:port-gadget}.
\end{lemma}
\begin{proof}
Fix a rounded seed that can be accepted on good validation, and its
unique reference $R_w=(s_k)$. Put
\[
 U_0=\{k:\dist(y_k,\mathcal D(s_k))>a_1\},
 \qquad |U_0|/|\Gamma|\leq\rho_{\rm acc}/a_1.
\]
Outside $U_0$, good flags unflag the port and its unique nearest logical
string is $s_k$. Within $U_0$, substitution and intermediate flag choices
may supply arbitrary values. In each deduplicated cone extend those
initial values to a whole raw word agreeing with $R_w$ outside $U_0$.
The raw uncertainty theorem gives a residual set $U_t$, depending only
on $U_0$, with arbitrarily small fixed density after sufficiently many
fixed rounds $t$. Its guarantee holds simultaneously for every such
extension. Thus different cones and correlated intermediate flags cause
no additional exceptional support. A corrected symbol is right off
$U_t$; a local proposal and edge flag are right off a fixed permutation
expansion of $U_t$.

Compute an edge flag at a uniform-marginal center. For a requested own
or neighboring encoding bit, compute the appropriate raw output symbol
and then the requested coordinate polynomial of $\mathcal D$. Matching
coordinates are uniform in the output average; the statistical requests
have the same center marginal. Each fixed permutation expansion
therefore costs at most its size times $|U_t|/|\Gamma|$. Set $t$ for
the already fixed list of high-level true roles and calls. No union over
all possible intermediate flag patterns is necessary.

The raw simulation uses at most $C_tL^{c_{\rm raw}}$ Boolean gates, with
exponent independent of $t$. A degree-at-most-ten Boolean polynomial has
at most $\sum_{j=0}^{10}\binom Lj=O((L+1)^{10})$ monomials. Compute each
monomial by a bounded-fanin AND tree and their sum by XOR gates expanded
in the fixed basis. Coefficients and the requested bit's description are
publicly generated. The extra bit-index and padding work has a fixed
polynomial exponent. Our initial choice of $c_{\rm allow}$ leaves
slack for this work. The number of raw ports and the multiplicative
factor may increase with $t$, but are now fixed. The port encoding and
its radius do not change.
\end{proof}

The completed port flags are now amplified for the finite list of
logical promises in validation and these correction cones. Conditional
on a sampled site, they use fresh tester randomness. Make the probability
that any selected invocation has a nonpublic input outside $\nu_B$
arbitrarily small. This protection concerns arbitrary rounded seeds,
including seeds undergoing validation, and does not require exclusion
of the geometric endpoint neighborhood. Uncertain flag results only
choose which already uncertain inputs are substituted.

\subsection{Small-support ideal tapes and complete branch counts}

\begin{lemma}[Averaged expander sampling]\label{out:walk}
The statistics needed above can be sampled with polynomial-in-$L$
random support per output coordinate, with any desired fixed confidence
in the average over output coordinates.
\end{lemma}
\begin{proof}
Write an output coordinate as $(b,k_0,z_0)$ with $b\in[4]$,
$k_0\in\Gamma$, and $z_0\in[S_0]$. Use fixed-length labeled walks in the
outer Cayley expander starting at $k_0$. Fresh bounded segments and
auxiliary uniform positions in $[S_0]$ supply all center and word
statistics. Averaging over uniform $k_0$ makes every walk position
stationary. If the mean-zero operator norm is at most $\lambda<1$,
bounded centered functions at times $s,t$ have covariance at most
$C\lambda^{|s-t|}$. Independent auxiliary randomness only changes the
bounded diagonal terms and replaces off-diagonal functions by their
conditional means. Therefore a batch of $q_s$ samples satisfies
\[
 \operatorname{Var}\left(\frac1{q_s}\sum_{j=1}^{q_s} Z_j\right)
 \leq\frac{C}{q_s^2}\sum_{s,t=1}^{q_s}\lambda^{|s-t|}
 \leq\frac{C(1+\lambda)}{(1-\lambda)q_s}.
\]
Chebyshev's inequality and a finite union give any fixed confidence by
choosing a sufficiently large fixed $q_s$. The argument permits distinct
bounded functions at the different slots and separate batches for
physical blocks.

Apply it to port rejection, the fixed global raw-checker and exceptional
indicators in validation, and each actual reference candidate's
statistics in the geometric procedure. At fixed rounded seeds those
reference vertices are fixed whenever good acceptance is possible.
Analyze all their roles before a numerical minimum or uncertain
acceptance chooses which ones to use. The concentration is not
conditioned on the realized answer path. Each walk has a fixed number
of fixed-degree label choices. Each remaining choice has polynomial-in-
$L$ support, and only a fixed number of choices is used. Their joint
support per output coordinate is consequently polynomial in $L$.
\end{proof}

The preceding routines now meet the local-access contract of
Definition~\ref{geo:access-contract}. Validation supplies fixed unique
references, average correction supplies the matching and statistical bit
requests, and the walk lemma supplies concentration for each underlying
true candidate. The geometric procedure reserves at most eight true word
roles and thirty-two piece roles, as enumerated in
Section~\ref{geo:candidates}. Missing roles are omitted; duplicate roles
may be retained.
For fixed $U$ outside the endpoint neighborhood and each fixed admissible
binary and scalar-quantization offset, it gives arbitrarily small
prescribed $4\E_{i,\mathrm{trial}}|\widehat v_i-v_i(U)|^2$, with $i$
uniform among the $4m$ components. All bit and flag errors and all
candidate statistics are analyzed before a numerical winner is selected.
On the resulting good event, Lemma~\ref{geo:stability} controls every
sufficiently near-minimizing candidate in each component before selection, so a winner correlated with
the requested component is harmless. The needed bit-error bounds remain
average bounds over the allocated requests.

Apart from the supplied calls, the procedure uses bounded comparison and
Boolean-answer processing, public rational numerical tables, and a
signed dyadic output in $[-B_1,B_1]$, including on every bad branch.
A fixed-width scalar-quantization grid with an independently randomized
offset suffices. Refining the finite grids of offsets changes comparison
thresholds, not the number of bins, answers, or slots. All these
statements concern the unscaled field.

This outer application uses the accuracy and finite-role clauses
\textup{(i)}--\textup{(iii)} of
Lemma~\ref{lem:geometric-procedures} with completed ideal Boolean
functional calls. Its optional physical-read clause \textup{(iv)} is
not applied to the full outer gadget expansion. The fixed count
$H_{\rm out}$ below counts direct outer read slots, and $B_{\rm out}$
counts ideal functional invocations. Theorem~\ref{thm:port-gadget} and
Lemma~\ref{out:routing} separately bound every physical read created by
substituting those invocations.

Applying parts \textup{(i)}--\textup{(iii)} of
Lemma~\ref{lem:geometric-procedures} at this ideal-oracle level, with
the preceding accuracy choices, we obtain ideal dyadic outputs satisfying,
for every eligible fixed $U$ and every fixed admissible offset,
\begin{equation}\label{out:ideal-accuracy}
 4\E_{i,\mathrm{tape}}\abs{\widehat v_i-v_i(U)}^2
       <(c_*\kappa/100)^2.
\end{equation}
All outputs remain in $[-B_1,B_1]$ on bad trials. The good-port failure
probability can independently be made arbitrarily small on this
high-level call list. It is useful to record the order explicitly:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item Fix geometric roles, mesh, scalar counts and numerical tolerances.
\item Choose validation confidence and raw correction rounds for this
finite logical-call list; fix the number of ports in every cone.
\item Amplify each completed flag for its prescribed failure probability.
Amplification adds tester reads, not new logical promises.
\item Only now unfold the complete outer answer tree, leaving functional
calls unexpanded and obtaining a fixed number $H_{\rm out}$ of direct
read slots and $B_{\rm out}$ of hypothetical functional invocations.
\item Choose $\upsilon$ for these $B_{\rm out}$ invocations and apply
Theorem~\ref{thm:port-gadget} with the now fixed arity and size factor.
\end{enumerate}
This order does not define a bit accuracy by a union bound over the very
answer tree created by amplifying that accuracy.

At the ideal-oracle level, unfold every flag pattern, substitution
pattern, quantized bin and hypothetical answer. Functional-call nodes
remain unexpanded: each contributes one completed Boolean oracle answer,
and its physical implementation is separately accounted for below.
Thus $H_{\rm out}$ counts only direct read slots and $B_{\rm out}$ counts
functional invocations. Supply randomness to unused nodes too. Earlier
answers that determine an invocation's function or addresses are fixed
publicly in that copy; they are not additional fragile input wires.
Thus all port functions remain Boolean functions on their declared raw
ports. A wrong numerical answer selects another reserved role or a
finite table case. It cannot become an arbitrary hidden word address.
Each hypothetical raw site is $\pi(k_0)$ for $\pi$ in a fixed finite set
$\mathcal P$: compose walk-label moves, raw-neighborhood permutations
and correction-cone moves. This set is independent of later offset-grid
refinement. Direct scalar and bit slots use matching $z_0$, independently
sampled uniform within-block positions, or the port tester's smooth
queries. For fixed $k_0$ and fixed offsets, summing their unconditioned
position densities over the finite complete tree gives a constant.

The tape law in the next lemma includes both late offset grids---the
binary threshold and the independent scalar-quantization offset---together
with all internal choices at the hypothetical nodes. The grids are fixed
before this final law is rationalized. The preceding accuracy estimates
hold for each fixed offset, and hence for this joint law; the finite grids
multiply its support bound only by a fixed factor.

\begin{lemma}[Rational tapes and their read loads]\label{out:tapes}
The true tape law can be replaced deterministically, at each output
coordinate, by a list of a common square length $T_{\rm tap}=\poly(L)$,
with total variation at most $\eta_0/S_0$, for any prescribed fixed
$\eta_0>0$. The lists use only choices in the original support. They
preserve \eqref{out:ideal-accuracy} and any bounded rare-event estimate
with arbitrarily small fixed slack. Across these lists, every base
physical coordinate is read by at most $C T_{\rm tap}$ direct
comparison slots in the complete unfolded outer trees.
\end{lemma}
\begin{proof}
Let $Q(L)$ bound the enumerable rational support of every tape law.
For a common square $T\geq Q(L)S_0/\eta_0$, round each mass $Tp_j$
down to an integer and distribute the remaining copies among nonzero
support entries. Each mass error is at most $1/T$, and total variation
is at most $Q(L)/T\leq\eta_0/S_0$. Floors, remainders and the ordering
of entries are computed with exact rational arithmetic in polynomial
time. The probability descriptions have polynomial bit complexity.
A function in $[0,A]$ changes expectation by at most $A$ times total
variation, proving the bounded-statistic assertions.

For the load assertion, explicitly list all potential direct slots
$\mathcal A$, including their fixed earlier-answer paths; set
$H=|\mathcal A|$. If a slot is omitted or supplied a public value, its
read subdistribution is dominated by that of its potential slot; it is
never renormalized on being selected. Fix a target $(b',k',z')$ and a
possible $\pi\in\mathcal P$. Its preimage $k_0=\pi^{-1}(k')$ is fixed.
For every fixed offset, the true law gives
\begin{equation}\label{out:direct-density}
 \sum_{z_0=1}^{S_0}
 \E[\text{number of direct reads at }(b',k',z')]
 \leq C,
\end{equation}
where $C$ is the sum of the unconditioned fixed-slot density constants.
Matching coordinates contribute one, uniform sampled coordinates
contribute their bounded counts, and tester queries contribute their
smoothness constants. Branches and seed roles add only a fixed factor.
A position-count random variable lies in $[0,H]$. Changing the law for
one output by total variation $\eta_0/S_0$ adds at most
$H\eta_0/S_0$ to its expectation; summing over $z_0$ adds only
$H\eta_0$. Sum the fixed preimages and physical roles and multiply by
$T_{\rm tap}$. This is the asserted $O(T_{\rm tap})$ load.
A constant total-variation tolerance without the $S_0$ denominator
would not give this bound.
\end{proof}

The occurrence bound \eqref{out:port-occurrence} for each substituted
gadget uses the same full-tree principle at its own scale. Its tape
approximation must control loads both on its internal word, of length
$m_{\rm in}$, and on an external port, of length $S_0$. Section~\ref{sec:inner-gadget}
therefore uses total variation $\eta/\max\{4m_{\rm in},S_0\}$; the
calculations \eqref{ig:external-load}--\eqref{ig:internal-load} give
$O(G_0/S_0)$ external occurrences per port position and $O(R_{\rm in})$
internal reads per base coordinate, where $R_{\rm in}$ is the gadget's
replica count and $G_0=4m_{\rm in}R_{\rm in}$.
This accounts for unequal polynomial block lengths without conditioning
on an executed branch.

\subsection{Cohorts, individual read routing, and gate realization}

At each base component $i\in[4m]$, allocate one cohort for every tape in
its list. A cohort has $K_c$ output replicas, where $K_c$ is a square
satisfying
\begin{equation}\label{out:cohort-size}
 \max\{1,C_gG_0\}\leq K_c\leq C_KG_0
\end{equation}
for a fixed $C_K$; increasing $C_g$ if necessary also covers all
designated-wire counts. The total number of replicas per component is
$R=T_{\rm tap}K_c$, again a square. Each cohort contains its complete
bounded list of functional invocations, sharing these invocations
among its $K_c$ outputs. For each invocation, assign its $W$ designated
wires to these outputs by round robin. Since $W\leq K_c$,
\begin{equation}\label{out:round-robin}
 \#\{\text{uses of any designated wire}\}
 \leq\lceil K_c/W\rceil\leq 2K_c/W.
\end{equation}
A fraction $\upsilon$ of wrong designated wires therefore affects at
most $2\upsilon$ of the cohort outputs. The remaining gates, called a
direct cone, are separately allocated for each output replica.

\begin{lemma}[Bounded physical read multiplicity]\label{out:routing}
Every physical replica can be made the input of at most a fixed number
of comparison occurrences, counting all outer and inner hypothetical
branches. The construction is deterministic and uses no shared rounded
bit between different comparison occurrences.
\end{lemma}
\begin{proof}
Fix a physical block and a raw site $k'$. For each of the fixed possible
address permutations, a functional port requesting $k'$ has a fixed
$k_0$ preimage. There are $S_0$ output bit positions and $T_{\rm tap}$
tapes at that preimage, and only a bounded number of invocation/port
roles. Thus at most $C S_0T_{\rm tap}$ invocation ports request $k'$.
By \eqref{out:port-occurrence}, any position in that site is used at most
\begin{equation}\label{out:gadget-load}
 C S_0T_{\rm tap}\,\frac{C_{\rm occ}G_0}{S_0}
       \leq C' T_{\rm tap}K_c=C'R
\end{equation}
times. Lemma~\ref{out:tapes}, followed by replication over the cohort,
gives $O(T_{\rm tap}K_c)=O(R)$ direct comparison occurrences at each
base coordinate. Enumerate all these occurrences and assign successive
ones round robin to its $R$ physical replicas. Each actual replica has
bounded resulting multiplicity. Each comparison is performed in its
own gate cone directly on that replica's $x$ value. In particular, an
error in one comparison wire is not broadcast to other inner cones.
\end{proof}

Write the physical replica values and their means as
\[
 X_{ir}=3x_{ir}-1,\qquad \bar X_i=R^{-1}\sum_rX_{ir},\qquad
 \norm{Y}_{\rm rep}^2=\frac1{mR}\sum_{i=1}^{4m}\sum_{r=1}^R Y_{ir}^2.
\]
A vector constant over replicas has replica norm equal to its base norm.
Let $J$ be orthogonal projection onto these constant-over-replica
vectors, so $JX=\bar X$ with the usual identification.

For each $i$, average the replica labels using the half-lazy
Margulis--Gabber--Galil operator $A_{\rm rep}$ on
$(\mathbb Z/q\mathbb Z)^2$, $q=\sqrt R$. Its sixteen equally weighted
labels consist of eight identity labels and the eight moves
\[
 (a\pm2b,b),\quad(a\pm(2b+1),b),\quad
 (a,b\pm2a),\quad(a,b\pm(2a+1)),
\]
with multiplicities retained. For every positive integer modulus the
normalized eight-move operator is symmetric with nonconstant numerical
radius at most $5\sqrt2/8$; see the primary formal proof
\cite[\texttt{mgg\_numerical\_radius}]{MGGAFP}. Thus
\begin{equation}\label{out:replica-gap}
 A_{\rm rep}J=J,\qquad
 \norm{A_{\rm rep}(I-J)Y}_{\rm rep}
       \leq(1-\kappa)\norm{(I-J)Y}_{\rm rep},
 \qquad \kappa=1/100.
\end{equation}
The modulus need not be prime. Loops and parallel labels are kept as
individual averaging inputs. The mean-zero space for modulus one is
zero. Every averaging connection has bounded multiplicity as well.

We specify the direct cones using exactly the gate types in
Definition~\ref{def:gcircuit}. This also specifies how public constants
and finite numerical processing are charged. Every needed constant
node is the fresh output of a Constant gate in the cone using it, unless
its value is incorporated as a legal Scale parameter or into a Boolean
truth table. A physical comparison at threshold $t\in[-1,2]$ is a
Constant gate with parameter $(t+1)/3$ and a Less gate on the physical
replica's $x$ node and that constant, with orientation chosen for the
required bit. Outside the physical range its answer is a public
constant. On a satisfied comparison, an ideal mean gap greater than
$4\omega$, an actual read deviation at most $\omega$, and
$\eps<\omega/1000$ leave a strict Less antecedent even after the
constant error: the node-scale gap is at least $\omega-\eps>\eps$.

Boolean finite tables, flag processing, and quantization-bin processing
use And, Or and Not with fan-in at most two. If their inputs are correct
near-bits, each gate produces the same correct near-bit interval,
without accumulating Boolean error. In particular the weak inequalities
at $\eps$ and $1-\eps$ are sufficient. A selection among hypothetical
invocations is implemented as a Boolean multiplexer: And each candidate
answer with its selector and Or the resulting terms. An inactive
selector is a low near-bit and forces that term low even if the unused
candidate answer is arbitrary in $[0,1]$. Thus unsuccessful unused
branches are shielded, without assuming that their outputs are bits.
All finite answer tables have fixed size, although their public entries
may depend on the tape and threshold.

Represent the ideal dyadic field value by
\begin{equation}\label{out:signed-value}
 \widehat v=B_1(p^*-n^*),\qquad
 p^*,n^*\in[0,1],\quad p^*n^*=0,
\end{equation}
using a fixed number of binary digits; enlarge the dyadic precision
within the earlier estimator slack. Scale the near-bit digits by their
dyadic weights and combine them by clipped Add gates to obtain $p,n$.
Compute the $x$-average by sixteen Scale gates of weight $1/16$ and a
bounded Add chain. All exact partial sums are in $[0,1]$. Choose a
positive rational step with
$t_\beta=\beta B_1/3\leq1$. Scale $p,n$ by $t_\beta$, add the positive
term to the computed average, then subtract the negative term. The last
output, or a final Copy output, is $x_{ir}$ itself. When the ideal signs
are exclusive this expression is exactly
\[
 \frac{1+\clip_{[-1,2]}((A_{\rm rep}X)_{ir}
                         +\beta\widehat v)}3.
\]
Indeed if the negative term vanishes the only possible clipping is at
the upper end; if the positive term vanishes it is at the lower end.
The affine change of coordinates commutes with the stochastic average.

\begin{lemma}[Coarse and fine direct-cone bounds]\label{out:direct-cone}
There is a fixed $C_{\rm ar}$, determined by the finite direct-cone
shape before $\beta$ is filled in, such that every entirely
$\eps$-satisfied direct cone obeys
\begin{equation}\label{out:coarse-pointwise}
 |X_{ir}-(A_{\rm rep}X)_{ir}|
       \leq2\beta B_1+C_{\rm ar}\eps.
\end{equation}
No correctness of its Boolean or functional inputs is required.
If its direct comparisons and selected invocation answers are correct
near-bits for the ideal run on $\bar X$, then also
\begin{equation}\label{out:fine-pointwise}
 |X_{ir}-\clip_{[-1,2]}((A_{\rm rep}X)_{ir}
                +\beta\widehat v_{ir}(\bar X))|
       \leq C_{\rm ar}\eps.
\end{equation}
\end{lemma}
\begin{proof}
All hidden nodes, including $p,n$, remain in $[0,1]$ even when their
logical values are wrong. Satisfied final Scale gates therefore make
each signed term at most $t_\beta+\eps$. The absolute change from the
computed average is at most their sum plus the final arithmetic errors.
The averaging chain is within $C\eps$ of its exact average. Returning
to physical units gives \eqref{out:coarse-pointwise}, after enlarging
$C_{\rm ar}$. For the fine bound, induction through the shielded
Boolean selections supplies the correct digits in
\eqref{out:signed-value}. Scale is nonexpansive and clipped Add and
Subtract are Lipschitz with respect to the sum of their input errors.
For an explicit bound, let $J_d$ be the number of dyadic digits in
one magnitude. A magnitude uses $J_d$ Scale and $J_d-1$ Add gates;
its error is at most $(3J_d-1)\eps$. The sixteen Scale and fifteen Add
gates for the average contribute at most $31\eps$. The two final
Scale gates and the final Add/Subtract pair therefore contribute a
total error at most $(6J_d+33)\eps$ in node units, or
$(18J_d+99)\eps$ in physical units. An optional final Copy adds
$3\eps$. The coarse bound's arithmetic error is at most $108\eps$
including that Copy. Thus $C_{\rm ar}=18J_d+108$ suffices for this
implementation, uniformly in all Scale parameters in $[0,1]$.
The exclusive-sign identity gives \eqref{out:fine-pointwise}.
\end{proof}

Every invocation owns a disjoint gate set, and every direct cone owns a
disjoint gate set. Internal outputs are fresh; designated and external
wires are read-only fan-outs. The cyclic physical replica output is
produced by the final gate of its own unique cone and by no other gate.
There is no promised free constant node. Let
\begin{equation}\label{out:gate-count}
 K_{\rm out}=4mR,
 \qquad |T|\leq C_{\rm tot}K_{\rm out}.
\end{equation}
The second assertion follows because a cohort has a bounded number of
invocations, each with $O(G_0)=O(K_c)$ gates, and $K_c$ direct cones of
bounded size. Each designated output is included in its invocation's
gate count. In particular, concentrating failures on extraction gates
receives the same charge as any other failures.

\subsection{All failure and spoilage charges}

We next choose the late offsets. Use a uniform rational grid in
$[2/5,3/5]$ for $\xi$ and, independently, a uniform rational grid modulo
the fixed scalar-bin width for the quantization offset. For every fixed
physical value, the probability of lying within $4\omega$ of a binary
threshold or any relevant shifted scalar boundary is at most
\begin{equation}\label{out:anticoncentration}
 C_{\rm bin}(D_{\rm grid}^{-1}+\omega),
\end{equation}
where $D_{\rm grid}$ is a lower bound on the two grid sizes and
$C_{\rm bin}$ depends only on the already fixed widths and bin count.
This follows by counting grid points in intervals of total length
$O(\omega)$, adding at most a fixed number of boundary grid points.
The number of comparison slots is unchanged when the grids are refined.
The ideal accuracy estimate is conditional on every allowed offset, so
refinement cannot invalidate it.

Call a routed comparison occurrence \emph{spoiled} if its mean value is
within $4\omega$ of its prescribed boundary or if its physical replica
value differs from its mean by more than $\omega$. Otherwise its input
has the prescribed ideal side with physical gap greater than $\omega$.
For direct cones, spoiled comparisons are charged to their output. For
functional invocations, count the occurrences themselves, including
all hypothetical branches.

\begin{lemma}[Deterministic charges for arbitrary assignments]
\label{out:charges}
There are fixed counting constants with the following properties.
For an assignment with at most $\delta|T|$ failed gates:
\begin{enumerate}[label=\textup{(\roman*)}]
\item At most a $C_{\rm tot}\delta$ fraction of direct cones contains a
failed gate.
\item Cohorts containing an invocation with more than $\delta_gG_0$
failed gates account for at most $C\delta/\delta_g$ of all outputs.
\item The total number of excessive-deviation comparison occurrences,
divided by $K_{\rm out}$, is at most
$C\norm{X-\bar X}_{\rm rep}^2/\omega^2$.
\item The averaged number of ambiguous gadget comparison occurrences,
divided by $K_{\rm out}$, is at most
$C(D_{\rm grid}^{-1}+\omega+\eta_0)$.
The fraction of direct cones with an ambiguous comparison has the same
form of bound. Here the averaging over tapes is the actual empirical
frequency of the circuit, and the bounds hold for every fixed mean
vector.
\item Cohorts containing an invocation with more than $\gamma_gG_0$
spoiled external occurrences account for at most $C/\gamma_g$ times
the total normalized gadget-spoil count.
\end{enumerate}
\end{lemma}
\begin{proof}
Disjointness of direct gate sets proves (i). The number of invocations
violating the threshold in (ii) is at most
$\delta|T|/(\delta_gG_0)$. Charging $K_c$ outputs for each and using
\eqref{out:cohort-size} and \eqref{out:gate-count} proves (ii), even if
the same cohort is charged repeatedly. For (iii), Markov's inequality
gives at most
$mR\norm{X-\bar X}_{\rm rep}^2/\omega^2$ excessive physical replicas.
Lemma~\ref{out:routing} bounds their total comparison multiplicity by
a constant. The normalization $K_{\rm out}=4mR$ only improves the bound.

For (iv), fix a port of an invocation at site $k$ and threshold $\xi$.
Equation~\eqref{out:port-occurrence} bounds its ambiguous occurrences by
\[
 C_{\rm occ}G_0\,
 \frac{\#\{z:|\bar X_{kz}-\xi|\leq4\omega\}}{S_0}.
\]
Every site is an image of a uniform $k_0$ under a member of the fixed
permutation list. Sum over that list and all invocation ports under the
unconditioned tape law, before selecting any answer branch. Averaging the offset and using
\eqref{out:anticoncentration} bounds the expected number per trial by
$C G_0(D_{\rm grid}^{-1}+\omega)$. The full occurrence bound is uniform
in threshold-dependent function data. Tape rationalization changes a
count bounded by $CG_0$ by at most $CG_0\eta_0/S_0$. Dividing each
trial's charge by its $K_c$ output replicas gives (iv). Direct slots use
the full-tree density argument in \eqref{out:direct-density}, and the
same grid count; their count per trial is bounded by $H_{\rm out}$.
This reasoning applies to every fixed eventual mean, rather than a
mean selected before construction. Finally, (v) is the same threshold
counting proof as (ii), with spoiled occurrences replacing failed gates.
\end{proof}

\subsection{Parameter order and synchronization before decisions}

We give an acyclic choice of all remaining constants. Reserve enough
slack in \eqref{out:ideal-accuracy} that after tape rationalization the
ideal error norm is at most $c_*\kappa/20$. Choose $q_0>0$ so small that
\begin{equation}\label{out:q0}
 20(B_1+1)\sqrt{q_0}<c_*\kappa/20.
\end{equation}
The geometric and statistical accuracies, correction rounds, and
completed flags are chosen as above to put the ideal good-port failure
probability below, for example, $q_0/20$. These are logical-call budgets.
After the complete outer tree is fixed, choose
$2B_{\rm out}\upsilon<q_0/20$ and obtain the gadget constants. Increasing
$K_c$ for these gadgets weights every trial equally and does not
magnify its ideal failure probability.

Now choose fine fixed offset grids, then a small positive rational
$\omega$, then a small tape tolerance $\eta_0$, and then a desired
synchronization radius $d_{\rm sync}>0$. By
Lemmas~\ref{out:tapes} and~\ref{out:charges}, they can be chosen so that,
whenever $\norm{X-\bar X}_{\rm rep}\leq d_{\rm sync}$, the sum of the
following empirical output fractions is less than $q_0/2$: failure of
the ideal good-port event, direct ambiguous/excessive comparisons,
and cohorts having an invocation with too many spoiled occurrences.
For example assign $q_0/20$ to each grouped term, with the
$C/\gamma_g$ factors included before choosing the responsible parameter.
Ideal squared error is bounded, so make $\eta_0$ small enough for its
reserved norm slack as well. This order is possible because all counts
and density constants are already fixed and independent of these late
grids and margins.

Choose a positive rational $\beta\leq\beta_0$ with
$\beta B_1/3\leq1$ and small enough that
$4\beta B_1/\kappa<d_{\rm sync}/2$. Finally choose positive rational
$\eps<1/10$ and $\delta<1$ so small that
\begin{gather}
 \eps\leq\eps_g,\qquad\eps<\omega/1000,
 \qquad C\delta/\delta_g<q_0/20,\label{out:epsilon-delta}\\
 \frac{4\beta B_1+2C_{\rm ar}\eps
             +6\sqrt{C_{\rm tot}\delta}}{\kappa}<d_{\rm sync},
 \label{out:coarse-budget}\\
 2C_{\rm ar}\eps+6\sqrt{C_{\rm tot}\delta}
                <\beta c_*\kappa/20.
 \label{out:fine-budget}
\end{gather}
There is positive slack in each inequality at $\eps=\delta=0$, so
fixed positive rational choices exist. If necessary decrease them for
any remaining direct numerical slack. All unstable events other than
failed direct cones, including the mapped extraction errors, now total
less than $q_0$.

The finitely many lower-size cutoffs are imposed after these choices by
increasing $C_{\rm base}$. These include $L_{\rm port}$, the finitely
many $L_B(a)$ for the chosen tester accuracies, and $L_g$ for the fixed
arity, size factor, and output loss. This does not create arity feedback. The raw
algorithm's exponent is independent of correction rounds; the chosen
port allowance has slack above that exponent and the degree-ten output
polynomial evaluation. For the fixed cones, increasing the additive
padding absorbs remaining multiplicative factors and cutoffs into a
single fixed size factor, even factor one if desired. Padding checks
and zero construction are $O(L)$ scans with uniform coefficients; their
implementation does not perform hidden work growing with
$C_{\rm base}$. Public computation of bit-polynomial coefficients may
have a larger fixed polynomial exponent than the circuit evaluating
the resulting monomials. Only the latter exponent is charged to the
functional size allowance. No parameter varies with $N$ after this
single choice.

\begin{lemma}[Coarse synchronization]\label{out:coarse-sync}
Every assignment satisfying all but $\delta|T|$ gates has
$\norm{X-\bar X}_{\rm rep}<d_{\rm sync}$, without any assumption that
its functional decisions are correct.
\end{lemma}
\begin{proof}
Apply \eqref{out:coarse-pointwise} to nonfailed direct cones. At failed
ones, both $X_{ir}$ and $(A_{\rm rep}X)_{ir}$ belong to $[-1,2]$, so
their difference is at most three. The factor four in the norm and
Lemma~\ref{out:charges}(i) give
\[
 \norm{X-A_{\rm rep}X}_{\rm rep}
 \leq4\beta B_1+2C_{\rm ar}\eps
                  +6\sqrt{C_{\rm tot}\delta}.
\]
On the other hand \eqref{out:replica-gap} and the reverse triangle
inequality give
$\kappa\norm{X-\bar X}_{\rm rep}
\leq\norm{X-A_{\rm rep}X}_{\rm rep}$.
Equation~\eqref{out:coarse-budget} proves the assertion. In particular,
failures at physical feedback outputs, threshold constants, or the
numeric averaging gates are already included; no decision is trusted
in establishing this estimate.
\end{proof}

\subsection{The clipped mean contradiction and universal decoding}

\begin{lemma}[Fine residual]\label{out:fine-residual}
If an assignment meets the gate-failure budget and
$\dist(\bar X,\mathcal E)>10\tau$, then
\begin{equation}\label{out:residual}
 E_{\rm res}:=
 \norm{X-\clip_{\mathcal B}(A_{\rm rep}X+\beta v(\bar X))}_{\rm rep}
                  <\beta c_*\kappa/2.
\end{equation}
Here clipping is componentwise on the replicas, and $v(\bar X)$ is
repeated identically over them.
\end{lemma}
\begin{proof}
Fix the actual mean $\bar X$ in all ideal tape experiments. Coarse
synchronization permits all spoilage charges already budgeted. Discard
cohorts with a high-failure or high-spoil invocation. In every remaining
invocation whose ideal branch-pattern inputs are all near ports or
public substitutions, Theorem~\ref{thm:port-gadget} applies. Mark all
cohort outputs receiving an incorrect designated wire from any such
invocation. Equations~\eqref{out:designated-count} and
\eqref{out:round-robin} bound their union by
$2B_{\rm out}\upsilon$ of the outputs. Do this marking for every
near-input hypothetical invocation before following the selected path.
Non-near-input invocations are irrelevant unless selected; on the ideal
good-port event a selected invocation necessarily is a near-input one.

Outside these marks and the other unstable events, every direct
comparison in the complete tree has the needed margin and every
selected invocation supplies its correct near-bit. Induct on the
ideal answer sequence: earlier reliable bits select its actual
substitution/function branch, the selected gadget is reliable, and
Boolean shielding removes arbitrary unused outputs. Thus the whole
selected direct computation equals its ideal Boolean run, and
\eqref{out:fine-pointwise} holds.

For an unstable but entirely satisfied direct cone, use only
\eqref{out:coarse-pointwise} and $|v_i(\bar X)|\leq B_1$:
its discrepancy from the clipped true-field update is at most
$3\beta B_1+C_{\rm ar}\eps$. This is proportional to $\beta$ apart
from numeric error. Failed direct cones use only the range bound three.
The empirical ideal error, clipping nonexpansiveness, and the total
unstable fraction at most $q_0$ therefore give
\[
 E_{\rm res}
 \leq\beta\bigl(c_*\kappa/20+6B_1\sqrt{q_0}\bigr)
      +2C_{\rm ar}\eps+6\sqrt{C_{\rm tot}\delta}
 <\beta c_*\kappa/2
\]
by \eqref{out:q0} and \eqref{out:fine-budget}. This includes arbitrary
concentration on extraction gates: above-budget invocations lose their
whole cohort; below-budget invocations lose their guaranteed fraction
of designated wires, and the lower bound in
\eqref{out:designated-count} prevents a concentration factor.
\end{proof}

\begin{lemma}[Clipping and replica means]\label{out:clip-mean}
Let $A$ be a stochastic replica operator satisfying
\eqref{out:replica-gap}, and let $c$ be constant across the replicas of
each component. Put $F=\clip_{\mathcal B}(AX+c)$ and
$E=\norm{X-F}_{\rm rep}$. Then
\[
 \norm{X-JX}_{\rm rep}\leq E/\kappa,\qquad
 \norm{JX-\clip_{\mathcal B}(JX+c)}_{\rm b}\leq E/\kappa.
\]
\end{lemma}
\begin{proof}
Write $Q=I-J$ and $g=\clip_{\mathcal B}(JX+c)$, a repeated vector.
The mean projection minimizes distance to repeated vectors, and
coordinate clipping is nonexpansive. Hence
\[
 \norm{QF}_{\rm rep}
 \leq\norm{F-g}_{\rm rep}
 \leq\norm{A(X-JX)}_{\rm rep}
 \leq(1-\kappa)\norm{QX}_{\rm rep}.
\]
Projecting $X-F$ gives
$\norm{QX}_{\rm rep}\leq E+(1-\kappa)\norm{QX}_{\rm rep}$,
which proves the first inequality. Taking means and then comparing to
$g$ gives
\[
 \norm{JX-g}_{\rm b}
 \leq\norm{J(X-F)}_{\rm rep}+\norm{JF-g}_{\rm rep}
 \leq E+(1-\kappa)\norm{QX}_{\rm rep}\leq E/\kappa.
\]
This argument does not exchange averaging and clipping.
\end{proof}

Apply Lemma~\ref{out:clip-mean} with $c=\beta v(\bar X)$ and the fine
residual. If the mean were farther than $10\tau$ from $\mathcal E$, it
would give
\[
 \norm{\bar X-\clip_{\mathcal B}(\bar X+\beta v(\bar X))}_{\rm b}
 <\beta c_* ,
\]
contrary to \eqref{out:geometry-displacement}. Thus every assignment
meeting the gate-failure budget satisfies
\begin{equation}\label{out:endpoint-proximity}
 \dist(\bar X,\mathcal E)\leq10\tau.
\end{equation}
Choose an endpoint witnessing this bound. Its first block is exactly
$E_w$ for a desired endpoint. Round the first block of $\bar X$ at
$1/2$. Each erroneous bit contributes at least $1/4$ to its squared
coordinate distance from $E_w$, and so the resulting word $y$ obeys
\[
 \dist(y,E_w)\leq4(10\tau)^2<\rho_{\rm dec}.
\]
The global decoder following \eqref{out:binary-encoding} recovers $w$.
The endpoint decoder in Theorem~\ref{thm:outer-representation} and
Lemma~\ref{lem:macrograph} returns a nonzero End-of-Line solution. It
does not require knowing which gates failed or which endpoint witnessed
\eqref{out:endpoint-proximity}.

This defines a deterministic decoder on every acceptable rational
assignment. Means and comparisons can be computed exactly: forming a
common denominator within a group increases its bit length by at most
the sum of its input denominator bit lengths, followed by polynomial
integer-arithmetic overhead. The port and raw algorithms are polynomial
time, with arbitrary fixed defaults on out-of-promise inputs. The
proved proximity places this actual input inside their guaranteed
radii. Hence decoding is polynomial in $N$ plus the total assignment
encoding length. The soundness argument itself applied to all real
assignments satisfying the failure budget.

\subsection{Complete output length, uniformity, and totality}

Every constituent scale is a fixed polynomial in $L$, and
$|\Gamma|=O(N\ell_N^{O(1)})$. Equations~\eqref{out:gate-count} and
\eqref{out:cohort-size} give, for some fixed exponent $b_1$,
\begin{equation}\label{out:size}
 |V|+|T|=O(mT_{\rm tap}K_c)
       =O(|\Gamma|S_0T_{\rm tap}G_0)
       =O(N\ell_N^{b_1}).
\end{equation}
This count includes every copied direct cone, every entire shared
invocation, its extraction gates, every Constant gate, and every
comparison in every hypothetical branch. Routing creates one connection
record per counted input occurrence; it is not an implicit description
of a larger circuit. Explicit node and gate identifiers require
$O(\log(N\ell_N^{b_1}))=O(\ell_N)$ bits. Field/index metadata, locally
generated function coefficients, and every rational parameter have
fixed polynomial-in-$L$ bit bounds. The offsets, bin widths,
$\beta,\eps,\delta$ and numerical precisions are fixed rational
constants. In particular, the threshold-input length qualification in
Theorem~\ref{thm:port-gadget} contributes only a fixed bound here.
If a representation repeats local parameter descriptions instead of
sharing them, multiplying \eqref{out:size} by their polynomial-in-$L$
length still changes only a fixed logarithmic exponent. Input/output
lists and delimiters obey the same bound. Thus there are a fixed
positive integer $A$ and fixed nonnegative integer $b$ for which the
complete output encoding has length at most $AN\ell_N^b$.

All tape masses and their integer multiplicities are deterministically
enumerable. So are every raw-neighborhood and correction-cone address,
every padded function, every inner invocation gate, every replica move,
and every round-robin routing record. Public outer encodings and the
source data are constructed by the polynomial-time algorithms already
proved. Polynomially many operations on polynomial-bit integers suffice
for all these tasks. Products and compositions of the finitely many
fixed polynomial bounds give one deterministic polynomial-time
reduction. It never lists the graph of all logical vertex labels. The
gate set is nonempty, all parameters lie in $[0,1]$, fan-in is at most
two, and the fresh-output construction enforces the required unique
output ownership despite the feedback cycles.

It remains to establish the finite totality assertion for arbitrary
generalized circuits, independently of this reduction.

\begin{lemma}[A fixed rational mesh satisfies every gate]
\label{out:totality}
For every fixed rational $0<\eps<1/10$, every generalized circuit in
Definition~\ref{def:gcircuit} has a rational assignment satisfying all
its gates, each coordinate having denominator
$q=\lceil20/\eps\rceil$. The complete assignment length is bounded by
one fixed polynomial with nonnegative integer coefficients in the
circuit encoding length.
\end{lemma}
\begin{proof}
Construct a continuous self-map of $[0,1]^V$. At an arithmetic gate's
unique output use its exact numerical operation. At a Less gate with
inputs $a,b$, use the clipped linear ramp that is zero when
$x_b-x_a\leq-\eps/2$ and one when
$x_b-x_a\geq\eps/2$. For Boolean gates define a continuous signal
$h:[0,1]\to[0,1]$ equal to zero on $[0,2\eps]$, equal to one on
$[1-2\eps,1]$, and linear between them. Use respectively
\[
 \max\{h(x_a),h(x_b)\},\qquad
 \min\{h(x_a),h(x_b)\},\qquad 1-h(x_a)
\]
for Or, And, and Not. Set coordinates not produced by gates to zero.
Unique output ownership makes these prescriptions a well-defined
continuous cube map. Brouwer's fixed-point theorem~\cite{Brouwer1911} gives a fixed point
$x^*$; no efficient computation of it is asserted.

Round every coordinate to the nearest point of
$\{0,1/q,\ldots,1\}$, obtaining $\widetilde x$ with
$|\widetilde x_v-x_v^*|\leq1/(2q)\leq\eps/40$.
Constant errors are at most $\eps/40$, Scale and Copy errors at most
$2\eps/40$, and Add and Subtract errors at most $3\eps/40$, using
parameter bounds and nonexpansive clipping. For a triggered strict
Less antecedent after rounding, its signed input gap before rounding
has the same sign and magnitude strictly greater than
$\eps-2\eps/40>\eps/2$. Its fixed-point output is therefore the
required endpoint zero or one, which rounding preserves.

If a weak Boolean low antecedent
$\widetilde x_a\leq\eps$ holds, then
$x_a^*\leq\eps+\eps/40<2\eps$, so $h(x_a^*)=0$.
Similarly $\widetilde x_a\geq1-\eps$ implies
$h(x_a^*)=1$. Every applicable Or, And, or Not conclusion follows
from its displayed exact formula, including equality at the weak
threshold. The corresponding fixed-point output is zero or one and
is unchanged by rounding. This verifies every gate implication.

The denominator $q$ is a fixed integer depending only on $\eps$.
There are at most a fixed multiple of $M+1$ explicitly listed nodes
in a length-$M$ input, and each output rational has a fixed bit bound.
Accounting also for order, indices if used, and delimiters, a
sufficiently large fixed integer $C$ makes
$p(M)=C(M+1)^2$ an upper bound for the entire assignment. This
polynomial has nonnegative integer coefficients and is independent of
the particular circuit and its parameter denominators.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
The preceding argument uses Theorem~\ref{thm:port-gadget}, whose construction
is proved in Sections~\ref{sec:inner-algebra}--\ref{sec:inner-gadget}, and
Lemma~\ref{lem:geometric-procedures}, proved in Section~\ref{sec:geometry}.
Use the fixed rational $\eps,\delta$ chosen in
\eqref{out:epsilon-delta}--\eqref{out:fine-budget} and the polynomial
$p$ from Lemma~\ref{out:totality}. That lemma proves totality, with all
gates satisfied. The explicit construction and \eqref{out:size} give
the claimed complete quasilinear encoding bound and a deterministic
polynomial-time reduction. Equations~\eqref{out:endpoint-proximity}
and~\eqref{out:tau}, followed by the global endpoint decoder, give the
required solution from every acceptable assignment, with the stated
polynomial running time. Every constant and algorithm has been fixed
independently of the input. This proves all four assertions.
\end{proof}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/06-outer-composition.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/07-inner-algebra.tex ====
\section{The inner path and its algebraic representation}\label{sec:inner-algebra}

We construct the algebraic part of the decision gadget promised in
Theorem~\ref{thm:port-gadget}.  Its graph consists of one directed path for
each tuple of read-only inputs.  Besides encoding these paths, the
construction arranges that every chart involving one read port has exactly
the same native layout and formula in a common one-port template.  This
literal correspondence, including all verification auxiliaries, will supply
the distinguished source in Section~\ref{sec:inner-gadget}.

\emph{All field, clock, dimension, and size notation is reset in this
section.}  In particular, the fields and the polynomial-size affine domain
below are independent of the outer index group.  The construction and its
proof do not use the outer circuit realization.  All graphs in this section
are described implicitly; we never enumerate their exponentially many
Boolean input tuples.

\subsection{Parameters and the extended path}\label{ia:path-parameters}

Fix the exponent allowance $c_{\rm allow}$ in
Theorem~\ref{thm:port-gadget}, and choose a fixed integer
$c_0>c_{\rm allow}+5$.  For a port length $L$, set
\begin{equation}
 n'=(L+2)^{c_0},\qquad
 h=\Theta\bigl((\log(n'+2))^{1/4}\bigr),\qquad
 H=\mathbb F_h,\qquad q=h^v,                         \label{ia:parameters}
\end{equation}
where $h$ is a deterministically chosen power of two and $v$ is a fixed
sufficiently large integer.  Let $\ell$ be the least positive integer
coprime to $v$ for which $h^\ell\ge(n')^3+3$, and put
\begin{equation}
 K_1=\mathbb F_{h^\ell},\quad K=\mathbb F_{q^\ell},\quad
 C=K_1^\times,\quad M=h^\ell-1,\quad D=\ell^2-1.
                                                               \label{ia:fields}
\end{equation}
Fix a generator $\gamma$ of $C$.  Since $\gcd(\ell,v)=1$, the subfields
$K_1$ and $\mathbb F_q$ of $K$ intersect in $H$ and generate $K$.
An $H$-basis of $K_1$ is therefore an $\mathbb F_q$-basis of $K$.
We use it throughout: one $K$ variable has $\ell$ coordinates over
$\mathbb F_q$, and its $H^\ell$ digit grid is precisely $K_1$.

An interval of $v$ consecutive integers contains an integer congruent to
$1$ modulo $v$.  Thus the coprimality restriction changes the first size
threshold for $\ell$ by at most $v$, and
\begin{equation}
 \ell=\Theta(\log n'/\log\log n'),\qquad
 h=o(\ell),\qquad |K|=q^\ell=(n')^{O(1)}.          \label{ia:size-bounds}
\end{equation}
Choose $v$ large enough that $q$ dominates the fixed power of $D$ needed
by the scalar line--point input in Section~\ref{sec:inner-binary}, with
positive power slack.  In particular, $D/q\longrightarrow0$.  After $v$,
fix an integer $a>v$ and set
\begin{equation}
 B=\lceil q^{1/a}\rceil\le\ell                     \label{ia:radix-size}
\end{equation}
for all sufficiently large $L$.  Every fixed multiple of $\ell h$ is then
eventually below $D$.  All these choices precede the number of ports and
the particular Boolean function.

All finite sets used below have polynomial cardinality in $n'$.  Field
presentations, compatible embeddings, bases, generators, interpolation
matrices, and deterministic orderings can consequently be found by
exhaustive finite-field algebra in polynomial time in these cardinalities.
For instance, irreducible presentations and suitable generators can be
found by enumeration and their defining properties checked by finite
arithmetic.  No claim of polynomial time in the logarithm of a field's
cardinality is required.  Fix these conventions once for the common
template.  Finitely many smaller sizes can be handled by enlarging the
fixed lower cutoff as in the gadget statement.

Now fix an arbitrary constant number $k\ge1$ of ports and an arbitrary
constant size factor for the Boolean circuit of
$f:(\{0,1\}^L)^k\to\{0,1\}$.  Invocations with fewer ports are padded
by unused public-zero ports.  Normalize the circuit to a topological list
of gates with Boolean rules of degree at most two over characteristic two,
and finish with a separate output copy gate.  This costs only a constant
factor in circuit size.  The size cutoff may depend on $k$ and on the
fixed size factor.

The logical tape has read-only groups $s_i\in\{0,1\}^L$ for $i\in[k]$,
a work group $w$ indexed by $K_1$, and legacy bits $e_z$ indexed by
$z\in C$.  Each $s_i$ is embedded by the same fixed injection of $[L]$
into $K_1$; other grid entries are specified zeros.  Stages
$r=0,\ldots,M-1$ have labels $c=\gamma^r$.  Work is blank at stage zero.
During step $r=j-1\to j$, for $1\le j\le L$, copy $s_i[j]$ to a
distinct work location $w_{i,j}$ for every $i$ in parallel.  Thereafter
assign one new gate location per step, using earlier work locations or
public constants.  Reserve a fixed grid position $y_{\rm out}$ solely
for the final output assignment.  Pad remaining steps by idle work rules.
The number of work positions and nonidle steps is at most
\[
 kL+O(1)+O(L^{c_{\rm allow}}),
\]
with constants allowed to depend on the circuit size factor.  The choices
above make both counts fit below the available grid and stage sizes.

At stage $c$, require $e$ to be one-hot at $c^{-1}$.  Require every
unassigned work position to be zero, every assigned copy to equal its
specified input, and every assigned gate to satisfy its rule.  Consequently
each Boolean tuple $s=(s_i)_i$ has exactly one valid tape
$b=(w,e,(s_i)_i)$ at every stage.  Join consecutive stages by their
assignment steps, and disable the wrap step $\gamma^{-1}\to1$.

\begin{lemma}[Validity and a constant edit alphabet]\label{ia:validity}
The union of the stage-valid tapes is described by polynomially many
Boolean equations of degree at most $d=3$.  For fixed $k$, every macro
step and its inverse use a fixed finite alphabet of flips, whose
coefficients are determined by a bounded number of logical reads and
public rule bits.  Equations involving $s_i$ and other groups can be
written as the sum of an $(e,s_i)$ contribution and an $(e,w)$
contribution.  No equation requires a product involving two distinct
positive port groups.
\end{lemma}

\begin{proof}
Impose bitness on all logical coordinates, and
\begin{equation}
 e_z e_{z'}=0\quad(z\ne z'),\qquad
 \sum_{z\in C}e_z=1.                              \label{ia:one-hot}
\end{equation}
For each stage $c$, multiply all its work requirements by
$e_{c^{-1}}$.  Gate equations have degree at most two before this
multiplication, so the resulting degree is at most three.  A copy
requirement at an occupied position is specifically
\begin{equation}
 e_{c^{-1}}w_{i,j}-e_{c^{-1}}s_i[j]=0.             \label{ia:copy-equation}
\end{equation}
The requirements are vacuous at unselected stages.  At the one selected
stage, the prefix equations determine all assigned work positions in
topological order and force every other position to zero.  This proves
the characterization and the dependency assertion.

For start frame $c$, list $k$ work write slots and the two legacy slots
$c^{-1},(\gamma c)^{-1}$.  During a gate step only the first work slot
is active; use a zero unit vector for every inactive slot.  Write
$v_1(c),\ldots,v_{k+2}(c)$ for these public vectors in the logical tape
space, and for $\sigma\in\{0,1\}^{k+2}$ put
\begin{equation}
 h_\sigma(c)=\sum_{r=1}^{k+2}\sigma_r v_r(c).
                                                               \label{ia:flip-mask}
\end{equation}
At a valid pre tape, choose a work coefficient to be the old bit plus
its required new assignment; choose both legacy coefficients to be one.
The required assignment is a read input, a degree-at-most-two gate of
earlier work bits, or the unchanged value for an inactive slot.  At a
valid post tape, the inverse clears precisely the newly assigned
positions and flips the same legacy positions.  Earlier operands have
not changed, so the two operations are reciprocal.  There is a bounded
list of old-value, operand, and input reads for fixed $k$.  Public phase,
gate, activity, and wrap bits select their roles.  Define the same
coefficient rules on arbitrary Boolean read answers; at the disabled
wrap use an idle work proposal.  This gives a finite branch system
without any assumption that those arbitrary answers describe a valid
tape.
\end{proof}

\begin{definition}[Extended inner graph]\label{ia:extended-path}
An extended label is $(c,b,\eta,t)$, where $c\in C$,
$\eta\in\{0,1\}$, and $0\le t\le q-1$.  In pre mode $\eta=0$,
$b$ must be valid at stage $c$; in post mode $\eta=1$, it must be valid
at stage $\gamma c$.  Edges are the following:
\begin{enumerate}
\item in pre mode, increase $t$ by one until $q-1$;
\item at full pre progress, perform the enabled macro edit and change
to post mode, retaining the frame and full progress;
\item in post mode, decrease $t$ by one until zero;
\item at empty post progress, change to empty pre mode in frame
$\gamma c$, leaving $b$ unchanged.
\end{enumerate}
The fourth operation is included also for $c=\gamma^{-1}$; only the
second operation is disabled at that frame.
\end{definition}

\begin{lemma}[Endpoints and output]\label{ia:path-endpoints}
For each $s$, the extended labels with that read-only tuple form a
single directed path, without isolated vertices or self edges.  Its
unique source and immediate successor are respectively
\begin{equation}
 (\gamma^{-1},b_0(s),1,q-1),\qquad
 (\gamma^{-1},b_0(s),1,q-2),                      \label{ia:source-successor}
\end{equation}
where $b_0(s)$ has blank work and legacy bit $e_1=1$.  Its other
endpoint is the full pre label at stage $M-1$, and there
$w[y_{\rm out}]=f(s)$.
\end{lemma}

\begin{proof}
Replace the unique macro tape at stage $x$ by its post labels in frame
$\gamma^{-1}x$, in decreasing order of $t$, followed by its pre labels
in frame $x$, in increasing order of $t$.  The frame-advance edge joins
the two halves.  Each enabled macro edit joins the full pre end of one
such segment to the full post start of the next.  The only absent join
is the wrap join from the final stage to stage zero.  This gives exactly
the stated path and endpoints.  Every segment has more than one vertex,
so there are no isolated vertices; each edge changes a label component,
so there are no self edges.  The final work assignment and all its
ancestors have been completed at the last stage, proving the output
claim.
\end{proof}

\subsection{Chart conventions and lifted cyclic interpolation}
\label{ia:lifts-and-masks}

A native chart is an $\mathbb F_q$-coefficient polynomial on
$O(\ell)$ coordinates over $\mathbb F_q$, possibly with slots.  A
persistent chart has at most one slot axis of length $O(\ell)$; stacks
will have a bounded number of such axes.  A full rotating or projection
variable always occupies one whole $K$ label.  Native charts will later
be embedded in one common $K^u$, with unused coordinates ignored.

For slot lengths $t_1,\ldots,t_r$ and masks
$L_b\in\mathbb F_q^{t_b}$, write
\begin{equation}
 L\cdot F(Y)=\sum_{z_1,\ldots,z_r}
             \left(\prod_{b=1}^r L_b[z_b]\right)
             F_{z_1,\ldots,z_r}(Y).                \label{ia:contraction}
\end{equation}
There are $q^{O(\ell)}$ mask tuples.  To query any specified mask tuple
smoothly, choose independent uniform $A_b$, substitute
$L_b=A_b+(L_b-A_b)$, and expand the $2^r$ terms.  Each term has uniform
mask marginals, even conditional on the point and the requested tuple.
A nonzero tensor remains nonzero after independent random contractions
with probability at least $(1-1/q)^r$: at each step a nonzero linear
map has a kernel of relative size at most $1/q$.

We repeatedly use the elementary polynomial zero bound: a nonzero
polynomial of total degree $d_*$ on $\mathbb F_q^m$ vanishes on at most
a $d_*/q$ fraction of the grid.  For completeness, expand in the last
variable with leading coefficient of degree at most $d_*-r$ and last
variable degree $r$.  Inductively that coefficient vanishes on at most
$(d_*-r)/q$ of the earlier coordinates; elsewhere there are at most
$r$ roots in the last variable.  Induction on $m$ proves the bound.
We apply it also after adjoining the finitely many mask groups.
Thus any nonzero degree-$O(D+\ell h)$ chart identity, including a
whole slot list, is detected with constant probability.  An ignored
coordinate group is checked by comparing two independent replacements
of that group with common uniform other coordinates.  Its difference
is a polynomial; the same bound gives an exact alternative between
independence and constant rejection probability.  Throughout, smoothness
means bounded marginal density relative to uniform point and mask
coordinates.  Ignored coordinates are independently filled uniformly.

Ordinary interpolation on the $H$ digits extends a function on $K_1$
with individual degrees below $h$ and total degree at most
$\ell(h-1)$.  The punctured cyclic grid requires a different extension.

\begin{lemma}[Cyclic lift, including descent at zero]\label{ia:cyclic-lift}
For $0\le j<M$, write $j=\sum_{t=0}^{\ell-1}j_t h^t$ and define
\begin{equation}
 \widetilde j=\sum_{b=0}^{\ell-1}j_{vb\bmod\ell}q^b.
                                                               \label{ia:lift-exponent}
\end{equation}
Replacing $Z^j$ by $Z^{\widetilde j}$ in interpolation on $C$ gives a
linear extension to $K$, of coordinate degree at most $\ell(h-1)$.
For $\mathbb F_q$-valued data this extension has coefficients in
$\mathbb F_q$ in the chosen coordinates, including its value at zero.
It commutes with $Z\mapsto xZ$ for every $x\in C$.
\end{lemma}

\begin{proof}
Modulo $M$ we have $q^b=h^{vb}$ and $h^\ell=1$, hence
\[
 \widetilde j\equiv
 \sum_b j_{vb\bmod\ell}h^{vb\bmod\ell}=j\pmod M.
\]
The two monomials therefore agree on $C$, where the powers
$1,Z,\ldots,Z^{M-1}$ form the usual interpolation basis.  Each map
$Z\mapsto Z^{q^b}$ is linear over $\mathbb F_q$, so the coordinate
degree of the lifted monomial is at most $\sum_t j_t\le\ell(h-1)$.

Multiplication of $\widetilde j$ by $q$ modulo $q^\ell-1$ cyclically
rotates its base-$q$ digits.  Under \eqref{ia:lift-exponent}, this is
the permutation $j\mapsto qj\bmod M$.  The digit list all of whose
entries equal $h-1$ is excluded by $j<M$; exponent zero is consequently
the unique representative of residue zero on $C$.  If the original
interpolant is $\sum_j a_jZ^j$ and its values belong to $\mathbb F_q$,
uniqueness of interpolation on $C$ gives
\[
 a_{qj\bmod M}=a_j^q.
\]
The digit rotation then proves that its lift $F$ satisfies
$F(z)^q=F(z)$ for every $z\in K^\times$.  At zero, $F(0)=a_0$ and
$a_0^q=a_0$, so the same assertion holds.

The coordinate degree is below $q$ after our cutoff.  Conjugating the
coordinate coefficients by the $q$-power map leaves the values on
$\mathbb F_q^\ell$ unchanged.  Uniqueness of interpolation with
individual degrees below $q$ forces each coefficient to be fixed by
that conjugation, and thus to lie in $\mathbb F_q$.  Finally
$x^{\widetilde j}=x^j$ for $x\in C$, which proves the scaling assertion
on every basis monomial.  These arguments apply coefficientwise in any
additional independent variables.
\end{proof}

\subsection{Canonical certificates for linear conditions}
\label{ia:linear-certificates}

Partial-sum polynomial certificates also appear in Rubinstein's
construction~\cite[Sections~6.2--6.3]{Rubinstein2016Settling}.
Here each input chart has its own canonical certificates, and contributions
from different charts meet only in a tested identity.  We require these
recipes to remain linear in their input chart and, in rotating calls, to
respect cyclic scaling of the designated variables even when the final
condition fails.

\begin{lemma}[Smooth linear verification]\label{ia:linear-tool}
Let a fixed finite list of primary native charts, each scalar or with
one $O(\ell)$ slot axis, be subject to an explicit
$\mathbb F_q$-linear condition $\mathcal S$ on their full evaluation
arrays.  Its matrix has polynomially many rows and columns, with
logarithms $O(\ell\log q)$.  Coefficient and special-grid conditions
may be expressed by public linear matrices.  A nonrotating call may
adjoin a separately enforced primary $1$ for affine terms.

In a rotating call, at most one full variable in each primary is
designated to scale by a common $x\in C$.  Assume $\mathcal S$ is
invariant under this simultaneous pullback and imposes the lifted
shape of Lemma~\ref{ia:cyclic-lift} in every rotating variable.  No
arbitrary noninvariant affine term is allowed in such a call.

The condition has a canonical verification construction with the
following properties.
\begin{enumerate}
\item A fixed number of auxiliary families is stored \emph{separately
for each primary block}.  Each has native dimension $O(\ell)$ and at
most one slot axis of length $O(\ell)$.  Its canonical arrays are
linear in that primary's arrays, also when $\mathcal S$ fails.
\item Honest input coordinate degree $d_{\rm in}$ produces auxiliary
degree $O(d_{\rm in}+\ell h)$.  A fixed mixture of bounded, smooth
linear field probes is perfectly complete on legal canonical tuples
within the degree budgets.  On genuine degree-at-most-$D$ polynomial
tuples, either $\mathcal S$ holds and every auxiliary is canonical, or
the mixture rejects with a fixed positive probability.
\item In a rotating call each auxiliary has a new designated full
variable $R$, and rotating its primary by $x$ changes the auxiliary by
$R\mapsto xR$.  Its canonical recipe has the lifted shape in $R$
whenever its primary has the required shape, even if final rows fail.
All canonical arrays and public maps are polynomial-time constructible.
\end{enumerate}
\end{lemma}

\begin{proof}
We give the construction, uniqueness, degree, and probe arguments.

\paragraph{Why arbitrary scaling factors preserve the linear condition.}
In a rotating call substitute $RZ$ for the designated variable of a
primary and denote the resulting chart by $f_R$.  An unchanged primary
stays unchanged.  We first show that legal input arrays satisfy
$\mathcal S$ for every $R\in K$, rather than only for $R\in C$.
Extend scalars in the evaluation space and its linear subspace
$\mathcal S$ from $\mathbb F_q$ to $K$.  Since $|C|=M$ is odd, it is
invertible in characteristic two.  The character projections
\[
 \Pi_j=\frac1M\sum_{x\in C}x^{-j}\rho(x)
 \qquad(0\le j<M)
\]
for the simultaneous scaling action $\rho$ decompose the space and
preserve every invariant subspace.  On the imposed lift shape, the
character $j$ is represented by the single exponent $\widetilde j$
in every rotating primary.  Nonrotating components occur in character
zero.  Substitution by $R\ne0$ therefore acts by the scalar
$R^{\widetilde j}$ on each character space, and preserves the extended
subspace.  At $R=0$ it projects onto character zero, which also preserves
the subspace.  Scaled evaluations of our $\mathbb F_q$-valued chart
are still $\mathbb F_q$-valued; intersecting with the original
evaluation space, or using the original defining rows, proves the
assertion over $\mathbb F_q$.

A failed original row fails at $R=1$, so it remains a nonzero polynomial
condition.  Tests requiring the substitution sample $R$ uniformly from
$K^\times$; this changes a uniform density by at most
$(1-|K|^{-1})^{-1}$ and keeps primary point marginals uniform.  The
substitution is bilinear in coordinate variables and at most doubles
total degree.  In a nonrotating call omit $R$ everywhere below.

\paragraph{Separate row contributions and partial sums.}
Pad the row set by zero rows to $H^\rho$, with $\rho=O(\ell)$.
For one primary block with $m_0$ native $q$-coordinates, put $m'=vm_0$.
An $H$-basis of $\mathbb F_q$ gives a packing map
\[
 w_{\rm pack}:\mathbb F_q^{m'}\longrightarrow\mathbb F_q^{m_0}.
\]
This is a surjective $\mathbb F_q$-linear map whose restriction to
$H^{m'}$ is a bijection onto $\mathbb F_q^{m_0}$.  In particular it
sends uniform inputs to uniform outputs.  For each primary slot $s_0$,
interpolate that block's public matrix entries as
$A_{s_0}(I,X)$ on $H^\rho\times H^{m'}$, with individual degrees below
$h$.  Define, for this block only,
\begin{align}
 p_{m'}(R,I,X)
   &=\sum_{s_0}A_{s_0}(I,X)
                 f_{s_0,R}(w_{\rm pack}(X)),       \label{ia:linear-top}\\
 p_j(R,I,X_1,\ldots,X_j)
   &=\sum_{x_{j+1},\ldots,x_{m'}\in H}
      p_{m'}(R,I,X_1,\ldots,X_j,x_{j+1},\ldots,x_{m'})
      \quad(0\le j<m').                           \label{ia:partial-sums}
\end{align}
The top identity requests the primary with the public slot mask
$(A_{s_0}(I,X))_{s_0}$, made smooth by additive splitting.  This is a
linear probe; the matrix entries are not unknown operands.

The final condition is the tested identity
\begin{equation}
 \sum_{\text{primary blocks }b} p_{0,b}(R,I)=0.
                                                               \label{ia:final-row-sum}
\end{equation}
\emph{No chart for this mixed sum is stored.}  On legal data it
vanishes on every $I\in H^\rho$ and every $R\in K$.  Its individual
$I$ degrees are below $h$, and its honest $R$ coordinate degrees are
below $q$, so this is a polynomial identity.  Conversely, its
vanishing at $R=1$ and at the row grid, with correct partial sums,
is exactly the original matrix condition.

\paragraph{Division enforces all partial sums uniquely.}
For $1\le j\le m'$ introduce quotient and remainder charts and impose
\begin{align}
 p_j&=(X_j^h-X_j)Q_j+X_j^{h-1}p_{j-1}+E_j,
                                                        \label{ia:division}\\
 p_{j-1}&\text{ is independent of }X_j,
 \qquad \deg_{X_j}E_j\le h-2.                       \label{ia:division-bounds}
\end{align}
In characteristic two,
\[
 \sum_{x\in H}x^r=0\quad(0\le r<h-1),\qquad
 \sum_{x\in H}x^{h-1}=1.
\]
Here the $r=0$ identity is $h=0$ in the field.  The coefficient of
$X_j^{h-1}$ in the remainder modulo $X_j^h-X_j$ is thus precisely the
sum of the original polynomial over $X_j\in H$.  Euclidean division
by this monic polynomial uniquely fixes $Q_j$, the top remainder
coefficient $p_{j-1}$, and $E_j$.  Starting at
\eqref{ia:linear-top}, downward induction fixes all certificates and
gives \eqref{ia:partial-sums}.  Independence from later trailing
coordinates follows in the same induction.

To test the degree restriction without concentrated reads, write
\begin{equation}
 G_j(R,I,X_{\ne j},U)
   =\sum_{b=0}^{h-2}E_{jb}(R,I,X_{\ne j})U_b,
 \qquad U\in\mathbb F_q^{h-1},                     \label{ia:coefficient-chart}
\end{equation}
where $E_{jb}$ is the coefficient of $X_j^b$ in $E_j$.  Check
independence from $X_j$, addition in $U$, and homogeneity under a
uniform nonzero scalar $\lambda\in\mathbb F_q$.  Sampling a nonzero
scalar makes $\lambda U$ uniform; zero homogeneity already follows
from addition.  If these identities are exact, $G_j$ is pointwise
$\mathbb F_q$-linear in $U$, and uniqueness of reduced interpolation
makes its polynomial linear in $U$.  Now check
\begin{equation}
 E_j(R,I,X)=G_j(R,I,X_{\ne j},(1,X_j,\ldots,X_j^{h-2})).
                                                               \label{ia:coefficient-evaluation}
\end{equation}
The special $U$ tuple is queried as two uniform additive summands.
Only \emph{after} linearity has been established in the exact-soundness
case analysis do we use the $O(D+h)$ degree bound for this substitution.
The earlier linearity discrepancies have degree $O(D)$ and themselves
have constant gap.  Equation~\eqref{ia:coefficient-evaluation} then
enforces \eqref{ia:division-bounds} and fixes $G_j$ uniquely.

\paragraph{A whole list is checked at once.}
Sampling a single recurrence index would lose a factor $m'$.  Instead
choose a cyclic permutation $\omega$ of the $m'$ scalar coordinates
so that coordinate $j$ of $\omega^jX$ is one fixed coordinate of $X$.
Store
\begin{equation}
 \overline p_j(R,I,X)=p_j(R,I,\omega^jX),
 \qquad
 p_{j-1}(R,I,\omega^jX)
      =\overline p_{j-1}(R,I,\omega X),             \label{ia:recurrence-frames}
\end{equation}
and store $Q_j,E_j,G_j$ in the equation-$j$ frame, with $U$ unchanged.
Both public powers in \eqref{ia:division}, and the predecessor
independence axis in its corresponding frame, are now common across
all $j$.  A uniform mask on the recurrence-index list checks all its
equations with boundedly many point probes.  Endpoint slices and
predecessor shifts only change the requested linear mask, which is
split smoothly.  The coefficient families use the same $U$ group in
every slot.  If $m'=0$, retain only the unsummed top contribution and
introduce no division family.

\paragraph{Degree, covariance, and exact gap.}
The top degree is at most
\begin{equation}
 2d_{\rm in}+(\rho+m')(h-1).                       \label{ia:tool-degree}
\end{equation}
Monic division by $X_j^h-X_j$, reduction, coefficient extraction, and
finite summation do not increase total degree.  Multiplication by a
linear $U_b$ costs at most one.  This proves the honest degree bound.
There is a fixed number of families per primary; the growing recurrence
index is one $O(\ell)$ slot axis, and the additional $U$ domain has
$h-1\le\ell$ coordinates.

First test ignored coordinates and the independence and linearity
devices.  Unless they hold identically, one has a nonzero polynomial
discrepancy of degree $O(D)$ and constant rejection probability.
Once they hold, every remaining displayed discrepancy has degree
$O(D+\ell h)=o(q)$ and a failed slot identity survives a random
contraction with constant probability.  Exact identities give the
uniqueness just proved and then \eqref{ia:final-row-sum}.  A fixed
mixture of test types therefore has a uniform positive gap.  Perfect
completeness concerns canonical tuples within the budgets; it does not
claim that every legal primary of degree exactly $D$ has auxiliaries
of degree at most $D$.

For covariance even on invalid inputs, expand a lifted primary after
substitution over $K$ as
\[
 f_R=\sum_{j=0}^{M-1}R^{\widetilde j}v_j
\]
in its evaluation arrays.  All subsequent operations in
\eqref{ia:linear-top}--\eqref{ia:coefficient-chart} are the same fixed
linear maps on that primary, independent of $R$.  This includes
division and coefficient extraction, since honest-degree evaluation
arrays determine the required polynomials by interpolation.  These
maps preserve the exponent list, and rotating the input by $x\in C$
replaces $R$ by $xR$.  No final zero-row identity is used.  In particular
every canonical certificate is defined even when its primary belongs
to data failing the final condition.

Finally, all point substitutions used in these tests are coordinate
permutations, surjective packing, nonzero scaling, fresh independent
coordinate replacement, or additive splitting.  They have bounded
point marginals.  Requested masks are split.  All arrays have
polynomial size, and every construction operation is finite linear
algebra on them.  This proves the remaining claims.
\end{proof}
\subsection{Persistent charts and their dependency tags}
\label{ia:persistent}

Every persistent chart has a canonical recipe $F(c,b;Y)$, possibly
with one slot axis.  At each fixed frame $c$, both its values and its
coordinate coefficients have an assigned multilinear Boolean degree
$\kappa_F\le d=3$ in $b$, after reducing by $b_j^2=b_j$.  This
logical degree is different from the coordinate degree bounded by
$D$.  All recipes are defined for every Boolean tape, including tapes
that fail validity.  A recipe either is independent of $c$, or has
one designated full variable whose scaling realizes $c\mapsto xc$.

Give each family a tag in $\{0,1,\ldots,k\}$.  A tag-$0$ recipe
depends only on $c,w,e$ and public data; that public data may include
the circuit, work addresses, and the port index.  A tag-$i$ recipe
depends only on $c,s_i,e$ and public data fixed by the standardized
one-port template.  It may ignore $s_i$ if it is a certificate created
inside that template.  Call tags compatible if they are equal or one
of them is zero.  After fixing a finite Boolean transition branch,
tag-$0$ update operands will all have tag $0$, and tag-$i$ update
operands will have compatible tags.

\paragraph{Monomials and the separated validity calls.}
Use the ordinary full-grid extension for $W_w$ and each $W_{s_i}$,
and the cyclic lift for $W_e$.  Include the constant chart $1$,
tested against its prescribed value.  For all ordered lists of length
at most $d$ from $\{w,e\}$, and separately from $\{s_i,e\}$ for
each $i$, include
\begin{equation}
 W_{g_1,\ldots,g_j}(b;Y_1,\ldots,Y_j)
    =\prod_{r=1}^j W_{g_r}(b;Y_r).                 \label{ia:monomial-charts}
\end{equation}
Repeated types use independent variables.  Share the legacy-only
types.  A monomial containing $s_i$ has tag $i$; all other monomials
have tag $0$.  Enforce each nontrivial product by factorization into
its shorter prefix and last factor on independent native variables,
together with all ignored-coordinate requirements.

Apply the nonrotating linear tool as follows.
\begin{enumerate}
\item One tag-$0$ validity call uses the monomials from $\{w,e\}$
and $1$.  It imposes the order-one extension shapes, bitness on the
grids, the one-hot requirements \eqref{ia:one-hot}, and all blank and
gate requirements of Lemma~\ref{ia:validity}.  Use one work-requirement
row for each pair (stage, work grid position), with a zero row when
that position is an occupied copy.  Other conditions have fixed
lists of rows.  Circuit data affect matrix entries only.
\item For each $i$, a separate copy/port call imposes the shape,
padding, and bitness of $W_{s_i}$, and the copy rows indexed by
$(c,j)$ for $c\in C$ and $1\le j\le L$.  The row is zero when
the stage index of $c$ is less than $j$.  Otherwise it compares
\begin{equation}
 W_{e,s_i}(c^{-1},\iota(j))
     -W_{e,w}(c^{-1},w_{i,j})=0,                  \label{ia:separate-copy-row}
\end{equation}
where $\iota$ is the common input injection.  Primary-list, row-order,
and padding conventions are identical in every such call.  The
matrix blocks for port-dependent primaries depend only on these
common stage and source-bit indices.  Work destinations occur only
in the separate $W_{e,w}$ matrix block.  Auxiliaries for port-dependent
blocks receive tag $i$; auxiliaries for the work-side block receive
tag $0$.
\end{enumerate}
For example, bitness is the linear row
\[
 W_{g,g}(y,y)-W_g(y)=0
\]
on represented grid entries.  Although its two grid arguments
coincide, it is verified as a matrix row, not by a concentrated
unknown-product query.  The same representation handles every
degree-at-most-three Boolean validity term.  Extension shapes are
linear evaluation/interpolation conditions; the cyclic shape is
$\mathbb F_q$-linear by Lemma~\ref{ia:cyclic-lift}.

If these conditions and the factorization identities hold exactly
on genuine polynomials, order-one grid entries are bits, their
shapes fix their full extensions, products fix higher monomials,
and the rows give exactly a valid union-of-stages tape.  The linear
tool fixes each certificate separately.  The initial auxiliary
assigned degrees can all be bounded by $d$, or by the smaller
degree of their primary when useful.  Since every certificate map
is linear in one primary, the same degree statement holds on
arbitrary Boolean tapes even when a final validity row is nonzero.

Include also a tag-$0$, point-constant output chart
\begin{equation}
 F_{\rm out}(b)=w[y_{\rm out}].                    \label{ia:output-chart}
\end{equation}
A separate nonrotating linear call ties it to the designated grid
entry of $W_w$ and enforces its ignored coordinates.  Give it and
the associated certificates assigned degree one.  This output chart
does not need a critical-address projection for transitions.

\paragraph{Frame and clock-times-monomial charts.}
Let $J_c(X)$ be the cyclic lift of the indicator of $c^{-1}$ on
$C$.  It has tag $0$, assigned degree zero, and
\begin{equation}
 J_{xc}(X)=J_c(xX).                                \label{ia:clock-covariance}
\end{equation}
Tie it to the legacy chart by
\begin{equation}
 W_e(X)=J_c(\gamma^\eta X).                        \label{ia:clock-legacy}
\end{equation}
Below, $\eta$ is enforced as a point-constant bit.  Equation
\eqref{ia:clock-legacy} is then a finite choice of two smooth
polynomial identities.  One-hot legacy validity determines a unique
$c$ and the required pre/post relation, and fixes $J_c$.  For every
monomial $W$, including $1$, include
\begin{equation}
 T_W(c,b;X,Y)=J_c(X)W(b;Y),                         \label{ia:clock-monomial}
\end{equation}
with independent arguments.  It has the same tag and degree tier as
$W$, and rotates only $X$.  Enforce it by factorization.

\begin{lemma}[Clock-indexed linear formulas]\label{ia:clock-formula}
Suppose for each $y\in C$ that $L_y$ is an explicitly given
$\mathbb F_q$-linear map from selected monomial evaluation arrays
of logical degree at most $k_0$ to the full native array of a scalar
or one-slot output.  There is a canonical chart, lifted in $Z$, with
\begin{equation}
 G(c,b;z,\cdot)=L_{cz}(W(b))\quad(z\in C).         \label{ia:clock-formula-value}
\end{equation}
It rotates only $Z$, and it and its verification auxiliaries have
assigned degree at most $k_0$.  A rotating linear-tool call verifies
them.  The recipes and their covariance hold for all Boolean $b$.
\end{lemma}

\begin{proof}
Use the needed $T_W$ inputs and the output $G$ as primaries.  Impose
their lifted shapes in the designated variables and the linear rows
\begin{equation}
 G(z,\cdot)=\sum_{x\in C}L_{x^{-1}z}(T(x,\cdot))
 \quad(z\in C).                                  \label{ia:clock-convolution}
\end{equation}
Rows also range over every output evaluation position and slot.
On canonical $T$ inputs, only $x=c^{-1}$ contributes, giving
\eqref{ia:clock-formula-value}.  The lift shape then determines the
whole chart uniquely.  The row condition is invariant under the
simultaneous replacement $G(z)\mapsto G(u_0z)$ and
$T(x)\mapsto T(u_0x)$: setting $x'=u_0x$ changes the map index to
$(x')^{-1}u_0z$.  The shape conditions are invariant too, so
Lemma~\ref{ia:linear-tool} applies.  Every input has logical degree
at most $k_0$; interpolation and all per-primary certificate operations
are linear and preserve that degree.  Covariance follows first on
$C$ and then by uniqueness of the lift, and for the auxiliaries by
the last part of the linear tool.  No validity row on $b$ was used.
Public clock-dependent terms are supplied through $T_1=J_c$, keeping
the rotating condition linear.
\end{proof}

Apply this lemma to a fixed list, for fixed $k$, of critical addresses
needed by either transition direction.  A critical output chart has
only the variable $Z$ and gives at $z\in C$ the required logical read
or rule bit for absolute frame $cz$.  For each port, make a separate
tag-$i$ read chart that reads $s_i[j]$ during initial copy step $j$
and is identically zero on every other step.  This is a standardized
degree-one call independent of all subsequent computation.  Work
reads have separate tag-$0$ charts, defined as zero when inactive;
there are a bounded number per write slot for old and operand reads.
Phase, gate, activity, and wrap-disable bits use tag-$0$ degree-zero
calls.  Fix their numbers and tiers by the rule basis and write-slot
bound, padding inactive roles.  Their public values and work address
maps may depend on the circuit.

Every auxiliary of a tag-$0$ clock call has tag $0$.  Every auxiliary
of a standardized tag-$i$ clock call has tag $i$, even an auxiliary
for a shared legacy-only or public $T$ primary in that call.  The
shared primary itself retains its base tag and formula.  This
distinction between an input's tag and the ancestry of its
per-call certificate is part of the construction.

A fixed-frame tape edit must update every persistent chart, including the
certificates introduced to verify other charts.  Its effect on a chart
$F$ is the difference
\[
 F(c,b+h_\sigma(c);Y)-F(c,b;Y).
\]
For a positive-tier chart we store this difference in a new chart with a
clock variable $Z$, whose restriction at $Z=1$ gives the required change.
Taking a Boolean difference lowers the logical degree.  We can also store the
changes of this new chart and its certificates, stopping after at most
three levels.  The construction below supplies these polynomial charts;
the projection stacks in Section~\ref{ia:projection-stacks} will realize
the restriction at $Z=1$ without a concentrated point read.

\begin{lemma}[Derivative closure for all persistent families]
\label{ia:derivative-closure}
The persistent system can be closed under every required fixed-frame
Boolean edit, including edits to verification auxiliaries, with a
fixed number of families for fixed $k$.  Each has at most one
$O(\ell)$ slot axis, native dimension $O(\ell)$, honest coordinate
degree $O(\ell h)$, and assigned logical degree at most three.  The
closure and its polynomial-time canonical recipes are defined on
invalid Boolean tapes as well.
\end{lemma}

\begin{proof}
Let $F$ have positive assigned tier $k_0$.  For every required flip
mask, retain its native variables $Y$ and slots, adjoin a new full
variable $Z$, and prescribe on the cyclic grid
\begin{equation}
 G_F^\sigma(c,b;z,Y)
   =F(cz,b+h_\sigma(cz);Y)-F(cz,b;Y),\qquad z\in C.
                                                               \label{ia:derivative}
\end{equation}
Extend in $Z$ by the cyclic lift.  Both parent terms are evaluated
in frame $cz$; the parent's existing argument coordinates remain
exactly $Y$.  A tag-$0$ parent uses the full flip alphabet
\eqref{ia:flip-mask}.  For a tag-$i$ parent, only the two legacy
coefficients matter: the input bits $s_i$ are never written and
work bits are absent from its recipe.  Use the fixed four masks
\begin{equation}
 h_\chi(y)=\chi_1\mathbf 1_{e_{y^{-1}}}
            +\chi_2\mathbf 1_{e_{(\gamma y)^{-1}}},
 \qquad \chi\in\{0,1\}^2,                        \label{ia:port-flip-mask}
\end{equation}
independent of $k$ and $i$.

For a squarefree monomial, a fixed Boolean translation gives
\begin{equation}
 \prod_{j\in S}(b_j+h_j)-\prod_{j\in S}b_j
   =\sum_{T\subsetneq S}\left(\prod_{j\in T}b_j\right)
                         \left(\prod_{j\in S\setminus T}h_j\right).
                                                               \label{ia:degree-drop}
\end{equation}
Thus \eqref{ia:derivative} has logical degree at most $k_0-1$.
Lifting its new variable adds at most $\ell(h-1)$ to its coordinate
degree.  For each absolute frame $y$, output evaluation point, and
inherited slot, the derivative is a known linear combination of
grid monomials of degree at most $k_0-1$: from $\{w,e\}$ for tag
$0$, or from $\{s_i,e\}$ for tag $i$.

These coefficients are computed from the \emph{canonical parent
recipe}, not from the presently supplied parent table.  An explicit
method is bounded-degree Boolean interpolation.  For a multilinear
function $P$ of degree at most $r$ on a universe of $N$ bits, its
coefficient on $S$, $|S|\le r$, is
\begin{equation}
 [b_S]P=\sum_{T\subseteq S}(-1)^{|S|-|T|}P(\mathbf 1_T).
                                                               \label{ia:boolean-inversion}
\end{equation}
In characteristic two the signs are all one.  Use this on the
right side of \eqref{ia:derivative}, which is computable on every
Boolean argument, including translated invalid tapes.  There are
only $\sum_{j\le r}\binom Nj=\poly(n')$ required subsets, because
$r\le3$ and $N=\poly(n')$.  For tag $i$ the interpolation universe
is the common ordered list of its $L$ source bits and $M$ legacy
bits, with no work coordinates.  Interpret the resulting coefficients
as public maps $L_y$ on the permitted monomial arrays.

Apply Lemma~\ref{ia:clock-formula} using only inputs through tier
$k_0-1$.  The output derivative and every newly introduced auxiliary
receive that tier and the parent's tag.  In particular a certificate
for a shared tag-$0$ $T$ input in this positive call receives the
positive ancestry tag.  The derivative rotates only its new variable,
since on the grid
\[
 G_F^\sigma(u_0c,b;z,Y)=G_F^\sigma(c,b;u_0z,Y),
\]
and lift uniqueness extends this to $K$.  Its older parent variables
are not additionally scaled under a frame change.  Each verification
auxiliary instead rotates its own new $R$, as proved by the linear
tool.

Repeat this operation for every added positive-tier family, including
derivatives and all their proof families.  Every descendant call
strictly lowers its assigned tier, whether or not its actual degree
was already smaller.  The recursion ends after at most three levels.
A family has only finitely many children for fixed $k$.  A derivative
inherits its parent's single slot axis; a verification auxiliary uses
its own recurrence-index axis rather than tensoring it with primary
slots.  Consequently the number of families is fixed, their native
dimensions and slot lengths remain $O(\ell)$, and their coordinate
degrees remain $O(\ell h)$.  Tier-zero charts require no derivative
for a fixed-frame Boolean edit.

All maps have polynomial construction cost: there are polynomially
many frames, evaluations, slots, monomials, and rows, and only a
fixed number of recursive levels.  Finite sums, monic division,
coefficient extraction, and interpolation are fixed linear operations
on each primary array.  They neither create a further logical product
nor import another primary's dependency.  This remains true when the
final row sum fails, by Lemma~\ref{ia:linear-tool}.  The asserted
invalid-tape recipes and their degree bounds follow as well.
\end{proof}

\begin{lemma}[Literal positive-port ancestry]\label{ia:port-ancestry}
There are deterministic local call and ancestry conventions such that
the entire tag-$i$ construction in an actual $k$-port graph is
identical, after renaming $s_i$ to $s$, to the positive-tag construction
in the one-port template with the same $L,n'$ and field parameters.
The agreement includes primary lists, row indexing and zero padding,
packing maps, all auxiliary and derivative families, native dimensions,
slot orders, and canonical arrays whenever $(c,e,s_i)=(c,e,s)$.
It holds independently of the actual work destinations, $k$, and $f$.
\end{lemma}

\begin{proof}
Choose row lists, their order, and zero padding separately for each
call's local template.  Do not choose a row dimension by maximizing
over all calls in the actual graph.  In each copy/port call use the
same input-shape rows, bitness rows, and $(c,j)$ copy rows, and the
same full primary-list conventions.  Although the work-side matrix
depends on $w_{i,j}$, its dimensions do not.  This makes all
port-dependent primary blocks literal copies of their dummy blocks.

For a concrete example, consider $W_{e,s_i}$ in
\eqref{ia:separate-copy-row}.  Its matrix $A_{e,s}(I,X)$ knows the
stage and the source position $\iota(j)$, and has no reference to
$w_{i,j}$.  That destination appears only in the separate
$A_{e,w}$ block.  The top formula for its positive certificate is
\[
 p_{m',e,s}(I,X)
       =A_{e,s}(I,X)W_{e,s_i}(w_{\rm pack}(X)),
\]
with the evident slot contraction when needed.  Its row padding,
packing, recurrence permutation, recurrence-index order, and
coefficient-vector domain are therefore exactly those of the
one-port construction.  Each partial sum, quotient, remainder, and
coefficient chart uses that block alone.  The equality with the
work-side contribution is tested only in \eqref{ia:final-row-sum};
there is no mixed stored chart from which work addresses could
enter this positive family.

The critical source-bit calls agree directly, since they use the
same initial-copy address rule and the same zero rule thereafter.
For the remaining families, induct on positive ancestry.  If a
parent's recipe and native layout agree, its derivative uses exactly
the same restricted mask \eqref{ia:port-flip-mask}.  Its values on
every subset in \eqref{ia:boolean-inversion}, including translated
invalid inputs, are identical in the two templates.  Hence its maps
$L_y$ are identical.  Make each derivative call separately by parent
type and mask, with the fixed allowed lower-tier monomial list, full
output-equality rows, shape rows, deterministic order, and local
padding.  Its output and every per-primary auxiliary then agree
literally by the linear-tool formulas.  This includes auxiliaries
whose primary is a shared legacy-only or constant $T$ chart: the
input formula is common and the auxiliary's own call matrix is the
standardized positive one.  These auxiliary families are themselves
parents in the induction.  Strictly decreasing tiers terminate it.

Keep descendants from distinct standardized calls separate, without
coalescing them differently with circuit-dependent tag-$0$ families.
Additional full work masks and additional tag-$0$ calls then create
separate branches and do not change any positive branch.  The
construction does not use validity to establish this correspondence.
\end{proof}

\begin{lemma}[A common dimension and inherited full labels]
\label{ia:placements}
The common point domain $K^u$ and all per-chart dimension bounds can
be fixed before $k$.  Persistent charts can be placed in it so that
every derivative preserves all its parent's labels and slot order
and adds an unused whole label for its new variable.  Corresponding
positive families have identical placements.  For each fixed $k,L$,
all actual graph family and slot layouts can be fixed independently
of the size-bounded circuit's values and wiring.
\end{lemma}

\begin{proof}
Initially a monomial uses at most $d$ full variables and a $T$ chart
one more.  Base linear calls have native dimension $O_d(\ell)$.
Their row counts are bounded by $q^{C\ell}$ for an absolute fixed
$C$ independent of $k$: the work grid, clock grid, input length, and
stage count are at most $h^\ell$, and shape conditions are polynomial
in the corresponding full evaluation sizes.  Occupied-copy calls
are separate, so their number does not enter an individual row
dimension.  They use fixed stage/source-index lists.

For a derivative call on a parent of $m_0$ native coordinates and
one possible slot axis, its output, shape, and clock-equality rows
have size polynomial in $q^{m_0+C\ell}$, with a fixed exponent.
Its permitted monomial input list is fixed by its tag and tier.
Different masks have separate calls.  Therefore $\rho/\ell$, the
new native dimension divided by $\ell$, and the new slot length
divided by $\ell$ are bounded by a fixed multiple of one plus the
corresponding parent bounds.  Digit unpacking contributes the fixed
factor $v$; the coefficient-vector domain contributes at most
$h-1\le\ell$ coordinates.  Iterating through the at most three
tier drops gives an absolute bound $C_{\rm dim}\ell$, independent
of the number of calls or ports.

Choose $u$ to accommodate this bound and at most $d+1$ additional
whole labels along an inherited derivative-axis chain.  Put every
full $K$ variable in a whole label, and pack remaining native
$q$-coordinates into additional labels, ignoring their padding.
An initial chart or a new verification auxiliary is a placement
root; unrelated roots may reuse labels according to their native
templates.  A derivative retains exactly its parent's placements
and chooses a deterministically first unused whole label for its
new variable.  Repeat along that chain.  No labels have to be
globally disjoint across all ports: tests between unrelated charts
assemble their native arguments separately and fill ignored
coordinates uniformly.  Enforce all ignored-coordinate conditions.
These deterministic choices depend only on local templates and
ancestry, proving positive-family agreement.

For tag-$0$ calls at fixed $k,L$, use the full uniform
(stage, work-position) validity list, fixed critical-read roles and
tiers, the full fixed flip alphabet for every parent, and the
allowed monomial, shape, and output lists fixed by domain and tier.
Only matrix entries and rule values vary with the circuit.  This
makes the entire actual layout independent of circuit wiring, as
claimed.
\end{proof}
\subsection{Projection stacks and their exact tests}\label{ia:projection-stacks}

Critical reads and tape edits require algebraic restrictions of charts
with one or more full labels set to $1$.  Reading these special points
directly would not preserve a uniform point marginal.  Instead we
express each restriction as a sum of invertible affine pullbacks and
store the affine-pulled values in slots.  The progress coordinate $t$
specifies which of these slots are active.  At full progress, summing
over the new stack axes will give the required restriction; changing
progress will use bounded slot shifts and affine pulls.

Give each critical address output its clock-formula variable as a
projection axis.  Other initial persistent charts, including all
verification auxiliaries, start with no projection axes.  A derivative
inherits its parent's axes, with their exact labels, and adds its new
outer variable.  New verification auxiliaries again start with no
axes.  Denote the resulting set by $A_F$; its size is at most $d+1$.

Fix a generator $g$ of $\mathbb F_q^\times$.  For a full label
$\alpha$, let $D_\alpha$ be pullback by
\begin{equation}
 Z_\alpha\longmapsto 1+g(Z_\alpha-1).              \label{ia:dilation}
\end{equation}
This is an invertible $K$-affine map on the common domain.

\begin{lemma}[Projection by full-label dilations]\label{ia:projection}
If the coordinate degree in label $\alpha$ is below $q-1$, then
\begin{equation}
 \sum_{r=0}^{q-2}D_\alpha^r F
       =F\big|_{Z_\alpha=1},                      \label{ia:projection-identity}
\end{equation}
where the right side ignores that label.  Dilations on distinct
full labels commute.
\end{lemma}

\begin{proof}
Center the $\mathbb F_q$ coordinates of $Z_\alpha$ at the coordinate
vector of $1\in K$.  Decompose $F$ into parts homogeneous of degree
$r$ in the centered coordinates.  Their sums under all nonzero
scalar dilations have multiplier
\[
 \sum_{j=0}^{q-2}g^{jr}
   =\begin{cases}1,&r=0,\\0,&0<r<q-1.\end{cases}
\]
For $r=0$ this is $q-1=1$ in characteristic two, and for positive
$r$ it is a geometric sum with ratio different from one.  Thus
only the constant homogeneous part remains.  Pullbacks on disjoint
labels commute by direct substitution.
\end{proof}

\paragraph{Mode and progress marker.}
Represent $\eta$ by a point-constant scalar chart.  Enforce constancy,
then membership in $\mathbb F_2\cdot1$ by random additive binary
forms annihilating $1$.  A value outside this subspace is detected
with probability $1/2$; on the subspace, a fixed form sending $1$
to $1$ reads the bit.

Let $N=B^a$.  Represent progress by a point-constant marker tensor
$M^{\rm mark}_j$, with $a$ radix axes of length $B$, indexing
$0\le j<N$.  At $t=0$ it is zero; at $t>0$ it is one-hot at $t-1$.
After constancy is enforced, test with independent uniform separable
masks $L,L'$ the identity
\begin{equation}
 (L\cdot M^{\rm mark})(L'\cdot M^{\rm mark})
                  =(LL')\cdot M^{\rm mark},       \label{ia:marker-identity}
\end{equation}
where each axis of $LL'$ is an entrywise product.  Expanding in the
separate mask variables, exactness says
$M_j^{\rm mark}M_{j'}^{\rm mark}=0$ for $j\ne j'$ and
$(M_j^{\rm mark})^2=M_j^{\rm mark}$ for each $j$.  Hence the tensor
is zero or has exactly one coefficient equal to one.  A nonzero
discrepancy is a polynomial of bounded mask degree and is detected
with constant probability.  Enforce zero support for
\begin{equation}
 q-1\le j<N.                                      \label{ia:marker-padding}
\end{equation}
This range is a union of at most $a+1$ radix rectangles: compare
digits from the most significant end, separating the first larger
digit and the boundary itself.  Masked restrictions check every
rectangle with boundedly many probes.  The remaining possibilities
are exactly $t\in\{0,\ldots,q-1\}$.  Both mode and marker have
tag $0$.

For each persistent $F$ and each nonempty $S\subseteq A_F$, include
a stack with the same tag, point placements, and inherited slots as
$F$, and $a$ additional radix slot axes for each $\alpha\in S$.
Its intended formula is
\begin{equation}
 C^S_{F,\mathbf j}(c,b,\eta,t;Y)=
 \begin{cases}
 \displaystyle\left(\prod_{\alpha\in S}
           D_\alpha^{t-1-j_\alpha}\right)F(c,b;Y),
             &j_\alpha<t\text{ for every }\alpha\in S,\\[2pt]
 0,           &\text{otherwise},
 \end{cases}
 \qquad C_F^\varnothing=F.                         \label{ia:stack-formula}
\end{equation}
Indices run through $0,\ldots,N-1$ on each integer axis.

\begin{lemma}[Canonical stacks and bounded radix probes]\label{ia:stacks}
For a legal marker, the stacks are uniquely characterized by a fixed
number of types of bounded-probe polynomial identities.  These tests
are perfectly complete and have constant exact gap on
degree-at-most-$D$ polynomials.  Their only unknown products are
marker times smaller-stack products, with independent full operand
point marginals and compatible tags.  All slot shifts, restrictions,
and boundary slices have bounded separable-mask expansions.
\end{lemma}

\begin{proof}
For each nonempty $S$, fix one $\alpha\in S$ and require
\begin{equation}
 C^S_{F,(j_\alpha,\mathbf j_-)}
  =D_\alpha C^S_{F,(j_\alpha+1,\mathbf j_-)}
      +M^{\rm mark}_{j_\alpha}
                 C^{S\setminus\{\alpha\}}_{F,\mathbf j_-},
                                                               \label{ia:stack-recurrence}
\end{equation}
with the upper boundary at $j_\alpha=N$ defined to be zero.
For fixed $\mathbf j_-$ this is triangular downward in $j_\alpha$.
If the marker is zero its only solution is zero.  If it is one-hot
at $t-1$, its solution is
\[
 C^S_{F,(j_\alpha,\mathbf j_-)}
   =\begin{cases}
     D_\alpha^{t-1-j_\alpha}
             C^{S\setminus\{\alpha\}}_{F,\mathbf j_-},
                       &j_\alpha<t,\\
     0,                 &j_\alpha\ge t.
    \end{cases}
\]
Induction on $|S|$ gives exactly \eqref{ia:stack-formula}.

Here are explicit slot formulas for the shift, including boundary
handling.  Use little-endian radix digits
$j=\sum_{s=0}^{a-1}j_sB^s$ and let
$\lambda_L(j)=\prod_{s=0}^{a-1}L_s[j_s]$.  With out-of-range
coefficients zero,
\begin{align}
 \sum_{j=0}^{N-1}\lambda_L(j)C_{j+1}
   &=\sum_{r=0}^{a-1}L^{(r,+)}\cdot C,
                                                        \label{ia:radix-plus}\\
 L_s^{(r,+)}[d]
   &=\begin{cases}
      L_s[B-1]\mathbf1_{d=0},&s<r,\\
      \mathbf1_{d\ge1}L_r[d-1],&s=r,\\
      L_s[d],&s>r,
     \end{cases}                                      \nonumber\\
 \sum_{j=0}^{N-1}\lambda_L(j)C_{j-1}
   &=\sum_{r=0}^{a-1}L^{(r,-)}\cdot C,
                                                        \label{ia:radix-minus}\\
 L_s^{(r,-)}[d]
   &=\begin{cases}
      L_s[0]\mathbf1_{d=B-1},&s<r,\\
      \mathbf1_{d\le B-2}L_r[d+1],&s=r,\\
      L_s[d],&s>r.
     \end{cases}                                      \nonumber
\end{align}
Indeed the first pullback mask on index $k>0$ is
$\lambda_L(k-1)$.  Partition $k$ by its least digit $r$ which is
nonzero; precisely the lower digits borrow.  The second pullback
is $\lambda_L(k+1)$ for $k<N-1$; partition by the least digit below
$B-1$.  Overflow is absent in both formulas.  Slicing at zero is
a single separable restriction, and interval restrictions use the
bounded rectangle decomposition used in \eqref{ia:marker-padding}.
Tensoring these expansions over the bounded number of stack axes
still gives a bounded number of separable masks.  Split every
requested mask additively as in \eqref{ia:contraction}.

Contract the entire identity \eqref{ia:stack-recurrence} over all
slots.  The last term factors into a contraction of the marker on
axis $\alpha$ and a contraction of the smaller stack on its other
axes and inherited slots.  In the exact-soundness analysis marker
constancy has already been imposed, so its full point can be sampled
independently of the smaller stack point.  This retains perfect
completeness on canonical data.  If the recurrence is not an
identity, its discrepancy has point degree at most $2D$ and bounded
mask degree, giving constant rejection probability.  Honest stacks
are affine pullbacks of persistent charts or zero, so they have the
same honest coordinate degree.  Marker tag $0$ is compatible with
every parent tag.  This proves all claims.
\end{proof}

\subsection{Exact validation and locally encoded evolution}
\label{ia:validation-and-evolution}

\begin{lemma}[Exact algebraic validity]\label{ia:exact-validator}
For each fixed $k$ there is a fixed finite mixture of bounded-probe
tests with perfect completeness on canonical extended labels and a
uniform constant gap on genuine degree-at-most-$D$ polynomial
families.  Passing all identities is equivalent to being the unique
canonical tuple of a valid extended label.  The gap is independent
of $L$ past the fixed cutoff and uniform over the allowed circuit
coefficients.  Linear probes have bounded point and mask density.
Every unknown product has compatible tags and can be queried with
independent full point marginals for its two operands.
\end{lemma}

\begin{proof}
The tests just constructed constitute a fixed number of family
types for fixed $k$; growing slot lists are checked by whole
contractions.  We prove exact soundness in an order that licenses
all substitutions.  First consider ignored-coordinate requirements,
mode and marker constancy, and the independence and coefficient
linearity devices of every linear tool.  If one fails, its direct
test has constant gap.  Otherwise these are exact polynomial
identities.  Monomial factorization and the nonrotating base calls
then force a Boolean tape, all its monomial extensions, its
union-of-stages validity, its output chart, and the canonical
per-primary certificates.  Mode and marker membership give unique
$\eta$ and $t$.  The legacy identity \eqref{ia:clock-legacy}
forces the unique frame $c$ and fixes $J_c$; clock-monomial
factorizations fix all $T_W$.

Now apply the clock-formula calls in construction order.  Their
uniqueness fixes each critical output, each derivative, and every
auxiliary.  A derivative's rows were computed from a prescribed
parent recipe; the proof never treats an unverified parent table
as an additional oracle for those rows.  The strictly decreasing
tier construction is therefore acyclic.  Finally the triangular
stack identities give \eqref{ia:stack-formula}.  Hence exactness
of all identities gives precisely a valid label's canonical tuple.
Conversely every valid label satisfies every identity for every
random choice in the support.

If exactness fails, the first failed identity in this ordering has
point degree $O(D+\ell h)$ after the stated preliminary devices
and bounded degree in its mask variables.  The polynomial zero and
contraction bounds give a uniform positive rejection probability.
Taking the minimum over the fixed mixture gives the asserted gap.
Honest degrees are $O(\ell h)<D$ eventually, also after any later
fixed-factor enlargement of the point domain.  The constants in
these degree bounds and in the finite family list depend on fixed
templates, not on the values of public circuit coefficients.

For the product claim, the only possibilities are monomial
factorizations, $J_cW$ factorizations, marker membership, and marker
times smaller stack.  In a factorization, choose the two native
operand arguments independently, assemble the output's relevant
groups from them, and independently fill all ignored coordinates.
Even repeated operand types then have independent full operand
points.  The relevant output groups are independent uniform too.
For marker products use independent points, justified by the
earlier constancy step.  Every such pair has compatible tags by
construction.  Bitness, diagonal grid entries, copy equalities,
and all other correlated grid uses are linear matrix rows, so they
require no concentrated unknown-product query.  The linear tool's
packing, permutation, and splitting probes have bounded individual
point density.  Dependent or prescribed masks, including those
inside products, are smoothed by fresh independent axis splittings.
These observations prove the stated probe interface.
\end{proof}

\begin{lemma}[All neighbor coordinates evolve by bounded affine pulls]
\label{ia:evolution}
For canonical valid data, edge existence and the transition branch
in either direction are determined using only mode, empty/full
marker comparisons, and boundedly many critical projected bits at
full progress.  Equality decisions can have any prescribed fixed
error probability by a fixed number of repetitions.  Once the
finite Boolean branch is fixed, every requested contraction in the
neighbor tuple is a bounded public linear sum of contractions in
the current tuple at invertible $K$-affine pullbacks of the common
point.  Its coefficients and slot pulls depend on the branch,
requested masks, and public data, but not on the point, row index,
or further hidden values.  Tag-$0$ updates use only tag $0$;
tag-$i$ updates use only compatible types.
\end{lemma}

\begin{proof}
We give formulas for each transition, including the boundaries.
Write $C^{S,t}_{F,\mathbf j}$ for the stack at progress $t$.

\paragraph{Progress increase and decrease.}
On increasing from $0$ to $1$, initialize every nonempty stack by
\begin{equation}
 C^{S,1}_{F,\mathbf j}
       =\mathbf1_{\mathbf j=\mathbf0}F,
 \qquad M^{\rm mark,new}_j=\mathbf1_{j=0}.         \label{ia:increase-zero}
\end{equation}
For $0<t<q-1$, and $\mathbf j$ in the padded radix range, let
$Z(\mathbf j)=\{\alpha\in S:j_\alpha=0\}$ and put
$k_\alpha=0$ for $\alpha\in Z(\mathbf j)$ and
$k_\alpha=j_\alpha-1$ otherwise.  Then
\begin{equation}
 C^{S,t+1}_{F,\mathbf j}
    =\left(\prod_{\alpha\in Z(\mathbf j)}D_\alpha\right)
           C^{S,t}_{F,\mathbf k},\qquad
 M^{\rm mark,new}_j=M^{\rm mark,old}_{j-1},       \label{ia:increase-positive}
\end{equation}
with zero padding.  For $j_\alpha>0$, the old exponent is
$t-1-(j_\alpha-1)=t-j_\alpha$, as required; for $j_\alpha=0$,
one extra dilation gives exponent $t$.  Old inactive coefficients
remain zero.  Partitioning by $Z(\mathbf j)$ and using the radix
pull formulas implements this with boundedly many mask operations.
For a decrease from $t>0$, the formula is simply
\begin{equation}
 C^{S,t-1}_{F,\mathbf j}
     =C^{S,t}_{F,(j_\alpha+1)_{\alpha\in S}},\qquad
 M^{\rm mark,new}_j=M^{\rm mark,old}_{j+1}.       \label{ia:decrease}
\end{equation}
Its exponent is $t-1-(j_\alpha+1)=t-2-j_\alpha$ and its support
is $j_\alpha<t-1$.  In particular $t=1$ produces empty stacks
and a zero marker.  Persistent data remain fixed during these
progress transitions.

\paragraph{Critical readings and full-progress edits.}
At $t=q-1$, for a critical chart $F$ with projection axis $\alpha$,
\begin{equation}
 \sum_{j=0}^{N-1}C^{\{\alpha\}}_{F,j}
     =\sum_{j=0}^{q-2}D_\alpha^{q-2-j}F
     =F\big|_{Z_\alpha=1}.                       \label{ia:critical-projection}
\end{equation}
All-ones radix masks perform this sum.  A critical output ignores
every other native group, so the result is a constant-in-point
logical or public rule bit.  Lemma~\ref{ia:validity} therefore
determines the edit mask from finitely many such bits.

Fix that mask and direction.  For every positive-tier parent $F$
and every $S\subseteq A_F$, including $S=\varnothing$, add the
increment
\begin{equation}
 \Delta C^S_{F,\mathbf j}
    =\sum_{r=0}^{N-1}
      C^{S\cup\{\alpha_{\rm new}\}}_{G_F^\sigma,(\mathbf j,r)}.
                                                               \label{ia:edit-increment}
\end{equation}
For a positive port parent use its corresponding legacy mask
$\chi$.  For active old indices, commuting all inherited-label
dilations with projection on the new label gives
\begin{align}
 \Delta C^S_{F,\mathbf j}
   &=\left(\prod_{\alpha\in S}D_\alpha^{q-2-j_\alpha}\right)
          G_F^\sigma(c,b;1,Y)                    \label{ia:edit-calculation}\\
   &=\left(\prod_{\alpha\in S}D_\alpha^{q-2-j_\alpha}\right)
          \bigl(F(c,b+h_\sigma(c);Y)-F(c,b;Y)\bigr).
                                                        \nonumber
\end{align}
If any inherited index is inactive, every summand is zero.  This
is precisely the new stack minus the old one, including the empty
stack, which is the persistent chart itself.  The recursion supplied
the necessary lower-tier derivative for \emph{every} positive-tier
parent, including proof and derivative charts, so all coordinates
update simultaneously from old data.  Tier-zero charts stay fixed
under the Boolean change.  The marker is unchanged and the mode
changes as prescribed.  Reverse edits obey the same calculation
with the inverse branch's coefficients.

\paragraph{Frame change and edge flags.}
At empty progress all stacks are zero.  For frame advance, a
rotating persistent chart is pulled back on its one designated
variable by multiplication by $\gamma$; for reverse advance use
$\gamma^{-1}$.  A nonrotating persistent chart stays fixed.  The
empty stacks remain zero, and the mode changes.  In particular,
an auxiliary with native variables $(R,I,X,U)$ rotates only $R$;
its packing, division, and coefficient formulas are not recomputed
by the update.

The successor rules are: pre increases unless full; at full pre
perform the enabled edit, or report the absent wrap edge; post
decreases unless empty; at empty post advance the frame.  The
predecessor rules are: full post performs the inverse enabled edit,
or reports the absent wrap edge; other post reverses a decrease;
positive-progress pre reverses an increase; empty pre reverses the
frame advance.  Empty must also be distinguished for initialization
in \eqref{ia:increase-zero}.  These are all finite cases.  Comparing
the marker with the public zero tensor or the public full tensor
uses random separable masks and binary forms, with a constant
inequality-detection gap.  A fixed number of repetitions makes
either comparison as reliable as prescribed.  Mode and critical
constant bits are read with forms taking $1$ to $1$.  No progress
index scan or proposed-target validity check is needed for edge
flags on the valid inner path.

Finally, all point maps in these formulas are identities, full-label
dilations \eqref{ia:dilation}, or full-label multiplications by
$\gamma^{\pm1}$, and their bounded compositions.  They are
invertible $K$-affine maps.  In \eqref{ia:edit-increment}, the child
retains the parent's placements and has an already available extra
whole label; the sum projects that label to one and makes it
irrelevant to the target coordinate.  Thus a requested update has
the form
\begin{equation}
 L\cdot F^{\rm new}(Y)
    =P_L(Y)+\sum_{r=1}^{R_*}a_r
                         (L_r\cdot F_r)(T_rY),    \label{ia:affine-update}
\end{equation}
where $R_*$ is bounded, $P_L$ is publicly computable, $a_r$ and
$L_r$ are fixed by the requested masks and branch, and $T_r$ is
an invertible $K$-affine map.  No $a_r$ or slot pull depends on
$Y$ or on a later row index.  The inherited tags in
Lemma~\ref{ia:derivative-closure}, and the public fixed branch,
give the tag assertions.
\end{proof}

\begin{remark}[Native tests and encoded updates have different maps]
\label{ia:testing-versus-evolution}
The arbitrary $\mathbb F_q$ coordinate permutations and surjective
digit packing in Lemma~\ref{ia:linear-tool} need not be $K$-affine.
They are used to construct certificates and to test systematic
target-point values with bounded density.  They are not maps for
propagating an encoded parity row or a line-coefficient array.
Equation~\eqref{ia:affine-update} is the separate evolution
interface for those encodings.  For example, the proof chart
$F(c,b;R,I,X,U)$ changes frame only through its whole $R$ label,
and changes Boolean tape through its stored child's full-label
projection.  Its canonical division is not rerun on the requested
encoded symbol.  This distinction permits the binary construction
to use ordinary systematic representations for all native tests
while transporting neighbor updates through its common affine
action and linear row encoding.
\end{remark}
\begin{lemma}[Source data, original indices, and Boolean degree]
\label{ia:source-data}
At each of the two labels in \eqref{ia:source-successor}, every
tag-$0$ chart and stack is publicly computable.  Every tag-$i$
chart and stack has exactly the value of its corresponding
positive-tag one-port-template chart at the same native point,
slot, and mask indices, evaluated on input $s_i$.  These statements
hold for the source and its first successor separately, with
their indicated progresses $q-1$ and $q-2$.

At either label, every prime-field coordinate function of every
chart value or coordinate coefficient has multilinear degree at
most three in the read inputs.  Any common public linear
representation of these arrays preserves this bound and their
original-index correspondence.  A binary bilinear form of two
such represented values has degree at most six.  Its coefficients
are computable in polynomial time by bounded-degree Boolean
interpolation.
\end{lemma}

\begin{proof}
At both labels, $c=\gamma^{-1}$, work is blank, $e_1=1$ and other
legacy bits are zero, the mode is post, and progress is the public
value specified in \eqref{ia:source-successor}.  Tag-$0$ persistent
recipes depend only on these values and public circuit data, so
they are computable.  Their derivatives might evaluate a translated
tape with nonblank work, but every flip vector is then public and
the translated work is public too.  Such a tape need not be valid:
the canonical recipe is evaluated by its finite-array operations,
and never by solving for a valid circuit computation.  In
particular a work-side copy certificate uses its own work primary
and public matrix, not a stored sum involving $s_i$.  Recursively,
all its derivative and proof arrays remain public.  Stack
dilations at either fixed progress preserve this conclusion.

Positive persistent recipes and layouts agree by
Lemma~\ref{ia:port-ancestry}, and their placements agree by
Lemma~\ref{ia:placements}.  The projection-axis choices, inherited
labels, $q,a,B$, and radix slot conventions also agree.  Formula
\eqref{ia:stack-formula} therefore gives identical stacks at each
of the two progresses, including every descendant proof stack.
This proves a literal map of original algebraic indices, not
merely equality up to unspecified isomorphisms or comparable sizes.
Applying the same public row extension, restriction to an ordered
line, extraction of its coefficients, slot contraction, or basis
map to corresponding arrays preserves both values and indices.
Later balancing by repeated copies may use different multiplicities;
it does not alter this statement about original indices.

At a fixed frame every persistent value and coefficient has
assigned degree at most three in the Boolean tape.  At the present
labels all tape bits except the read inputs are fixed public bits.
Fixed-progress stacks are fixed affine pullbacks or zero.  Their
degree is thus at most three in the read inputs as well.  A public
linear operation over $\mathbb F_q$, or over any fixed public
extension field followed by prime-field coordinates, is
$\mathbb F_2$-linear on these functions.  It cannot increase
multilinear degree.  For two degree-three coordinate vectors, a
binary bilinear form is a sum of products of two such coordinate
functions and hence has degree at most six.  Copies do not alter
degree.  Evaluate the specified canonical bit on supports of size
at most six and apply \eqref{ia:boolean-inversion}; the number of
evaluations and each canonical finite-array computation are
polynomial.  This proves the coefficient-computation assertion.
\end{proof}

\subsection{The complete inner algebra interface}\label{ia:interface}

\begin{lemma}[Inner algebra interface]\label{lem:inner-algebra-interface}
Fix $c_{\rm allow}$.  The field and grid parameters
\eqref{ia:parameters}--\eqref{ia:radix-size}, the common domain
$K^u$, and the one-port positive family templates can be chosen
before the port count and the function.  For every fixed $k$ and
fixed circuit size factor, and all sufficiently large $L$, a
uniform polynomial-time construction has the following properties.
\begin{enumerate}
\item \textbf{Paths and endpoints.}
The graph on all port tuples is the graph of
Definition~\ref{ia:extended-path}.  On each fixed tuple $s$ it is
one directed path with source and first successor
\eqref{ia:source-successor}, no isolated vertices or self edges,
and a sink at full pre stage $M-1$.  The point-constant output
chart \eqref{ia:output-chart} equals $f(s)$ at the sink.

\item \textbf{Canonical polynomial arrays and their size.}
Each valid extended label has a unique canonical tuple
$\mathcal A(c,b,\eta,t)$ of $\mathbb F_q$-coefficient polynomials
on $K^u$.  Persistent native dimensions are at most
$C_{\rm dim}\ell$, with $C_{\rm dim}$ independent of $k$, and
each persistent family has at most one $O(\ell)$ slot axis.
Stack families have a bounded number of slot axes, each of length
$O(\ell)$.  Their honest coordinate degrees are $O(\ell h)<D$.
For fixed $k$, the number of families is fixed and every full
evaluation table, fully masked table, public matrix, and indexing
map has polynomial size and is polynomial-time constructible.
The layouts for fixed $k,L$ are independent of the circuit's
values and wiring.  The construction does not enumerate all
Boolean tapes or graph vertices.

\item \textbf{Exact validity with smooth binary-form probes.}
There is a fixed finite mixture of bounded-probe algebraic tests,
perfectly complete on canonical valid tuples.  For every tuple
of genuine degree-at-most-$D$ polynomials, either it is a
canonical valid tuple or the rejection probability is at least
$\theta_{\rm alg}>0$.  For fixed $k$ this gap and the probe density
bounds are uniform in $L$ and in the allowed circuit coefficients.
Each linear probe has bounded point and mask marginals.  Every
unknown product has compatible tags and independent full point
marginals for its two operands, also for repeated types.
The tests require only a bounded number of additive binary forms
and binary bilinear forms of hidden field values, followed by a
bounded number of Boolean operations.  They do not require reading
the full binary expansion of a hidden field element.

\item \textbf{Invalid tapes and finite derivative closure.}
Canonical recipes exist also for arbitrary Boolean $b$ at every
specified $c,\eta,t$, whether or not that label is valid.  They
retain their coordinate bounds, logical degree at most three,
per-primary dependencies, and designated frame covariance.
Every positive-tier persistent family, including every
verification auxiliary, has all its required lower-tier
derivatives.  The recursion terminates in at most three tier
drops.  A nonvalid tuple so obtained has the same exact rejection
gap; no alternative certificate can hide a failed row condition.

\item \textbf{Neighbor evolution.}
On valid data, both edge flags and finite transition branches use
only mode, empty/full marker comparisons, and boundedly many
critical projected bits at full progress.  At any desired fixed
reliability, these distinctions use boundedly many binary probes.
After a branch is fixed, every neighbor contraction has the
form \eqref{ia:affine-update} with boundedly many invertible
$K$-affine pulls of the common point.  Coefficients and slot
pulls are independent of point and row position.  Tag-$0$
updates use tag $0$ only; positive updates use compatible tags.
The formulas update all proof data simultaneously.  Their
fixed-branch algebraic identities remain valid on canonical
invalid Boolean inputs, without asserting that the resulting
proposed label is valid.

\item \textbf{Standardized ports and source access.}
Every positive-tag family has the literal per-call ancestry and
original-index correspondence of Lemma~\ref{ia:port-ancestry},
including all descendants, full-label placements, projection
axes, and stacks.  At the source and first successor, the
tag-$0$ part is public, and tag $i$ agrees with the one-port
template on $s_i$.  The source Boolean-degree bounds are three
under public linear representations and six after one binary
bilinear pairing, as in Lemma~\ref{ia:source-data}.  These
templates and bounds are fixed before actual $k$ or any later
correction accuracy is chosen.
\end{enumerate}
\end{lemma}

\begin{proof}
The path and endpoint assertions are
Lemmas~\ref{ia:validity} and~\ref{ia:path-endpoints}.  The linear
tool, clock-formula construction, derivative closure, and common
placements give the persistent arrays.  Lemma~\ref{ia:stacks}
gives their stacks.  At fixed $k$, all depths and family counts
are fixed, while all dimensions and total mask lengths are
$O(\ell)$.  Consequently every domain and mask space has size
$q^{O(\ell)}=\poly(n')=\poly(L)$.  Row and coefficient arrays
have the same polynomial bounds.  Their construction consists
of the explicit finite-array operations and bounded-degree
interpolation above.  In particular, increasing the number of
separate calls does not change the already chosen common $u$.

Lemma~\ref{ia:exact-validator} proves the exact gap and all point
and tag restrictions.  Here is its conversion to the asserted
binary-form interface.  A nonzero element of $\mathbb F_q$ is
detected by a uniformly random additive form
$\mu:\mathbb F_q\to\mathbb F_2$ with probability $1/2$.
Applying $\mu$ to a field equality turns its public linear terms
into additive forms, since
\[
 \mu\left(\sum_r a_r x_r\right)
      =\bigoplus_r(\mu\circ[a_r])(x_r),
 \qquad [a_r](x)=a_rx.
\]
For a product, the requested bit is the binary bilinear form
$(x,y)\mapsto\mu(xy)$.  Random annihilating forms handle mode
membership as already specified.  A bounded field test therefore
uses a bounded number of such bits and Boolean operations,
losing at most a fixed factor in its gap.  Prescribed masks and
forms can be split by fresh additive randomness.  Their public
descriptions have polynomial size; even the space of all binary
bilinear forms on $\mathbb F_q$ has cardinality
\[
 2^{(\log_2 q)^2}=\poly(n'),
\]
since $\log q=O(\log\log n')$.  A bounded sequence of all the
displayed random choices consequently has polynomial support.

The invalid-tape assertion follows from the per-primary
canonical formulas and the finite-difference construction, not
from successful final rows.  If a label is invalid, a validity
or frame/legacy condition fails and exact soundness rejects;
uniqueness of the division certificates prevents a different
proof table from canceling the failed condition.
Lemma~\ref{ia:evolution} proves all fixed-branch update formulas,
including for canonical invalid tapes.  Frame changes occur
only when stacks are empty, so no commutation between centered
dilations and frame scaling is needed.  Finally
Lemmas~\ref{ia:port-ancestry}, \ref{ia:placements}, and
\ref{ia:source-data} give the exact standardized source
correspondence and its Boolean-degree bounds.  This proves
every part of the interface.
\end{proof}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/07-inner-algebra.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/08-inner-binary.tex ====
\section{Binary testing and average correction at a fixed radius}
\label{sec:inner-binary}

We convert the polynomial arrays of Section~\ref{sec:inner-algebra} into
binary words.  The correction guarantee is averaged over the requested
encoded coordinates.  Its radius is fixed before the requested accuracy
and the density of those coordinates are specified.  This quantifier
order is essential when the one-port construction is subsequently used
many times.

All notation in this section is inner notation.  In particular, the
groups below are polynomial-size affine groups and are unrelated to the
outer index group.  Fix one of the padded templates of
Lemma~\ref{lem:inner-algebra-interface}.  We call each polynomial family
in its canonical tuple, together with its slot axes, a \emph{track}.
We use its fields
\(\F_q\subset K=\F_{q^\ell}\), common target domain \(K^u\), degree
bound \(D=\ell^2-1\), and fixed number of slot-mask axes per track.
The integer \(u\) and the field conventions are common to the actual
and one-port templates.  Honest chart degrees, including the constant
factor enlargement below, are \(O(\ell h)<D\) for sufficiently large
size.  The field-size choice already made in
Section~\ref{ia:path-parameters} ensures \(q/D\longrightarrow\infty\).
Constants in this section may depend on the fixed template, but are uniform
in the public circuit within that template.

\subsection{Rectangular coordinates and the common row code}
\label{ib:rectangular-code}

The algebraic neighbor formulas use invertible affine maps of the target
\(K^u\). We will add parity rows so that local row decoding can reduce
errors, and use a group action to move a requested value among columns
without changing it. The rectangular coordinates below make this action
transitive on point and ordered-line columns. Target affine maps commute
with it, so neighbor updates remain column permutations after row encoding.

Choose a constant integer \(r>2u\), \(r\ge3\).  A rectangular point
is \(I=(P,z)\), where \(P(x)=Ax+b\) maps \(K^r\) affinely to
\(K^u\) and \(z\in K^r\).  Retain points with \(\rank A=u\).
For an ordered \(\F_q\)-line
\[
 I_t=(P_0+tP_1,z_0+tz_1),\qquad t\in\F_q,
\]
retain it precisely when the matrix obtained by stacking \(A_0,A_1\)
has rank \(2u\) and \(z_1\ne0\).  Every point on such a line is
retained, since a dependence in \(A_0+tA_1\) would give a dependence
in the stacked matrix.  A sequential row-exposure bound shows that
the omitted point and line fractions are \(o(1)\) as \(\abs K\to
\infty\); for example rank failure for an \(s\)-by-\(r\) uniform
matrix is at most \(\sum_{j=0}^{s-1}\abs K^{j-r}\).

Write
\[
 U(K)=K^r\rtimes\mathrm{SL}_r(K),\qquad
 \mathcal H=U(K)\times U(K),\qquad
 (g_1,g_2)(P,z)=(P\circ g_1^{-1},g_2z).
\]
This is an affine action on the rectangular \(\F_q\)-coordinates,
and acts on ordered lines without changing their parameter.  There
is one orbit of retained points and one orbit of retained ordered
lines.  Indeed, right special-linear transformations are transitive
on full-rank matrices with \(u\) rows, and on those with \(2u\)
rows: first align by a general-linear transformation and then adjust
its determinant on an unused complementary dimension.  Translation
in the argument of \(P\) adjusts the one or two offsets because the
corresponding matrix has full row rank.  Independently, the second
affine factor is transitive on points and on ordered pairs of a base
point and a nonzero direction.  All retention conditions are
invariant.  In particular the point at each fixed parameter of a
uniform retained ordered line is uniform on retained points.

An invertible affine target map \(d:K^u\to K^u\) induces
\((P,z)\mapsto(d\circ P,z)\).  It commutes with \(\mathcal H\),
preserves both retained domains, and preserves line parameters.  A
target chart \(f\) is represented by \(f(P(z))\); this substitution
increases its coordinate degree by at most a factor two.

\begin{lemma}[Uniform affine-group expansion]
\label{ib:affine-expansion}
There is a fixed symmetric labelled multiset
\(u_1,\ldots,u_{d_e}\) in each \(\mathcal H\), with \(d_e\ge8\),
whose averaging operator has norm at most \(1/64\) on mean-zero
functions.  The labels, group operations, and complete group tables
are deterministically constructible in polynomial time in the inner
size.  The degree and gap are independent of the fields.
\end{lemma}

\begin{proof}
Here are the two primary group-theoretic inputs and their exact use.
Ershov--Jaikin-Zapirain~\cite[Theorem~1.1]{EJZ2010} prove property
\((T)\) for \(\mathrm{EL}_r(R)\) for every finitely generated
associative unital ring \(R\) and \(r\ge3\).  Their generating-set
bound in Section~6.1 applies to the elementary matrices with entries
in a fixed ring-generating set containing 1.  In particular, their
equations (6.1)--(6.2) give the positive Kazhdan constant
\[
 \kappa_r=\bigl[8(48\sqrt2+2\sqrt{3r})\bigr]^{-1}
\]
for
\(\{e_{ij}(1),e_{ij}(t):i\ne j\}\) in
\(\mathrm{SL}_r(\F_2[t])\); the equality with the elementary group
holds because \(\F_2[t]\) is Euclidean.  No restriction excluding
characteristic two occurs in this theorem.

Shalom~\cite[Theorem~3.4 and Corollary~3.5]{Shalom1999}, applied to the discrete
commutative ring \(R=\F_2[t]\), bounds the displacement of every
translation in \(R^2\rtimes\mathrm{SL}_2(R)\) by 484 times the
maximum displacement on the basis translations and
\(e_{12}(1),e_{12}(t),e_{21}(1),e_{21}(t)\).  Signs coincide in
characteristic two.  This is the homogeneous displacement form of
the corollary's tolerance statement, obtained by scaling and taking
limits.  It bounds the translation sector; it alone does not assert
expansion of the full affine group.

Set \(\Lambda=R^r\rtimes\mathrm{SL}_r(R)\) and let \(S\) consist
of all basis translations and the elementary matrices just listed
for all ordered coordinate pairs.  Embed Shalom's rank-two group on
each such pair.  Since a translation is a product of \(r\)
coordinate translations, every unitary representation satisfies
\[
 \sup_{a\in R^r}\norm{av-v}
       \le 484r\max_{s\in S}\norm{sv-v}.
 \tag{\ref{ib:affine-expansion}.1}
\]
Let \(V_N\) be the translation-invariant subspace.  Its orthogonal
projection commutes with \(\Lambda\), since the translations are
normal.  On \(V_N^\perp\), the minimum-norm point of the closed
convex translation-orbit hull is zero.  The displayed bound therefore
gives
\(\norm v\le484r\max_{s\in S}\norm{sv-v}\) on this subspace.
On \(V_N\), a representation without group-invariant vectors is a
representation of \(\mathrm{SL}_r(R)\) without invariant vectors,
so the preceding Kazhdan bound applies.  Orthogonally combining the
two bounds gives a positive Kazhdan constant \(\kappa\) for
\((\Lambda,S)\).  Explicitly, if \(\kappa_r\) is the quoted
 linear-group constant, one may take
\(\kappa=((484r)^2+\kappa_r^{-2})^{-1/2}\).

The set \(S\) generates \(\Lambda\): the third-index elementary
commutator multiplies entries, entries in a root subgroup add, and
conjugating basis translations gives arbitrary coordinate
translations.  Choose a field presentation
\(K=R/(p(t))\) by a monic irreducible polynomial.  Elementary
generation over a field shows that the quotient map surjects onto
\(U(K)\).  Pulling back its mean-zero regular representation
preserves the Kazhdan bound.  If \(d=\abs S\), the half-lazy walk
has Dirichlet form
\[
 \langle v,(1-M)v\rangle
   =\frac{1}{4d}\sum_{s\in S}\norm{sv-v}^2
   \ge\frac{\kappa^2}{4d}\norm v^2
\]
on mean-zero vectors, and nonnegative spectrum.  On the product
group, choose a factor uniformly and take this step there.  The
commuting tensor-factor operators give another uniform positive
gap and nonnegative spectrum.  A sufficiently large fixed power has
mean-zero norm at most \(1/64\).  Keep all walk labels, including
identities and coincident group elements, as a symmetric multiset;
repeat labels if necessary to ensure \(d_e\ge8\).  All matrices
have fixed dimensions over polynomial-size fields, so enumeration
and every stated computation are polynomial-time finite-array
operations.
\end{proof}

Form a bipartite graph with both vertex sets \(\mathcal H\), and
edges
\(\mathcal R=[d_e]\times\mathcal H\), where \((i,g)\) joins
left \(g\) to right \(u_i g\).  Its cross-part normalized second
singular norm is \(\lambda\le1/64\).  Put
\(c_e=\lfloor d_e/4\rfloor\).

\begin{lemma}[Systematic row extension]
\label{ib:row-extension}
There is a polynomial-size extension field \(J/\F_q\) and two
\(c_e\)-by-\(d_e\) matrices \(H_L,H_R\) over \(J\) whose every
\(c_e\)-column minor is nonzero, such that the edge code
\begin{align}
 \sum_i(H_L)_{bi}U_{i,g}&=0,\label{ib:left-row}\\
 \sum_i(H_R)_{bi}U_{i,u_i^{-1}x}&=0\label{ib:right-row}
\end{align}
has a unique linear extension \(E\) from labels \(i>2c_e\) to all
labels.  The same holds with symbols in any finite-dimensional
\(J\)-space.  Each local code has distance at least \(c_e+1\),
and \(E\) commutes with every right shift
\((R_hU)_{i,g}=U_{i,gh}\).
\end{lemma}

\begin{proof}
Treat the \(2c_ed_e\) matrix entries as indeterminates.  The square
system matrix on parity labels \(1,\ldots,2c_e\) has nonzero
determinant polynomial: specialize left row \(b\) to select label
\(b\), and right row \(b\) to select label \(c_e+b\).  This gives
a permutation matrix, including when some generators coincide.
Multiply this determinant by all maximal minors of both local
matrices.  Each factor is a nonzero polynomial; their product is
nonzero and has total degree at most
\[
 2c_e\abs{\mathcal H}
     +2c_e\binom{d_e}{c_e}.
\]
Take the least extension \(J/\F_q\) of cardinality exceeding this
bound.  It has polynomial cardinality.  A nonzero polynomial of
smaller total degree than \(\abs J\) cannot vanish everywhere on
\(J^{2c_ed_e}\).  Exhaustive search over this constant number of
entries, testing the determinant and minors, therefore finds a
specialization in polynomial time.  Inverting the parity matrix
constructs \(E\).

Column independence proves local distance \(c_e+1\), also for
vector symbols: a nonzero vector-valued local word supported on at
most \(c_e\) columns would contradict that independence after any
nonzero coordinate projection.  Right shifts permute left stars
and right stars in \eqref{ib:left-row}--\eqref{ib:right-row} and
preserve the systematic labels.  Uniqueness proves \(ER_h=R_hE\).
\end{proof}

Choose this field, these matrices, and their deterministic ordering
from the common point and field parameters alone, independently of
the number or identities of tracks.  Every track uses this one
encoder.  On systematic rows prescribe, slotwise,
\[
 Y_{i,g}(I)=f(P'(z')),\qquad (P',z')=gI,
\]
and extend columnwise by \(E\).  Equivariance yields
\begin{equation}
 Y_{i,gh}(I)=Y_{i,g}(hI)
 \quad\hbox{for all rows, including parity rows.}
 \label{ib:covariance}
\end{equation}
Every parity-slot function is a degree-in-bound polynomial over
\(J\).  At an ordered retained line store its actual restriction,
using coefficients through degree \(D\) and padding by zero.
These coefficients obey the same row constraints and covariance.
The commuting target maps act by the same column permutations on
point tables, line tables, and their encoded parity rows.
In particular, covariance gives
\(Y_{i,gh}(h^{-1}I)=Y_{i,g}(I)\) for a point or an ordered line \(I\).
Thus changing the row and making the inverse change to the column
preserves the requested symbol. This is the movement among columns that
will accompany local row decoding.

\subsection{Binary blocks and smooth probes}
\label{ib:binary-blocks}

For each track there are point and line blocks, both denoted \(Y\)
with their type understood.  A point position specifies a row edge,
a retained point, one \(\F_q\)-vector mask per slot axis, and an
additive binary form \(\mu:J\to\F_2\).  Its bit is \(\mu\) of
the slot contraction.  A line position additionally specifies masks
\(A,B\in\F_q^\ell\) for the coefficient index.  If \(Y(t)\)
is already contracted on the original slots, the bit is
\begin{equation}
 \mu\left(\sum_{j,l=0}^{\ell-1}A[j]B[l]
                     [t^{j+\ell l}]Y(t)\right).
 \label{ib:line-bit}
\end{equation}
This uses \(D+1=\ell^2\).  Evaluation at a specified parameter
uses the separable masks \((t^j)_j\) and \((t^{\ell l})_l\),
with the usual convention \(t^0=1\).

The short two-mask coefficient representation in \eqref{ib:line-bit}
is adapted, through the inherited polynomial-size construction, from
Amireddy--Behera--Srinivasan--Sudan--Willumsgaard
\cite[Section~3.1, Definition~3.1 and Claim~3.2]{ABSsw26}.
This attribution concerns the coefficient encoding and its evaluation
masks, not the correction and evolution machinery developed here.

Fix an \(\F_q\)-linear map \(\pi:J\to\F_q\) restricting to
the identity on \(\F_q\).  For each compatible ordered pair of
point-track types, include a product block \(Z\).  Compatibility
means that one tag is zero or both positive tags agree.  A type may
be paired with itself, with two independent positions.  An original
\(Z\) position consists of the two complete edge/point/slot-mask
positions, \(b_1,b_2\in J\), and a binary bilinear form \(Q\)
on the \(\F_2\)-coordinates of \(\F_q\).  For the two contracted
symbols the bit is
\begin{equation}
 Q\bigl(\pi(b_1Y_1),\pi(b_2Y_2)\bigr).
 \label{ib:product-bit}
\end{equation}
There are no line-product blocks and no products of distinct
positive tags.

Give every block equal weight.  If its original length is \(L_b\),
repeat each original bit uniformly until every block has length
\(\prod_b L_b\).  Every raw query chooses a uniform repetition
copy.  Let \(b_\#\) be the block count, \(m_{\rm in}\) the total
length, and \(\mathcal W(v)\) the canonical word for an extended
label \(v\).  These lengths are polynomial in the inner size:
there are constantly many blocks, \(q^{O(\ell)}\) slot masks,
fixed-dimensional matrix domains over \(K\), polynomial-size
\(J\), and \(2^{(\log_2q)^2}\) bilinear forms \(Q\).
Here \(\log q=O(\log\log n')\), so the last quantity is also
polynomial in \(n'\).  The product used for balancing has constantly
many polynomial factors.  Lengths are fixed within a padded template
independently of the circuit coefficients.  Positive-tag original
indices and canonical formulas coincide with the corresponding
one-port indices; repetition counts need not coincide.

We will repeatedly use the following explicit smoothing operation.
For a specified additive argument \(a\), choose uniform \(r\)
and write \(a=r+(a+r)\).  Multilinearity expands a requested value
into a bounded XOR of terms, each uniform in that argument.
Split different axes independently.  This applies to slot masks,
coefficient masks, bit forms, \(Q\), and, when needed, the two
multiplier axes.  Each term has uniform marginals jointly across
its independently split axes, conditional on the entire specified
request.  Pull back all public scalars and other linear maps before
splitting.  This remains true for zero coefficients; a zero term
may instead be omitted and supplied as public zero.

To probe a target value at \(y\in K^u\), choose a uniform row
group \(g\), full-rank \(A\), and \(z\), set
\(P(x)=Ax+y-Az\), and use the systematic position
\begin{equation}
 (i,g,g^{-1}(P,z)).
 \label{ib:systematic-probe}
\end{equation}
For uniform \(y\), this is jointly uniform in row and retained
point within its systematic label; every fiber has the same number
of \((A,z)\) choices.  A density bound for \(y\) therefore gives
the same bound for this joint distribution.  Independently chosen
target arguments can use independent representations.  An additive
form on a target \(\F_q\)-value extends to \(J\), for example by
composition with \(\pi\).  These procedures are defined on every
raw word and are smooth before any replacement or soundness claim.

\subsection{Dimension-independent replacement lemmas}
\label{ib:replacement-tools}

The tester's soundness proof replaces arbitrary binary data by symbol
tables and then by polynomial arrays. The two replacement lemmas below
make the error cost independent of the number of field coordinates and
slot entries.

We state precisely the low-degree input being used.  The corrected
Theorem~13 of Friedl--Sudan~\cite{FS1995} says that for every
\(\eta>0\) there is \(C_\eta<\infty\) such that, if
\(\abs{\F}\ge C_\eta D\), \(D\ge1\), and a function
\(f:\F^m\to\F\) and degree-at-most-\(D\) univariate claims
\(p_{x,h}\) satisfy
\[
 e=\Pr_{x,h\in\F^m,\,t\in\F}
       [p_{x,h}(t)\ne f(x+th)]\le \tfrac18-\eta,
\]
then a polynomial \(p:\F^m\to\F\) of total degree at most \(D\)
satisfies \(\dist(f,p)\le2e\).  The variables in this probability
are independent and uniform, including the possible direction \(h=0\);
the field-size constant does not depend on \(m\).  We use
\(\eta=1/16\).  Neither a vector-valued conclusion nor a restriction
on the line domain is part of this cited statement.

\begin{lemma}[Vector and deleted-domain low-degree replacement]
\label{ib:vector-ld}
Let \(V\) be any finite-dimensional \(\F_q\)-space.  In the preceding
field regime, line--point disagreement tending to zero for
\(V\)-valued point functions and degree-in-bound line claims implies
point distance tending to zero from a single \(V\)-valued polynomial
of total degree at most \(D\), uniformly in both dimensions.
The conclusion remains true if the point and ordered-line domains
omit fractions tending to zero, every retained line lies entirely in
the retained point domain, and the point at a uniform parameter on a
uniform retained line is uniform on retained points.  In this case the
mean fraction of line coefficient vectors differing from the
polynomial's restrictions also tends to zero.
\end{lemma}

\begin{proof}
First use the full domains.  For every \(\alpha\in V^*\), scalar
projection and the cited theorem give a polynomial \(p_\alpha\) with
\(\dist(\alpha f,p_\alpha)\le2e\).  Distinct scalar polynomials in
this degree range differ at a fraction at least \(1-D/q\) of points.
Consequently, when \(6e<1-D/q\), comparison with
\((\alpha+\beta)f\) forces
\[
 p_{\alpha+\beta}=p_\alpha+p_\beta,
 \qquad p_{a\alpha}=a p_\alpha \quad(a\in\F_q).
\]
The second identity uses at most \(4e\) disagreement; the assertions
with a zero form or scalar follow in the same way.  Coefficientwise
finite-dimensional duality now gives one \(V\)-valued polynomial
\(p\) for which \(\alpha p=p_\alpha\) for all \(\alpha\).
A uniform \(\alpha\) detects each nonzero vector with probability
\(1-1/q\), so
\[
 (1-1/q)\dist(f,p)
 =\E_\alpha\dist(\alpha f,p_\alpha)\le2e.
\]
There is no factor depending on \(\dim V\).

For deleted domains, extend the point function arbitrarily and give
every added line any degree-in-bound claim, for example zero.  The
full incidence disagreement is at most the retained-line disagreement
plus the omitted-line fraction.  The full-domain conclusion therefore
gives vanishing error also on retained points.  On any retained line
whose claim differs from the restriction of \(p\), a nonzero scalar
projection of their difference is a univariate polynomial of degree
at most \(D\).  The two claims disagree at a fraction at least
\(1-D/q\) of its parameters.  Averaging the original test error and
the point replacement error, using the asserted uniform point
marginal, bounds the fraction of such lines by their sum divided by
\(1-D/q\).  This tends to zero and proves the coefficient claim.
\end{proof}

The addition-test argument below uses the majority self-correction method
of Blum, Luby, and Rubinfeld~\cite[Generic Linear Self-Testing]{BLR1993}.
We include it together with the field-homogeneity, multiple-axis, and
slice-averaging extensions needed for our vector alphabets.

\begin{lemma}[Axis replacement]
\label{ib:axis-replacement}
For elementary binary vector groups, the addition test has a
dimension-independent high-agreement replacement by an additive map,
also with a vector-group alphabet.  For \(\F_q\)-spaces of
characteristic two, addition and scalar-homogeneity tests give
replacement by an \(\F_q\)-linear map.  For any fixed number of axes,
testing these identities in each axis at uniform other arguments gives
replacement by a multilinear map as the failure probability tends to
zero, uniformly in every domain and alphabet dimension.  These
conclusions also hold in mean over additional slice parameters.
\end{lemma}

\begin{proof}
Suppose first that \(f:G\to A\) fails addition with probability
\(e\), where both groups have exponent two.  For fixed \(x\) put
\(D_x(y)=f(x+y)+f(y)\).  The addition identities on the pairs
\((x+y,y+z)\) and \((y,y+z)\) imply \(D_x(y)=D_x(z)\).
Each pair is uniform when \(y,z\) are independent uniform elements,
so the collision probability is at least \(1-2e\), for every \(x\).
The most frequent value \(L(x)\) therefore has mass at least
\(1-2e\).  The identity
\[
 D_{x+y}(z)=D_x(y+z)+D_y(z)
\]
and three majority bounds imply \(L(x+y)=L(x)+L(y)\) when
\(6e<1\): at least one \(z\) realizes all three majority values.
If \(f(x)\ne L(x)\), the conditional addition failure at \(x\)
is at least \(1-2e\).  Thus
\(\dist(f,L)\le e/(1-2e)\).

For the field version, transfer the homogeneity test to \(L\).
The scalars \(a\) satisfying \(L(ax)=aL(x)\) identically form an
additive subgroup of \(\F_q\).  If it is proper, at least half the
nonzero scalars fail; for each failing scalar its nonzero additive
discrepancy is nonzero on at least half the inputs.  The homogeneity
test with uniform nonzero scalar would then have failure at least
\(1/4\).  This contradicts failure tending to zero after a vanishing
replacement cost.  Hence \(L\) is field-linear.

Induct on the number \(j\) of axes.  Markov's inequality, with a
threshold tending slowly to zero, makes all but a vanishing fraction
of first-axis slices good for the remaining \(j-1\) tests.  Replace
good slices by multilinear maps and the remaining slices by zero.
This changes a vanishing fraction of values.  Regard the resulting
slices as a map into the vector space of multilinear maps on
\(j-1\) axes.  A nonzero such map is nonzero on at least
\(2^{-(j-1)}\) of uniform inputs: successively contract each axis,
using that a nonzero linear map has a kernel of relative size at most
\(1/2\).  Thus vanishing first-axis test failure on values gives
vanishing failure as equalities in this vector-space alphabet.
The one-axis result finishes the induction.  The same Markov cutoff
over any additional parameters proves the mean version.
\end{proof}

\subsection{Testing the binary representation}
\label{ib:testing}

\begin{proposition}[Smooth tester and distance]
\label{ib:tester}
For each fixed template, the canonical encodings of valid extended
labels have constant relative distance.  They admit a perfectly
complete bounded-bit tester with the following uniform gap: for every
fixed \(a>0\), there are \(\theta(a)>0\) and a size cutoff such that
rejection below \(\theta(a)\) implies distance less than \(a\) from
one canonical valid word.  Each potential raw read on each fixed
answer branch has bounded density relative to the balanced word,
without conditioning on execution of that branch.  Consequently
rejection is at most a fixed constant times distance from a canonical
valid word.
\end{proposition}

\begin{proof}
Take a positive fixed mixture of the test families described below,
including each present block type.  Their query counts are bounded,
and all requested nonuniform masks are implemented by the splitting
operation above.  We prove the gap by a sequential replacement
argument.  Consider any sequence of increasing sizes and words whose
rejection tends to zero, allowing the public circuits to vary in the
fixed template.  We show distance tending to zero from a canonical
valid word.  A bounded smooth test retains vanishing rejection after
a vanishing-distance change, by the union bound over its potential
reads; for a branching test use its entire bounded answer tree.

\paragraph{Copies, symbols, and slot tensors.}
Compare two independent random copies of a uniform original bit.
If the fraction of ones among its copies is \(p\), the test rejects
with probability \(2p(1-p)\), which is at least
\(\min(p,1-p)\).  Replacing all copies by their majority costs
vanishing distance.  In each point or line \(Y\) block test
additivity in \(\mu\).  Lemma~\ref{ib:axis-replacement} and binary
double duality replace the block, in mean over other indices, by
Hadamard tables of genuine \(J\)-symbols.  Next test addition and
\(\F_q\)-homogeneity in every slot-mask axis, including the two
coefficient-mask axes for a line.  Test symbol equality by uniform
\(\mu\), which detects any inequality with probability \(1/2\);
pull field scalars onto this form before smoothing its raw accesses.
The same replacement lemma makes all these symbols honest
multilinear slot tensors at each edge and point or line, at vanishing
mean cost.  No global polynomial property has yet been asserted.

\paragraph{Point and line polynomials.}
At a uniform edge, retained ordered line, and parameter, compare the
point symbol with line evaluation, using uniform original slot masks
and a uniform bit comparison.  Split the two specified evaluation
masks in \eqref{ib:line-bit}.  A nonzero symbol discrepancy has a
constant binary detection probability, so the line--point
disagreement tends to zero in mean.  For most edge and original-mask
tuples apply Lemma~\ref{ib:vector-ld}, with \(V=J\) as an
\(\F_q\)-space, to obtain a degree-at-most-\(D\) rectangular
polynomial.  Use zero on the exceptional tuples.  The lemma's
line-coefficient conclusion also replaces the line tables by the
restrictions of these polynomials at vanishing cost.

The selected polynomials need not yet be coherent between original
mask tuples.  Transfer the previous multilinearity identities to
them using their mean point error.  A false identity between
degree-in-bound polynomials is nonzero on at least
\(1-D/q-o(1)\) of retained points.  Thus the identities fail with
vanishing probability as equalities in the vector space of
degree-in-bound polynomials.  Apply
Lemma~\ref{ib:axis-replacement} in that vector-space alphabet, in
mean over edges, and use zero on exceptional edges.  Changing a
vanishing fraction of mask tuples changes a vanishing fraction of
point and line bits.  We now have genuine polynomial slot families
at every row, and their exact line restrictions.

\paragraph{Covariance and common systematic data.}
Test \eqref{ib:covariance} on point symbols with independent uniform
\(g,h,I\), uniform masks, and a bit form.  Undo each row's chart:
\[
 F_{i,g}(I)=Y_{i,g}(g^{-1}I).
\]
If two such polynomial slot tuples differ, the discrepancy is seen
with constant probability, first on a retained point, then by
successive nonzero mask contractions, then by the bit form.  Since
\(g\) and \(gh\) are independent uniform elements, for each fixed
label and track the squared frequencies of the uncharted tuples
sum to \(1-o(1)\).  Their maximum frequency is therefore
\(1-o(1)\).  Replace other rows by this predominant tuple, charted
back.  This changes vanishing point and line distance and enforces
covariance exactly, including on lines.

Compare uncharted tuples of the finitely many systematic labels,
using a shared uniform retained point and masks and independent row
groups.  The same constant gap forces an exactly common polynomial
tuple \(F\) on all systematic labels of a track.  Test that its
contracted values lie in \(\F_q\) by a random binary form on
\(J/\F_q\), pulled back to \(J\) and split.  The quotient is an
\(\F_q\)-space.  If a polynomial coefficient were outside
\(\F_q\), its quotient would be a nonzero vector polynomial and
would have a constant detection gap.  Consequently every systematic
coefficient is in \(\F_q\).

\paragraph{Factoring through the target and enforcing parity.}
Choose uniform \(y\in K^u\) and two conditionally independent
generic representations as in \eqref{ib:systematic-probe}.  Compare
their values, with independent row groups.  Each individual marginal
is smooth; this test is a linear comparison and makes no product
query on the correlated pair.  For a slot coefficient the uncharted
identity tested is
\begin{equation}
 F(A,y-Az,z)=F(A',y-A'z',z').
 \label{ib:factor-target}
\end{equation}
The difference has coordinate degree at most \(2D\), and the
full-rank restrictions omit an \(o(1)\) fraction.  A failed
polynomial identity therefore has constant detection probability,
also after the bounded slot contractions.  Hence
\eqref{ib:factor-target} holds identically.  Fix any full-rank
\(A'\) and any \(z'\).  Its right side defines a target polynomial
\(f(y)\) of degree at most \(D\) with \(\F_q\) coefficients.
Substitute \(y=Az+b\) to obtain the exact factorization
\(F(P,z)=f(P(z))\).

Test the left and right star equations at uniform stars and retained
points, with slot masks and random \(J\)-bit forms; pull row
coefficients onto these forms.  In an exactly covariant family, a
failed polynomial star relation fails on every right-shift translate
of that star, with an invertible affine change of argument.
Polynomial, mask, and bit detection thus give a constant rejection
gap for any failure.  All star identities hold once the rejection is
sufficiently small.  Lemma~\ref{ib:row-extension} now forces the
exact parity extension of the systematic target data and its line
restrictions.

\paragraph{Product blocks.}
In each present \(Z\) block test additivity in \(Q\) and replace
at vanishing cost by exact \(Q\)-linear maps, in mean over all its
other indices.  Sample a uniform independent pair of
edge/point/slot-mask indices and uniform multipliers.  Test the
product identity for
\(Q(x,x')=\alpha(x)\alpha'(x')\), with independent uniform
binary forms \(\alpha,\alpha'\) on \(\F_q\).
On the \(Z\) side split only \(Q\), keeping these already uniform
other indices fixed.  On each \(Y\) side pull the multiplier and
\(\pi\) into the requested bit form and split that form.  Compare
with the Boolean product of the two resulting bits.

The \(Y\)'s are already genuine symbols.  The difference between
a wrong \(Q\)-linear map and their true product is a nonzero binary
matrix.  A uniform outer-product form detects it with probability
at least \(1/4\), by two successive nonzero contractions.  The
fraction of wrong maps therefore tends to zero.  Replace them by
\eqref{ib:product-bit}.  This works separately for every compatible
pair and never invokes a missing product block.

\paragraph{Algebraic validity.}
Finally run the algebraic validator of
Lemma~\ref{lem:inner-algebra-interface} on the recovered target
polynomials, using systematic representations.  Linear target
probes need only their individual bounded-density marginal.  For
each product probe that lemma gives compatible tags and independent
uniform full target arguments.  Represent these independently in
rectangular coordinates.  The corresponding \(Z\) query therefore
has the required joint density.  Identity multipliers on
\(\F_q\)-symbols and the demanded bilinear form implement the
field-level product; split the necessary slot, form, and multiplier
axes for its raw reads.  The exact algebraic gap forces canonical
extended validity.  The native linear maps used by the validator,
including coordinate packing, need not commute with the row action:
they are used here only to specify systematic target arguments.

All replacements have vanishing total binary cost, so the original
word is at vanishing distance from a canonical valid word.  If the
stated fixed-gap conclusion failed, for each integer \(j\) one
could choose a size at least \(j\) and a word of distance at least
\(a\) with rejection below \(1/j\), contradicting this sequence
argument.  Each constituent test is perfectly complete for every
random choice in its support.  All its raw marginals are smooth by
the displayed representations and splitting; this also proves the
claimed linear upper bound on rejection near a canonical word.

For distance, different logical tapes differ in an order-one grid
polynomial.  Its difference is nonzero on a constant fraction of
uniform target arguments, hence on a constant fraction of systematic
row/point positions; a bit form detects the symbol difference with
probability \(1/2\).  A concerned block has fixed positive weight.
If the logical tape agrees, different valid extended labels differ
in mode or in the progress marker, because legacy and mode fix the
frame.  These constant slot tensors likewise have a fixed
nonzero-contraction gap.  This gives a constant lower bound on
relative distance.
\end{proof}

\subsection{Value-independent uncertainty supports}
\label{ib:uncertainty}

The correction argument uses uncertainty sets rather than the actual
wrong symbols at intermediate stages.  This permits separate
consistent choices in each bounded dependency cone.

The alternating local decoding strategy is classical; see
Z\'emor~\cite[Section~III-A, Lemma~5 and Theorem~6]{Zemor2001}.
The proofs below supply the vector-space, repeated-edge,
value-independent-support, and finite-cone refinements needed here,
rather than invoking that result verbatim.

For a \(Y\) block, a column fixes a retained point or line and all slot
masks, including line coefficient masks. At that column, varying the
edge gives a row codeword of \(J\)-symbols in the reference encoding.
The binary data for one such symbol form its Hadamard table, including
all repetition copies; we call this its \emph{inner table}. The first
lemma controls uncertainty in these row-code positions without examining
the uncertain symbols' values.

\begin{lemma}[One-axis support contraction]
\label{ib:one-axis-support}
For the row code above, set
\[
 t_* =\lfloor c_e/2\rfloor+1,\qquad
 \vartheta=t_*/d_e\ge1/8,\qquad \zeta_*=1/256.
\]
Starting with an uncertain edge set of density
\(\zeta\le\zeta_*\), a constant number of alternating local
decoding half-rounds leaves an arbitrarily small fixed uncertain
edge density.  The resulting support depends only on the starting
uncertainty set and contains every possible output error for every
assignment equal to the reference codeword off that set.  This holds
for symbols in any finite-dimensional \(J\)-space.
\end{lemma}

\begin{proof}
At a star decode to the unique local codeword within
\(t_*-1=\lfloor c_e/2\rfloor\) errors, if it exists; otherwise
output zero on that star.  Distance \(c_e+1\) gives uniqueness.
This decoder uses a bounded number of linear symbol operations and
symbol-equality decisions: enumerate the information sets of size
\(d_e-c_e\), solve using the complementary invertible minor,
and count agreements with each candidate.  There are constantly
many such choices.

On a half-round mark a star bad if it meets at least \(t_*\)
currently uncertain edges, and declare all edges incident to bad
stars possibly uncertain afterward.  Every other star decodes
correctly, regardless of the values on uncertain edges.  The first
bad-star fraction is at most \(\zeta/\vartheta\le8\zeta\).
If \(x\) and \(y\) are consecutive bad-star fractions on opposite
sides, the new bad stars receive at least a \(\vartheta\) fraction
of their incident edges from the preceding bad stars.  Expander
mixing with multiplicities gives
\begin{equation}
 \vartheta y\le xy+\lambda\sqrt{xy}.
 \label{ib:mixing-contraction}
\end{equation}
When \(x\le\vartheta/2\), either \(y=0\), or division and
squaring imply
\[
 y\le(2\lambda/\vartheta)^2x\le x/16.
\]
Since \(8\zeta_*\le\vartheta/2\), induction applies at every
subsequent half-round.  The current edge-support fraction equals
the latest bad-star fraction.  A constant number of half-rounds
therefore gives any prescribed fixed residual density.  Every
support operation used only sets of uncertain positions, proving
the value-independent assertion.
\end{proof}

For a \(Z\) block, a column fixes both retained points, their slot
masks, and \(Q\). Its row position is a joint pair of edges, and its
inner table consists of the multiplier pairs and repetition copies.
The reference multiplier tables obey row constraints along each edge
axis, but the two axes use different scalar actions. We verify these
actions before applying a two-axis contraction.

For \(Z\), the relevant symbol alphabet is the additive group
\(\mathcal B_{\rm mul}\) of all binary bilinear functions
\(F:J\times J\to\F_2\).  It carries two commuting \(J\)-space
structures
\begin{equation}
 (a\cdot_1F)(b_1,b_2)=F(ab_1,b_2),\qquad
 (a\cdot_2F)(b_1,b_2)=F(b_1,ab_2).
 \label{ib:two-actions}
\end{equation}
Additivity in each argument and the field laws verify the vector
space axioms and commutation.  Distinct such functions differ on
at least \(1/4\) of uniform multiplier pairs.  Fixing the two
outer points, the original slot masks, and \(Q\) makes the true
multiplier tables across edge pairs a two-axis codeword.  For
example the first-axis relation with coefficients \(a_j\) is
\[
 \sum_j Q\bigl(\pi(b_1a_jY_{1,j}),\pi(b_2Y_2)\bigr)
 =Q\bigl(\pi(b_1\sum_j a_jY_{1,j}),\pi(b_2Y_2)\bigr)=0.
\]
The second-axis relation is identical with the roles exchanged.
The next lemma permits precisely these two possibly different space
structures.

\begin{lemma}[Two-axis support contraction and cone consistency]
\label{ib:two-axis-support}
Consider an array on two edge axes whose reference obeys the row
code in each slice, with possibly different \(J\)-space structures
on the same additive symbol group for the two axes.  Put
\(\zeta_{**}=\zeta_*^2/4\).  From initial uncertainty density at
most \(\zeta_{**}\), a constant number of first-axis half-rounds
followed by a constant number of second-axis half-rounds leaves any
prescribed fixed residual density.  The uncertainty support is
value-independent.  For one output position, the same guarantee
holds when arbitrary values are chosen on initially uncertain
positions specifically for its finite dependency cone, provided
each repeated initial position receives one consistent value inside
that cone.
\end{lemma}

\begin{proof}
First-axis slices with uncertainty density greater than
\(\zeta_*\) occupy at most a \(\zeta_*/4\) fraction.  Keep these
whole stripes uncertain.  On all other slices use
Lemma~\ref{ib:one-axis-support} to make residual density at most
\(\delta_1\).  In all but \(2\delta_1/\zeta_*\) of the
second-axis slices, the additional residual outside these stripes
has density at most \(\zeta_*/2\).  Such a slice starts with
uncertainty at most \(3\zeta_*/4<\zeta_*\).  Contract it to
\(\delta_2\), leaving all other second-axis slices uncertain.
The final density is at most
\[
 2\delta_1/\zeta_*+\delta_2,
\]
which is arbitrarily small.  Intermediate arrays need not satisfy
both slice constraints: each phase is compared to the fixed
reference and uses its appropriate space structure only.

A fixed number of rounds and constant degree give a finite
dependency cone.  Extend any consistent assignment in that cone
arbitrarily to the other uncertain initial positions, retaining the
reference elsewhere.  The global support statement applies to this
extension and determines the same output.  Separate cones may use
different values on their uncertain inputs; no common assignment
between cones is needed.
\end{proof}

\subsection{Bit simulation of a finite decoding cone}
\label{ib:cone-simulation}

\begin{lemma}[Inner-table flags and arbitrary-form prediction]
\label{ib:inner-table-prediction}
There are constants \(0<\nu_1<\nu_2<1/64\), independent of the
number of row rounds, with the following properties.  For a table
of bits, including its actual repetition copies, one can distinguish
distance at most \(\nu_1\) from a legal inner table from distance
greater than \(\nu_2\) from every legal inner table, with any
prescribed fixed reliability and bounded smooth queries.  The legal
inner tables are either Hadamard tables of \(J\)-symbols or binary
bilinear functions on \(J\times J\).  Inside radius \(\nu_2\)
the legal symbol is unique, and any specified bit of it can be
predicted with arbitrarily high fixed reliability by bounded queries
with uniform inner-index and copy marginals.  The specified form or
multiplier pair may be any function of previously exposed data;
the assertion holds conditional on those data, using fresh randomness.
\end{lemma}

\begin{proof}
Random-copy agreement and binary addition, or the two-axis binary
tests, are perfectly complete.  Lemma~\ref{ib:axis-replacement},
together with the majority replacement of copies, gives a
dimension- and copy-count-independent positive rejection lower
bound \(\gamma_2\) at distance greater than a fixed \(\nu_2\)
from every legal inner table.  If this failed, a sequence of such
tables with rejection tending to zero would contradict that lemma.
Near a legal table, smoothness bounds rejection by \(C_2\) times
distance, with an absolute \(C_2\).  Choose
\(\nu_1<\min(\nu_2,\gamma_2/(4C_2))\).  Estimating rejection
from independent trials and comparing with \(\gamma_2/2\)
distinguishes the two promised cases with any fixed reliability;
Chebyshev's inequality suffices to choose a bounded sample count.
The intermediate annulus may be assigned either flag.

Distinct Hadamard tables differ on half their forms.  Distinct
bilinear functions differ on at least a quarter of multiplier
pairs.  Repetition preserves these distances, proving uniqueness
within \(\nu_2\).  For a specified Hadamard form \(\mu\), choose
uniform \(\alpha\), query random copies at \(\alpha\) and
\(\mu+\alpha\), and XOR.  Each query is uniform in its inner
index, so one trial errs with probability at most \(2\nu_2\).
For a bilinear table at specified \((b_1,b_2)\), choose independent
uniform \(r_1,r_2\) and use
\begin{align*}
 F(b_1,b_2)={}&F(r_1,r_2)+F(b_1+r_1,r_2)\\
             &+F(r_1,b_2+r_2)+F(b_1+r_1,b_2+r_2).
\end{align*}
Each pair is jointly uniform, with a fresh random copy, and the
error probability is at most \(4\nu_2<1/16\).  Independent
repetition and majority give any desired fixed error.  These bounds
hold for every specified bit, so remain valid after conditioning
on all prior data if the splits and copies are fresh.
\end{proof}

\begin{lemma}[Consistent bit simulation]
\label{ib:bit-simulation}
Fix a finite cone of one- or two-axis row decoding, a reference satisfying
the corresponding row constraints, and a requested bit of the output
symbol. For analysis, declare an initial position uncertain when its
received inner table is more than \(\nu_1\) from its reference table,
and propagate this set by the support rules above. With bounded bit
queries the requested bit can be simulated to any fixed error probability
\(\sigma>0\), on a consistent assignment of initially uncertain
symbols that agrees with the reference at every initial table
within \(\nu_1\) of its true inner table. If the output position lies
outside the value-independent support, the predicted bit equals its
reference bit except with probability \(\sigma\).  All hidden operations are bit equalities
and requested bits; field coefficients and address computations
are public.  The routine is bounded and defined on arbitrary words.
\end{lemma}

\begin{proof}
List the cone's initial positions and deduplicate repeated positions
within that cone.  Flag each distinct table once by
Lemma~\ref{ib:inner-table-prediction}, calling the far outcome
flagged, and assign flagged positions the zero symbol.  At an
unflagged table within \(\nu_2\) of a legal table, assign its
unique nearby symbol.  Choose the flag reliability so that, except
with a prescribed small probability, every table within \(\nu_1\)
of its true symbol is unflagged and no table farther than \(\nu_2\)
from all legal symbols is unflagged.  On this event the assignment
is well-defined and consistent inside the cone.  Certain inputs
receive their true symbols: a competing symbol within \(\nu_2\)
would contradict inner distance.  The remaining assignments,
including arbitrary choices in the gap annulus, are permitted by
Lemmas~\ref{ib:one-axis-support} and~\ref{ib:two-axis-support}.

There is no need to recover the assigned symbols in full.  Unfold
the finite symbol-level decision tree for information-set decoding,
agreement counts, and fallback-zero outcomes.  On each fixed branch
every candidate, comparison, and output is a linear expression in
the assigned initial symbols.  For two axes these expressions are
sums of compositions of the two scalar actions
\eqref{ib:two-actions}.  Their coefficients are obtained publicly
from the fixed local matrices and the finite branch choices.

A false symbol equality is detected by a fresh uniform Hadamard
form with probability \(1/2\), or by a fresh uniform multiplier
pair with probability at least \(1/4\).  Each resulting bit of a
linear expression reduces, by additivity and scalar pullback, to
a bounded number of specified bits of the initial assigned tables.
Use Lemma~\ref{ib:inner-table-prediction} on unflagged inputs and
public zero on flagged inputs.  Independent equality trials make
the chance of missing a false equality arbitrarily small; an exact
equality fails only if one of its input-bit predictions fails.
The requested final output bit is handled in the same manner.

The order of choices prevents circular error bookkeeping.  First
fix the row rounds and the complete finite symbol-level tree.
Next choose flag and equality-detection budgets for its bounded
number of possible decisions.  This fixes the number of
initial-bit predictions needed in each tested expression.  Then
choose their smaller error budgets and their repetitions.  Given
successful flags, every later specified bit has the uniform
worst-request guarantee of the previous lemma conditional on prior
answers, using fresh splits after all scalar pullbacks.  A union
bound, or conditioning successively on the preceding correct
symbol-level decisions, bounds total simulation error by
\(\sigma\).  We do not choose the same budgets by union-bounding
over the later amplified bit tree.  After all choices that full
tree is nevertheless finite and bounded.  When a promise fails,
the same finite comparisons and fallback-zero rules still define
an output; only correctness is relinquished.
\end{proof}

\subsection{Arbitrarily accurate average correction}
\label{ib:average-correction-section}

For a point \(Y\) request its \emph{outer index} means its row
edge together with its retained point; for a line \(Y\) request
it means its edge and retained ordered line.  For \(Z\) it means
the \emph{joint pair} of edge/point indices.  A request law has
density at most \(C_{\rm req}\) within its block when its outer
index probability is at most \(C_{\rm req}\) times the corresponding
uniform probability.  Every other requested mask, form, or multiplier
may be an arbitrary public function of the request.

Call a binary word \emph{structurally canonical} if it is obtained from
the canonical recipes of Lemma~\ref{lem:inner-algebra-interface} at a
specified frame, mode, progress, and arbitrary Boolean tape, followed by
the point, line, row-extension, product, and repetition constructions
above. The tape need not satisfy the final validity conditions. Such a
word nevertheless satisfies the row-code, covariance, and multilinear
mask identities and has exactly the prescribed product tables.

\begin{proposition}[Average correction from one fixed radius]
\label{ib:average-correction}
For every fixed template there is \(\rho>0\) such that the following
holds for all sufficiently large sizes.  Let a received balanced
word \(w\) satisfy \(\dist(w,\mathcal W)\le\rho\), where
\(\mathcal W\) is structurally canonical; validity of its label
is not needed here.  For every fixed \(C_{\rm req}<\infty\) and
\(\eta>0\), a bounded-bit algorithm predicts requests with outer
density at most \(C_{\rm req}\) with average error at most \(\eta\).
It has smooth potential raw reads on every fixed hypothetical answer
branch.  The radius is independent of \(C_{\rm req}\), \(\eta\),
the row-round count, and the subsequent amplification count.
\end{proposition}

\begin{proof}
Use the constants \(\nu_1,\nu_2\) already chosen, and put
\(\zeta=\min(\zeta_*,\zeta_{**})\). Use the column and inner-table
descriptions in Section~\ref{ib:uncertainty}: rows are edges for \(Y\) and edge pairs
for \(Z\). Call an initial row position uncertain if its received inner
table is more than \(\nu_1\) from its reference table.

Each block has error fraction at most \(b_\#\rho\).  If
\(u_{\rm unc}\) is the uncertain edge fraction in a uniform
column, or pair fraction for \(Z\), then
\begin{equation}
 \E u_{\rm unc}\le\frac{b_\#\rho}{\nu_1}.
 \label{ib:uncertain-average}
\end{equation}
Call columns with \(u_{\rm unc}>\zeta\) bad.  Their fraction is
at most \(b_\#\rho/(\nu_1\zeta)\).  On every other column,
choose enough rounds in the support lemmas to leave uncertainty
fraction at most \(\delta'>0\), whose value will be fixed later.
No residual-support claim is made on bad columns.

For a \(Y\) request, independently split all its slot masks,
including the line coefficient masks, and expand.  Keep its
specified Hadamard form as the inner bit request.  For each expanded
term choose independent uniform \(h\in\mathcal H\) and replace
the row/outer pair by
\begin{equation}
 (i,g,I)\longmapsto(i,gh,h^{-1}I).
 \label{ib:random-shift}
\end{equation}
Covariance preserves the true value.  For a \(Z\) request split
both slot-mask lists and \(Q\), expand, and shift its two row/outer
indices independently by \eqref{ib:random-shift}.  Keep the
specified multipliers as its inner bit request.  Let \(T_0\) be
a fixed common upper bound on the expansion count, determined by
the template, before any accuracy choices.

For each fixed original request, every term has a uniform column:
the relevant \(\mathcal H\)-action is transitive and all its
column-mask axes have been independently randomized.  In \(Z\)
the two independent shifts give the uniform pair of outer points.
Choose the radius once and for all so that
\begin{equation}
 \frac{T_0b_\#\rho}{\nu_1\zeta}\le\frac1{64}.
 \label{ib:fixed-radius}
\end{equation}
Then, for every individual request, the chance that any term uses
a bad column is at most \(1/64\).  This conclusion does not assume
that the shifted row is independent of the shifted outer point.

For every fixed shift, \eqref{ib:random-shift} is a permutation
of the complete row/outer index; the independent pair of shifts
is a permutation of the complete \(Z\) index.  Thus averaging
over an outer-density-\(C_{\rm req}\) request law preserves that
density bound.  Expanded column masks are uniform conditional on
the indices.  The mean chance of hitting the good-column residual
support in any of the terms is consequently at most
\(T_0C_{\rm req}\delta'\).  By Markov's inequality, outside an
exceptional fraction at most \(8T_0C_{\rm req}\delta'\) of
original requests, the conditional sum of those residual-hit
probabilities is at most \(1/8\).

Bit-simulate every term's row cone by
Lemma~\ref{ib:bit-simulation}, with total simulation error at most
\(1/64\).  On any nonexceptional request, the full randomized
XOR trial therefore errs with probability at most
\[
 \frac1{64}+\frac18+\frac1{64}=\frac5{32}<\frac14.
\]
Cone-specific assignments are sufficient because the support bounds
cover every consistent assignment in each cone.  Choose
\(\delta'\) so that \(8T_0C_{\rm req}\delta'\le\eta/2\).
This determines a constant row-round count.  Repeat the entire
trial independently at the fixed request, including shifts, flags,
and all predictions, and take majority.  A bounded number of
repetitions makes its error at each nonexceptional request at most
\(\eta/2\).  Adding the exceptional fraction proves average error
at most \(\eta\) from the fixed radius \eqref{ib:fixed-radius}.
This proves an average guarantee, and makes no assertion of
arbitrarily accurate correction at every prescribed outer index.
Smoothness is proved in the next lemma.
\end{proof}

\begin{lemma}[All hypothetical read slots are smooth]
\label{ib:all-branch-density}
For the tester, the corrector of
Proposition~\ref{ib:average-correction}, and the finite linear
compositions used below, use their stated bounded outer-index request
laws, with joint pair density for \(Z\). Freeze any hypothetical sequence
of hidden answer bits.  Under the original, unconditioned public random tape
and request law, the raw-read subdistribution of every potential
slot has bounded density relative to its balanced input block.
After all accuracy counts have been fixed, the sum of these bounds
over the complete finite answer tree is a constant independent of
the size.  Defaults, wrong flags, and wrong equality answers satisfy
the same assertion.  Public numerical thresholds that only select
among these fixed schedules do not change the bound.
\end{lemma}

\begin{proof}
First enumerate every potential initial row-cone \emph{path slot},
including all candidate uses, before deduplication.  A term's
initial covariance shift is the joint permutation
\eqref{ib:random-shift}.  A fixed row-star path subsequently changes
the edge label among finitely many possibilities and multiplies
\(g\) on the left by a fixed group element.  A right-star move
from label \(i\) to \(j\), for example, uses
\(g\mapsto u_j^{-1}u_i g\).  Each such map is a permutation
within a fixed edge-label class, so costs only a fixed label factor
in a density bound.  In a two-axis cone the same reasoning applies
separately to each axis and preserves the joint pair bound.
Consequently each enumerated path slot has a fixed density bound.

Deduplication, representative selection, and omission of unused
terms only decide whether an already enumerated slot is read.
Its resulting unconditioned subdistribution is dominated by the
original slot distribution.  We never condition and renormalize
on the event that this slot was chosen as a representative or that
its flag had a specified outcome.  Summing the bounds of all
potential slots, including unexecuted ones, bounds their complete
occurrence count.  A flagged, failed, or unused zero value is public
zero and requires no placeholder query at a special address.

On a frozen symbolic branch, all information-set interpolation
matrices depend only on \(H_L,H_R\) and public graph indices.
The relevant inverses exist by their maximal-minor property.
The coefficients of all selected linear expressions, including
the compositions of \eqref{ib:two-actions}, can therefore be
computed publicly.  Hidden work is only the bounded equality and
output-bit calls described in Lemma~\ref{ib:bit-simulation}.
Their specified forms and multiplier pairs may depend on every
frozen answer, public coefficient, and preceding random choice.
Fresh splits \emph{after} these pullbacks give uniform raw inner
forms or jointly uniform multiplier pairs conditional on all that
information.  Zero coefficients can be split as well, or omitted.
Fresh uniform copies give the required copy marginal.  Wrong
answers change the selected expression, but introduce no new type
of row-cone path.

The tester's schedules were explicitly smooth before replacements.
The neighbor compositions below use finite lists of commuting
affine column permutations and changes between fixed block types,
so the same argument adds only fixed type factors.  Dropping a
public operand takes a marginal of a joint law, which also
preserves a density bound.  Finally freeze all simulation and
amplification counts, unfold the full raw bit tree, and sum these
finitely many bounds.  This tree may be large, but is constant at
the chosen accuracy.  A later threshold offset that selects its
branches changes neither its slot count nor these public address
maps.  The assertion concerns the unconditioned schedules for
all branches, not correctness on unsuccessful branches.
\end{proof}

The request statement also applies to a subdistribution within a
type: pad its missing mass by the uniform law, which increases a
density bound by at most 1, and charge the original subdistribution
by the resulting error bound.  A bounded list of type or branch
choices is handled by adding their bounds and error budgets.
This does not normalize on a rare branch.  Likewise, if a mixed
product has a public operand, a joint density
\(p(x,y)\le C/(\abs X\abs Y)\) implies
\begin{equation}
 \sum_y p(x,y)\le C/\abs X.
 \label{ib:public-marginal}
\end{equation}
The surviving \(x\) is the complete edge/point index, not only
its point.  The requested form
\(u\mapsto Q(\pi(b_1u),\pi(b_2v_{\rm public}(y)))\) may depend
on the discarded index; it does not select a new surviving index.
One may forget an actual repetition-copy index and choose a fresh
uniform copy in the corresponding one-port block.  This changes
only the fixed block-weight factor, not the original-index bound.

All public supports used here have polynomial cardinality for fixed
accuracy.  Groups, retained points and lines, representation fibers,
fields, masks, forms, and copy sets are polynomial-listable; only a
constant number of independent draws is used.  Their sampling laws
are rational.  Public indices, coefficient operations, and the
bounded Boolean answer tables can be enumerated in polynomial time.
Neither a hidden field symbol nor a hidden array is processed as one
unit-cost query.

\subsection{Neighbor bits and global port recovery}
\label{ib:neighbors-recovery}

\begin{proposition}[Local flags and encoded neighbors]
\label{ib:neighbors}
Within a sufficiently small fixed radius of a valid canonical word,
one can decide its incident-edge flags with arbitrarily high fixed
reliability, and supply its own bits and those of every existing
neighbor with arbitrarily small average error for bounded-density
requests.  The number of bit probes is bounded at each accuracy.
The complete hypothetical answer tree, including wrong decisions
and missing-edge defaults, satisfies
Lemma~\ref{ib:all-branch-density}.
\end{proposition}

\begin{proof}
Reduce \(\rho\) if needed to lie below half the distance of
Proposition~\ref{ib:tester}; a nearby valid label is then unique.
Use average correction at uniform systematic edge/point positions
to read the mode and the needed critical-bit projections of
Lemma~\ref{lem:inner-algebra-interface}.  At full progress the
all-ones contraction of the relevant singleton stack is a constant
polynomial equal to its critical Boolean read.  Use a specified
additive form taking 1 to 1; its actual raw queries are smoothed by
the corrector.  Such uniform systematic requests have bounded
outer density even though the logical read itself has a sparse
public address.

Compare the progress marker with its public empty and full tensors
by uniform separable masks and bit forms at systematic points.
On the true constant tensors a false equality has a fixed gap by
successive nonzero contractions.  Corrected probes and sufficiently
many repetitions distinguish these cases with arbitrary fixed
reliability.  Together with mode and the full-progress critical
bits, including the wrap predicate, these are exactly the finite
decisions required by the path transition rules.  Explicitly, a
pre successor increases progress unless full; at full it performs
the enabled edit or has a missing wrap edge.  A post successor
decreases progress unless empty; at empty it advances the frame.
For predecessors, full post reverses the edit or has the missing
wrap edge, other post reverses a decrease, positive-progress pre
reverses an increase, and empty pre reverses a frame advance.
An increase also distinguishes empty progress for initialization.
There is no scan of the full progress index and no proposed-tape
validity test.

Fix the true Boolean transition branch of an existing edge.  The
algebraic interface expresses every updated contraction as a
bounded sum of public terms and public scalar multiples of old
contractions at invertible \(K\)-affine pullbacks of the common
target.  Slot pulls and scalars are independent of the target point
and row.  The pullbacks commute with the rectangular action, so
they commute through the common row encoder as column permutations.
Public initialization terms have publicly computable actual
parity values.  Thus each requested point \(Y\) bit is an XOR
of boundedly many input \(Y\) requests with specified masks and
forms at bounded-density indices.  For line \(Y\), the affine
map preserves the line parameter, so the identical argument applies
to the separable coefficient contractions
\eqref{ib:line-bit}.  The derivative update uses the separately
stored child and its summed new-axis stack; it does not recompute
native packing, polynomial division, or coefficient extraction
from a requested line or parity symbol.

For a \(Z\) bit insert the two \(Y\) update sums into
\eqref{ib:product-bit} and expand by bilinearity.  Two variable
terms give an available input \(Z\) block by tag compatibility;
absorb public coefficients into multipliers or masks.  The two
row/point coordinates undergo separate affine column permutations,
with their rows fixed.  This preserves their joint density even
when the two input types coincide: their two positions are still
distinct coordinates of the product domain, never identified into
a diagonal.  A term with one public operand is an arbitrary
specified binary linear form of the other \(J\)-symbol, and
\eqref{ib:public-marginal} gives the necessary complete surviving
row/point bound.  Here the public operand is its actual encoded
\(J\)-symbol, including parity values, not an unencoded target
value.  Two public operands give a public bit.

Apply Proposition~\ref{ib:average-correction} to every needed term
with a suitably smaller accuracy.  Each output type and fixed
branch has a bounded list of transformations, including radix
pulls and zero terms.  Charge their error and density bounds by
their sum before selecting a term or branch.  Charge the finite
decision errors separately, and analyze each relevant correct
branch with its true decisions fixed before intersecting the good
events.  Wrong decisions still select from the same public lists;
missing or unused values are public zero.  Therefore the
all-hypothetical-branch smoothness assertion follows from
Lemma~\ref{ib:all-branch-density}.
\end{proof}

\begin{proposition}[Deterministic global read-port recovery]
\label{ib:global-recovery}
At a sufficiently small fixed balanced distance from a canonical
word, every order-one logical read port can be recovered exactly
in deterministic polynomial time.  In particular this applies to
the one-port template.  No constant-query claim is made for this
global decoder.
\end{proposition}

\begin{proof}
At every systematic edge/point position in the port's order-one
point block, read the entire Hadamard table including copies and
choose a nearest \(J\)-symbol, breaking ties publicly.  Whenever
the table error is less than \(1/4\), the selected symbol is true.
Markov's inequality bounds the fraction of incorrect systematic
symbols by a fixed multiple of balanced word error, accounting for
the block weight and the systematic row fraction.

Map these symbols to their represented target arguments using the
known row action.  All fibers have equal size: for a fixed target
\(y\), any full-rank \(A\) and any \(z\) determine the unique
offset \(y-Az\), and the number of choices is the same for every
row group.  Take a plurality in each fiber.  Every wrong plurality
requires at least half its fiber's symbols to be wrong, so this
loses at most a factor two.  Project outputs outside the subfield
to any fixed \(\F_q\)-estimate, which cannot spoil a correct
symbol, and take another plurality over ignored target coordinates.
After reducing the fixed radius, we have an estimate
\(\widehat f:\F_q^\ell\to\F_q\) of the native order-one
polynomial with symbol error at most a sufficiently small fixed
\(\alpha>0\).

For each desired grid point \(x\), enumerate all directions
\(z\in\F_q^\ell\).  Along the \(q\)-word
\((\widehat f(x+tz))_{t\in\F_q}\), the expected number of
errors over \(z\) is at most
\(1+(q-1)\alpha\): every nonzero \(t\) makes \(x+tz\) uniform.
Let
\[
 t_{\rm RS}=\left\lfloor\frac{q-D-1}{2}\right\rfloor.
\]
For sufficiently small fixed \(\alpha\) and large size, Markov's
inequality shows that a strict majority of directions have at most
\(t_{\rm RS}\) errors, since \(D/q\to0\).

Use the Berlekamp--Welch reconstruction method in the linear-system
presentation of Gemmell and Sudan~\cite[Appendix~A]{GemmellSudan1992}.
Specifically, uniquely decode each line by solving the linear equations
\[
 N(t)=E(t)\widehat f(x+tz)\quad(t\in\F_q),\qquad
 E\text{ monic of degree }t_{\rm RS},\quad
 \deg N\le D+t_{\rm RS}.
\]
On a good direction, the error locator, padded by any monic factor
to degree \(t_{\rm RS}\), provides a solution with
\(N=Ef_{\rm line}\).  Every solution has this equality: the
difference \(N-Ef_{\rm line}\) has degree at most
\(D+t_{\rm RS}\) and vanishes at at least
\(q-t_{\rm RS}>D+t_{\rm RS}\) correct positions.
Divide, verify degree and agreement, and use public zero on a
failure.  The strict majority of returned values at parameter zero
is exactly \(f(x)\).  Repeat for all port grid indices.
All domains, tables, directions, fields, and linear systems here
are polynomial-size, so the complete algorithm is deterministic
polynomial time.
\end{proof}

\subsection{The binary interface}

\begin{lemma}[Inner binary interface]
\label{lem:inner-binary-interface}
For the inner algebraic families of
Lemma~\ref{lem:inner-algebra-interface}, there is a deterministic
polynomial-time binary encoding with the following properties,
uniform over the circuits in each fixed padded template.
\begin{enumerate}[label=(\roman*)]
\item Its balanced blocks are the point and line \(Y\)'s and the
compatible point-product \(Z\)'s above.  Canonical valid extended
labels have constant distance, and admit the perfectly complete
bounded-bit tester and fixed-gap proximity conclusion of
Proposition~\ref{ib:tester}.
\item One radius \(\rho>0\), fixed by the template before any
request accuracy or density, permits every fixed average error
\(\eta>0\) for requests of any fixed bounded outer-index density.
For \(Z\), the requirement is a joint pair-density bound.  Other
masks, multipliers, and forms may be arbitrary request-dependent
ones.  Finite type subdistributions are charged without
renormalization.  Mixed products with a public operand may be
reduced to arbitrary linear forms using
\eqref{ib:public-marginal}.
\item In a small fixed neighborhood of a valid label, incident
flags and current or existing-neighbor encoded bits have the
bounded-query reliability and averaged accuracy in
Proposition~\ref{ib:neighbors}.  Every routine is total with
bounded decisions and public-zero defaults.
\item After all fixed accuracy counts are chosen, all hypothetical
raw read slots in the full answer tree have a finite summed
density bound as in Lemma~\ref{ib:all-branch-density}.  This
includes potential cone paths before deduplication, unsuccessful
flags and comparisons, and arbitrary pulled-back forms.  All
coefficient and address computations are public, all remaining
hidden processing uses boundedly many bits, and all random
supports and rational sampling tables are polynomial-time
constructible.
\item The common row encoder, field presentations, positive-tag
original layouts, and fixed-label formulas are identical in
corresponding actual and one-port families.  Repetition copies
may differ and are sampled afresh in the receiving family.
At a fixed distinguished source or its first successor, a
positive \(Y\) bit has Boolean degree at most three in that
port's read input, and a same-positive-tag \(Z\) bit has degree
at most six.  An order-one port has the deterministic polynomial
global recovery of Proposition~\ref{ib:global-recovery}.
\end{enumerate}
In particular, the radius for the one-port template can be fixed
before a later actual port-count bound is chosen.  The actual
template's own radius may depend on that bound through its number
of blocks.
\end{lemma}

\begin{proof}
The construction and the preceding propositions prove
(i)--(iv) and global recovery.  For (v), field and row-code choices
were made solely from the common point parameters.  The positive
charts, including per-primary verification auxiliaries and their
tagged descendants, have the literal common-index ancestry of
Lemma~\ref{lem:inner-algebra-interface}.  Columnwise linear
extension and fixed-parameter line restriction preserve that
identity.  Passing through an original index and choosing a fresh
copy therefore suffices even when balanced multiplicities differ.

At either of the two specified fixed labels, legacy, work, frame,
mode, and progress are public, and every positive persistent recipe
has Boolean degree at most three in its port input.  Public matrix
operations, division by fixed monic polynomials, coefficient
extraction, slot contraction, fixed affine substitutions, line
restriction, \(J\)-linear row extension, coordinate changes, and
additive bit forms are all \(\F_2\)-linear in these hidden-data
functions.  None raises that degree.  The one additional binary
bilinear operation in \eqref{ib:product-bit} gives degree at most
six.  Finally \eqref{ib:fixed-radius} uses only the template's
\(T_0,b_\#,\nu_1,\zeta\); the later requested density and
accuracy affect only row rounds and repetitions.  Applying it
first to the fixed one-port template proves the last assertion.
\end{proof}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/08-inner-binary.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/09-inner-gadget.tex ====
\section{The fixed port and its robust decision gadget}\label{sec:inner-gadget}

We now prove Theorem~\ref{thm:port-gadget}. Throughout this section all
fields, groups, block lengths, and path labels are those of the inner
construction of Sections~\ref{sec:inner-algebra} and~\ref{sec:inner-binary}.
In particular, its polynomial-size affine group is unrelated to the outer
index group. We first construct the port independently of the eventual
arity. We then fix an actual arity and construct a generalized subcircuit
for one Boolean decision. The subcircuit will use the geometric procedures
of Lemma~\ref{lem:geometric-procedures}, whose proof is independent of this
section. Word distances in this section are relative Hamming distances;
vector distances use the displayed normalized block norm.

\subsection{A port fixed before the arity}

Fix the exponent allowance $c_{\rm allow}$ and the common field, point,
slot, and row-code parameters supplied by
Lemmas~\ref{lem:inner-algebra-interface} and
\ref{lem:inner-binary-interface}. In particular these parameters are
chosen from $n'=(L+2)^{c_0}$, with fixed $c_0>c_{\rm allow}+5$, before
specifying the number of input ports or the actual Boolean circuit.
Construct the one-port instance of the inner path with the prescribed
initial copy phase and, afterwards, a fixed trivial computation. Include
all its positive-tag validity, critical-address, derivative, and
verification-auxiliary families, as well as its tag-zero families. This is
the \emph{dummy template}. Write $a_s$ for its distinguished source and
$a_s^+$ for its immediate successor. Their common tape has blank work,
stage-zero legacy, input $s$, frame $\gamma^{-1}$, and post mode. Their
progress values are respectively $q-1$ and $q-2$. Define
\begin{equation}\label{ig:port-definition}
 \mathcal D(s)=\bigl(\mathcal W_{\rm dum}(a_s),
                         \mathcal W_{\rm dum}(a_s^+)\bigr),
 \qquad S_0=2m_{\rm dum}.
\end{equation}
Both halves use the entire balanced binary word of the same dummy template.

\begin{proposition}[Fixed dummy port]\label{prop:dummy-port}
The map in~\eqref{ig:port-definition} has polynomial length, constant
relative Hamming distance $d_{\mathcal D}>0$, and the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every output bit is a multilinear polynomial in $s$ over $\F_2$ of
 degree at most six. Its coefficients, and the whole canonical encoding,
 are computable deterministically in time polynomial in $L$.
\item There is a perfectly complete tester using a bounded number of bit
 reads and bounded Boolean processing. Its public randomness is uniformly
 enumerable with polynomial-size support and rational probabilities of
 polynomial bit length. Every fixed branch has unconditioned read
 subdistributions of bounded density relative to $[S_0]$. There is a
 constant $C_B$ such that proximity $a$ to a port word implies rejection
 at most $C_Ba$. For every fixed $a>0$, there is a fixed
 $\theta_B(a)>0$ such that rejection less than $\theta_B(a)$ implies
 proximity less than $a$ to some $\mathcal D(s)$, for all sufficiently
 large $L$.
\item There are a deterministic polynomial-time exact decoder at a fixed
 positive Hamming radius and a fixed
 $0<\nu_B<d_{\mathcal D}/4$. From any word within $\nu_B$ of
 $\mathcal D(s)$, either half supplies bits of its canonical word with
 any fixed requested average accuracy under any fixed bounded-density
 law on the original edge and outer indices. For a product block this
 law is on the joint pair of indices. Requested masks and additive
 forms may depend arbitrarily on the public request. The read procedures
 have unconditioned fixed-branch smoothness, including their unsuccessful
 branches.
\end{enumerate}
The dummy template, its decoder radius, and $\nu_B$ are all fixed before
any later actual arity, circuit-size factor, or correction-round count.
They are not decreased when those constants increase.
\end{proposition}

\begin{proof}
The inner distance assertion applies separately to each half. Distinct
inputs give distinct labels already in the order-one input track, so it
gives the claimed distance for their concatenation.

For testing, mix the ordinary valid-label testers on the two halves with
the following additional checks. At systematic point probes compare the
legacy chart with the public stage-zero polynomial, the mode with post,
and the progress marker with its specified one-hot tensor, using the
source or successor progress value as appropriate. Uniform slot masks
and additive binary forms give smooth bit versions of these checks; a
specified form can also be split into fresh uniform summands. On exact
valid labels, an incorrect location condition has a fixed positive
rejection gap by the low-degree or constant-tensor distance. Correct
conditions force the blank-work tape, the specified progress and mode,
and frame $\gamma^{-1}$. Finally compare the two order-one input point
tracks at identical original systematic edge/point indices and identical
uniform bit forms, choosing copies independently in the two halves.
Different inputs give a fixed positive discrepancy gap. The two reads
have smooth marginals individually; they do not require a product block.
Each mixture weight is a fixed positive constant.

Here is the fixed-gap implication explicitly. Let $g>0$ be a common lower
bound for the rejection of any violated location or equality requirement
on exact pairs of valid labels, with the mixture weights included. Let
$C$ bound the change in these tests' rejection under changes to a fraction
of the whole port. Given $a>0$, choose
\[
 0<a_0<\min\{a,g/(4C)\}.
\]
Ordinary soundness supplies a positive rejection threshold forcing each
half within $a_0$ of an exact valid label. Take the total tester threshold
small enough to imply both of these assertions and also smaller than
$g/2$. The pair of nearby exact labels is at distance less than $a_0$
from the received port. If any required exact condition were wrong,
smoothness would make the received rejection at least
$g-Ca_0>g/2$, a contradiction. Thus the exact pair is precisely
$\mathcal D(s)$ and the received distance is less than $a$. This defines
$\theta_B(a)$. Perfect completeness and the upper bound $C_Ba$ follow by
charging every read whose answer differs from the exact port word. All
choices and computations have the bounded, polynomially enumerable
properties asserted in the two inner interfaces.

At the two fixed labels all logical inputs except $s$ are public.
Canonical persistent values and their coordinate coefficients have
logical degree at most three, even on invalid Boolean tapes. Their
verification certificates use public linear maps on one primary array:
multiplication by the public test matrix, summation, division by a public
monic polynomial, coefficient extraction, and interpolation. These
operations preserve degree. Fixed-progress stack formation, restriction
to a point or a line, line-coefficient extraction, slot contraction, row
extension, and additive bit forms are linear over the prime field. Thus a
$Y$ bit has degree at most three. A $Z$ bit is a binary bilinear form of
two such operands and has degree at most six. Repeating bits for balance
changes nothing. The degree bound is independent of the allowance and
of all future template choices.

For completeness, the coefficient algorithm need not manipulate a
large symbolic expression. For a specified bit function $b$, evaluate
the canonical dummy word on $1_T$ for all $T\subseteq[L]$ with
$|T|\le6$. Its coefficient at the squarefree monomial indexed by $A$ is
\[
 c_A=\sum_{T\subseteq A}b(1_T)\quad\text{in }\F_2,
 \qquad |A|\le6.
\]
Boolean multilinear inversion and the degree bound give every
coefficient; there are $O((L+1)^6)$ evaluations, each polynomial-time.

Use the deterministic order-one recovery from
Lemma~\ref{lem:inner-binary-interface} on the first half. A port error
fraction $r$ causes an error fraction at most $2r$ there, so a fixed
sufficiently small $r$ gives exact recovery of $s$. Finally choose
$\nu_B<d_{\mathcal D}/4$ with $2\nu_B$ inside the dummy average-correction
radius and, if necessary, the just established recovery radius.
A port within $\nu_B$ corrupts either half by at most $2\nu_B$.
The dummy corrector therefore answers the stated requests from that
half; its read density relative to the whole port loses only a factor
two. Its radius is independent of the requested accuracy and density
constant. Every constant used here belongs to the one-port template.
\end{proof}

\subsection{Exact original-index correspondence and virtual source bits}

Now fix an actual arity $k$, a circuit-size factor, and a Boolean circuit
$f$ covered by Theorem~\ref{thm:port-gadget}, padding unused inputs with
public zero ports. Fix words $y_i$ with
\begin{equation}\label{ig:fixed-ports}
 \dist\bigl(y_i,\mathcal D(s_i)\bigr)\le\nu_B.
\end{equation}
For a public zero port all bits of $\mathcal D(0^L)$ are computed publicly.
The true graph used below is the single actual-template path for the
fixed tuple $s=(s_1,\ldots,s_k)$, rather than the union over possible
input tuples. Write $w_*(s)$ and $w_*^+(s)$ for its source and immediate
successor. These have the same public frame, tape locations, mode, and
progress as their dummy counterparts, but their input-dependent encoding
bits must be supplied from the ports.

\begin{lemma}[Source correspondence and virtual access]\label{ig:virtual-source}
For either $w_*(s)$ or $w_*^+(s)$, every tag-$i$ point or line $Y$
operand agrees with its dummy counterpart for $s_i$ at the same original
indices, after renaming the port tag. The same is true of a $Z$ block
whose two operands both have tag $i$. Tag-zero data at either label are
publicly computable. Consequently source and successor encoded-bit
requests with bounded original-index density admit arbitrarily accurate
average answers from~\eqref{ig:fixed-ports}, with smooth external reads.
For mixed products the requisite density is joint density on the complete
edge/point indices before marginalization. Actual and dummy balanced-copy
multiplicities need not agree.
\end{lemma}

\begin{proof}
Apply Lemma~\ref{ia:source-data} at the two labels in
\eqref{ia:source-successor}. Both have frame $\gamma^{-1}$, blank work,
stage-zero legacy, and post mode; their progresses are respectively
$q-1$ and $q-2$. At either progress the lemma identifies each positive
algebraic array with its one-port counterpart at the same native point,
slot, and mask indices. This includes verification auxiliaries and all
derivative descendants: Lemma~\ref{ia:port-ancestry} fixes their separate
call layouts independently of work destinations, the number of ports,
and the actual circuit. The placements and fixed-progress stack formulas
are common as well.

The common row field, row matrices, systematic extension, and point/line
domains were fixed before $k$. Applying these same linear operations
preserves the equality at the original encoded indices, for the source
and its first decrement separately. The actual template may balance its
block types with different copy counts. Forgetting its copy index and
choosing a fresh uniformly random dummy copy preserves the original index
and costs only the fixed actual block-weight factor.

The same source-data lemma makes every tag-zero array public at these
labels. Its conclusion includes derivatives evaluated on public shifted
work, even when the shifted Boolean tape is invalid: canonical certificate
recipes are defined there and do not solve a final validity equation.
Thus no work-side certificate needs to evaluate $f(s)$ or a stored sum
with a port-dependent primary. The actual encoded $J$-symbols of these
public arrays, including their parity values, are computed with the
actual template's common row encoder.

The only remaining case is a product with tags $(i,0)$ or $(0,i)$.
Write $x$ for the positive operand's entire original edge/point index and
$z$ for the public operand's entire index. After the indicated contractions
and public scalar multipliers its bit is, in one orientation,
\begin{equation}\label{ig:mixed-form}
 Q\bigl(\pi(b_i u(x)),\pi(b_0 v_{\rm pub}(z))\bigr)
       =\mu_{z,b_i,b_0,Q}\bigl(u(x)\bigr),
\end{equation}
where $\mu_{z,b_i,b_0,Q}$ is a specified additive binary form. The dummy
$Y$ corrector permits precisely such arbitrary forms, including dependence
on both requested positions and all the public coefficients. It retains
$x$ at the same original index. If the request law satisfies
\[
 p(x,z)\le\frac{C}{|\mathcal X|\,|\mathcal Z|},
 \qquad\text{then}\qquad
 \sum_zp(x,z)\le\frac{C}{|\mathcal X|}.
\]
This proves the required density for the dummy $Y$ request on both its
edge and its point. A same-positive-tag $Z$ request retains both original
indices and their joint density; even when its types coincide, no pair
is replaced by a diagonal. A $(0,0)$ product is public. There are no
products of two distinct positive tags. Proposition~\ref{prop:dummy-port}
now supplies all the answers. Fixing a source/successor choice, orientation,
or block type costs a fixed factor. All error and density bounds are
summed before any later answer-dependent selection; no density bound is
conditioned on that selection.
\end{proof}

\subsection{Validation of the true path and numerical access}

Let $m=m_{\rm in}$ be the length of one actual-template word, and let
$b_\#$ be its number of equally weighted original binary block types.
The path labels have fixed separation $d_*>0$ in the normalized Euclidean
norm. Constants from this point may depend on the actual arity,
circuit-size factor, and desired output-loss fraction $\upsilon>0$.
They will not alter $\mathcal D$ or $\nu_B$.

The ordinary binary tester recognizes the union of paths over all input
tuples. Before using a seed for geometry, we must ensure that its nearby
label belongs to the path for the externally supplied tuple $s$.
We do this by comparing each input track with its corresponding port.

\begin{lemma}[Validation at the fixed external tuple]\label{ig:validation}
There are fixed $\rho_{\rm in},\rho_{\rm in,comp}>0$ and bounded-read
validation routines for an internal binary seed $z\in\{0,1\}^m$ such
that, with arbitrarily small prescribed failure probability, acceptance
implies distance less than $\rho_{\rm in}$ to a unique label on the
true-$s$ path, and distance less than $\rho_{\rm in,comp}$ to a true-path
label implies acceptance. These radii are chosen before the reliability
of the routines. For each fixed seed that can be accepted on good
decisions its reference label is fixed independently of the decision
randomness. Current and actual incident-neighbor bits and existence flags
can subsequently be supplied with arbitrary fixed average accuracy for
the request laws of Lemma~\ref{lem:geometric-procedures}.
\end{lemma}

\begin{proof}
Choose $\rho_{\rm in}$ within the whole-word uniqueness and actual-template
correction radii. Estimate rejection of the ordinary tester for the union
of all port tuples. For each port $i$, also sample the order-one point
track $W_{s_i}$ of $z$ at uniform systematic original edge/point indices
and uniform forms, and compare it with the bit at the identical original
index of the corrected dummy source half of $y_i$. Choose a random actual
copy for the direct read. This input track is unrotating and uses the
same extension at every actual path label.

On an exact valid label the ideal mismatch is zero if its $i$th port is
$s_i$, and at least a constant $p_s>0$ otherwise: the nonzero order-one
polynomial differs on a constant fraction of represented arguments, and
a uniform additive form detects a nonzero difference with probability
$1/2$. We can choose one $p_s$ for the fixed templates. A balanced error
fraction $a'$ in $z$ changes a comparison rate by at most $C'_k a'$,
where $C'_k$ includes the systematic and block-weight density factors.
Choose
\[
 0<a'<\rho_{\rm in},\qquad C'_k a'<p_s/8.
\]
Use half the ordinary soundness threshold for $a'$ as the rejection
cutoff, and $p_s/2$ as each mismatch cutoff. Good statistics with fixed
positive slack imply that an accepted seed is $a'$-close to an exact
valid label. The mismatch gap then forces every port of that label to
be $s_i$. Conversely, smoothness of the ordinary tester and of the
comparison probes makes all rates lie strictly below their cutoffs when
the seed is within a sufficiently small fixed
$\rho_{\rm in,comp}>0$ of a true-path label.

First fix sample counts that concentrate the underlying ideal rates at
the required slack; independent samples and Chebyshev's inequality
suffice. Next choose the dummy correction accuracy so that the bounded
number of supplied comparison bits fail with the desired total
probability. This order leaves $a'$, $\rho_{\rm in,comp}$, and $\nu_B$
unchanged. On good acceptance, the ordinary correction and neighbor
procedures of Lemma~\ref{lem:inner-binary-interface} apply to the fixed
unique reference. They return the true incident flags, including the
disabled wrap flag, and true encoding bits at existing neighbors with
arbitrary average accuracy. Analyze each possible reference of the fixed
seed before intersecting with acceptance; one never conditions a
smoothness estimate on that event. On unsuccessful branches every routine
still has a bounded Boolean output or a fixed default.
\end{proof}

The point-vector space for the gadget is
\begin{equation}\label{ig:point-space}
 \mathcal B=[-1,2]^{4m},\qquad
 \|U\|_{\rm b}^2=\frac1m\sum_{i=1}^{4m}U_i^2.
\end{equation}
Among its first-block coordinates select the systematic point $Y$ bits
of the constant output track $F_{\rm out}$, with forms $\mu$ satisfying
$\mu(1)=1$, including all their balanced copies. At the unique sink all
these bits equal $f(s)$. At least half the row labels are systematic,
half the forms take $1$ to $1$, and the track has weight $1/b_\#$.
Consequently the selected fraction among all $4m$ coordinates is at least
\begin{equation}\label{ig:selection}
 c_{\rm sel}=\frac1{16b_\#}.
\end{equation}
Choose the geometric parameter $\tau>0$ so small that
\begin{equation}\label{ig:tau}
 \frac{(5\tau)^2}{(2/5)^2}<\rho_{\rm in,comp},
 \qquad
 \frac{16(10\tau)^2}{c_{\rm sel}}<\frac{\upsilon}{4}.
\end{equation}
A binary disagreement after rounding a physical block at any threshold
in $[2/5,3/5]$ incurs squared error at least $(2/5)^2$ from the true bit.
Thus the first inequality gives the geometric completeness condition.
Obtain fixed $B_1,c_*,\beta_0>0$ from
Lemma~\ref{lem:geometric-procedures} and its geometric lower bound,
increasing $B_1$ to an integer if necessary. Write $\mathcal E$ for the
endpoint set of the true path. It excludes the top of the distinguished
vertical segment; here every member of $\mathcal E$ has first block equal
to the unique sink encoding. In particular,
\begin{equation}\label{ig:geometric-bound}
 \|v(U)\|_\infty\le B_1-1,
 \qquad
 \|\clip_{\mathcal B}(U+\beta v(U))-U\|_{\rm b}\ge c_*\beta
\end{equation}
whenever $\dist(U,\mathcal E)>10\tau$ and $0<\beta\le\beta_0$.

We verify the numerical access contract rather than presupposing access
to hidden circuit values. There are at most eight true word roles: each
of two seed roles and its two possible neighbors, and the special source
and successor. Four incident edge roles contribute four straight pieces
and three internal fillets each; with two possible seed junctions and
the special vertical and its bottom fillet there are at most 32 piece
roles. Lemma~\ref{ig:virtual-source} always supplies the special roles,
even when neither seed validates. The other roles use
Lemma~\ref{ig:validation} and the actual neighbor routines. Their requests
are full-word uniform samples for statistics or the coordinate matching
the requested field component. Condition on a fixed output block type,
or sum over the finite type list, to obtain the required edge/outer laws,
including joint laws for $Z$ blocks. Mode, progress, and critical decisions
use their own uniform systematic marginals.

Fix the geometric mesh, numerical tolerances, scalar sample counts, and
high-level role list first. Allocate calls for the possible roles and
matching components before any numerical minimum or branch is selected.
For a fixed input and rounding threshold, each seed has at most one
reference that good validation could accept, so there are only boundedly
many underlying actual candidates. Independent uniform-coordinate
samples concentrate all their bounded statistics, including physical
block statistics, without conditioning on which candidate wins. Set the
required bit and decision error budgets for these high-level calls next,
and only then fix the lower-level correction and repetition counts.
A numerical winner chooses a reserved role and mesh case; it cannot turn
a hidden answer into a new arbitrary word address.

Raw scalar samples can be quantized on fixed-width grids with public
random offsets. Their finite range and fixed widths give finitely many
answers. All public field, address, coefficient, mesh, reciprocal,
square-root, and trigonometric calculations are made during the
reduction, to the fixed required precision. A finite table indexed by
the bounded answers supplies the required numerical operations. The
procedures return bounded values on arbitrary unsuccessful trials as
well. Refining an offset grid later changes the thresholds and public
table entries, but not the bin count, word-address primitives, or number
of answer slots.

It follows from Lemma~\ref{lem:geometric-procedures}, with these explicit
oracles, that for $\kappa=1/100$ we can fix a finite unscaled estimator
whose signed output has the form
\begin{equation}\label{ig:ideal-estimator}
 \widehat v_i(U,y)=B_1(p_i^*-n_i^*),\qquad
 p_i^*,n_i^*\in[0,1]\text{ fixed-precision dyadics},\quad p_i^*n_i^*=0,
\end{equation}
and, whenever $\dist(U,\mathcal E)>10\tau$,
\begin{equation}\label{ig:ideal-error}
 4\E_{i,t}\bigl|\widehat v_i(U,y)-v_i(U)\bigr|^2
        <(c_*\kappa/100)^2.
\end{equation}
Here $i$ is uniform on $[4m]$, $t$ is the true public tape, and the
inequality holds conditional on each fixed admissible internal rounding
threshold and scalar offset. Rounding the signed output to its fixed
dyadic grid is included in the accuracy. All choices to this point are
independent of $\beta$ and of any external comparison margin.

\subsection{Full tapes, deterministic copies, and read routing}

We require bounds for every read actually wired into the circuit. Unfold
all hypothetical Boolean, validation, equality, and quantization answers,
assigning fresh public randomness to unexecuted nodes as well. After the
accuracy choices above this is a fixed finite tree. Let $H$ bound the
number of its raw comparison occurrences. A branch may select an
unsuccessful seed, a wrong flag, a wrong transition type, or a default;
these choices do not license a new addressing primitive.

More explicitly, list each potential initial row-cone path before
removing repeated positions. On a frozen symbolic branch the covariance
map $(g,I)\mapsto(gh,h^{-1}I)$ is a joint permutation; a further row-star
step is a fixed left multiplication of the edge coordinate, with a change
among boundedly many edge labels. On product blocks the two axes undergo
these transformations separately. Neighbor proposals use a finite list
of the affine pullbacks from the inner algebra. Omissions, deduplication,
and representative choices only select subdistributions of the listed
paths, without renormalization. A flagged zero is a public value and
needs no artificial dummy read at a concentrated address. Information-set
inverses and linear-expression coefficients are public functions of the
frozen branch and indices. Fresh additive splitting is performed after
all form or multiplier pullbacks, so arbitrary answer-dependent requested
forms still give the uniform inner marginals. These facts are exactly
the unconditioned guarantees of
Lemma~\ref{lem:inner-binary-interface}; they apply whether the answers
on the frozen branch are correct or not.

These inner routines satisfy the additional physical-read
hypothesis~\ref{geo:access-finite} of
Definition~\ref{geo:access-contract}. We therefore invoke part~(iv) of
Lemma~\ref{lem:geometric-procedures} at the physical-read level: each
completed inner bit or decision already has a bounded physical answer
tree with the stated unconditioned branch marginals.

For a fixed internal offset choice, sum these bounds over all potential
slots and geometric roles. With $i$ uniform on $[4m]$, the expected number
of reads of a specified internal base coordinate is at most $C/(4m)$,
and of a specified position of any one external port is at most $C/S_0$,
for a fixed constant $C$. Internal direct reads use uniform statistic
positions or matching output positions. Virtual external reads use the
original-index correspondence and the marginal calculation
in~\eqref{ig:mixed-form}. The constants $H,C$ are uniform over the
function and over the external threshold $\xi$ for the fixed bounds,
and remain unchanged by later offset-grid refinement. These are counts
on the whole tree, not smoothness conditioned on execution.

Choose finite uniform rational grids for the internal common binary
threshold $\xi_{\rm in}\in[2/5,3/5]$ and the scalar offsets modulo the
already fixed quantization widths. If $D_{\rm grid}$ is a lower bound
on their sizes, then for a fixed scalar value the probability of being
within $4\omega_{\rm in}$ of a binary threshold or one of its relevant
shifted boundaries is at most
\begin{equation}\label{ig:ambiguity}
 O\bigl(D_{\rm grid}^{-1}+C_{\rm width}\omega_{\rm in}\bigr),
\end{equation}
where $C_{\rm width}$ is fixed by the widths and finite range. Sum the
conditional slot-density bounds over all base positions at every offset,
and then over the bounded slot list. Thus the true-law fraction of cones
having any such ambiguous internal comparison is uniformly as small as
desired, for every fixed mean array, by refining the grids and decreasing
the positive rational $\omega_{\rm in}$. External comparisons are not
included here; their spoil budget is a separate hypothesis.

For each output coordinate $i$, the full tape law, including the offsets
and conditional random choices at all hypothetical nodes, has enumerable
polynomial support and rational probabilities of polynomial bit length.
A fixed product or finite tree of polynomial-size choices still has
these bounds. Choose a common square $R$ and replace each law by an equal
list of $R$ tapes with total variation at most
\begin{equation}\label{ig:tv}
 t_{\rm tv}=\frac{\eta_{\rm tv}}{\max\{4m,S_0\}}.
\end{equation}
Indeed floor $R$ times each probability and distribute the remaining
integer counts within the support. If $P(L)$ is a common support bound,
choosing a square $R\ge P(L)\max\{4m,S_0\}/\eta_{\rm tv}$ suffices.
A square within a fixed factor of this polynomial bound is available,
and the integer and rational operations cost polynomial time. The bound
and $R$ can be common to all padded functions in the stated class.
Use the $r$th listed tape at physical replica $(i,r)$ and put
\begin{equation}\label{ig:gscale}
 G_0=4mR.
\end{equation}
By bounded outputs, sufficiently small fixed $\eta_{\rm tv}$ preserves
from~\eqref{ig:ideal-error} the empirical bound
\begin{equation}\label{ig:empirical-error}
 \left(\frac1{mR}\sum_{i,r}
       |\widehat v_{ir}(U,y)-v_i(U)|^2\right)^{1/2}
       \le c_*\kappa/20,
\end{equation}
as well as the desired rare-event slack in~\eqref{ig:ambiguity}.
This estimate holds for every fixed $U$: the tape lists depend only on
the public laws, and total variation controls every bounded function of
the tape, including its error on that $U$.

For clarity, the unequal-size load calculation is
\begin{align}
 \#\{\text{reads of external position }j\}
 &\le G_0\left(\frac C{S_0}
            +\frac{H\eta_{\rm tv}}{\max\{4m,S_0\}}\right)
 \le (C+H\eta_{\rm tv})\frac{G_0}{S_0},
 \label{ig:external-load}\\
 \#\{\text{reads of internal base coordinate }j\}
 &\le G_0\left(\frac C{4m}
            +\frac{H\eta_{\rm tv}}{\max\{4m,S_0\}}\right)
 \le (C+H\eta_{\rm tv})R.
 \label{ig:internal-load}
\end{align}
For one position the count on a tape lies in $[0,H]$, so changing its
law by total variation $t_{\rm tv}$ changes its expectation by at most
$Ht_{\rm tv}$, which proves these inequalities. They hold even when
$m/S_0$ grows as an arbitrarily large fixed power of $L$.
Route the internal occurrences round-robin to the $R$ replicas of their
base coordinate. Every replica is then read a bounded number of times.
Every external occurrence is exposed separately for the caller to route;
\eqref{ig:external-load} proves the constant $C_{\rm occ}$ in
Theorem~\ref{thm:port-gadget}.

\subsection{An explicit generalized subcircuit}

Let $R=b^2$ and identify replica labels with $(\mathbb Z/b\mathbb Z)^2$.
On each base coordinate use the average $A$ of the eight labeled moves
\[
 (u\pm2v,v),\quad (u\pm(2v+1),v),\quad
 (u,v\pm2u),\quad (u,v\pm(2u+1))
\]
and eight identity labels, each with weight $1/16$. The all-modulus
Margulis--Gabber--Galil bound, in the primary formal development
\cite[\texttt{mgg\_numerical\_radius}]{MGGAFP}, bounds the normalized
upper mean-zero eigenvalue of the eight-move operator by $5\sqrt2/8$.
The operator is symmetric with spectrum in $[-1,1]$. Half-laziness
therefore yields
\begin{equation}\label{ig:replica-gap}
 \|AD\|\le(1-\kappa)\|D\|,\qquad \kappa=1/100,
\end{equation}
on replica deviations, since $(1+5\sqrt2/8)/2<1-\kappa$.
The statement includes every positive integer $b$ and retains loops
and repeated labels. No prime-modulus choice or randomized graph search
is needed.

Each physical replica is a node $x_{ir}\in[0,1]$, interpreted as
$X_{ir}=3x_{ir}-1\in[-1,2]$. Its own \emph{direct cone} produces it
using the prescribed 16 averaging inputs and its fixed estimator tape.
All gates of different direct cones are distinct; they may read the
same existing node as input. The estimator uses only the nine gate types
of Definition~\ref{def:gcircuit}, as follows.

For every raw comparison occurrence create its own Constant threshold
node and a Less gate, oriented to give the desired high bit. Internal
binary thresholds and quantization boundaries are translated from
physical coordinates by $a\mapsto(a+1)/3$. Out-of-range constant cases
use Boolean constants. An external occurrence reads its individually
routed external node and compares it with a Constant at $(\xi+1)/3$.
Threshold gates are inside the cone and are never shared between cones.
All public zero bits are produced by Constant gates.

Unfold finite answer-dependent choices and implement their Boolean
answers and finite lookup outputs by truth-table circuits over And, Or,
and Not. A multiplexer, for example, is
$(c\mathbin{\mathrm{And}}a)\mathbin{\mathrm{Or}}
 ((\mathrm{Not}\,c)\mathbin{\mathrm{And}}b)$.
Copy gates may provide fan-out, although unrestricted input fan-out
already suffices. Induction through these circuits preserves a correct
near-bit with the same tolerance: weak Boolean antecedents apply to
inputs at most $\eps$ or at least $1-\eps$. This costs a fixed number
of gates because the complete answer tree and output precision have
already been fixed. In particular, polynomial-size field elements and
indices are computed publicly to choose wires, coefficients, and tables;
they are not hidden strings being processed by a bounded Boolean cone.

Apply Scale with parameter $1/16$ to each averaging input and accumulate
with Add to obtain their mean. Use Scale and Add to sum the fixed dyadic
weights of the signed-output bits, obtaining nodes $p,n\in[0,1]$ for
$p^*,n^*$ in~\eqref{ig:ideal-estimator}. For a positive rational $\beta$
chosen below, put $t_\beta=\beta B_1/3\le1$. Scale $p,n$ by
$t_\beta$, add the positive term to the averaging node, and subtract
the negative term, in that order, to produce $x_{ir}$. All parameters
lie in $[0,1]$. Exact partial sums for averages and dyadic expansions
are at most one; hence their clipped Add realization is exact on ideal
inputs. Clipped sums and differences are Lipschitz in their inputs, so
there is a fixed arithmetic constant $C_{\rm ar}$ with the following
two implications whenever the entire direct cone is $\eps$-satisfied.
First,
\begin{equation}
 |X_{ir}-(AX)_{ir}|
   \le2\beta B_1+C_{\rm ar}\eps.
       \label{ig:coarse-direct}
\end{equation}
Under the same hypothesis, if all its raw comparisons give the ideal
near-bits, then
\begin{equation}
 \left|X_{ir}-\clip_{[-1,2]}
       \bigl((AX)_{ir}+\beta\widehat v_{ir}\bigr)\right|
   \le C_{\rm ar}\eps.
       \label{ig:fine-direct}
\end{equation}
Here are explicit error counts. Sixteen Scale gates and fifteen Add
gates produce an averaging node $\widetilde a$ with
$|\widetilde a-a|\le31\eps$, where $a=(Ax)_{ir}$.
For
\[
 H(a,p,n)=\max\{\min\{a+t_\beta p,1\}-t_\beta n,0\},
\]
the two final Scale gates, Add, and Subtract give
$|x_{ir}-H(a,p,n)|\le35\eps$. Since
$|H(a,p,n)-a|\le2t_\beta$, this proves
\eqref{ig:coarse-direct} with arithmetic error $105\eps$ and requires
no estimator correctness. If the raw comparisons give the ideal near-bits,
pad each signed dyadic expansion to $J$ bits. Its $J$ Scale gates and
$J-1$ Add gates reconstruct each of $p^*,n^*$ within
$(3J-1)\eps$: at most $J\eps$ comes from its input bits and
$(2J-1)\eps$ from the arithmetic gates. Every exact partial sum is
at most one. The map $H$ is 1-Lipschitz in $a$ and $t_\beta$-Lipschitz
in each of $p,n$. Since $p^*n^*=0$,
\[
 3H(a,p^*,n^*)-1
   =\clip_{[-1,2]}\bigl((AX)_{ir}
                         +\beta B_1(p^*-n^*)\bigr).
\]
Thus both estimates hold with $C_{\rm ar}=18J+99$. This calculation
also covers intermediate saturations.

For each selected first-block coordinate and each replica add a separate
extraction pair: a Constant at $1/2$ in $x$ coordinates and a Less gate
comparing that constant with $x_{ir}$. Its output is the designated
near-bit. These gates are not used in the feedback. Let $W$ be the number
of designated outputs. The construction has
\begin{equation}\label{ig:gate-count}
 c_{\rm sel}G_0\le W\le G_0,
 \qquad |T_{\rm invocation}|\le C_gG_0
\end{equation}
for a fixed $C_g\ge1$ including every extraction gate and every hypothetical
branch. Each node is the output of at most one gate, and cycles arise only
through the prescribed physical input nodes. Gate listing, tape
construction, address calculations, and all parameters other than the
supplied threshold cost polynomial time and polynomial bit length in
$L$, with fixed exponents for the chosen bounds. Uses of an arbitrary
rational $\xi$ additionally cost its supplied bit length (and polynomial
time in that length). The gate topology and count do not depend on that
length or on the external physical margin.

\subsection{Robustness under arbitrary failed gates}

We give the choice order and prove the universal assignment guarantee.
For a replica array put
\begin{equation}\label{ig:replica-norm}
 \|Y\|_{\rm rep}^2=\frac1{mR}\sum_{i,r}Y_{ir}^2,
 \qquad \bar X_i=\frac1R\sum_rX_{ir}.
\end{equation}
Repeated mean vectors have precisely the norm~\eqref{ig:point-space}.
Projection onto means and onto deviations are orthogonal contractions.
The maximum norm of an array bounded coordinatewise by three is six;
this explains the constants in the failure charges below.

\paragraph{Parameter choices.}
Choose $q_{\rm bad}>0$ so that
\begin{equation}\label{ig:qbad}
 6B_1\sqrt{q_{\rm bad}}<c_*\kappa/10,
\end{equation}
and choose $\gamma_g>0$ much smaller than $q_{\rm bad}$. Choose the
internal grids, the positive $\omega_{\rm in}$, and the tape-TV tolerance
so that, for every fixed means, the empirical fraction of cones with an
ambiguous internal comparison is less than $q_{\rm bad}/4$.
The full-tree constants $H,C$ are already fixed. By
\eqref{ig:internal-load}, let $D_{\rm read}$ bound the number of routed
internal comparison inputs using any actual replica. Choose
$d_{\rm sync}>0$ sufficiently small that
\begin{equation}\label{ig:sync-choice}
 \frac{D_{\rm read}d_{\rm sync}^2}{4\omega_{\rm in}^2}
       <q_{\rm bad}/4,
 \qquad
 \frac{64d_{\rm sync}^2}{c_{\rm sel}}<\upsilon/4.
\end{equation}
If $\|X-\bar X\|_{\rm rep}\le d_{\rm sync}$, the first bound and
Markov's inequality charge fewer than $q_{\rm bad}/4$ of the cones for
any internal routed read deviating more than $\omega_{\rm in}$ from its
base mean. Choose grouped budgets with positive slack, so that this
charge, the ambiguity charge, and the $\gamma_g$ external charge sum
to less than $q_{\rm bad}$.

Now fix a positive rational $\beta\le\beta_0$ such that
$t_\beta\le1$ and $4\beta B_1/\kappa<d_{\rm sync}/2$.
Finally choose positive rational $\eps_g,\delta_g$ with
\begin{gather}
 \eps_g<\min\{1/100,\omega_{\rm in}/1000\},\qquad
 \delta_g/c_{\rm sel}<\upsilon/4,
 \label{ig:last-small}\\
 \frac{4\beta B_1+2C_{\rm ar}\eps_g+6\sqrt{\delta_g}}
         {\kappa}<d_{\rm sync},\qquad
 2C_{\rm ar}\eps_g+6\sqrt{\delta_g}
         <\beta c_*\kappa/10.
 \label{ig:epsilon-delta}
\end{gather}
All inequalities can hold because each right side has positive slack
when the later constants are decreased. None of these choices uses the
subsequently supplied external margin $\omega>0$. That margin imposes
only the additional restriction $\eps<\omega/1000$ on the actual gate
tolerance, alongside $\eps\le\eps_g$.

Fix any assignment of values in $[0,1]$ to all nodes with at most
$\delta_gG_0$ failed invocation gates. Assume that at most
$\gamma_gG_0$ of the external comparison occurrences are spoiled:
every other occurrence lies on the side of $\xi$ prescribed by the
fixed word $y_i$, with physical gap greater than $\omega$.
Different occurrences are not required to have equal physical values.
A direct cone is called failed if any of its own gates fails. Since its
gates are disjoint from those of other direct cones, their failed fraction
is at most $\delta_g$.

\paragraph{Coarse synchronization.}
First obtain synchronization without assuming any correct estimator
answer. Apply~\eqref{ig:coarse-direct} on the unfailed cones and the
physical range bound on the others. The triangle inequality, the
factor-four norm normalization, and~\eqref{ig:replica-gap} give
\begin{equation}\label{ig:coarse-sync}
 \kappa\|X-\bar X\|_{\rm rep}
 \le\|X-AX\|_{\rm rep}
 \le4\beta B_1+2C_{\rm ar}\eps+6\sqrt{\delta_g}
 <\kappa d_{\rm sync}.
\end{equation}
For the first inequality write $D=X-\bar X$ and use
$\|D\|\le\|D-AD\|+\|AD\|$. Mark as unstable a cone with an
ambiguous internal comparison at these fixed means, an excessive routed
internal deviation, or any spoiled external occurrence. The fraction of
unstable cones is less than $q_{\rm bad}$ by the choices above. An
external spoiled occurrence charges only its own cone; no repeated-value
consistency was used.

\paragraph{Fine residual and endpoint proximity.}
Suppose for a contradiction that
$\dist(\bar X,\mathcal E)>10\tau$. In a stable, unfailed cone,
each internal comparison has physical mean gap greater than
$4\omega_{\rm in}$ and input deviation at most $\omega_{\rm in}$.
Its transformed $x$ gap is therefore at least $\omega_{\rm in}$.
The Constant error and the strict Less tolerance are dominated by
$\eps\le\eps_g<\omega_{\rm in}/1000$, forcing the ideal near-bit.
Every external comparison likewise has $x$ gap greater than $\omega/3$;
its Constant and Less errors are dominated by $\eps<\omega/1000$.
Thus~\eqref{ig:fine-direct} applies with the empirical ideal estimator
on $(\bar X,y)$. On unstable but unfailed cones,
\eqref{ig:coarse-direct} and $\|v\|_\infty\le B_1$ give discrepancy
at most $3\beta B_1+C_{\rm ar}\eps$ from the clipped mean-field
update at $AX$. On failed cones that discrepancy is at most three.
Using~\eqref{ig:empirical-error}, obtain
\begin{align}
 E_{\rm in}
 &:=\bigl\|X-\clip_{\mathcal B}
              (AX+\beta v(\bar X))\bigr\|_{\rm rep}\notag\\
 &\le\beta\bigl(c_*\kappa/20+6B_1\sqrt{q_{\rm bad}}\bigr)
       +2C_{\rm ar}\eps+6\sqrt{\delta_g}
 <\beta c_*\kappa/2.
 \label{ig:refined-error}
\end{align}
The notation $\clip_{\mathcal B}$ here means coordinatewise clipping
on every replica, with $v(\bar X)$ repeated over replicas.

We keep the clipping inside the contraction argument. Put
\[
 F=\clip_{\mathcal B}(AX+\beta v(\bar X)),\qquad
 C_0=\clip_{\mathcal B}(\bar X+\beta v(\bar X)),
\]
where $C_0$ is constant on replicas. Nonexpansiveness and
\eqref{ig:replica-gap} show
\[
 \|F-C_0\|_{\rm rep}\le(1-\kappa)\|X-\bar X\|_{\rm rep},
 \qquad
 \|F-\bar F\|_{\rm rep}\le(1-\kappa)\|X-\bar X\|_{\rm rep}.
\]
Projecting $X-F$ onto deviations now gives
$\|X-\bar X\|_{\rm rep}\le E_{\rm in}/\kappa$.
Projecting onto means and comparing $\bar F$ with $C_0$ gives
\begin{align*}
 \|\bar X-C_0\|_{\rm b}
 &\le E_{\rm in}+(1-\kappa)\|X-\bar X\|_{\rm rep}\\
 &\le E_{\rm in}/\kappa<c_*\beta.
\end{align*}
This contradicts~\eqref{ig:geometric-bound}. No commutation of clipping
with taking means was asserted. Hence
$\dist(\bar X,\mathcal E)\le10\tau$, and its first block has
mean squared distance at most $(10\tau)^2$ from the sink word.

\paragraph{Designated outputs.}
It remains to account for adversarial concentration on designated
outputs. Among all $G_0$ replica indices, discard those in the first
block with mean discrepancy at least $1/4$ from the corresponding sink
bit, and those with replica deviation greater than $1/8$. The squared
bounds just proved and~\eqref{ig:coarse-sync} bound their respective
counts, with harmless slack for normalization, by
\[
 16(10\tau)^2G_0\quad\text{and}\quad64d_{\rm sync}^2G_0.
\]
At each remaining selected position, the physical replica differs by
less than $3/8$ from the bit $f(s)$, so its distance from the physical
threshold $1/2$ is greater than $1/8$. In $x$ coordinates the gap is
greater than $1/24$. Since $\eps\le\eps_g<1/100$, a satisfied
extraction pair returns the correct near-bit. Every extraction pair has
its own two gates, so at most $\delta_gG_0$ designated outputs are
charged for their own failed gates, even if every failed gate is
concentrated there. By~\eqref{ig:gate-count}, the incorrect fraction is
at most
\begin{equation}\label{ig:extraction-loss}
 \frac{16(10\tau)^2+64d_{\rm sync}^2+\delta_g}{c_{\rm sel}}
 <\upsilon.
\end{equation}
The choice of $\tau$, synchronization radius, and failure budget reserved
this divisor explicitly.

We have proved the assertion for every assignment satisfying the stated
failure and spoil budgets, with all failed gates otherwise arbitrary.
Taking $c_g=c_{\rm sel}$, together with
Proposition~\ref{prop:dummy-port},~\eqref{ig:external-load}, and
\eqref{ig:gate-count}, gives every assertion of
Theorem~\ref{thm:port-gadget}. The proof uses no part of the outer
circuitization: its only remaining independent input is the geometric
lemma proved next.
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/09-inner-gadget.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/10-geometry.tex ====
\section{Encoded paths and finite local displacement procedures}
\label{sec:geometry}

The encoded-path and displacement-field method follows the construction
in Rubinstein~\cite[Sections~3.1--3.2]{Rubinstein2016Settling}: current and successor
blocks trace a copy--toggle polygon, and radial interpolation moves from
the tangent through the inward and reverse-tangent directions to a fixed
vertical direction. Here we prove the geometric interface at the normalized
codeword scale, including rounded corners, displacement after clipping,
and finite local access with average errors. This section is independent
of both circuit realizations. Its hypotheses concern entire binary words and local access
to their bits; they do not depend on the algebraic interpretation of an
individual coordinate. In particular, in the inner application the graph
is the path for the fixed true input tuple, and the distinguished source
is accessed through the virtual procedures of Section~\ref{sec:inner-gadget}.

\subsection{The geometric statement and its parameters}
\label{geo:setup}

Let $G$ be a finite directed graph of indegree and outdegree at most one,
with no self-edge and no isolated vertex. Fix a source $w_*$ with an actual
successor. All other vertices missing an incoming or an outgoing edge are
called desired endpoints. Suppose that distinct vertices have encodings
$E_w\in\{0,1\}^m$ satisfying
\begin{equation}\label{geo:distance}
 \norm{E_u-E_v}_{\rm b}\ge d_*>0\qquad(u\ne v).
\end{equation}
We decrease $d_*$ to a positive rational at most one. On any number of
blocks of length $m$ use the inner product and norm
\begin{equation}\label{geo:norm}
 \langle U,V\rangle_{\rm b}=m^{-1}\sum_j U_jV_j,
 \qquad \norm{U}_{\rm b}^2=m^{-1}\sum_j U_j^2.
\end{equation}
An unqualified norm in this section means this norm. The infinity norm
is unnormalized. Write $\mathcal B=[-1,2]^{4m}$, and let
$\clip_{\mathcal B}$ denote coordinatewise clipping to this cube.
Clipping is nonexpansive for~\eqref{geo:norm}. Scalars occupying a block
mean constant repetitions.

\begin{theorem}[Encoded-path geometry]\label{thm:geometry}
Under these hypotheses, there is a union $\mathcal L\subset\mathcal B$
of directed, rounded paths and cycles, with desired endpoint set
$\mathcal E$, having the following properties. For every sufficiently
small positive rational $\tau$, there are a positive integer $B_1$ and
positive rational constants $c_*,\beta_0$, depending only on $d_*$ and
$\tau$, and a field $v:\mathcal B\to\mathbb R^{4m}$ such that
\begin{align}
 \norm{v(X)}_\infty&\le B_1-1,\label{geo:bounded-field}\\
 \norm{\clip_{\mathcal B}(X+\beta v(X))-X}
   &\ge c_*\beta
 \quad\text{if }\dist(X,\mathcal E)>10\tau,
       \quad 0<\beta\le\beta_0.\label{geo:displacement}
\end{align}
The field is continuous on the relative open set
$\{X\in\mathcal B:\dist(X,\mathcal E)>10\tau\}$.
Every $P\in\mathcal E$ has first block exactly $E_w$ for a desired
endpoint $w$. Consequently, if $\norm{X-P}\le10\tau$, rounding $X_1$
at $1/2$ gives relative Hamming distance at most $4(10\tau)^2$ from
$E_w$. The exceptional top of the distinguished vertical is excluded
from $\mathcal E$ and has field $-\mathbf e$, where
$\mathbf e=(0,0,0,1)$.

The initial choice of $\tau$ may impose any further fixed positive upper
bounds, including validation and global-recovery radii. All these choices,
and $B_1,c_*,\beta_0$, precede every numerical accuracy choice and the
choice of a step size $\beta$.
\end{theorem}

We prove the theorem before formulating the precise access contract for
its finite numerical implementation. In particular no global regularity
claim at desired endpoints will be needed.

\subsection{Polygons, fillets, and local projection}
\label{geo:curves}

For an edge $u\to v$, draw, with fourth block zero,
\begin{equation}\label{geo:polygon}
 \begin{aligned}
 &(E_u,E_u,0)\xrightarrow{\;2\;}(E_u,E_v,0)
                 \xrightarrow{\;3\uparrow\;}(E_u,E_v,1)\\
 &\qquad\xrightarrow{\;1\;}(E_v,E_v,1)
                 \xrightarrow{\;3\downarrow\;}(E_v,E_v,0).
 \end{aligned}
\end{equation}
An arrow label names the changing block; the up and down arrows record
the direction of the third block. This is a schematic of block changes,
not a distance-preserving planar drawing.
Concatenate these polygons at graph-vertex junctions. Put
$H_*=(E_{w_*},E_{w_*},0)$ and prepend the segment
$(H_*,2)\longrightarrow(H_*,0)$ to the path out of $w_*$.
We call the four segment types in~\eqref{geo:polygon} \emph{horizontal},
meaning that their fourth block is zero; this includes the two third-block
toggles. Every horizontal segment has length in $[d_*,1]$, and consecutive unit
directions are orthogonal: the successive varying blocks in a polygon
are $2,3,1,3$, a graph junction changes from block $3$ to block $2$,
and the special bottom changes from block $4$ to block $2$.

\begin{lemma}[Separation of original segments]\label{geo:separation}
There is $d_{\rm sep}>0$, depending only on $d_*$, such that distinct
nonconsecutive original segments are at distance at least $d_{\rm sep}$.
Their adjacency graph, with adjacency meaning consecutiveness, has no
triangle.
\end{lemma}
\begin{proof}
For binary words $a,b,c$ and $0\le s\le1$,
\begin{equation}\label{geo:binary-chord}
 \norm{(1-s)a+sb-c}\ge\min(s,1-s)\norm{a-b}.
\end{equation}
Indeed on a coordinate where $a,b$ differ, the chord coordinate is $s$
or $1-s$, whose distance from a binary coordinate is at least
$\min(s,1-s)$. Summing squares proves the claim. In particular, a
point on this chord within distance $z$ of a binary word is within
distance $z$ of one of the chord endpoints.

Consider first segments varying the same block. If their distance is
less than $d_*$, all fixed encoding blocks and repeated binary levels
agree. For block $2$, the tail agrees and determines the outgoing
edge. For block $1$, the head agrees and determines the incoming edge.
For block $3$, an upward segment has a fixed unequal pair $(E_u,E_v)$,
whereas a downward segment has an equal pair $(E_v,E_v)$; the two types
cannot have the same fixed pair. Within either type that pair determines
the segment, using uniqueness of the incoming edge for the downward
type. There is only one block-$4$ segment.

For two segments varying different blocks, choose points at distance
$z$. Each varying block of a horizontal segment is within distance $z$
of a fixed binary block of the other. By~\eqref{geo:binary-chord}, its
point is within distance $z$ of an endpoint of its original segment.
For the special vertical, its distance from a horizontal segment forces
its height to be at most $z$, so it is within $z$ of its bottom. The
two selected endpoints are therefore within $3z$. Distinct original
endpoints are separated by at least $d_*$: one of their encoding blocks,
binary third levels, or vertical levels differs. Taking $3z<d_*$ forces
the endpoints to coincide.

An unequal ordered pair $(E_u,E_v)$ occurs at the two internal toggle
corners of its own edge only, with the third level distinguishing them.
An endpoint $(E_v,E_v,1,0)$ belongs to the last internal corner of the
unique incoming edge. An endpoint $(E_v,E_v,0,0)$ belongs to at most
the incoming final segment and the outgoing first segment. If both
exist, they are consecutive. At $w_*$ there is no incoming graph edge,
and the special bottom takes exactly that role. The top belongs only
to the special vertical. Thus equality of endpoints of distinct
segments implies consecutiveness. For example $d_{\rm sep}=d_*/4$
is sufficient for the asserted strict-distance argument. The adjacency
components are paths and cycles; any cycle contains at least eight
segments because the graph has no self-edge. Adding a segment at a
path beginning creates no triangle.
\end{proof}

Choose a positive rational $a_0$ such that
\begin{equation}\label{geo:trim-choice}
 2a_0<d_{\rm sep}/2,\qquad4a_0<d_*,\qquad a_0<1/10.
\end{equation}
Trim arclength $a_0$ from both ends of every horizontal segment, whether
or not its outer graph junction is present. At a shared corner $C$ with
incoming and outgoing unit directions $u_0,v_0$, join the trim points by
\begin{align}
 Z(s)&=C-a_0u_0+a_0v_0
             +a_0(\sin s\,u_0-\cos s\,v_0),
             &0\le s\le\pi/2,\label{geo:fillet}\\
 T(s)&=\cos s\,u_0+\sin s\,v_0.\label{geo:fillet-tangent}
\end{align}
The arc has speed $a_0$ in the parameter $s$, and $T$ is its directed
unit tangent. Figure~\ref{fig:geo-fillet} shows this replacement in the
plane of its two orthogonal directions. Use the same fillet at the special
bottom, trimming the vertical there only. Its top is not trimmed. Unmatched horizontal trim
points remain endpoints. Let $\mathcal L$ be the resulting compact union
and set
\begin{equation}\label{geo:endpoint-set}
 \mathcal E=\{\text{unmatched endpoints of }\mathcal L\}
                \setminus\{(H_*,2)\}.
\end{equation}

The first three blocks stay in $[0,1]$: on a fillet the separate varied
block coordinates stay in the intervals traversed by their original
segments. The fourth block is zero except on the vertical and its bottom
fillet; on that fillet it is a repeated scalar in $[0,a_0]$.
Consequently $\langle T,\mathbf e\rangle\le0$ everywhere. Fix an
integer $C_t\ge2/d_*$. For arclength $\lambda$,
\begin{equation}\label{geo:tangent-bounds}
 \norm T_\infty\le C_t,\qquad
 \norm{dT/d\lambda}\le1/a_0,\qquad
 \norm{dT/d\lambda}_\infty\le2C_t/a_0.
\end{equation}
These derivative bounds hold on piece interiors and give the same
Lipschitz bounds across joins because the tangent is continuous there.

\begin{figure}[htbp]
\centering
\setlength{\unitlength}{1pt}
\begin{picture}(290,160)
\thinlines
\multiput(180,30)(-5,0){10}{\line(-1,0){2}}
\multiput(180,30)(0,5){10}{\line(0,1){2}}
\put(180,30){\circle*{3}}
\put(187,24){\makebox(0,0)[l]{$C$}}
\put(130,80){\circle*{3}}
\put(122,87){\makebox(0,0)[r]{$O$}}
\thicklines
\put(20,30){\line(1,0){110}}
\put(62,30){\vector(1,0){25}}
\put(74,18){\makebox(0,0)[t]{$u_0$}}
\qbezier(130,30)(139.946,30)(149.134,33.806)
\qbezier(149.134,33.806)(158.323,37.612)(165.355,44.645)
\qbezier(165.355,44.645)(172.388,51.677)(176.194,60.866)
\qbezier(176.194,60.866)(180,70.054)(180,80)
\put(180,80){\line(0,1){65}}
\put(180,103){\vector(0,1){24}}
\put(170,116){\makebox(0,0)[r]{$v_0$}}
\put(130,30){\circle*{3}}
\put(180,80){\circle*{3}}
\put(130,9){\makebox(0,0)[t]{$C-a_0u_0$}}
\put(188,80){\makebox(0,0)[l]{$C+a_0v_0$}}
\end{picture}
\caption{Rounding one orthogonal corner. The dashed legs are removed;
the directed path follows the retained straight portions and the fillet
from $C-a_0u_0$ to $C+a_0v_0$. Its center is
$O=C-a_0u_0+a_0v_0$ and its radius is $a_0$. This local geometry lies
in the orthonormal $(u_0,v_0)$ plane.}
\label{fig:geo-fillet}
\end{figure}

The numerical procedure will choose a curve point by minimizing estimated
distances. We therefore need more than a unique projection: a curve point
with small squared-distance excess must be close to the true projection,
and its tangent close to the projection's tangent, in every coordinate.
The next lemma gives these bounds before any candidate is selected.

\begin{lemma}[Local chains and strong excess]\label{geo:projection}
There is $\rho_{\rm loc}>0$, depending only on $d_*,a_0$, such that
two points $Z,Z'\in\mathcal L$ with $\norm{Z-Z'}<\rho_{\rm loc}$
are joined along one local straight or single-corner chain. Its length
$\Delta$ satisfies $\Delta\le\sqrt2\norm{Z-Z'}$.
For every sufficiently small $\tau>0$, if
\begin{equation}\label{geo:tube-hypothesis}
 r=\dist(X,\mathcal L)\le4\tau,\qquad
 \dist(X,\mathcal E\cup\{(H_*,2)\})>10\tau,
\end{equation}
there is a unique closest point $Z$, which is interior and satisfies
$\langle X-Z,T(Z)\rangle=0$. If $Z'\in\mathcal L$ has squared excess
$g=\norm{X-Z'}^2-r^2\le\tau^2$ and signed arclength difference $s'$
from $Z$ on that chain, then
\begin{equation}\label{geo:strong-excess}
 g\ge (s')^2/2.
\end{equation}
In particular,
\begin{align}
 \norm{Z'-Z}&\le\sqrt{2g},&
 \norm{Z'-Z}_\infty&\le C_t\sqrt{2g},\label{geo:point-error}\\
 \norm{T(Z')-T(Z)}&\le\sqrt{2g}/a_0,&
 \norm{T(Z')-T(Z)}_\infty&\le2C_t\sqrt{2g}/a_0.
 \label{geo:tangent-error}
\end{align}
The tangent at an endpoint competitor is its one-sided tangent.
\end{lemma}
\begin{proof}
Support a straight point by its original segment, and an arc point by
the two segments of its corner. An arc point is within $a_0$ of each
supporting segment. Choose $\rho_{\rm loc}+2a_0<d_{\rm sep}$.
Every across-point pair of supports is then identical or consecutive.
Each point's support set is itself a clique, so their union is a clique
and contains at most two segments by Lemma~\ref{geo:separation}. In the
one-segment case both points lie on its retained straight. In the
two-segment case they lie on the two retained straights and their
shared fillet. Opposite-end trims are disjoint by~\eqref{geo:trim-choice}.
This also rules out unintended identifications or intersections.

Along a two-direction chain, all tangent coefficients in its orthogonal
directions $u_0,v_0$ are nonnegative, and
$\langle T,u_0+v_0\rangle\ge1$. Integrating in the directed order gives
$\Delta\le\langle Z'-Z,u_0+v_0\rangle\le\sqrt2\norm{Z'-Z}$.
Reverse the order if necessary. A straight chain has equality between
length and endpoint distance.

Compactness gives a minimizer. The endpoint assumption makes every
minimizer interior, so differentiating squared distance along its
$C^1$ path gives the asserted perpendicularity. A competitor of the
stated excess satisfies
\[
 \norm{Z'-Z}\le4\tau+\sqrt{17}\tau<9\tau.
\]
Require $9\tau<\rho_{\rm loc}$. The local chain then has
$|s'|<13\tau$, and integration of~\eqref{geo:tangent-bounds} gives
\[
 Z'-Z=s'T(Z)+R,\qquad \norm R\le(s')^2/(2a_0).
\]
For $|s'|/(2a_0)<1$, expansion of squared distance and perpendicularity
therefore imply
\begin{align*}
 g&=\norm{Z'-Z}^2-2\langle X-Z,Z'-Z\rangle\\
  &\ge(s')^2\left[
      \left(1-\frac{|s'|}{2a_0}\right)^2-\frac r{a_0}\right]
   \ge(s')^2/2,
\end{align*}
where, for example, $\tau/a_0\le1/64$ makes the last inequality hold.
Applying this to a second minimizer proves uniqueness. Integrating the
unit tangent, its infinity bound, and its two Lipschitz bounds over
length $|s'|\le\sqrt{2g}$ proves~\eqref{geo:point-error} and
\eqref{geo:tangent-error}. All estimates are independent of $m$.
\end{proof}

At an unmatched beginning, the trimmed segment varies only block $2$;
at an unmatched ending it varies only block $3$. Their first blocks
are therefore exactly the encoding of the missing-neighbor vertex.
There are no isolated vertices requiring a separate convention. The
source's horizontal beginning is joined to the special vertical and
is not an endpoint of $\mathcal L$. This proves the endpoint assertion
in Theorem~\ref{thm:geometry}. If $P\in\mathcal E$ and $\norm{X-P}\le10\tau$, every wrongly
rounded first coordinate costs at least $1/4$ in squared error, proving
the Hamming bound in that theorem.

\begin{remark}[Local Lipschitz continuity of projection]
\label{geo:projection-lipschitz}
Suppose both $X$ and $Y$ satisfy~\eqref{geo:tube-hypothesis}, and
$\norm{X-Y}\le\tau/20$. Write $P,Q$ for their respective projections
onto $\mathcal L$ and $r_X,r_Y$ for their distances to $\mathcal L$.
With the same sufficiently small choice of $\tau$, one has
\begin{equation}\label{geo:projection-lipschitz-bound}
 \left(1-\frac{r_X+r_Y}{a_0}\right)\norm{P-Q}^2
 \le \langle X-Y,P-Q\rangle
 \le \norm{X-Y}\norm{P-Q}.
\end{equation}
In particular, the projection is locally $2$-Lipschitz on this tube.
This is a local assertion away from the specified endpoints.

Indeed, the conclusion is immediate if $P=Q$. Otherwise,
$\norm{P-Q}\le r_X+\norm{X-Y}+r_Y\le(8+1/20)\tau<\rho_{\rm loc}$.
Lemma~\ref{geo:projection} therefore puts $P,Q$ on one local chain,
whose arclength $s$ satisfies $s^2\le2\norm{P-Q}^2$.
Integrating~\eqref{geo:tangent-bounds} from either end expresses its
chord as the signed length times the tangent there, with remainder
of norm at most $s^2/(2a_0)$. Perpendicularity at $P$ and $Q$ gives
\[
 \big|\langle X-P,P-Q\rangle\big|
       \le\frac{r_Xs^2}{2a_0},\qquad
 \big|\langle Y-Q,P-Q\rangle\big|
       \le\frac{r_Ys^2}{2a_0}.
\]
Expanding $X-Y=(P-Q)+(X-P)-(Y-Q)$ proves the first inequality
in~\eqref{geo:projection-lipschitz-bound}; Cauchy--Schwarz proves
the second. Finally $r_X+r_Y\le8\tau$ and $\tau/a_0\le1/64$
make its left coefficient at least $7/8$, and in particular at least
$1/2$, proving the stated factor $2$.
\end{remark}

\subsection{A field with uniform displacement after clipping}

For the inside prescription, away from the desired endpoint neighborhood,
successive radial intervals interpolate from the directed tangent through
the inward normal and reverse tangent to the fixed fourth-block direction.
The four directional stages in~\eqref{geo:Phi} parallel
\cite[Equation~(1)]{Rubinstein2016Settling}; the projection and clipping estimates
below establish the bounds needed for the present curve.
A separate prescription above fourth-block mean level $1/2$ handles the
distinguished vertical. We check that the prescriptions agree and that
clipping preserves positive displacement away from the desired endpoints.
The normalized tube bound permits individual coordinates of $X$ to be
far from the curve, so raw field length alone does not prove this last
claim. We will bound the fraction of coordinates where clipping can
remove motion, and then handle the upper face separately.
\label{geo:field}

Fix $\tau$ satisfying Lemma~\ref{geo:projection}, and impose also
\begin{equation}\label{geo:tau-choice}
 \tau<1/100,\qquad6(C_t+1)\tau\le1/4.
\end{equation}
All further fixed validation or recovery upper bounds on $\tau$ are
imposed now. For $X=(X_1,X_2,X_3,X_4)$ put
$\mu=m^{-1}\sum_j(X_4)_j$. For a point $Z$ and a unit tangent $T$ let
$r=\norm{X-Z}$ and define
\begin{equation}\label{geo:Phi}
 \Phi(X;Z,T)=
 \begin{cases}
 (1-r/\tau)T+(Z-X)/\tau,&0\le r\le\tau,\\[2pt]
 -(r/\tau-1)T+\dfrac{2\tau-r}{\tau r}(Z-X),
                       &\tau\le r\le2\tau,\\[5pt]
 -\dfrac{3\tau-r}{\tau}T+\dfrac{r-2\tau}{\tau}\mathbf e,
                       &2\tau\le r\le3\tau,\\[5pt]
 \mathbf e,&r\ge3\tau.
 \end{cases}
\end{equation}
The values at shared boundaries agree. There is no reciprocal radius
in the formula used near zero.

For $\mu\le1/2$ and $\dist(X,\mathcal E)>10\tau$, use
$v(X)=\Phi(X;Z,T(Z))$ when $\dist(X,\mathcal L)\le3\tau$, and
$v(X)=\mathbf e$ otherwise. The top is at distance at least $3/2$
from this region, by the difference of fourth-block means; hence the
projection lemma applies. At the remaining inside points put $v(X)=0$.
For $\mu\ge1/2$, the default upward motion can be lost at the upper
face. We add motion toward the distinguished vertical, with coefficient
zero at the mean interface. Close to that vertical the tangent direction
is downward, including at its top. Set
\begin{equation}\label{geo:outside}
 \begin{aligned}
 Z_\mu&=(H_*,\mu),\qquad r_\mu=\norm{X-Z_\mu},\qquad
 h_\mu=\min\{1,2(\mu-1/2)\},\\
 v(X)&=\Phi(X;Z_\mu,-\mathbf e)+h_\mu(Z_\mu-X).
 \end{aligned}
\end{equation}

\begin{lemma}[Agreement at the mean interface]\label{geo:interface}
In $2/5<\mu<3/5$, the inside prescription extended to this slab equals
$\Phi(X;Z_\mu,-\mathbf e)$. The outside extension, using
$h_\mu=\max\{0,\min\{1,2(\mu-1/2)\}\}$, differs from it in norm by
at most $12|\mu-1/2|$ and in each component by at most $6|\mu-1/2|$.
Thus the definitions agree at $\mu=1/2$.
\end{lemma}
\begin{proof}
Every nonvertical piece, including the bottom fillet, has fourth height
at most $a_0<1/10$. Its distance to a point in the slab exceeds $3/10$,
by the mean inequality $\norm{X_4-z\mathbf1}\ge|\mu-z|$.
It cannot be in the $4\tau$ tube. The retained vertical contains all
heights of this slab, and
\[
 m^{-1}\sum_j((X_4)_j-t)^2
 =m^{-1}\sum_j((X_4)_j-\mu)^2+(t-\mu)^2
\]
shows that its nearest line point is $Z_\mu$. For $T=-\mathbf e$,
the last annulus of~\eqref{geo:Phi} is already identically
$\mathbf e$ at every $r\ge2\tau$. Consequently the same equality
holds even where the closest curve piece is far away and the inside
field is default. The desired endpoints have fourth height zero and
the top has height two, so neither endpoint exception occurs in the
slab. Each coordinate of $Z_\mu-X$ has magnitude at most three and
its norm is at most six. Multiplication by
$h_\mu\le2|\mu-1/2|$ proves both bounds.
\end{proof}

The projection is continuous wherever it is unique in the tube:
if $X_n\to X$, compactness makes every subsequential limit of their
projections a minimizer for $X$, hence the unique minimizer. Its tangent
is continuous along the local chains. Formula~\eqref{geo:Phi} matches
the default at $3\tau$, and Lemma~\ref{geo:interface} matches the mean
interface. This proves precisely the continuity claimed in
Theorem~\ref{thm:geometry}.

\begin{proof}[Proof of the displacement assertion in Theorem~\ref{thm:geometry}]
Choose $\beta_0>0$ satisfying all the upper bounds on $\beta$ used below.
\paragraph{Inside the tube and in the inside default region.}
First suppose $\mu\le1/2$ and $r=\dist(X,\mathcal L)\le3\tau$.
For $r\le\tau$ the raw field is the convex combination of $T$ and
the inward unit normal with coefficients $1-r/\tau,r/\tau$.
For $\tau\le r\le2\tau$ it is the convex combination of $-T$ and
the inward unit normal with coefficients $r/\tau-1,2-r/\tau$.
The two directions in each combination are orthogonal. At $r=0$
the field is $T$. In the third annulus the two unit directions are
$-T,\mathbf e$, whose inner product is nonnegative. Therefore in all
three annuli $\norm{v(X)}\ge1/\sqrt2$.

Write $v=b+w(Z-X)$, where $\norm b_\infty\le C_t+1$ and
$0\le w\le1/\tau$. All coordinates of $Z$ lie in $[0,1]$: for a
vertical projection its height is at most $\mu+3\tau<1$, and all
other pieces already have this property. Require
$\beta/\tau\le1/2$ and $\beta(C_t+1)\le1/4$.
The preliminary point $Y=X+\beta w(Z-X)$ lies in the box. If
$|X_i-Z_i|<1/2$, then $-1/2<Y_i<3/2$, and the additional displacement
$\beta b_i$ causes no clipping. The number of remaining coordinates,
divided by $m$, is at most $4r^2\le36\tau^2$. At each, the clipping
loss is at most $\beta(C_t+1)$ because $Y$ was in the box. The norm of
all loss is at most $6\tau\beta(C_t+1)\le\beta/4$. Subtracting it
from the raw displacement leaves at least $\beta/4$.

In the inside default case, let $p$ be the fraction of fourth coordinates
at most $7/4$. Since all entries are at least $-1$,
\[
 1/2\ge\mu\ge-p+(7/4)(1-p),\qquad p\ge5/11.
\]
For $\beta\le1/4$ these coordinates retain the full upward displacement,
giving norm at least $\beta\sqrt{5/11}$.

\paragraph{Above the mean interface.}
Now suppose $\mu\ge1/2$, and abbreviate $Z=Z_\mu$, $r=r_\mu$.
The fourth coordinates of $Z-X$ have mean zero, so they are orthogonal
to $\mathbf e$. Write $v=b\mathbf e+w(Z-X)$, with the scalar table
\begin{equation}\label{geo:outside-coefficients}
\begin{array}{c|cc}
\text{radius}&b&w\\\hline
0\le r\le\tau&-1+r/\tau&1/\tau+h_\mu\\
\tau\le r\le2\tau&r/\tau-1&2/r-1/\tau+h_\mu\\
r\ge2\tau&1&h_\mu
\end{array}
\end{equation}
and hence
\begin{equation}\label{geo:outside-norm}
 |b|\le1,\quad0\le w\le w_{\max}:=1/\tau+1,
 \quad b^2+w^2r^2\ge1/2.
\end{equation}
The last assertion follows from the two orthogonal components of the
first two radial regimes; adding $h_\mu$ increases the nonnegative
normal coefficient. In the last regime $b=1$. Require
$\beta w_{\max}\le1/2$. The move toward $Z$ is a convex move inside
the box, and horizontal coordinates undergo no further move or clipping.

If $b<0$, then $r<\tau$. The subsequent downward step cannot clip at
the upper boundary. Where $|(X_4)_j-\mu|<1/2$, the preliminary fourth
coordinate is nonnegative since $\mu\ge1/2$, and the following step
is at least $-\beta\ge-1$. Thus lower clipping can occur only on a
fraction at most $4\tau^2$ of the fourth coordinates. Its loss on each
is at most $\beta$, giving norm loss at most $2\tau\beta$. By
\eqref{geo:outside-norm} this leaves at least $\beta/4$.

It remains to treat $b\ge0$. Set
\[
 \sigma^2=m^{-1}\sum_j((X_4)_j-\mu)^2,\qquad
 H=\norm{(X-Z)_{1,2,3}},\qquad r^2=H^2+\sigma^2,
\]
and choose a positive rational
\begin{equation}\label{geo:eta-choice}
 \eta\le\min\left\{\frac\tau2,\frac1{4w_{\max}},
           \frac12\min\left(\frac12,\frac1{4w_{\max}}\right)\right\}.
\end{equation}
If $\sigma\ge\eta$, a uniform fourth coordinate satisfies
\[
 \sigma^2\le3\E|(X_4)_j-\mu|
        =6\E(\mu-(X_4)_j)_+.
\]
Put $c=\eta^2/12$. Since the positive part is at most three,
$\E(\mu-(X_4)_j)_+\ge\eta^2/6$ implies that a fraction at least
$\eta^2/36$ have $\mu-(X_4)_j\ge c$: otherwise its expectation
would be less than $c+3\eta^2/36=\eta^2/6$.
On those coordinates the room to the upper boundary is at least $c$.
Also $b+wc\ge c/12$. Indeed~\eqref{geo:outside-norm} and $r\le6$
imply that either $b\ge1/2$ or $w\ge1/12$; $c<1$ handles the first
alternative. For $\beta\le1$, clipping therefore retains displacement
at least $\beta c/12$ on that fraction. Its norm is at least
$\beta\eta^3/864$.

Suppose instead $\sigma<\eta$. If $wH\ge\tau/4$, the horizontal
displacement, which is not clipped, is at least $\beta\tau/4$.
Otherwise~\eqref{geo:outside-norm} gives
\[
 b^2\ge\frac12-w^2H^2-w^2\sigma^2
       \ge\frac12-\frac{\tau^2}{16}-w_{\max}^2\eta^2
       \ge\frac38,
\]
so $b\ge1/2$. If $\mu\le1$, let
$d_0=\min\{1/2,1/(4w_{\max})\}$. By~\eqref{geo:eta-choice}, at
least three quarters of the fourth coordinates have deviation from
$\mu$ at most $d_0$. Their upper room is at least $1/2$, and their
velocity is at least $b-w_{\max}d_0\ge1/4$. They retain at least
$\beta/4$ for $\beta\le1$, giving norm at least
$\sqrt3\beta/8>\beta/8$. If $\mu>1$, then $h_\mu=1$, $w\ge1$,
and $b\ge0$ in~\eqref{geo:outside-coefficients} requires $r\ge\tau$.
It follows that
\[
 wH\ge H\ge\sqrt{\tau^2-\eta^2}\ge\sqrt3\tau/2,
\]
contrary to the remaining case assumption. This exhausts all outside
points, including the upper face. At $(H_*,2)$, $r=0$ and the formula
gives $v=-\mathbf e$ exactly.

Take positive rational
$c_*\le\min\{1/8,\tau/4,\eta^3/864\}$ and positive rational
$\beta_0$ satisfying all restrictions just made. All coordinate
differences used in the formulas have magnitude at most three, so any
integer $B_1\ge C_t+6+3/\tau$ satisfies~\eqref{geo:bounded-field},
including the zero convention near inside endpoints. These choices
depend only on the previously fixed geometric constants. This completes
the proof of Theorem~\ref{thm:geometry}.
\end{proof}

\subsection{The local-access contract}
\label{geo:access}

We next state the information needed to approximate this particular
field. All probability statements below are for a fixed physical vector
$X$, averaged over a uniform physical component $I\in[4m]$ and the
public trial randomness. Write $J=1+((I-1)\bmod m)$ for its word
coordinate. Fix a common threshold $\xi\in[2/5,3/5]$ and round the
first two blocks at $\xi$. The two resulting binary words $y_1,y_2$
are fixed, although the procedures only read requested coordinates.
Any scalar-quantization offset may also be fixed throughout the following
contract and accuracy analysis.

\begin{definition}[Admissible local access]\label{geo:access-contract}
An \emph{oracle access implementation} satisfies parts (a)--(c) below
and the following operational condition. At its declared oracle level,
each supplied bit or decision routine uses a bounded tree of completed
Boolean answers, apart from public index computations. Every input,
including an input outside a correctness promise, gives Boolean answers
or public defaults. After its requested accuracies are fixed, the
oracle tree and its symbolic query roles are bounded independently of
$m$ and remain fixed under refinement of threshold-offset grids.
These conditions impose no constant bound on the physical circuit
realizing a completed oracle answer. Part (d) is an additional
\emph{physical-read hypothesis}, required only when a conclusion about
the fully expanded physical reads is explicitly invoked. All constants
are uniform in $m$, $X$, and admissible fixed offsets. Geometric and
validation radii and the reference rule below are fixed before
reliability and numerical accuracy choices. Operational, sample,
repetition, and read-density bounds may depend on the requested fixed
accuracies as well as the encoding construction.
\begin{enumerate}[label=\textup{(\alph*)},leftmargin=*]
\item\label{geo:access-validation}
Each seed has a validation decision whose failure probability can be
made any prescribed fixed positive number. For fixed $y_s$ there is
either no valid reference or a single fixed vertex $u_s$ such that, off
the validation failure event, acceptance identifies $u_s$. This
reference is independent of the actual random acceptance decision.
It is also independent of requested confidence and numerical accuracy.
If $\norm{X_s-E_u}\le5\tau$, off that failure event the seed is
accepted with reference $u$. Intermediate seeds with a proper reference
may be accepted or rejected. An accepted seed is used only through
local routines, not through a globally decoded vertex name.
\item\label{geo:access-bits}
For any accepted reference, incident-edge flags can be supplied with
arbitrarily small fixed error, and bits of its own and its actual
immediate neighbors' encodings can be supplied with any prescribed
fixed average error. The same bit guarantee holds for $E_{w_*}$ and
the encoding of its actual successor, even if neither seed is accepted.
These special words may be public or virtual.

The bit guarantee is unconditioned for each fixed relevant symbolic
role and branch: a request at a whole-word coordinate with density at
most a fixed constant times $1/m$ has the asserted error under its
given request law. It is not a guarantee conditional on selecting that
branch. Requests to different word operands may use the same complete
coordinate. The guarantee is required for the matching coordinate $J$
and for all preallocated statistical coordinates below. A finite union
of role or fixed block-type alternatives is allowed.
\item\label{geo:access-statistics}
For every finite family of bounded functions $f:[m]\to\mathbb R$
fixed by $X$, the rounded seeds' possible actual references, and their
actual candidate data, one can choose fixed sample counts so that each
sample average estimates $m^{-1}\sum_jf(j)$ to any prescribed fixed
additive tolerance with any prescribed fixed failure probability.
The law is again unconditioned under $(I,\text{trial})$, for each fixed
relevant role alternative. Every individual sampled word coordinate has
uniform marginal, or the prescribed bounded-density marginal supported
by the same bit-access guarantee. Corresponding word operands and
physical blocks are evaluated at matching coordinates.

Independent uniform samples satisfy this requirement. Other sample
schemes are permitted if their concentration statement holds for the
fixed underlying functions in this sense; samples and different batches
need not be independent after averaging over $I$.
\item\label{geo:access-finite}
\emph{Additional physical-read hypothesis.} After a fixed accuracy has
been specified, every supplied routine has a realization with a bounded
number of primitive physical read answers and bounded Boolean
processing, apart from public index computations. The bound is
independent of $m$. On every fixed branch of its complete physical
answer tree, including branches with wrong earlier answers, the
unconditioned marginal of each raw address in a physical domain of
size $M$ is at most $C/M$, with a fixed $C$. A branch that uses public
data need not make a physical read. These bounds hold
uniformly in admissible offsets, whose refinement changes comparison
constants and not address primitives or numbers of hidden-answer slots.
On each fixed symbolic answer branch, the address rule itself is
independent of the numerical values of these offsets.
\end{enumerate}
\end{definition}

Part~\ref{geo:access-bits} controls accuracy of a completed logical call
on a fixed valid reference. The optional
part~\ref{geo:access-finite} additionally controls physical reads on
every wired branch, whether or not it is accurate or executed. The
inner implementation of Section~\ref{sec:inner-gadget} satisfies this
additional hypothesis. The outer construction invokes only the oracle
contract: its ideal functional answers are subsequently replaced by
gadgets of polynomial size in the port length. Its fixed direct-read
count $H_{\rm out}$ and functional-invocation count $B_{\rm out}$,
followed by Theorem~\ref{thm:port-gadget} and
Lemma~\ref{out:routing}, account separately for that physical
substitution. We do not assert a size-independent physical read count
for the expanded outer construction. Neither application requires
per-coordinate correctness.

The physical completeness condition in part~\ref{geo:access-validation}
is obtained from a binary completeness radius $\rho_{\rm comp}>0$ by
the initial choice
\begin{equation}\label{geo:rounding-radius}
 \frac{(5\tau)^2}{(2/5)^2}<\rho_{\rm comp}.
\end{equation}
Indeed a rounded error relative to a binary coordinate costs squared
physical error at least $(2/5)^2$ for every allowed $\xi$. An
acceptance radius smaller than half the encoding's relative Hamming
distance makes its reference unique. In the inner application this
reference must lie on the path for the fixed true input tuple, exactly
as established by the validation in Section~\ref{sec:inner-gadget}.

\begin{remark}[Whole-word coordinates in the inner application]
A matching target is a whole encoding coordinate. In a product block
called $Z$, that coordinate includes both internal labels of the product.
Using the same whole coordinate in $E_u$ and $E_v$ preserves the supplied
joint request law; it does not replace the two internal labels of either
word by a diagonal pair.

For mixed public and variable products in the virtual source, the joint
density bound gives the marginal bound
\[
 p(x,y)\le\frac{C}{|A||B|}
 \quad\Longrightarrow\quad
 \sum_{y\in B}p(x,y)\le\frac C{|A|}.
\]
The product becomes an additive form on the variable operand. That form
may depend on the public operand and the discarded coordinate, but it
does not use that coordinate to select a new variable address.
Same-positive-tag products retain the whole pair, as required by
Lemma~\ref{lem:inner-binary-interface}. Lemma~\ref{ig:virtual-source}
proves this transport from the actual source and first successor to the
fixed dummy ports. Thus their virtual supply has exactly the
whole-coordinate access required by the contract.
\end{remark}

\begin{lemma}[Geometric procedures]\label{lem:geometric-procedures}
Fix the geometric constants of Theorem~\ref{thm:geometry} and an
oracle access implementation satisfying parts (a)--(c) and the
operational condition of Definition~\ref{geo:access-contract}.
For every fixed $a>0$ there is a finite local numerical schema with
properties (i)--(iii) below. Property (iv) additionally assumes the
physical-read hypothesis~\ref{geo:access-finite}.
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
\item It returns a signed value $\widehat v_I(X)$ on a fixed dyadic grid
inside $[-B_1,B_1]$. It terminates and is bounded on every input and
every trial. For every fixed $X$ with $\dist(X,\mathcal E)>10\tau$,
and conditional on each fixed admissible binary threshold and
scalar-quantization offset,
\begin{equation}\label{geo:average-accuracy}
 4\E_{I,\,\mathrm{trial}}
       |\widehat v_I(X)-v_I(X)|^2\le a^2.
\end{equation}
\item At fixed rounded seeds, correct decisions refer to at most eight
true-word roles and 32 actual piece roles. Meshes, scalar tolerances,
sample counts, and all statistical and final matching bit requests
are fixed before choosing the smaller bit-error budgets, and all final
matching requests are allocated before numerical minimization.
\item Apart from the supplied routines and public index computations,
the schema uses finitely many scalar comparisons and bounded Boolean
lookup. Its only word-coordinate address primitives are the matching
coordinate and the preallocated statistical coordinates. This remains
true on every hypothetical wrong or default branch.
If the oracle-level public spaces, probabilities, routine descriptions,
and address computations are uniformly enumerable with polynomial
support, time, and rational-probability bit complexity in a caller's
size parameter, the composed oracle-level schema has those properties
as well. This does not include physical expansion of an oracle call.
\item If in addition the physical-read
hypothesis~\ref{geo:access-finite} holds, then, once all accuracies and
implementation counts have been fixed, the fully unfolded physical
primitive tree has a fixed number $H$ of possible read slots. For each
physical domain of size $M$, its unconditioned
expected occurrence count at any fixed position, averaged over $I$
and public trials, is at most $C/M$, for a fixed sum $C$ of slot
density bounds. Both $H,C$ remain fixed under subsequent refinement
of threshold-offset grids. Under the corresponding polynomial
enumerability premise for these physical realizations, the fully
expanded physical schema also has the enumerability properties in
part (iii).
\end{enumerate}
Every choice concerns the unscaled field. In particular none depends
on a subsequently chosen $0<\beta\le\beta_0$.
\end{lemma}

\subsection{A finite list covering every close piece}
\label{geo:candidates}

From a seed reference $u$, reserve the encoding roles for $u$, its
predecessor, and its successor. Reserve its two possible incident edge
roles. For each actual incident edge list the four retained straight
segments and its three internal fillets. The trim convention uses the
same $a_0$ at both outer ends regardless of whether those ends are
shared, so these seven pieces need only their two endpoint words.
When both graph edges at $u$ exist, also list the graph-junction fillet
at $u$. The incoming direction there is the fixed downward third-block
direction, so the junction needs only $E_u$, the outgoing neighbor,
and the two flags. Apply this procedure to both seeds.

Always add the special retained vertical and its bottom fillet using
the distinguished source and its actual successor. These two pieces
are present even on a trial with no accepted seed. The ordinary
junction rule requires both graph edges and therefore never invents
another junction at the source, which has no graph predecessor.
There is no need to recognize the source by comparing hidden names.

This is the following explicit reservation schedule; duplicate roles
do no harm.
\begin{center}
\begin{tabular}{@{}lrr@{}}
\toprule
Reserved roles&Word roles&Piece roles\\
\midrule
Self, predecessor, successor for two seeds&6&0\\
Four incident edges, seven pieces each&0&28\\
Two ordinary junctions&0&2\\
Distinguished source and actual successor&2&0\\
Special vertical and bottom fillet&0&2\\
\midrule
Total upper bound&8&32\\
\bottomrule
\end{tabular}
\end{center}
On good validation and flag answers, the admitted roles are actual
pieces. A gray acceptance merely chooses whether to include one fixed
reference's list. For fixed $X,\xi$, all possible underlying actual
words and pieces in the table are fixed independently of later accuracy
choices and numerical decisions.

\begin{lemma}[Two-seed coverage]\label{geo:coverage}
Every non-special piece has either its first or its second block fixed
to an encoding of a vertex whose seed generates it. Consequently,
on the strong-completeness and flag good event, every piece at distance
at most $4\tau$ from $X$ is included in the reserved candidate list.
\end{lemma}
\begin{proof}
For $u\to v$, its first segment, first fillet, and upward toggle have
first block $E_u$. The fillet from that toggle to the block-$1$
segment, the block-$1$ segment, and its fillet to the downward toggle
have second block $E_v$. The downward toggle has both blocks $E_v$.
An ordinary graph-junction fillet at $u$ changes only blocks $3$ and
$2$, so its first block is $E_u$. In each case the stated vertex
generates the piece from its incident data. If the piece is within
$4\tau$ of $X$, its fixed block is within $4\tau$ of the corresponding
physical block. Part~\ref{geo:access-validation} of the contract
therefore accepts that seed with its true reference. Correct flags
include the piece. The two special pieces are included unconditionally.
Gray acceptance cannot remove a piece required by this argument; it
can only add or remove other actual pieces.
\end{proof}

\subsection{Numerical data and the order of their estimation}
\label{geo:numerical}

Put a finite parameter mesh, including endpoints, on each straight
piece and each angle interval $[0,\pi/2]$ of a fillet. All original
lengths are at most two, so a sufficiently fine fixed mesh gives
maximum arclength spacing at most any prescribed fixed $D>0$.
For the proof, a mesh entry means its exact point $Z'$ and its exact
tangent on its underlying actual piece. The implementation computes
approximations to these entries.

At most five endpoint-word distance statistics are needed: the four
ordinary edge roles and the special edge. Each estimates
\begin{equation}\label{geo:length-statistic}
 \ell_{uv}^2=m^{-1}\sum_{j=1}^m(E_u(j)-E_v(j))^2.
\end{equation}
Use the same coordinate in the two words. The true nonconstant
lengths are in $[d_*,1]$; clamp their estimated square roots to
$[d_*/2,2]$. The formulas for trim points, straight tangents, and
\eqref{geo:fillet}--\eqref{geo:fillet-tangent} then have only bounded
denominators separated from zero. Their coordinates are uniformly
continuous functions of the endpoint bits at that coordinate, the
lengths, and the finitely many fixed trigonometric mesh constants.
Thus correct endpoint bits and sufficiently accurate length and public
constant approximations give any desired component error, uniformly
in the coordinate and in $m$. A component error at most $z$ everywhere
gives full-vector norm error at most $2z$.

Also estimate the fourth-block mean $\mu$. For each of the at most
32 meshed roles, estimate each exact mesh point's squared distance by
the sum of its four block means,
\begin{equation}\label{geo:mesh-statistic}
 \norm{X-Z'}^2
   =\sum_{b=1}^4m^{-1}\sum_j(X_b(j)-Z'_b(j))^2.
\end{equation}
Finally reserve the corresponding outside-radius statistic for the
exact point $Z_\mu$. Physical samples and endpoint-bit samples in a
summand use the same $j$. One may use separate batches for every
statistic and alternative role. Before a numerical winner is selected,
reserve the final matching-coordinate requests for all eight word
roles, as well as the matching physical coordinate. Inactive requests
may later be omitted or use public defaults.

These data define the numerical estimator. Use the estimated mean to
choose the inside or outside prescription, with a fixed rule at equality.
For the inside prescription, select an admitted mesh entry with minimum
estimated squared distance, again with a fixed rule for ties. Clamp that
estimate at zero and take its square root. Substitute this scalar radius,
the computed tangent component, and the computed component of $Z'-X$
into the component formula~\eqref{geo:Phi}; its last regime gives the
default value. The radius is supplied by the statistic, not by a
full-vector norm computation. For the outside prescription, clamp the
outside squared-radius estimate at zero and take its square root. Use
this radius, the estimated mean, and the special matching bits in
\eqref{geo:outside}. Bounded clamping and final dyadic rounding
will make this a bounded procedure on every trial. We next prove that
the estimates are accurate and that the resulting component is stable
under approximate minimization and side selection.

Here is a precise reason that estimates based on earlier approximate
lengths still satisfy the required concentration statement. Fix $X$, the
rounded seeds, and one possible actual reference assignment. The exact
functions in~\eqref{geo:length-statistic}, $\mu$,
\eqref{geo:mesh-statistic}, and the outside-radius statistic are fixed
functions of the sample index; they do not depend on an estimated
length or a numerical winner. They have bounded ranges: a squared
physical difference is at most nine, and the sum of four such terms
is at most 36. Apply part~\ref{geo:access-statistics} to these exact
functions. On the event of correct sampled bits, earlier good length
estimates and good numerical constants perturb every implemented
summand by a uniformly small amount. For example if $|z-z'|\le t$
and $x,z,z'\in[-2,3]$, then
\[
 |(x-z)^2-(x-z')^2|
   =|z-z'||2x-z-z'|\le10t.
\]
Clamps keep all approximations in one such fixed range. The analogous
bound for replacing a physical read by its quantized representative
is the same. For the outside point, a good mean estimate changes all
fourth coordinates by at most its scalar error; the other coordinates
use special-word bits. It follows that concentration of the exact
statistics yields accuracy of all implemented statistics after these
small deterministic perturbations. No assertion conditioned on the
earlier estimates is being used.

For independent uniform sampling, this concentration needs no external
input. If $|f|\le M$, independence gives
$\operatorname{Var}(q^{-1}\sum_{j=1}^q f(K_j))\le M^2/q$.
Chebyshev's inequality gives failure probability at most
$M^2/(qz^2)$ at tolerance $z$; a union bound handles the fixed finite
list. In the outer application the batches are instead stationary
under the output-coordinate average. The concentration calculation in
Section~\ref{sec:outer-composition} supplies the same required small
failure probabilities for these fixed functions. The geometry uses
that conclusion, and does not condition on a gray acceptance pattern,
a realized mesh winner, or a particular output coordinate.

\begin{lemma}[Uniform component stability]\label{geo:stability}
Fix a positive desired component error $e$. Mesh resolution,
quantization width, scalar tolerances, and precision of public constants
can be chosen so that, whenever all relevant decisions, exact scalar
statistics, and requested bits are good, the local procedure's component
error is at most $e$ at every $X$ outside the endpoint neighborhood.
The bound holds for every output coordinate, even if the approximate
winner depends on that coordinate. It also holds with arbitrary tie
choices between numerically minimizing entries.
\end{lemma}
\begin{proof}
For the inside prescription, consider the selected mesh entry.
Let $z>0$ bound every squared-distance error, including the parameter,
bit, quantization, and scalar-statistic errors just discussed. Suppose
first that $r=\dist(X,\mathcal L)\le3\tau$. By
Lemma~\ref{geo:coverage}, a minimizing piece is present. A mesh point
at arclength at most $D$ from its projection has squared excess at most
$2rD+D^2$. Every admitted exact entry is on the actual curve, so none
has negative excess. The exact point $Z'$ underlying the approximate
winner therefore has
\begin{equation}\label{geo:winner-excess}
 0\le g=\norm{X-Z'}^2-r^2\le2rD+D^2+2z.
\end{equation}
Choose $D,z$ first small enough that the right side is at most
$\tau^2$, and then still smaller as required. Equations
\eqref{geo:point-error}--\eqref{geo:tangent-error} show that both its
point and tangent approach those of the true projection in infinity
norm, uniformly. Correct endpoint bits at the final matching coordinate
also make its computed point and tangent close to their exact values.
Clamping a squared-radius estimate at zero and taking square root is
uniformly continuous on the fixed bounded interval. Consequently its
estimated radius approaches $r$.

Each component of~\eqref{geo:Phi} is a bounded continuous expression
in that radius, the tangent component, and the components of $Z-X$.
The formulas match at $\tau,2\tau,3\tau$, and the only reciprocal
radius is protected by $r\ge\tau$ in its regime. Safe clamping can be
used at approximate regime boundaries. Uniform continuity on these
fixed compact ranges proves the desired component error. Crucially,
the infinity-norm bounds hold for every good candidate satisfying
\eqref{geo:winner-excess}; they therefore also hold when the chosen
candidate is correlated with the requested coordinate.

If the true inside radius is $r>3\tau$, the true field is default.
Every actual candidate entry has distance at least $r$. Good squared
distance estimates can yield a nondefault computed expression only
arbitrarily near the $3\tau$ cutoff as $z$ tends to zero. Near that
cutoff the coefficient of $T$ tends to zero and the coefficient of
$\mathbf e$ tends to one, uniformly on the clamped coordinate ranges.
Thus the computed value approaches default. If its radius estimate is
at least $3\tau$, it is already default. Coverage far from the curve
is not needed in either case.

For the outside prescription, use the special bits, the estimated mean
and outside squared radius in~\eqref{geo:outside}. Correct special
matching bits and good statistics make all its scalar and coordinate
arguments close to their exact values. The same protected denominators
and compact-range continuity apply, and $h_\mu$ is continuous. This
includes the special top, where the exact value is $-\mathbf e$.

Finally consider the choice between the inside and outside computations.
If it differs from the side of the true mean, a good
estimate forces $|\mu-1/2|$ to be small. In the interface slab all
endpoints remain far, and the only possible close piece is the always
included vertical. The inside computation is therefore valid for the
inside prescription extended to that slab. The outside computation is
valid for its extension using the clamped $h_\mu$. By
Lemma~\ref{geo:interface}, the two target values differ componentwise
by at most $6|\mu-1/2|$. Adding that error proves stability even with
the wrong side choice. These cases include exact mean, radius, and
minimum ties, without requiring a reliable hidden-real comparison at
zero gap.
\end{proof}

\subsection{Average accuracy and finite answer trees}
\label{geo:finite-schema}

\begin{proof}[Proof of Lemma~\ref{lem:geometric-procedures}]
\paragraph{Oracle accuracy with fixed references.}
Fix a desired $a>0$. Choose a component error $e>0$ and failure budget
$p>0$ so that
\begin{equation}\label{geo:accuracy-budget}
 4e^2+16B_1^2p<a^2.
\end{equation}
First choose the mesh, scalar tolerances, quantization width, and public
numerical precision for Lemma~\ref{geo:stability}, with slack for final
output rounding. Fix scalar sample counts that give its concentration
events a total failure probability below a chosen part of $p$.
The eight word roles, 32 piece roles, five possible length statistics,
mean, mesh-distance statistics, and outside-radius statistic now
determine a fixed number $N_{\rm call}$ of high-level completed calls. This count
includes all sampled bit requests and all matching final bit requests
over the possible relevant true references and flag alternatives.

Make validation and edge-flag errors small enough for their total budget.
For every remaining high-level bit call require average error smaller
than the allocated part of $p/N_{\rm call}$. Analyze each call on its possible
fixed reference before conditioning on acceptance or on the numerical
winner. Parts~\ref{geo:access-validation}--\ref{geo:access-statistics}
of the access contract and a union bound then charge the event of
\emph{any} required bit, decision, or statistical error by at most
$p$. Independence of these events is unnecessary. A fixed reference
does not have to be selected on every good trial; the unconditioned
analysis simply pays for its possible calls in advance. Validation
failures, including any acceptance with no proper reference, are
charged separately. The two special-word roles are charged in exactly
the same way when their supply is virtual.

On the complement of the charged failure event, component error is
at most $e$ by Lemma~\ref{geo:stability} and the planned final rounding.
On every outcome clamp the numerical answer to $[-B_1,B_1]$ and round
to a sufficiently fine fixed dyadic grid there. Detected invalid
numerical inputs may return any fixed bounded default. Undetected
incorrect choices are already included in the failure event. The true
field satisfies~\eqref{geo:bounded-field}, so the component error on
an arbitrary outcome is at most $2B_1$. Hence
\[
 4\E|\widehat v_I(X)-v_I(X)|^2
    \le4e^2+16B_1^2p<a^2.
\]
This proves~\eqref{geo:average-accuracy} unconditionally over the
output-coordinate and trial law. It does not assert correctness
conditioned on a rare selected coordinate or branch. The proof began
with arbitrary fixed admissible offsets, so it applies to each such
pair, and therefore also after averaging any chosen laws on them.

\paragraph{Finite oracle implementation.}
Here are the finite implementation and raw-address details. Fix a small
rational bin width $h>0$ and compare each physical scalar read to
thresholds $a+kh+\zeta$ on a fixed interval larger than $[-1,2]$,
where $\zeta$ is an offset modulo $h$. Clamp at the endpoints.
Representatives can depend just on the bin number, on a fixed rational
grid, and are within $O(h)$ of the input for every $\zeta$. This
error was included in the earlier tolerance choice. Comparisons whose
answers are forced by the allowed physical range can be public.
Seed bits use the common threshold $\xi$. Each offset may eventually
range over a fine finite rational grid. No numerical assertion above
requires a continuous offset distribution.

There are now only finitely many bits and bin indices in the geometric
computation. Public rational approximations to the fixed mesh sine and
cosine values, protected length normalizations, clamped square roots,
and all other numerical expressions can be tabulated to the chosen
precision. Bounded Boolean lookup implements the resulting answer
function. In particular neither an exact curve minimum nor an
unbounded hidden vertex name or transcript is processed by the local
Boolean computation. Public index arithmetic and any caller-specific
field or interpolation calculations are performed when constructing
the corresponding fixed branch and its wiring.

Wrong validation, flag, or type answers can invoke a reserved symbolic
routine on an unsuitable seed or select another reserved role. Wrong
length, radius, side, or minimum answers select a fixed mesh or lookup
case. None of these answers supplies a new word-coordinate address:
statistical coordinates remain the preallocated coordinates and final
data remain at $J$. Inactive or nonsensical roles may use public-zero
defaults, which require no query to a distinguished physical position.
Consequently direct scalar reads have the matching or prescribed
statistical marginals on every fixed hypothetical oracle branch. This
proves the structural assertion of part (iii), without inspecting the
physical realization of a supplied oracle answer. A constant number
of choices from polynomially enumerable oracle-level public spaces
has polynomial support; a constant product of rational probabilities
of polynomial bit complexity has polynomial bit complexity. Public
indices, oracle descriptions, and bounded answer tables retain the
same polynomial construction bound. This proves the oracle-level
enumerability assertion of part (iii).

\paragraph{Physical read load under the additional hypothesis.}
For part (iv), now assume the additional physical-read
hypothesis~\ref{geo:access-finite}. The supplied routines then have
its all-branch primitive bounds, even on unsuitable seeds.

Only at this point, after all requested accuracies and the repetitions
implementing them have been fixed, unfold the full lower-level answer
tree. Give unused possible nodes their prescribed public random choices
as well. It has a fixed finite number $H$ of potential primitive slots.
For a domain of size $M$, let $C_\ell/M$ bound the probability that
slot $\ell$ reads any specified position, for its fixed symbolic branch
under the unconditioned public law. A slot not using this domain has
$C_\ell=0$. Summing gives
\begin{equation}\label{geo:raw-load}
 \E_{I,\,\mathrm{trial}}
       [\text{number of potential reads of position }j]
 \le\frac{\sum_{\ell=1}^H C_\ell}{M}
 =\frac C M.
\end{equation}
This counts all wired possibilities rather than conditioning on the
executed path. Omitting an inactive occurrence only reduces its
unconditioned mass. In particular this argument does not require
accuracy on wrong branches. The high-level count $N_{\rm call}$ used for the
correctness union bound preceded amplification; the larger count $H$
is used for raw load and subsequent gate-failure accounting. It is
not fed back into the choice of the accuracy that determines $H$.

Refining offset grids changes the values of thresholds for a fixed
finite set of comparisons. It changes neither the bin width and bin
count, nor the mesh, nor the symbolic answer-tree slots, nor a
fixed branch's address primitive. Thus $H$ and $C$ remain uniform
under that refinement. Under the stated polynomial enumerability
premise for the physical routines, a constant number of their public
choices, also at unused tree nodes, again has polynomial support and
rational-probability bit complexity. Their public addresses and bounded
answer tables are constructed in polynomial time. This proves the
physical enumerability assertion of part (iv) and completes the proof.
\end{proof}

\begin{remark}[Late margins under physical realization]\label{geo:late-margins}
Assume the additional physical-read
hypothesis~\ref{geo:access-finite}. Its grid uniformity explains the
margin order for the fully expanded physical schema in part (iv).
Once the full number of physical comparison slots is fixed, a
uniform $K$-point threshold grid on an interval of length $L_0>0$
satisfies, for every fixed scalar $x$,
\[
 \Pr(|x-\xi|\le d)\le2/K+4d/L_0.
\]
For an offset grid modulo $h$, the probability of lying within $d$
of any bin boundary is at most $4/K+8d/h$. To see these bounds,
intersect the relevant interval of length $2d$ with the equally spaced
grid; each interval contains at most its length divided by the mesh
spacing plus two points. Modulo $h$ there are at most two intervals
after wraparound. These enlarged constants cover the endpoints.

Fix an output coordinate and all nonthreshold public choices on every
hypothetical branch. Each possible raw address is then fixed under
the address contract, regardless of the eventual branch. A union bound
over their fixed finite list, first choosing the offset grids fine
enough and then a positive rational margin small enough, makes the
probability of any small-gap comparison as small as prescribed,
uniformly in $X$. No search over offsets or enlargement of the
hidden-answer tree is involved. These margin losses are separate from
\eqref{geo:average-accuracy}; the latter already holds for each fixed
admissible offset. Occurrence-dependent perturbations within the margin
are allowed because the full list was charged before following any
branch. For an application using only the oracle contract, this
argument does not account for physical comparisons inside substituted
oracle gadgets. The outer construction uses its own direct-read and
functional-boundary margin analysis in
Section~\ref{sec:outer-composition}, together with the port theorem
and its separate physical substitution count.
\end{remark}

The geometric procedures are now proved independently of both circuit
constructions. Together with the port construction in
Section~\ref{sec:inner-gadget}, they establish the two interfaces used in
Section~\ref{sec:outer-composition}, completing the proof of
Theorem~\ref{thm:main}.
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/10-geometry.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/11-weak-nash.tex ====
\section{Consequences for games and markets}\label{sec:weak-nash}

Theorem~\ref{thm:main} supplies the positive failed-gate premise in
the cited reductions to equilibrium problems. We first put its
circuits into the forms those reductions use. This preparation, together
with the finite-witness checks below, makes the transfers precise for
our gate conventions and complete binary encodings. The applications
then concern weak and relative Nash equilibria, allocation and market
equilibria, and stationary equilibria in discounted games. Their
approximation requirements differ, so we specify each target relation
before stating its consequence.

\subsection{Circuit preparation for the applications}

The Boolean tolerance in Definition~\ref{def:gcircuit} is nonmonotone.
We therefore remove Boolean gates before reducing numerical tolerances
or controlling fan-out. The two stages of the following lemma are used
separately below.

\begin{lemma}[Common circuit interfaces]\label{lem:application-circuits}
There are fixed positive rational constants $\eta,\gamma,\tau,\gamma_2$
and deterministic polynomial-time constructions which, from any
End-of-Line instance $I$, produce circuits $G_{\rm nb}$ and $G_2$
using only Constant, Scale, Copy, Add, Subtract, and Less. Their parameters
lie in $[0,1]$, and every used node has a defining gate. The circuit
$G_2$ has fan-out at most two.

Every assignment of $G_{\rm nb}$ satisfying all but a $\gamma$
fraction of its gates at tolerance $\eta$, and every assignment of
$G_2$ satisfying all but a $\gamma_2$ fraction at tolerance $\tau$,
can be converted to an assignment of $R(I)$ meeting the tolerances of
Theorem~\ref{thm:main}. For rational assignments the conversion and the
resulting End-of-Line decoder take time polynomial in the input length
plus the total assignment encoding length.
\end{lemma}

\begin{proof}
Let $\eps_c,\delta_c$ be the constants in Theorem~\ref{thm:main}, and
let $G=(V,T)=R(I)$.

\paragraph{Removing Boolean gates.}
First give every used node a defining gate, if necessary. A node used as
an input but not defined by a gate can be given the self-copy gate
$\mathrm{Copy}(v;v)$, which every assignment satisfies. There are at
most $2|T|$ such nodes, because the original gates have at most $2|T|$
input positions. Thus the new gate count is at most $3|T|$; reducing
the permitted failed-gate fraction by a factor three compensates.
Unused nodes may be omitted and later assigned zero.

On this gate list, use the strong-Boolean relation of Schuldenzucker
and Seuken at the same tolerance $\eps_c$. Their Proposition~2 shows
gatewise that a strong-Boolean $\eps_c$ solution satisfies the
weak-Boolean relations in Definition~\ref{def:gcircuit}, since
$\eps_c<1/10<1/4$. Their Lemma~1 replaces each Boolean gate by a
constant-size gadget with no Boolean gates, such that an
$\eps_c/2$ solution restricts to a strong-Boolean $\eps_c$ solution.
Their Section~6 records the same reduction with a positive failed-gate
fraction: the constant-size gadgets lose only a constant factor in
that fraction~\cite[Proposition~2, Lemma~1, and Section~6]{SchuldenzuckerSeuken2019}.
The comparison gate in that paper has the opposite orientation from
$\mathrm{Less}$ here, so we swap its two inputs. The new parameters
remain bounded rationals.

Consequently there are fixed positive rationals
$\eta=\eps_c/2$ and $\gamma$ and a polynomial-time constructed circuit
$G_{\rm nb}$ using only Constant, Scale, Copy, Add, Subtract, and Less
such that any assignment satisfying its gates at tolerance $\eta$ with
at most a $\gamma$ fraction of failures yields an assignment of $G$
satisfying the original $\eps_c,\delta_c$ bounds. All scale factors
are at most one. This no-Boolean relation is monotone in its numerical
tolerance. We use that monotonicity only after this conversion; at no
point is a weak-Boolean gate reclassified at a larger tolerance.

\paragraph{Fan-out reduction with fixed slack.}
We spell out the precision and failure accounting in the fan-out
steps of~\cite[proof of Theorem~2, pp.~4--5]{BPR2016}.
Suppose $G_{\rm nb}$ has $m$ gates, put
$\gamma_0=\min\{\gamma,1/4\}$, and set
\[
 D=\lceil 8/\gamma_0\rceil,\qquad
 \eta_1=\eta/3,\qquad \gamma_1=\gamma_0/4.
\]
There are at most $2m$ outgoing wire occurrences in total, hence at
most $\gamma_0m/4$ gates with fan-out at least $D$. Delete those gates.
For each wire from a deleted output into a retained gate, supply an
independent $\mathrm{Constant}(0)$ gate. The resulting circuit has
at most $3m$ gates and fan-out less than $D$.

Given an assignment to this circuit, restore each deleted output as
zero. A failed new constant can spoil only its one receiving gate. A
new constant satisfied at tolerance $\eta_1$ changes its receiving
input by at most $\eta_1$ under restoration. For a retained arithmetic
gate satisfied at tolerance $\eta_1$, the residual after changing its
two inputs increases by at most $2\eta_1$; this uses the scale bound
one. For Less, an original input gap greater than $3\eta_1$ remains
greater than $\eta_1$ after those two changes, so its required output
also remains valid at tolerance $3\eta_1=\eta$. Therefore, if $f_1$
gates failed before restoration, at most
$\gamma_0m/4+f_1$ original gates can fail at tolerance $\eta$.
The bound $f_1\leq\gamma_1(3m)$ makes this at most $\gamma_0m$.
This also shows why a smaller numerical tolerance is used when the
replacement constants are made exact. The large-multiplier step in
the cited construction is unnecessary for our bounded scale factors.

Next route the $d>2$ outgoing wires of each remaining gate through
a balanced binary tree of Copy gates. One may use at most $2d$ new
gates and depth at most
\[
 h=\lceil\log_2 D\rceil+1.
\]
If the preceding circuit has $m_1$ gates, there are at most $2m_1$
wire occurrences, so the new fan-out-two circuit has at most $5m_1$
gates. Use the fixed parameters
\[
 \tau=\frac{\eta_1}{2(2h+1)},\qquad
 \gamma_2=\frac{\gamma_1}{5D}.
\]
When every Copy gate on a route is $\tau$-satisfied, its leaf differs
from the root by at most $h\tau$. The same arithmetic and comparison
check shows that a retained gate satisfied at tolerance $\tau$
becomes satisfied at tolerance at most $(2h+1)\tau<\eta_1$ when its
inputs are restored to the roots. A failed Copy gate has fewer than
$D$ descendant receiving gates. Charging these receivers, and charging
each failed retained gate to itself, loses at most $D$ retained gates
per failed gate. Thus at most a $\gamma_2$ fraction of failures in
the new circuit produces at most a $\gamma_1$ fraction in the preceding
one. All depths, blowups, and tolerance losses are fixed independently
of $I$.

Denote the final fan-out-two circuit by $G_2$. All instance maps have
polynomial bit cost: they copy bounded rational parameters and introduce
only fixed rational constants. Their solution maps restrict assignments,
restore zero values, or copy existing coordinates, so they also have
polynomial bit cost on rational inputs.

The resulting assignment of $G$ need not obey the particular witness
polynomial $p(M)$ in Definition~\ref{def:gcircuit}. This causes no
restriction on the decoder. The argument after
\eqref{out:endpoint-proximity} proves soundness for every real assignment
meeting the gate-failure budget, and gives a deterministic decoder for
rational assignments in time polynomial in $N$ plus their total encoding
length. Applying it proves the stated interface.
\end{proof}

\subsection{Weak approximate Nash equilibria}

Babichenko, Papadimitriou, and Rubinstein reduce positive-gap generalized
circuits to weak approximate Nash search, where a fixed fraction of
players may have deviation gain larger than a fixed positive
error~\cite[Definition~4 and Theorem~2]{BPR2016}.
Lemma~\ref{lem:application-circuits} supplies the required circuit
interface. We retain the normalization and finite-witness details of the
transfer.

An $n$-player two-action polymatrix game is specified by an undirected
interaction graph and, for each edge, a pair of rational $2$-by-$2$
payoff matrices. Each player chooses action $0$ or $1$ and receives the
sum of its payoffs on incident edges. We use matrices with entries in
$[0,1/3]$ and graphs of maximum degree at most three, so every total
payoff lies in $[0,1]$. The encoding explicitly lists all players,
edges, and matrices, so its length is at least $n$. For a product
mixed-strategy profile $\sigma$,
write
\[
 \operatorname{reg}_i(\sigma)
 =\max_{a\in\{0,1\}}u_i(a,\sigma_{-i})-u_i(\sigma)
\]
for player $i$'s possible gain from deviation. A rational profile is
encoded by the binary rational probabilities $\sigma_i(1)\in[0,1]$.

\begin{corollary}[Fixed-gap weak approximate Nash hardness]
\label{cor:weak-nash}
There are fixed positive rational constants $\eps_{\rm w},\delta_{\rm w}$
and a fixed polynomial $p_{\rm w}$ for which the following search problem
is PPAD-hard under deterministic polynomial-time many-one search
reductions. Given an explicitly encoded rational two-action polymatrix
game of maximum degree at most three with the normalization above and
complete binary length $L$, find a rational product profile of total
encoding length at most $p_{\rm w}(L)$ such that
\begin{equation}\label{eq:weak-nash}
 \bigl|\{i:\operatorname{reg}_i(\sigma)>\eps_{\rm w}\}\bigr|
 \leq \lfloor\delta_{\rm w}n\rfloor .
\end{equation}
No further regret bound is imposed on the players counted on the left.
\end{corollary}

\begin{proof}
Use the fan-out-two circuit of Lemma~\ref{lem:application-circuits},
with numerical tolerance $\tau$ and failure allowance $\gamma_2$.

\paragraph{The game reduction.}
For a fan-out-two generalized circuit, the gate construction in
Rubinstein~\cite[Section~7]{Rubinstein2018}, used in
the forward reduction of~\cite[Theorem~2]{BPR2016}, has two actions per
player and maximum degree at most three. If the players of a gate
gadget use only actions within $\xi_0$ of a best response, their node
probabilities satisfy that gate at tolerance $2\xi_0$. Choose a fixed
positive rational $\xi_0<\tau/2$. These bounds imply our no-Boolean
gate relations at tolerance $\tau$: an antecedent for Less at
tolerance $\tau$ is also an antecedent at tolerance $2\xi_0$, and
the arithmetic and output bounds are stronger. The scale gadget's
output $\min\{\alpha x,1\}$ equals $\alpha x$ here because $\alpha\leq1$.
Each gate uses a fixed number of players, and a player belongs to only
a fixed number of gate gadgets.
Hence there are fixed integers $C_1,C_2$ such that the game has at most
$C_1$ times as many players as circuit gates and each player not
meeting the well-supported condition can spoil at most $C_2$ gates.
Choose a fixed positive rational
\[
 \delta_{\rm w}\leq\frac{\gamma_2}{C_1C_2}.
\]
The decoded circuit assignment then meets its failure budget whenever
all but a $\delta_{\rm w}$ fraction of players meet that condition.

The gadget matrices have rational entries bounded in absolute value
by a fixed rational $B\geq1$: their variable scale parameters lie in
$[0,1]$, and all other gadget constants are fixed. Replacing every
payoff entry $a$ by $(a+B)/(6B)$ puts the entries in $[0,1/3]$.
For each player this adds an action-independent constant and scales
every deviation gain by $\lambda=1/(6B)$. Put $\xi=\lambda\xi_0$.
Thus the needed well-supported tolerance in the normalized game is
the fixed positive rational $\xi$.

Choose a fixed rational $t>0$ such that
$t(t+13)<\xi$, and put $\eps_{\rm w}=t^2$. We use the local pruning
argument of Rubinstein~\cite[Lemma~5]{Rubinstein2018}; this is the lemma
cited with an earlier number in the proof of BPR's Theorem~2. For every
player whose regret is at most $t^2$, remove actions whose payoff is
more than $(1+1/t)t^2$ below its best response and renormalize.
Leave the other players unchanged. The regret bound ensures that the
removed probability is less than one. The lemma's calculation, with
incoming degree at most three and total payoffs in $[0,1]$, shows that
every retained action of each formerly happy player is within
\[
 t(t+1+4\cdot3)=t(t+13)<\xi
\]
of a best response after all these changes. Unchanged unhappy
neighbors contribute no change to that calculation. The fraction of
players not meeting the well-supported condition therefore does not
increase. This gives the required circuit assignment from any profile
satisfying~\eqref{eq:weak-nash}.

\paragraph{Rational witnesses and decoding.}
All instance maps above have polynomial bit cost. The auxiliary
constants are fixed rationals, and the game matrices copy bounded
rational gate parameters through fixed-size gadgets. For a rational
profile, its regrets require only a fixed number of rational operations
per player. The pruning threshold is rational, so exact comparisons,
deletion of actions, and renormalization have polynomial bit cost.
The extracted node probabilities, restrictions, and restored zero
values consequently form a rational assignment of polynomial total
encoding length.

Lemma~\ref{lem:application-circuits} decodes this assignment in its
total encoding length, so the particular witness polynomial $p(M)$ of
Definition~\ref{def:gcircuit} imposes no additional condition here.

Finally, the finite target search relation has a polynomial witness
bound. Round the probabilities of a mixed Nash equilibrium to a grid
of denominator $q$, where the fixed integer $q$ is greater than
$3/\eps_{\rm w}$. With entries in $[0,1/3]$ and degree at most three,
each pure-action payoff changes by at most $1/q$ when the neighboring
probabilities are rounded. Changing the player's own probability
changes its payoff by at most another $1/q$. Its resulting regret is
therefore at most $3/q<\eps_{\rm w}$. A fixed polynomial $p_{\rm w}$
can encode these $n$ rational probabilities. Satisfaction of
\eqref{eq:weak-nash} is checked by rational arithmetic in polynomial
time. The composed instance map and solution decoder establish the
stated search reduction.
\end{proof}


\subsection{Relative approximate bimatrix equilibria}

For a rational bimatrix game $(A,B)$ with $A\geq0$ and $B\leq0$
entrywise, a pair of mixed strategies $(x,y)$ is a relative
$\eps_{\rm r}$-approximate Nash equilibrium if
\begin{equation}\label{eq:relative-nash}
 x^{\mathsf T}Ay\geq(1-\eps_{\rm r})\max_i(Ay)_i,
 \qquad
 x^{\mathsf T}By\geq(1+\eps_{\rm r})\max_j(x^{\mathsf T}B)_j.
\end{equation}
The plus sign in the second inequality reflects the column player's
nonpositive payoffs: each player's loss is at most $\eps_{\rm r}$
times the absolute value of its best-response payoff. These are
conditions on expected payoffs, not on every action in the support.

\begin{corollary}[Relative approximate Nash hardness]
\label{cor:relative-nash}
There are a fixed rational $0<\eps_{\rm r}<1$ and a fixed polynomial
$p_{\rm r}$ such that the following search problem is PPAD-complete
under deterministic polynomial-time many-one search reductions.
Given an explicitly encoded rational bimatrix game $(A,B)$ with
$A\geq0$, $B\leq0$, and complete binary length $L$, find a rational
mixed-strategy profile of total encoding length at most $p_{\rm r}(L)$
satisfying~\eqref{eq:relative-nash}.
\end{corollary}

\begin{proof}
Apply the implication from Conjecture~2 to Theorem~3 in
\cite[Section~4]{BPR2016}. This published chain passes through a
bipartite degree-three family of weak approximate Nash instances.
Theorem~\ref{thm:main} supplies its fixed-tolerance circuit premise on
a bounded-rational subclass, and Lemma~\ref{lem:application-circuits}
makes the gate-basis and fan-out preparation explicit. No change of
weak-Boolean tolerance is needed. The remaining polynomial reduction
gives hardness for a fixed positive relative error;
choosing a smaller positive rational error only strengthens the
requirements. Its marginal and conditional-probability decoder uses
polynomial-time rational arithmetic. Composing with the circuit
decoder after~\eqref{out:endpoint-proximity}, which is polynomial in
the total rational assignment length, gives the required search
reduction.

For membership, an exact Nash equilibrium of the same game satisfies
\eqref{eq:relative-nash}. Exact rational bimatrix Nash belongs to
PPAD, and every rational bimatrix game has an exact equilibrium of
polynomial encoding length~\cite[Theorem~1.1 and Section~2]{CDT2007Preprint}.
Choose $p_{\rm r}$ to accommodate that bound. The displayed inequalities
are checked by polynomial-time rational arithmetic, so the stated
finite search problem is total and belongs to PPAD.
\end{proof}


\subsection{Indivisible course allocation}

We also apply the course-allocation implication of Babichenko,
Papadimitriou, and Rubinstein~\cite[Theorem~4 and Appendix~B]{BPR2016}.
There are $M\geq1$ courses with nonnegative integer capacities $q_j$
and $N\geq1$ students. Each student's acceptable bundles are explicitly
listed in strict preference order, with the empty bundle as a final,
always-available choice; let $k$ be the maximum bundle size.
At prices $p_j\geq0$ and budgets $b_i>0$, each student receives its
most-preferred bundle $x_i\in\{0,1\}^M$ satisfying $p\cdot x_i\leq b_i$.
Budish's clearing convention and guaranteed threshold are
\cite[Definition~1 and Theorem~1]{Budish2011}
\[
 z_j=\begin{cases}
 \sum_i x_{ij}-q_j,&p_j>0,\\
 \max\{\sum_i x_{ij}-q_j,0\},&p_j=0,
 \end{cases}
 \qquad
 \alpha_* = \frac{\sqrt{M\min\{2k,M\}}}{2}.
\]
For positive budgets, their Gini coefficient is
\[
 \operatorname{Gini}(b)
 =\frac{\sum_{i,h}|b_i-b_h|}{2N\sum_i b_i}.
\]

\begin{corollary}[Low-Gini course-allocation hardness]
\label{cor:course-allocation}
There are a fixed rational $0<\gamma_{\rm c}<1/2$ and a fixed
polynomial $p_{\rm c}$ such that the following search problem is
PPAD-complete under deterministic polynomial-time many-one search
reductions. Given the explicit lists and capacities above, of total
binary length $L$, find rational prices and positive rational budgets,
together with the demanded bundles, of total encoding length at most
$p_{\rm c}(L)$, satisfying
\[
 \norm{z}_2\leq\alpha_*,\qquad
 \operatorname{Gini}(b)\leq\gamma_{\rm c}.
\]
\end{corollary}

\begin{proof}
Use the no-Boolean, fan-out-two circuits of
Lemma~\ref{lem:application-circuits}, with their fixed positive numerical
and failed-gate tolerances. The published reduction in
\cite[Appendix~B.2--B.3]{BPR2016} applies to this bounded-rational
subclass: after common normalization of prices and budgets, any
invalid circuit decoding forces the Gini coefficient to be at least a fixed
positive constant. Choose $\gamma_{\rm c}$ strictly below that
bound. Median normalization and price extraction have polynomial bit
cost on rational outputs. Composing with the circuit decoder after
\eqref{out:endpoint-proximity} therefore gives the hardness reduction,
with time polynomial in the input and total witness length.

For membership, Othman, Papadimitriou, and
Rubinstein~\cite[Appendix~A, Remarks~1--2]{OPR2014} prove that computing
an allocation at this clearing threshold with budgets in
$[1,1+\beta]$ belongs to PPAD in the explicit-list model, for every
fixed rational $\beta>0$. Choose a rational $0<\beta<\gamma_{\rm c}$.
Such budgets have Gini coefficient at most $\beta/2<\gamma_{\rm c}$;
existence is guaranteed by~\cite[Theorem~1]{Budish2011}.
The budget interval selects solutions for membership and is not an
additional requirement of the target problem.

Polynomial-size rational witnesses can also be seen directly.
Fix such an allocation and its zero-price pattern, and cap prices at
$2+\beta$ without changing demands. Keep the budget interval, these
price bounds and affordability of the chosen bundles. Introduce a
common slack $s\in[0,1]$ for the unaffordability of every strictly
preferred listed bundle and for the positivity of every positive
price. Maximizing $s$ over the resulting bounded rational polytope
has a positive optimum and a rational vertex optimum of polynomial
bit length. Demands and the price-sign pattern, hence clearing error,
are preserved. All target conditions are checked by rational
arithmetic and enumeration of the explicit lists; the clearing test
is equivalently $4\sum_jz_j^2\leq M\min\{2k,M\}$.
\end{proof}


\subsection{Markets with approximately optimal bundles}

The market reduction of Deligkas, Fearnley, Hollender, and
Melissourgos~\cite{Fisher2026} uses a ternary circuit intermediary.
We record it first; it will also supply the input to the single-controller
game reduction below.

A Pure-Circuit instance consists of explicit finite lists of nodes and
gates. An assignment $z$ gives each node a value in $\{0,1,\bot\}$.
A gate $\mathrm{NAND}(u,v;w)$ requires $z_w=0$ if $z_u=z_v=1$,
and $z_w=1$ if either input is zero; otherwise it imposes no condition.
A gate $\mathrm{PURIFY}(u;v,w)$ requires at least one of $z_v,z_w$
to lie in $\{0,1\}$, and requires $z_v=z_w=z_u$ whenever
$z_u\in\{0,1\}$.
Each gate uses three distinct nodes, and each node is the output of
exactly one gate. We use the version in which each node is also an input
to exactly one gate~\cite[Section~2.2.1]{Fisher2026}.

\begin{lemma}[The input-once Pure-Circuit interface]
\label{lem:application-pure-circuit}
There are a fixed positive rational $\delta_F$ and a deterministic
polynomial-time reduction from End-of-Line to NAND/PURIFY circuits of
the form just defined. Every ternary assignment satisfying all but a
$\delta_F$ fraction of the gates yields an End-of-Line solution in
polynomial time.
\end{lemma}
\begin{proof}
Apply \cite[Theorem~2.2 and Lemma~A.2]{Fisher2026} to the
no-Boolean, fan-out-two circuits of
Lemma~\ref{lem:application-circuits}. The published bridge gives a fixed positive failure allowance and makes
every node an input to exactly one gate. Its solution decoder returns
rational unary-count values, to which the decoder in
Lemma~\ref{lem:application-circuits} applies. All comparisons and
arithmetic have polynomial bit cost. No change of tolerance for a
weak-Boolean gate is required.
\end{proof}

We now define the simple market families used in the reduction.
There is one unit of each good, and utilities have the separable form
\[
 u_i(x_i)=\sum_j\min\{a_{ij}x_{ij},b_{ij}\},
 \qquad a_{ij}\geq0,\quad b_{ij}\in[0,\infty].
\]
All finite data are rational and explicitly encoded. Each buyer has
nonzero utility for only a bounded number of goods, each good has
nonzero utility for only a bounded number of buyers, and the ratio of
nonzero slopes for each buyer is bounded. These bounds are constants
for the family. Every buyer has at least one uncapped term of positive
slope. Fisher buyers have budget one. In the exchange version, every
buyer initially owns $1/|B|$ units of every good, where $B$ is the set
of buyers; in particular the endowments are identical and the economy
is strongly connected.

For errors $\eps_{\rm m},\delta_{\rm m}>0$, an approximate equilibrium
consists of prices $p_j\geq0$ and bundles $x_{ij}\geq0$. Each bundle
must be affordable and give its buyer at least $1-\delta_{\rm m}$
times the maximum utility attainable within that budget; each good
satisfies $\abs{\sum_i x_{ij}-1}\leq\eps_{\rm m}$. In the exchange
case the budget is the value of the buyer's endowment at $p$.
Affordability is exact. Outputs are rational prices and bundles of
polynomial encoding length; the rational-witness formulation is
discussed in~\cite[Section~2.1]{Fisher2026}.

\begin{corollary}[Approximate market equilibrium hardness]
\label{cor:market-hardness}
For every fixed $0<\eps_{\rm m}<1/9$, there is a fixed
rational $0<\delta_{\rm m}<1$ such that finding an
$(\eps_{\rm m},\delta_{\rm m})$-approximate equilibrium in the simple
Fisher markets above is PPAD-hard under deterministic polynomial-time
many-one search reductions. The same holds for their exchange-market
versions with identical endowments and strongly connected economies.
\end{corollary}

\begin{proof}
Use the circuits of Lemma~\ref{lem:application-pure-circuit}.
For each fixed $\eps_{\rm m}<1/9$,
\cite[Proposition~4.4]{Fisher2026} supplies a fixed positive $\delta_{\rm m}$ and a polynomial-time
reduction to the stated Fisher markets. Composing its solution decoder
with the circuit decoders proves the Fisher assertion of
\cite[Theorem~4.1]{Fisher2026} without its conjectural premise.
For rational $\eps_{\rm m}$, the other constants may be chosen rational
within the reduction's strict margins. For a nonrational fixed
$\eps_{\rm m}$, choose a rational clearing tolerance strictly between
$\eps_{\rm m}$ and $1/9$ and use its reduction: every solution with
the smaller clearing error also satisfies the larger one.
The Pure-Circuit bridge returns rational unary-count values;
for rational market witnesses, all comparisons and decoding have
polynomial bit cost. The decoder after~\eqref{out:endpoint-proximity}
applies in the total length of the resulting rational assignment,
without requiring the particular bound $p(M)$ in
Definition~\ref{def:gcircuit}.

For the exchange markets, normalize prices so that
$\sum_jp_j=|B|$. Every buyer's endowment then has value one, so the
same prices and bundles give a Fisher equilibrium with precisely the
same two errors. The positive-slope uncapped terms ensure that the
price sum is nonzero at any approximate equilibrium. This is the
transfer in~\cite[Appendix~B and Theorem~B.1]{Fisher2026}; normalization
also preserves polynomial rational encoding length.
\end{proof}

\subsection{Stationary equilibria in discounted games}

Daskalakis, Golowich, and Zhang~\cite{DGZ2023} obtain conditional
hardness for stationary equilibria whose incentive constraints are
averaged over an initial distribution. Their premise also follows
from Theorem~\ref{thm:main}.

A finite two-player discounted stochastic game has explicitly encoded
rational transition probabilities, rewards, and initial distribution
$\mu$. A stationary joint policy $\pi$ assigns a distribution on action
pairs to each state, sampled afresh at every visit. Write $\pi_{-i}$
for the other player's marginal policy and use normalized values
\[
 V_i^\pi(\mu)=(1-\gamma)\E_{\mu,\pi}
                 \sum_{t\geq0}\gamma^t r_i(s_t,a_t).
\]
The policy is an $\eps$-stationary Markov coarse correlated equilibrium
(CCE) when
\[
 \sup_{\sigma_i}
 \bigl(V_i^{\sigma_i\times\pi_{-i}}(\mu)-V_i^\pi(\mu)\bigr)
 \leq\eps\qquad(i=1,2),
\]
where $\sigma_i$ ranges over stationary policies chosen independently
of any action recommendation. These constraints use the supplied
$\mu$, rather than requiring the bound at every state. An
$\eps$-stationary Nash equilibrium has the same supplied-$\mu$
incentive bound and additionally requires $\pi$ to be a product policy.

\begin{corollary}[Stationary stochastic-game hardness]
\label{cor:stationary-game-hardness}
For some fixed rational $\eps_{\rm s}>0$, finding an
$\eps_{\rm s}$-stationary Markov CCE is PPAD-hard under deterministic
polynomial-time many-one search reductions, even for two-player
turn-based general-sum games with discount $\gamma=1/2$, rewards in
$[-1,1]$, and at most two actions for the active player at each state.
The inactive player's action affects neither rewards nor transitions.
The same hardness holds for $\eps_{\rm s}$-stationary Nash equilibria
with the same supplied-initial-distribution convention. Outputs are
rational policy tables of polynomial encoding length.
\end{corollary}

\begin{proof}
Use the no-Boolean circuit $G_{\rm nb}$ of
Lemma~\ref{lem:application-circuits}, with its fixed positive rational
tolerance $\eta$ and failure allowance $\delta_{\rm nb}$ (the constant
$\gamma$ of that lemma). The target basis in
\cite[Definition~8 and Appendix~D]{DGZ2023} consists of exact zero/one
constants, clipped linear gates, and comparison gates. For Scale and
Copy, repeat the input with coefficient pairs $(\alpha/2,\alpha/2)$
and $(1/2,1/2)$, respectively; Add and Subtract use $(1,1)$ and
$(1,-1)$. A Constant$(\alpha)$ uses its own exact-one node followed
by the scaled linear gate. Retain Less: each strict source comparison
antecedent implies the corresponding weak target antecedent, and the
target output error is smaller. Choose a positive rational target
tolerance below $\min\{\eta,1/12\}$. Each original gate uses at most
two target gates, and each failed target gate can spoil only its own
original gate. A target failure fraction at most $\delta_{\rm nb}/2$ therefore
suffices. This proves the positive-gap premise in
\cite[Conjecture~10]{DGZ2023}; no weak-Boolean tolerance is changed.

Apply the second assertion of~\cite[Theorem~4]{DGZ2023}. Its turn-based
construction has the stated discount, reward bounds and binary
choices. Taking statewise marginals of a joint policy in such a game
gives a product policy with the same values and unilateral deviation
gains, so its approximate stationary CCE decodes to an approximate
stationary Nash equilibrium with the same error. Choose a smaller
positive rational error if needed.
The fixed reduction parameters can be chosen rational with slack.
For rational policy tables, policy evaluation solves rational linear
systems, and the gate decoder and restrictions have polynomial bit
cost. The decoder after~\eqref{out:endpoint-proximity} then applies in
the total rational assignment length. Polynomial-size rational
approximate policy tables exist by rounding stationary Nash policies
at this fixed discount, as in~\cite[Section~2.5]{DGZ2023}.
\end{proof}

\subsection{Restricted stationary-game families}

The preceding consequence also supplies the input for the triangular
zero-sum construction of Park, Zhang, and Ozdaglar~\cite{PZO2025}.
Here the interaction graph concerns the whole auxiliary function,
not just immediate rewards: for every $V:S\to\mathbb R$, each
\[
 Q_i^V(s,a)=r_i(s,a)+\gamma\sum_{s'}P(s'\mid s,a)V(s')
\]
must be a sum of functions of $(s,a_i,a_j)$ over neighbors $j$ of $i$.
This is the Q-function separability of~\cite[Definition~1]{PZO2025}.
Use the same normalized values and supplied-initial-distribution CCE
inequalities as above, now for all three players, with $\pi_{-i}$ the
joint marginal of the other two players.

\begin{corollary}[Triangular zero-sum stationary-game hardness]
\label{cor:triangle-stationary-hardness}
For some fixed rational $\eps_{\triangle}>0$, finding an
$\eps_{\triangle}$-stationary Markov CCE is PPAD-hard under deterministic
polynomial-time many-one search reductions even for rational three-player
zero-sum games with Q-function-separable interactions on the fixed
triangle $K_3$, discount $1/2$, and each player's stage reward in $[-1,1]$.
The hard games are turn-based, with at most two actions for the active
player and a third player with only one action. Outputs are rational
joint-policy tables of polynomial encoding length; incentives are
required only from the supplied rational initial distribution.
\end{corollary}

\begin{proof}
Apply the triangle construction in~\cite[Appendix~C, Case~1]{PZO2025}
to Corollary~\ref{cor:stationary-game-hardness}.
Write $c(s)\in\{1,2\}$ for the state controller and suppress inactive
actions. Add a one-action third player and retain states, transitions
and initial distribution. Specify oriented edge rewards before scaling by
\[
 (b_{12},b_{13},b_{23})=
 \begin{cases}
 (-r_2,r_1+r_2,0),&c(s)=1,\\
 (r_1,0,r_1+r_2),&c(s)=2,
 \end{cases}
 \qquad b_{ji}=-b_{ij}.
\]
Each term depends only on the actions at its edge. Divide every edge
reward by two. The resulting total rewards are
$(r_1/2,r_2/2,-(r_1+r_2)/2)$, lie in $[-1,1]$, and sum to zero.
For every test function $V$, the continuation term in $Q_i^V$ depends
only on $a_{c(s)}$. Allocate it to edge $\{i,c(s)\}$ when $i\ne c(s)$,
and to any incident edge when $i=c(s)$. This proves the required
separability on $K_3$ for every player and every $V$.

The state process of any new-game policy agrees with that of its
first-two-player restriction in the original game. Each original
player's normalized value and deviation gain in the new game is half
the corresponding quantity in the original game; the dummy has no
deviation. Thus error
$\eps_{\triangle}=\eps_{\rm s}/2$ suffices at the unchanged initial
distribution. The primary source uses unnormalized values, which are
twice our normalized values at discount $1/2$; the stated error
calculation uses our convention throughout. All new data use only
rational addition and halving, and restriction has polynomial bit cost.
Polynomial-size rational witnesses follow by the same fixed-discount
rounding argument as above.
\end{proof}

The single-controller construction of Farina, Kontogiannis, Panageas,
and Pollatos~\cite{FKPP2026} gives a different restricted family.
One player controls transitions at every state, while both players'
actions may affect rewards. Joint policies need not be product policies.

\begin{corollary}[Single-controller stationary-game hardness]
\label{cor:single-controller-hardness}
For some fixed rational $\eps_{\rm sc}>0$, finding an
$\eps_{\rm sc}$-stationary Markov CCE is PPAD-hard under deterministic
polynomial-time many-one search reductions for rational two-player
single-controller games with discount $1/16$, rewards in $[0,1]$,
and at most two actions per player per state. In the hard instances,
the initial law is uniform on the circuit-output variable states of
the cited construction and gives no mass to its absorbing sinks.
Outputs are rational joint-policy tables of polynomial encoding length.
The same assertion, with the same error, holds for every fixed rational
$\gamma\in[1/16,1)$.
\end{corollary}

\begin{proof}
Lemma~\ref{lem:application-pure-circuit} supplies input-once
NAND/PURIFY circuits with a fixed positive failed-gate allowance
$\delta_F$. Convert each NAND$(u,v;w)$ to
\[
 \mathrm{NOT}(u;a),\qquad \mathrm{NOT}(v;b),\qquad
 \mathrm{OR}(a,b;w),
\]
using fresh private nodes $a,b$, and keep PURIFY unchanged.
These are the ternary relations on $\{0,1,\bot\}$
of~\cite[Definition~2.6]{FKPP2026}: if $u=v=1$, the NOT outputs are
both zero and OR forces $w=0$; if either input is zero, its NOT output
is one and OR forces $w=1$, even when the other input is $\bot$.
In all other cases NAND imposes no condition. PURIFY has the same
relation in both bases.
Every node remains the input of exactly one gate, and outputs remain
uniquely defined. An original failed gate charges a failure in its own
disjoint replacement block. Since $m$ original gates give at most $3m$
new gates, a failed-gate allowance $\delta_F/3$ suffices. No numerical
tolerance of a weak-Boolean gate is changed.

This establishes the premise of~\cite[Conjecture~1]{FKPP2026}.
Apply its Theorem~5.5, choosing the fixed reduction parameters and a
positive rational equilibrium error within the strict bounds in its
equations~(23)--(24). Its states consist of circuit-variable states
and constantly many absorbing sinks; the initial law is uniform
precisely on the output-variable states. For a rational joint policy,
equation~(8) of that paper decodes a node from the reward player's
marginal $p$: output zero for $p\leq1/4$, one for $p\geq3/4$, and
$\bot$ otherwise. Marginalization, comparison and restriction have
polynomial bit cost. Follow these by the Fisher unary-count decoder
and the decoder after~\eqref{out:endpoint-proximity}.
At fixed discount, rounding stationary equilibrium policies with
slack gives polynomial-size rational witnesses; policy evaluation and
best-response optimization are rational finite-state discounted MDP
computations. No collapse of correlated policies to Nash policies is
used in this reduction.

Finally, the extension in~\cite[Remark~5.7]{FKPP2026} preserves the
normalized incentive inequalities exactly. Put $\beta=1/16$ and, for
a fixed rational $\gamma\in[\beta,1)$, set
\[
 \tau=\frac{\beta(1-\gamma)}{\gamma(1-\beta)},\qquad
 \widetilde P=(1-\tau)I+\tau P.
\]
For every stationary joint or unilaterally deviated policy $\pi$,
\[
 I-\gamma\widetilde P^\pi
 =\frac{1-\gamma}{1-\beta}(I-\beta P^\pi).
\]
Thus $(1-\gamma)(I-\gamma\widetilde P^\pi)^{-1}r_i^\pi$
equals $(1-\beta)(I-\beta P^\pi)^{-1}r_i^\pi$.
Rewards, actions and initial law are unchanged, and the lazy transition
remains rational and single-controller with polynomial encoding length.
\end{proof}

\subsection{Hylland--Zeckhauser equilibria}

The reduction of Braverman, Liu, Xue, and Zhou~\cite{HZ2026}
also yields constant-error hardness for one-sided allocation with
unit demand. An instance has $n$ agents, $n$ unit-supply goods, and
an explicitly encoded rational utility matrix $u\in[0,1]^{n\times n}$.
In their approximate Hylland--Zeckhauser (HZ) equilibrium
\cite[Definition~2.1]{HZ2026}, an allocation $x\geq0$ and prices
$p\geq0$ satisfy
\[
 \sum_i x_{ij}=1,\qquad \sum_j x_{ij}=1,\qquad \min_jp_j=0,
\]
and, for each agent $i$,
\[
 p\cdot x_i\leq1+\eps,\qquad
 u_i\cdot x_i\geq
 \max\{u_i\cdot y:y\geq0,\ \sum_jy_j=1,\ p\cdot y\leq1\}-\eps.
\]
Thus clearing and unit demand are exact; the additive errors concern
expenditure and utility, with comparison budget one.

\begin{corollary}[Approximate HZ equilibrium hardness]
\label{cor:hz-hardness}
There are a fixed rational $\eps_{\rm h}>0$ and a fixed polynomial
$p_{\rm h}$ such that finding an $\eps_{\rm h}$-HZ equilibrium with
rational prices and allocation of total encoding length at most
$p_{\rm h}(L)$ is PPAD-hard under deterministic polynomial-time
many-one search reductions, where $L$ is the input encoding length.
\end{corollary}

\begin{proof}
Use the fan-out-two circuit of Lemma~\ref{lem:application-circuits},
with its fixed positive numerical tolerance $\tau$ and failure
allowance $\gamma_2$. Its no-Boolean gate relations agree with
\cite[Table~1]{HZ2026}, so no Boolean tolerance is changed.
The reduction underlying~\cite[Theorem~3.2]{HZ2026} has two stages:
Lemma~3.4 of that paper converts these circuits to bounded-degree
threshold games, and Lemma~3.25 converts those games to HZ instances.
The construction uses a fixed integer $m$ and rational market data,
with HZ error $m^{-40}$. In that construction, each threshold-game
vertex $v$ has an associated group $G_{v,1}$ of goods. The first
decoder assigns $v$ the value $\min\{1,m^2p(G_{v,1})\}$, where
$p(G_{v,1})$ is the minimum price in that group. This decoder, the
threshold-game-to-circuit decoder, and the subsequent circuit decoders
have polynomial bit cost on rational witnesses. Composing
with the decoder after~\eqref{out:endpoint-proximity} therefore gives
an End-of-Line solution in time polynomial in the input and total
rational witness length.

For completeness, polynomial-size rational HZ witnesses preserve the
exact equalities above. Start from a normalized exact equilibrium
$(x,p)$, as in~\cite[Lemma~B.2]{HZ2026}; summing budgets and column
sums gives $\sum_jp_j\leq n$.
Round prices upward on a rational grid of mesh at most
$\eps_{\rm h}/4$, retaining zero prices. Express $x$ as a convex
combination of at most $n^2$ permutation matrices and round its
weights, preserving nonnegativity and total weight one, so each row
changes in $\ell_1$ norm by at most $\eps_{\rm h}/[4(n+1)]$.
A denominator of order $n^3/\eps_{\rm h}$ suffices. Row and column
sums remain exact, expenditures increase by less than $\eps_{\rm h}$,
and utilities decrease by less than $\eps_{\rm h}$. Since prices only
increase, the comparison optimum cannot increase. These rational
witnesses have a fixed polynomial encoding bound; their utility
conditions are checked by rational linear programming.
\end{proof}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/11-weak-nash.tex ====

==== BEGIN The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/12-extensions.tex ====
\section{Further algebraic and binary representations}
\label{sec:extensions}

The constructions above use separate algebraic representations at the
outer and inner scales, and retain only the binary products needed for
functional evaluation. We record two extensions of these representations.
The first applies the inner algebraic charts directly to the End-of-Line
macrograph. The second allows unrestricted binary products and checks
proposed moves. These extensions are independent of the proof of
Theorem~\ref{thm:main} and the consequences in Section~\ref{sec:weak-nash}.

\subsection{A polynomial chart representation of the End-of-Line macrograph}
\label{ia:eol-variant}

The polynomial-chart construction also realizes the full reciprocal
End-of-Line macrograph, including its other paths and cycles, without
the standardized-port requirement. We record this separate application
of the same algebraic tools. Its size
is polynomial; the quasilinear outer representation used in the main
reduction remains the construction of Sections~\ref{sec:outer-codes}--
\ref{sec:outer-representation}.
All field and chart notation in this subsection is local to this separate
instantiation. The fields, allowance, and one-port templates fixed in
Section~\ref{sec:inner-algebra} retain their original meaning outside
this subsection.

\begin{proposition}[Full macrograph polynomial representation]
\label{ia:eol-algebra-interface}
For a normalized End-of-Line instance of encoding length \(N\), one
can construct in deterministic polynomial time a cyclic macrograph and
its progress-path extension with the following properties.
\begin{enumerate}[label=(\roman*)]
\item The graph has indegree and outdegree at most one, no isolated
vertices or self edges, and a public exceptional source with a public
successor. Every other endpoint yields an End-of-Line answer from
its logical tape in polynomial time. Other paths and cycles are
retained.
\item Its valid extended labels have unique canonical polynomial
chart tuples on one fixed-dimensional domain \(K^u\). There are a
fixed number of chart families, native dimensions \(O(\ell)\),
a bounded number of \(O(\ell)\) slot axes, and honest coordinate
degree \(O(\ell h)<D\). All full evaluation, mask, coefficient,
and indexing arrays have polynomial size. The construction does
not enumerate the graph's vertices.
\item The tuples have a perfectly complete bounded binary-form
algebraic validator with a constant exact gap on genuine
degree-at-most-\(D\) polynomials. Linear probes are smooth; each
unknown product uses two independent full point marginals. Canonical
recipes with the same degree and covariance properties are defined
on every Boolean logical tape at every specified frame, mode, and
progress. An invalid label so represented has the same exact gap.
\item Mode, empty/full progress, and the critical bits for a proposed
edit admit bounded smooth decisions at any fixed reliability. Once
the true decision branch is fixed, either proposed canonical tuple has the
bounded affine update formulas of~\eqref{ia:affine-update}, including
its full proof data, whether or not the proposed target is valid.
The coefficients are independent of point and row position, and
the updates commute with common linear row encodings and ordered-line
coefficient encodings.
\end{enumerate}
Here the fields and all fixed constants are chosen for this whole
construction. No common one-port source or radius-before-arity
conclusion is asserted. Target validity is tested after prediction,
as in Proposition~\ref{ib:proposal-validation}; it is not presumed
by an affine update formula.
\end{proposition}

\begin{proof}
\paragraph{Cyclic schedule and Boolean validity.}
Use the normalized transcript positions and the two orders of
Section~\ref{macro:transcripts}, renaming their length \(T=O(N)\).
For this variant keep two content tapes \(A,B\); their occupied
prefix lengths are specified by the public stage, so no occupation
coordinates are needed. With the notation \(Q_x,U_y,V_y\) from
that section, use the following four stage conditions:
\[
\begin{array}{c|l}
 0\le s\le T
   & A=Q_x,\quad B\text{ a forward prefix of length }s
                           \text{ from }S(x),\\
 T\le s\le2T
   & B=Q_y,\quad A\text{ a reconstruction prefix of length }2T-s
                           \text{ from }y,\\
 2T\le s\le3T
   & B\text{ a valid full transcript},\quad
                    A=\operatorname{pref}_{\pi_{\mathrm f},s-2T}(B),\\
 3T\le s\le4T
   & A\text{ a valid full transcript},\quad
                    B=\operatorname{pref}_{\pi_{\mathrm f},4T-s}(A).
\end{array}
\]
The boundary descriptions agree. At \(s=T\), the first gives
\(A=Q_x,B=Q_y,y=S(x),P(y)=x\); hence reconstruction from \(y\)
gives exactly \(A\). Conversely, a complete valid reconstruction
from \(y\) gives \(x=P(y)\), \(S(x)=y\), and the same pair.
At \(2T\), \(3T\), and \(4T\) the descriptions respectively say
that \(A\) is empty, that \(A=B\) is full, and that \(B\) is empty.
The last is the stage-zero hub condition.

Choose a fixed polynomial scale \(n_G=C_G(N+2)^2\), with \(C_G\)
large enough that \(n_G\ge4T+3\) and all fixed lower cutoffs and
degree inequalities below hold, and use~\eqref{ia:parameters}--
\eqref{ia:radix-size} with \(n'\) replaced by \(n_G\). In
particular \(h^\ell\ge n_G^3+3\), \(M=h^\ell-1\), and
\(C=K_1^\times=\langle\gamma\rangle\). Pad stages
\(4T,\ldots,M-1\) by the hub condition, and identify stage \(s\)
with \(c=\gamma^s\) by enumerating the group. No discrete logarithm
is needed. All hub-to-hub steps, including the wrap to stage zero,
are allowed.

The forward tape proposals are, in order: fill the next forward
\(B\) position; erase the last reconstruction \(A\) position;
copy the next \(B\) position into \(A\); erase the last copied
\(B\) position; or do nothing at a padded hub. The reverse proposals
are respectively erasure, local reconstruction, erasure, copying,
and the identity. A fill uses its computed bit even if the new
terminal comparison fails. An actual edge is exactly a proposed
pair whose two stage conditions hold. Prefix uniqueness from
Section~\ref{macro:transcripts} makes the operations reciprocal
on such pairs: earlier operands or the unchanged full tape determine
each erased bit. Thus both degrees are at most one.

Add a legacy vector \(e\) on \(C\), one-hot at \(c^{-1}\), and
write \(b=(A,B,e)\). Impose bitness, the one-hot equations
\eqref{ia:one-hot}, and, for each stage \(c\), multiply every tape
constraint at that stage by \(e_{c^{-1}}\). Gate constraints have
degree at most two; blank, copy, and terminal-comparison constraints
are linear. The resulting \(O(MT+M^2)\) rows have degree at most
three and describe exactly the union of valid stage tapes. Their row and
column index sets are polynomial in \(N\), hence require only
\(O(\ell)\) digit coordinates in the linear verification tool.
Padding other tape coordinates by zero to full digit grids preserves
these bounds. A proposal changes at most one content coordinate and the
two legacy coordinates \(c^{-1},(\gamma c)^{-1}\). Use these three
unit vectors, or a zero content vector at hubs, as a fixed binary
flip alphabet. The coefficient of the content flip is the old bit
plus its required new value. Either direction needs only the old
bit, at most two operands or one copy bit, and finitely many public
phase and rule bits. The same rule defines proposals on every
Boolean tape, without a validity call.

\paragraph{Endpoints and progress paths.}
A forward prefix can always be shortened, so a missing successor
is its first failed terminal comparison. With \(A=Q_x\), put
\(y=S(x)\). Then \(P(S(y))\ne y\) and \(P(y)=x\). If \(y=0\),
the source promises would give \(x=P(0)=0\) and \(S(0)=0\), a
contradiction. A reconstruction prefix can always be shortened in
the forward direction; a missing predecessor gives
\(y=\operatorname{in}_S(B)\) and \(S(P(y))\ne y\). Copy and hub
stages have both neighbors. Thus there are no isolated vertices.
The only zero exception is \(B=Q_0\) with its longest valid
reconstruction prefix from zero. This transcript exists because
\(P(S(0))=0\), and its maximal length \(k_*<T\) follows from
\(S(P(0))=S(0)\ne0\). Its stage is \(2T-k_*\), and its successor
shortens the prefix or starts copying if \(k_*=0\). Direct circuit
evaluation computes this source and successor.

For each macro tape at stage \(c\), introduce the path
\[
 (\gamma^{-1}c,b,1,q-1),\ldots,(\gamma^{-1}c,b,1,0),
 (c,b,0,0),\ldots,(c,b,0,q-1).
\]
Attach each actual macro edge from the full pre end of one path
to the full post start of the next. This is the four-operation
extension of Definition~\ref{ia:extended-path}, with actual
macroedge validity in place of that definition's disabled-wrap
rule. The replacement paths are disjoint, so they preserve all
components, endpoints, and degree bounds. Every edge changes a
mode or progress coordinate. If the exceptional macro source is
\((c_*,b_*)\), its extended source and successor are
\((\gamma^{-1}c_*,b_*,1,q-1)\) and
\((\gamma^{-1}c_*,b_*,1,q-2)\). At another endpoint, scanning the
legacy vector and transcript gives the appropriate nonzero vector
above in polynomial time.

\paragraph{Canonical arrays without positive tags.}
Use the ordinary order-one lifts for \(A,B\), the cyclic lift of
Lemma~\ref{ia:cyclic-lift} for \(e\), and the constant chart. Include
\emph{every} ordered monomial type list of length at most three
in these groups, with independent argument groups even for repeated
types. There are constantly many lists. Recursive factorization
fixes the monomials from the order-one charts. Apply
Lemma~\ref{ia:linear-tool} to their extension, zero-padding, bitness,
and cubic validity rows. A coincident-grid product in a validity
row is a linear condition on a tensor entry, so its verification
uses that lemma rather than a diagonal unknown-product query.
This fixes all its canonical certificates uniquely.

Adjoin \(J_c\), the clock-times-monomial charts, and the clock/legacy
identity~\eqref{ia:clock-legacy}. The clock-indexed construction
of Lemma~\ref{ia:clock-formula} provides every critical logical bit
and every public rule bit for both proposal directions. Their
addresses are indexed by the absolute frame \(cz\); they do not
require reading the unknown frame as a binary word. All these
canonical formulas are defined for arbitrary Boolean \(b\), even
when a final validity row fails.

For every persistent chart of positive assigned Boolean degree,
including every verification auxiliary, and each of the eight
flip masks, apply the derivative construction~\eqref{ia:derivative}
using the same frame in both terms. The top Boolean degree cancels,
so the derivative and all certificates introduced to verify it
have degree at most one less. Construct the coefficient maps from
the parent's specified canonical recipe by
\eqref{ia:boolean-inversion}, and repeat on every new positive-degree
family. After at most three drops the process stops. It updates
the certificates themselves and does not assume their validity.
The number of families is fixed. Each persistent family has at
most one \(O(\ell)\) slot axis, and finitely many applications of
the degree estimate~\eqref{ia:tool-degree} leave coordinate degree
\(O(\ell h)<D\). Give all inherited projection axes consistent
full \(K\) labels and reserve a new label for each derivative
axis; a fixed common \(K^u\) suffices. No positive-tag reuse or
one-port identification is needed.

\paragraph{Validation, affine updates, and size.}
Use exactly the mode, radix marker, and projection stacks of
Lemma~\ref{ia:stacks}. Their triangular recurrences uniquely fix
all coefficients. At full progress, the singleton-stack sums
\eqref{ia:critical-projection} supply the constant-polynomial
critical values for the finite proposal rule. For every persistent
chart and every stack subset, the summed derivative stack in
\eqref{ia:edit-increment} gives its exact change at the fixed frame.
These simultaneous identities hold on every Boolean tape. Increasing
and decreasing progress use~\eqref{ia:increase-zero}--
\eqref{ia:decrease}; frame advancement occurs only with empty stacks
and uses the designated scaling by \(\gamma\). Thus the complete
proposed target, including every proof chart, obeys the bounded
affine formulas. Public coefficients and mask pulls are independent
of target point and row; a common linear row extension and
line-coefficient extraction commute with them.

The exact-gap argument of Lemma~\ref{ia:exact-validator} now applies
to this finite family: first enforce shape, independence, and
linearity, then monomial and clock identities, then all unique
linear certificates, and finally the marker and stack recurrences.
If every identity holds, there is one valid tape and its canonical
tuple. Otherwise a nonzero polynomial discrepancy, after the stated
earlier linearity conditions, is detected with a fixed positive
probability by uniform points, separable masks, and binary forms.
Products of nonconstant monomial factors use their independent
argument groups; clock products use a fresh independent group;
marker products are sampled at independent points after constancy
is enforced. Ignored coordinates are filled independently. Hence
all unknown products have independent full point marginals, and
all linear probes retain the smoothness of the linear tool.
On an invalid Boolean tape the recipes, covariance, and degree
bounds still hold, but some final identity fails with the same gap.

Finally, \(M,|K|,q^{O(\ell)}\), the validity matrices, and all
mask spaces have polynomial size in \(N\). More explicitly,
\(T\le M\), the stage rows number \(O(M^2)\), and the logical
monomials of degree at most three number \(O(M^3)\). All later
row and column sets have size a fixed power of these quantities
or of \(|K|\); their base-\(h\) digit dimensions are therefore
\(O(\ell)\). The fixed-depth degree estimates remain
\(O(\ell h)<D\) after the chosen cutoff. Boolean interpolation
through degree three, every array operation, and every fixed-depth
certificate construction consequently take polynomial time. No step enumerates Boolean assignments
or graph vertices. This proves all the asserted costs and interfaces.
\end{proof}

\subsection{Unrestricted products and validation of proposed moves}
\label{ib:unrestricted-variants}

We return to the binary construction and notation of
Section~\ref{sec:inner-binary}. The restriction to compatible tags is
needed for the standardized one-port source in
Section~\ref{sec:inner-gadget}. The binary methods
also give the following variant when that source requirement is absent.
Its constants are fixed for the entire chosen chart family.

\begin{proposition}[All ordered products and full logical recovery]
\label{ib:all-pairs-interface}
Fix an inner algebraic template, and replace the compatible-pair list
by the list of \emph{all} ordered pairs of point-track types in
\eqref{ib:product-bit}. The two positions remain independent even
when the types coincide; no line-product blocks are added. Rebalance all blocks by the repetition construction of
Section~\ref{ib:binary-blocks}. This encoding has polynomial length and a
uniform polynomial-time construction, and has the following properties.
\begin{enumerate}[label=(\roman*)]
\item Valid canonical words have constant distance and the perfectly
complete smooth tester and proximity guarantee of
Proposition~\ref{ib:tester}. A structurally canonical word whose
label is invalid has rejection probability at least one fixed
\(\theta_{\rm inv}>0\).
\item One fixed radius permits arbitrary fixed average accuracy for
bounded-density requests, with joint pair density for products,
arbitrary public masks and forms, and all-hypothetical-branch
smoothness as in Proposition~\ref{ib:average-correction} and
Lemma~\ref{ib:all-branch-density}.
\item From a sufficiently small fixed neighborhood of any structurally
canonical word, its \emph{entire} logical tape
\(b=(w,e,(s_i)_i)\) is recoverable exactly in deterministic polynomial
time, whether or not that tape is valid.
\end{enumerate}
The constants and the radius may depend on the whole fixed template
and its block list. No common one-port layout or radius fixed before
arity is asserted for this variant.

More generally, the same binary transfer applies to a supplied fixed
finite chart template with the field, degree, slot, and polynomial-size
bounds used in Section~\ref{sec:inner-binary}, subject to these explicit
conditions: its canonical
recipes are defined also on invalid logical tapes; its bounded
binary-form algebraic validator has perfect completeness and an exact
constant gap on genuine degree-in-bound polynomial tuples; its linear
probes have bounded point marginals and its products have independent
full point marginals; and its order-one grid charts and constant
mode/marker tensors separate labels. Full logical recovery requires
order-one grid charts for every logical coordinate. This transfer
assumes these algebraic properties; it does not assert that the
fixed-input path of Section~\ref{sec:inner-algebra} realizes an
arbitrary macrograph.
\end{proposition}

\begin{proof}
There are still only constantly many blocks, each of polynomial
size, so the balancing product remains polynomial. The point and
line construction, rectangular action, and common row extension do
not use tags. In the proof of Proposition~\ref{ib:tester}, all
replacements before the product step concern individual \(Y\)
tracks. For each added product block, the additivity test in \(Q\)
and the two independent uniform operand positions give exactly the
same product replacement: a false binary matrix is detected by a
uniform outer-product form with probability at least \(1/4\).
Taking a positive fixed mixture over the enlarged finite block list
therefore gives the same vanishing-rejection replacement argument.
The algebraic validator is then applied to the common target tuple;
every requested point product is now present. Its exact gap forces
validity. On an invalid structurally canonical word, no replacement
is necessary: the final validator component already rejects with a
fixed positive probability. Its mixture weight and the fixed losses
in binary-form simulation define \(\theta_{\rm inv}\). Distinct
logical or mode/marker data retain the positive distance witnessed
by their \(Y\) blocks, with the new fixed block weights.

For correction, the two-axis support and inner-table arguments
apply to any ordered pair of point tracks. They use independent
row coordinates and the two row-code actions, not tag compatibility.
Use the new block count in \eqref{ib:uncertain-average} and choose
one radius satisfying \eqref{ib:fixed-radius} for this block list.
The bad-column bound, support contraction, finite-cone consistency,
and amplification then prove the same average guarantee. Each pair
of affine shifts remains a permutation of the full pair domain,
even for a repeated type. The proof of
Lemma~\ref{ib:all-branch-density} consequently applies to every
added block, with a new finite summed density bound. In particular,
no product request is converted to a diagonal pair.

For full recovery, apply the proof of
Proposition~\ref{ib:global-recovery} to each order-one chart
\(W_w,W_e,W_{s_i}\). The work and port charts use ordinary grid
extensions, and the legacy chart uses its degree-in-bound cyclic
lift. In each case nearest Hadamard symbols, equal-fiber plurality,
and plurality over ignored coordinates produce a native polynomial
estimate of sufficiently small constant symbol error. For each
logical grid point, the complete enumeration of lines and the
verified Berlekamp--Welch systems in that proof recover its exact
value by a strict majority. The argument uses degree and table
structure, not the final validity equations. There are polynomially
many grid points and a fixed number of charts. Reducing the radius
by the fixed block-weight factors makes all these decodings valid
simultaneously. The same proof applies to the supplied general
chart template under the stated separation and grid hypotheses.
\end{proof}

\begin{proposition}[Prediction before target validation]
\label{ib:proposal-validation}
Use the unrestricted representation of
Proposition~\ref{ib:all-pairs-interface}. Suppose a valid current
label has a prescribed proposal chosen by finitely many decisions
which are recoverable, to any fixed reliability, by bounded smooth
bit procedures. On a valid current label, once its true decision
branch is fixed, suppose the full canonical proposed tuple is a
definite structurally canonical tuple, possibly invalid, whose
contractions are bounded sums of public terms and old contractions
at invertible \(K\)-affine target pulls. For each branch these pulls
must belong to a fixed bounded public list, independent of the
requested outer indices and mask or form values. Masks may select
terms from that list and determine their slot pulls and forms.
Require bounded slot expansions, coefficients independent of target
point and row, compatibility with the common row and line encodings,
and publicly computable actual encoded values for public terms.
On hypothetical wrong branches only these bounded schedules and
smooth defaults are required; a canonical target identity is asserted
only when the branch predicates hold.
All branch lists, supports, and public computations are to have the
uniform polynomial-size bounds of Section~\ref{sec:inner-binary}.

From a sufficiently small fixed neighborhood of the current valid
word, its proposed word can be predicted to arbitrary fixed average
accuracy on bounded-density requests \emph{before} deciding target
validity. That validity can then be decided with arbitrary fixed
reliability. If an actual-edge predicate is exactly target validity
together with specified public or similarly recoverable guards,
this also decides that edge flag. All routines are bounded on every
received word and retain the complete hypothetical-tree smoothness
bound. No reciprocity of a general proposed graph is inferred from
target validity alone.
\end{proposition}

\begin{proof}
Fix the true decision branch of the nearby current label, charging
the selector's error separately. For a point or line \(Y\) request,
commute the proposed update through the common row encoder and,
for a line, through restriction and coefficient contraction.
This gives a bounded XOR of old \(Y\) requests with publicly
pulled-back forms, plus public bits. Each map in the fixed public
list induces a commuting column permutation on the retained point
and ordered-line domains, preserving bounded outer-index density.
If masks select among these maps, sum their unconditioned slot
bounds before selection; the list is bounded independently of the
request. Apply average correction with the error divided among
these boundedly many slots.

For a proposed \(Z\) bit, expand both updated symbols by bilinearity.
Every two-variable term is present because all ordered point-track
pairs were included. Each side undergoes its own affine column
permutation, preserving the joint density of the pair. A term with
one public actual \(J\)-symbol is an arbitrary linear form of the
other symbol, and \eqref{ib:public-marginal} supplies its complete
surviving row/point density. Two public operands give a public bit.
Thus the proposal is predicted using only the current close word;
its target need not be valid or supplied as a received word.

Now simulate the tester on the proposed word by supplying each bit
from this predictor with fresh internal randomness. First fix the
selector error below \(\theta_{\rm inv}/16\) and its complete
bounded tree. Unfold that tree and the bounded tester tree. Each
fixed hypothetical tester request has bounded density, so next
choose the per-prediction errors to sum to less than
\(\theta_{\rm inv}/16\) over all these slots. The simulated
rejection probability differs from the true one by less than
\(\theta_{\rm inv}/8\). A valid proposal has
true rejection zero; an invalid structural proposal has rejection
at least \(\theta_{\rm inv}\). Independent repeated simulations
and the threshold \(\theta_{\rm inv}/2\) therefore give any
specified fixed reliability. Include the stated guards separately
when interpreting the result as an edge flag.

All error estimates use the unconditioned request law on each
fixed branch, followed by a union bound over the complete bounded
tree. They do not renormalize after a successful comparison or
condition on a rare selected branch. Wrong decisions and failed
comparisons still select only from the supplied finite lists;
public-zero defaults and fresh splitting preserve the raw-read
bounds of Lemma~\ref{ib:all-branch-density}. At fixed accuracy,
the number of simulations and all their supports are bounded and
polynomially constructible, as required.
\end{proof}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//sections/12-extensions.tex ====

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  eprint = {0809.4095},
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}

@article{Shalom1999,
  author = {Shalom, Yehuda},
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}

@misc{Helfgott2013,
  author = {Helfgott, H. A.},
  title = {The Ternary {Goldbach} Conjecture Is True},
  howpublished = {\url{https://arxiv.org/abs/1312.7748v2}},
  year = {2013},
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  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  doi = {10.48550/arXiv.1312.7748},
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  note = {Version 2 (2014), Main Theorem, Section 1.1, p. 2; conclusion in Section 7.4}
}

@misc{Xylouris2011,
  author = {Xylouris, Triantafyllos},
  title = {On {Linnik}'s Constant},
  howpublished = {\url{https://arxiv.org/abs/0906.2749v1}},
  year = {2009},
  eprint = {0906.2749},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  doi = {10.48550/arXiv.0906.2749},
  url = {https://arxiv.org/abs/0906.2749v1},
  note = {Diplomarbeit in German, titled \emph{{\"U}ber die Linniksche Konstante}; Theorem 1.1, p.~9}
}

@article{BG2008,
  author = {Bourgain, Jean and Gamburd, Alex},
  title = {Uniform Expansion Bounds for {Cayley} Graphs of {$\mathrm{SL}_2(\mathbb{F}_p)$}},
  journal = {Annals of Mathematics},
  volume = {167},
  number = {2},
  pages = {625--642},
  year = {2008},
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  url = {https://annals.math.princeton.edu/2008/167-2/p07},
  note = {Theorem 1 and the discussion on p. 626. \url{https://doi.org/10.4007/annals.2008.167.625}}
}

@article{MGGAFP,
  author = {Karayel, Emin},
  title = {Expander Graphs},
  journal = {Archive of Formal Proofs},
  year = {2023},
  month = mar,
  url = {https://isa-afp.org/entries/Expander_Graphs.html},
  note = {Isabelle formalization. Theory \path{Expander_Graphs_MGG},
    function \path{mgg_graph_step} and lemma \path{mgg_numerical_radius};
    normalization in \path{Expander_Graphs_Definition} and
    \texttt{expansionD2} in \path{Expander_Graphs_Eigenvalues}.
    \href{https://isa-afp.org/browser_info/current/AFP/Expander_Graphs/Expander_Graphs_MGG.html}{Online formal development}.
    Accessed September 25, 2026}
}

@article{Margulis1973,
  author = {Margulis, G. A.},
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}

@article{GabberGalil1981,
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@article{HLW2006,
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}

@article{Papadimitriou1994,
  author = {Papadimitriou, Christos H.},
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}

@article{Rubinstein2018,
  author = {Rubinstein, Aviad},
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}

@inproceedings{Rubinstein2016Settling,
  author = {Rubinstein, Aviad},
  title = {Settling the Complexity of Computing Approximate Two-Player {Nash} Equilibria},
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  pages = {258--265},
  publisher = {IEEE},
  year = {2016},
  doi = {10.1109/FOCS.2016.35},
  eprint = {1606.04550},
  archivePrefix = {arXiv},
  primaryClass = {cs.CC},
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    (29 August 2016): Theorem 1.2, Sections 1.1 and 1.3,
    Lemmas 3.3--3.4, and Sections 5--6.
    \href{https://arxiv.org/pdf/1606.04550v2}{Full version}}
}

@misc{Golowich2026,
  author = {Golowich, Noah},
  title = {The Fine-Grained Complexity of Approximate {Nash} Equilibrium and Free Games},
  year = {2026},
  eprint = {2609.07136},
  archivePrefix = {arXiv},
  primaryClass = {cs.GT},
  doi = {10.48550/arXiv.2609.07136},
  howpublished = {\url{https://arxiv.org/abs/2609.07136v2}},
  note = {Version 2, submitted 9 September 2026.
    Theorem 1.1, Definition 2.1, and Section 2.2}
}

@article{Brouwer1911,
  author = {Brouwer, L. E. J.},
  title = {{\"U}ber {Abbildung} von {Mannigfaltigkeiten}},
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  pages = {97--115},
  year = {1911},
  doi = {10.1007/BF01456931},
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}

@misc{SchuldenzuckerSeuken2019,
  author = {Schuldenzucker, Steffen and Seuken, Sven},
  title = {Monotonic and Non-Monotonic Solution Concepts for Generalized Circuits},
  year = {2019},
  eprint = {1907.12854},
  archivePrefix = {arXiv},
  primaryClass = {cs.CC},
  doi = {10.48550/arXiv.1907.12854},
  howpublished = {\url{https://arxiv.org/abs/1907.12854v1}},
  note = {Version 1, 30 July 2019. Propositions 1--2, Lemma 1, and Sections 4 and 6}
}

@misc{CDT2007Preprint,
  author = {Chen, Xi and Deng, Xiaotie and Teng, Shang-Hua},
  title = {Settling the Complexity of Computing Two-Player {Nash} Equilibria},
  year = {2007},
  eprint = {0704.1678},
  archivePrefix = {arXiv},
  primaryClass = {cs.GT},
  doi = {10.48550/arXiv.0704.1678},
  howpublished = {\url{https://arxiv.org/abs/0704.1678v1}},
  note = {Version 1. Theorem 1.1, Sections 2 and 4.2, and Lemma 5.6; locators refer to this preprint}
}

@article{DGP2009,
  author = {Daskalakis, Constantinos and Goldberg, Paul W. and Papadimitriou, Christos H.},
  title = {The Complexity of Computing a {Nash} Equilibrium},
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  year = {2009},
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  url = {https://www.cs.ox.ac.uk/people/paul.goldberg/papers/SICOMP-dgp09.pdf},
  note = {Section 5.1, Lemmas 5.2--5.5. \url{https://doi.org/10.1137/070699652}}
}

@article{BLR1993,
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}

@article{Benes1964Permutation,
 author={Bene{\v{s}}, V. E.},
 title={Permutation Groups, Complexes, and Rearrangeable Connecting Networks},
 journal={Bell System Technical Journal}, volume={43}, number={4},
 year={1964}, month=jul, pages={1619--1640},
 doi={10.1002/j.1538-7305.1964.tb04102.x},
 url={https://archive.org/download/bstj43-4-1619/bstj43-4-1619_text.pdf},
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}

@article{GemmellSudan1992,
 author={Gemmell, Peter and Sudan, Madhu},
 title={Highly Resilient Correctors for Polynomials},
 journal={Information Processing Letters}, volume={43}, number={4},
 year={1992}, pages={169--174},
 doi={10.1016/0020-0190(92)90195-2},
 url={https://people.csail.mit.edu/madhu/papers/1992/gemmell.pdf},
 note={\href{https://people.csail.mit.edu/madhu/papers/1992/gemmell.pdf}{Author manuscript}, Appendix A, p. 7, presents the Berlekamp--Welch decoder. \url{https://doi.org/10.1016/0020-0190(92)90195-2}}
}

@article{CooleyTukey1965,
  author = {Cooley, James W. and Tukey, John W.},
  title = {An Algorithm for the Machine Calculation of Complex {Fourier} Series},
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    \href{https://janmr.com/files/papers/cooley-tukey.pdf}{Article scan}}
}

@article{Bennett1973,
 author={Bennett, C. H.},
 title={Logical Reversibility of Computation},
 journal={IBM Journal of Research and Development}, volume={17},
 year={1973}, month=nov, pages={525--532},
 doi={10.1147/rd.176.0525},
 url={https://www.cs.princeton.edu/courses/archive/fall04/cos576/papers/bennett73.html},
 note={Reversible cleanup discussion, pp. 525--526. \url{https://doi.org/10.1147/rd.176.0525}}
}

@inproceedings{DGZ2023,
  author = {Daskalakis, Constantinos and Golowich, Noah and Zhang, Kaiqing},
  title = {The Complexity of {Markov} Equilibrium in Stochastic Games},
  booktitle = {Proceedings of the Thirty Sixth Conference on Learning Theory},
  series = {Proceedings of Machine Learning Research},
  volume = {195},
  pages = {4180--4234},
  year = {2023},
  publisher = {PMLR},
  url = {https://proceedings.mlr.press/v195/daskalakis23a.html},
  note = {Theorem 4, Conjecture 10 and Appendix D use the published numbering.
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}

@misc{HZ2026,
  author = {Braverman, Mark and Liu, Jingyi and Xue, Eric and Zhou, Chenghan},
  title = {Hardness of Approximate {Hylland--Zeckhauser} Equilibria},
  year = {2026},
  eprint = {2606.00951},
  archivePrefix = {arXiv},
  primaryClass = {cs.GT},
  howpublished = {\url{https://arxiv.org/abs/2606.00951v2}},
  note = {Version 2, 9 June 2026. Definition 2.1, Table 1, Theorem 3.2,
    and Lemmas 3.4 and 3.25}
}

@inproceedings{PZO2025,
  author = {Park, Chanwoo and Zhang, Kaiqing and Ozdaglar, Asuman},
  title = {Multi-Player Zero-Sum {Markov} Games with Networked Separable Interactions},
  booktitle = {Advances in Neural Information Processing Systems},
  volume = {36},
  year = {2023},
  eprint = {2307.09470v3},
  archivePrefix = {arXiv},
  primaryClass = {cs.GT},
  url = {https://papers.nips.cc/paper_files/paper/2023/hash/75c411b0a06fa9e78f2a516b57b2ce62-Abstract-Conference.html},
  note = {Numbered references use the full version, arXiv:2307.09470v3
    (12 July 2025): Definition 1 and Appendix C, Case 1.
    \href{https://arxiv.org/abs/2307.09470v3}{Full version}}
}

@misc{FKPP2026,
  author = {Farina, Gabriele and Kontogiannis, Andreas and Panageas, Ioannis and Pollatos, Vasilis},
  title = {The Complexity of Computing Coarse Correlated Equilibria in {Markov} Games with a Single Controller},
  year = {2026},
  eprint = {2607.10897v1},
  archivePrefix = {arXiv},
  primaryClass = {cs.GT},
  howpublished = {\url{https://arxiv.org/abs/2607.10897v1}},
  note = {Version 1. Definition 2.6, Theorem 5.5 and Remark 5.7}
}

@article{Budish2011,
  author = {Budish, Eric},
  title = {The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes},
  journal = {Journal of Political Economy},
  volume = {119},
  number = {6},
  pages = {1061--1103},
  year = {2011},
  doi = {10.1086/664613},
  url = {https://ericbudish.org/wp-content/uploads/combinatorial_assignment_problem.pdf},
  note = {Definition 1 and Theorem 1, p.~1069.
    \href{https://ericbudish.org/wp-content/uploads/combinatorial_assignment_problem.pdf}{Published article}}
}

@article{OPR2014,
  author = {Othman, Abraham and Papadimitriou, Christos H. and Rubinstein, Aviad},
  title = {The Complexity of Fairness through Equilibrium},
  journal = {ACM Transactions on Economics and Computation},
  volume = {4},
  number = {4},
  pages = {20:1--20:19},
  year = {2016},
  eprint = {1312.6249v3},
  archivePrefix = {arXiv},
  primaryClass = {cs.GT},
  note = {The Appendix A locator uses arXiv:1312.6249v3
    (30 September 2014).
    \href{https://arxiv.org/pdf/1312.6249v3}{Full version}}
}

@misc{ABSsw26,
  author = {Amireddy, Prashanth and Behera, Amik Raj and Srinivasan, Srikanth and Sudan, Madhu and Willumsgaard, Sophus Valentin},
  title = {A Simple Algebraic Proof of the {PCP} Theorem},
  year = {2026},
  eprint = {2608.17429},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.17429v1},
  howpublished = {\url{https://arxiv.org/abs/2608.17429v1}},
  note = {Section 3.1, Definition 3.1 and Claim 3.2}
}

@article{Zemor2001,
  author = {Z{\'e}mor, Gilles},
  title = {On Expander Codes},
  journal = {IEEE Transactions on Information Theory},
  volume = {47},
  number = {2},
  pages = {835--837},
  year = {2001},
  url = {https://courses.grainger.illinois.edu/cs598sgt/sp2015/zemor.pdf},
  note = {Section III-A, Lemma 5 and Theorem 6. \url{https://courses.grainger.illinois.edu/cs598sgt/sp2015/zemor.pdf}}
}
==== END The-PCP-for-PPAD-conjecture-a-quasilinear-reduction//references.bib ====