==== TABLE OF CONTENTS ====
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//figures/negative-contraction.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//figures/stack-pattern.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//main.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//preamble.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/01-introduction.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/02-ordinary.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/03-negative.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/04-caps.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/05-stack.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/06-geometry.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/07-estimates.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/08-indices.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/09-analysis.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/10-gluing.tex
Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//references.bib
===========================
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//figures/negative-contraction.tex ====
\begin{tikzpicture}[font=\small,line cap=round,line join=round]
\begin{scope}[xshift=2.4cm,yshift=1.9cm,scale=.92]
\coordinate (a) at (-1.6,1);
\coordinate (b) at (0,1.8);
\coordinate (c) at (1.6,1);
\coordinate (d) at (1.2,-1.4);
\coordinate (e) at (-1.2,-1.4);
\coordinate (istart) at (0,-1.4);
\coordinate (iend) at (.8,1.4);
\fill[gray!7] (a)--(b)--(iend)--(istart)--(e)--cycle;
\fill[blue!7] (iend)--(c)--(d)--(istart)--cycle;
\draw[thick] (a)--(b)--(c)--(d)--(e)--cycle;
\draw[thick,dashed] (istart)--(iend);
\node[right] at (.45,.2) {$I$};
\node at (-.5,.15) {$\Delta_l$};
\node at (.95,-.55) {$\Delta_r$};
\node[above left] at (-.8,1.4) {$Z_3$};
\node[above right] at (.8,1.4) {$Z_1$};
\node[left] at (-1.4,-.2) {$M_{03}'$};
\node[right] at (1.4,-.2) {$M_{14}'$};
\node[below] at (0,-1.4) {$M_{04}'$};
\node at (0,-2.2) {Pentagon $P$};
\end{scope}
\begin{scope}[xshift=6.6cm,yshift=2.4cm,xscale=.94]
\begin{scope}
\clip (0,0) .. controls (1.3,1.5) and (4.3,1.5) .. (5.6,0)
.. controls (4.3,-1.5) and (1.3,-1.5) .. cycle;
\fill[gray!7] (0,-1.6) rectangle (5.6,1.6);
\fill[blue!15] (1.9,-1.6) rectangle (3.7,1.6);
\end{scope}
\draw[thick] (0,0) .. controls (1.3,1.5) and (4.3,1.5) .. (5.6,0)
.. controls (4.3,-1.5) and (1.3,-1.5) .. cycle;
\draw[densely dashed] (1.9,.99) .. controls (1.65,.5) and (1.65,-.5) .. (1.9,-.99);
\draw[densely dashed] (3.7,.99) .. controls (3.95,.5) and (3.95,-.5) .. (3.7,-.99);
\draw[thick] (2.8,0) ellipse [x radius=.24,y radius=1.125];
\node at (1.05,.1) {$\Delta_l$};
\node at (4.55,.1) {$\Delta_r$};
\node[above] at (2.8,1.15) {$\beta$};
\draw[thin,->] (0,-1.85)--(5.8,-1.85);
\foreach \x/\name in {0/a,1.9/u,2.8/t_0,3.7/v,5.6/b} {
\draw (\x,-1.77)--(\x,-1.93);
\node[below] at (\x,-1.94) {$\name$};
}
\draw[densely dotted] (0,0)--(0,-1.75);
\draw[densely dotted] (1.9,-.99)--(1.9,-1.75);
\draw[densely dotted] (2.8,-1.125)--(2.8,-1.75);
\draw[densely dotted] (3.7,-.99)--(3.7,-1.75);
\draw[densely dotted] (5.6,0)--(5.6,-1.75);
\node at (2.8,-2.75) {Merged $\infty$ side and its sphere};
\end{scope}
\end{tikzpicture}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//figures/negative-contraction.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//figures/stack-pattern.tex ====
\begin{tikzpicture}[
block/.style={draw,minimum width=9mm,minimum height=8mm,inner sep=2pt},
arrow/.style={-{Stealth[length=1.6mm]}},font=\small]
\node (left) at (0,0) {$C_-$};
\node[block] (b1) at (1.35,0) {$B$};
\node[block,fill=blue!7] (h) at (2.7,0) {$H$};
\node[block] (b2) at (4.05,0) {$B$};
\node[block] (b3) at (5.65,0) {$B$};
\node[block] (b4) at (7.25,0) {$B$};
\node (dots) at (8.85,0) {$\cdots$};
\node (right) at (10.2,0) {$C_+$};
\draw[arrow] (left)--(b1);
\draw[arrow] (b1)--(h);
\draw[arrow] (h)--(b2);
\draw[arrow] (b2)--node[below] {$\phi$}(b3);
\draw[arrow] (b3)--node[below] {$\phi$}(b4);
\draw[arrow] (b4)--node[below] {$\phi$}(dots);
\draw[arrow] (dots)--(right);
\draw[decorate,decoration={brace,amplitude=4pt}]
(0.85,.65)--(4.55,.65)
node[midway,above=5pt] {one positive macro};
\draw[densely dashed] (1.35,-.4)--(1.35,-.85)
node[below,font=\scriptsize] {$J_j$};
\draw[densely dashed] (4.05,-.4)--(4.05,-.85)
node[below,font=\scriptsize] {$J_{j+1}$};
\draw[decorate,decoration={brace,amplitude=4pt,mirror}]
(1.35,-1.35)--(4.05,-1.35)
node[midway,below=5pt,font=\scriptsize] {positive cell $P$};
\draw[decorate,decoration={brace,amplitude=4pt,mirror}]
(0.85,-2.05)--(7.7,-2.05)
node[midway,below=5pt] {four interval blocks per positive bridge};
\end{tikzpicture}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//figures/stack-pattern.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Purely cosmetic surgery on knots in the three-sphere}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove the purely cosmetic surgery conjecture for knots in the three-sphere.
Distinct Dehn surgery slopes on a nontrivial smooth knot never give
orientation-preservingly homeomorphic manifolds.
\end{abstract}
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-ordinary}
\input{sections/03-negative}
\input{sections/04-caps}
\input{sections/05-stack}
\input{sections/06-geometry}
\input{sections/07-estimates}
\input{sections/08-indices}
\input{sections/09-analysis}
\input{sections/10-gluing}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//main.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//preamble.tex ====
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=1.05in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{microtype,booktabs,longtable,array,enumitem}
\usepackage{graphicx,xcolor,tikz}
\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing}
\usepackage{xurl}
\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{convention}[theorem]{Convention}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\F}{\mathbb F}
\newcommand{\CP}{\mathbb{CP}}
\newcommand{\RP}{\mathbb{RP}}
\newcommand{\SU}{\operatorname{SU}}
\newcommand{\U}{\operatorname{U}}
\newcommand{\SO}{\operatorname{SO}}
\newcommand{\PU}{\operatorname{PU}}
\newcommand{\Spinc}{\operatorname{Spin}^{c}}
\newcommand{\PD}{\operatorname{PD}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\ad}{\operatorname{ad}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\ind}{\operatorname{ind}}
\newcommand{\coker}{\operatorname{coker}}
\newcommand{\Hol}{\operatorname{Hol}}
\newcommand{\Scal}{\operatorname{Scal}}
\newcommand{\Vol}{\operatorname{Vol}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\grad}{\operatorname{grad}}
\setlist{topsep=4pt,itemsep=2pt,parsep=0pt}
\setlength{\emergencystretch}{2em}
\hypersetup{pdftitle={Purely cosmetic surgery on knots in the three-sphere},pdfauthor={OpenAI}}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//preamble.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
Let $K$ be a smooth knot in the oriented three-sphere, and let
$E_K=S^3\setminus\operatorname{int}(\nu K)$ be its exterior.
The meridian $\mu$ and preferred longitude $\lambda$ identify the unoriented
slopes on $\partial E_K$ with $\Q\cup\{\infty\}$: the primitive class
$p\mu+q\lambda$, considered up to simultaneous sign, represents $p/q$.
Write $S^3_{p/q}(K)$ for the corresponding Dehn filling, with the orientation
extending that of $E_K$.
Two distinct slopes giving orientation-preservingly homeomorphic fillings
are called \emph{purely cosmetic}.
\begin{theorem}\label{thm:main}
Let $K\subset S^3$ be a smooth nontrivial knot. For any
$r,s\in\Q\cup\{\infty\}$, an orientation-preserving homeomorphism
\[
S^3_r(K)\cong S^3_s(K)
\]
implies $r=s$.
\end{theorem}
Theorem~\ref{thm:main} resolves the purely cosmetic surgery conjecture for
knots in the three-sphere. The homeomorphism in its statement is a map of
the entire filled manifolds; it need not preserve a surgery core or the
knot exterior. The orientation hypothesis is essential to the question:
orientation-reversing, or \emph{chirally cosmetic}, coincidences are a
different phenomenon.
\subsection{The remaining case}
Dehn surgery converts the peripheral information of a knot into a closed
three-manifold. The cosmetic surgery problem asks how much of the filling
slope can be recovered from that oriented manifold. For knot exteriors,
this is an instance of the cosmetic filling problem formulated by
Gordon~\cite[Conjecture~6.1]{Gordon1991} and recorded as Bleiler's
Problem~1.81(A) in Kirby's list~\cite{Kirby1997}.
Bleiler, Hodgson, and Weeks developed the terminology and distinguished
fillings related by a symmetry of the exterior from those that are
not~\cite{BleilerHodgsonWeeks1999}. Here the homeomorphism between
fillings is arbitrary. Gordon and Luecke's theorem that nontrivial
surgery on a nontrivial knot cannot produce $S^3$
\cite[Theorem~2]{GordonLuecke1989} settles the meridional case.
Several theories have supplied obstructions to a cosmetic pair of finite slopes.
Boyer and Lines obtained an Alexander-polynomial obstruction from
surgery formulae for classical three-manifold invariants~\cite{BL90}.
The rational surgery formula of Ozsv\'ath and Szab\'o~\cite{OS11},
Wang's genus-one theorem~\cite{Wang06}, and Wu's exclusion of
same-sign purely cosmetic pairs~\cite{Wu11} established a Heegaard
Floer approach. Ni and Wu strengthened the restriction to opposite
slopes~\cite[Theorem~1.2]{NiWu2015}. Using graded Floer information
through immersed curves, Hanselman restricted the possibilities to
$\pm2$ or $\pm1/q$ for a positive integer $q$, and showed that the
$\pm2$ case requires genus two~\cite[Theorem~2]{Hanselman2023}.
Other obstructions and special cases clarify the range of the problem.
Ichihara and Wu used the Jones polynomial~\cite{IchiharaWu2019},
and Detcherry developed a quantum obstruction~\cite{Detcherry2026}.
Tao settled nontrivial cable and composite knots
\cite{TaoCables2019,TaoConnected2022}, and Stipsicz and Szab\'o
settled nontrivial pretzel knots~\cite{StipsiczSzabo2021}.
Kotelskiy, Lidman, Moore, Watson, and Zibrowius proved an equivariant
version for strongly invertible knots, in which the diffeomorphism
commutes with the extended involutions
\cite{KotelskiyLidmanMooreWatsonZibrowius2024}.
Futer, Purcell, and Schleimer combined knot invariants, hyperbolic
estimates, and verified computations to establish the conjecture
through $19$ crossings and for the one-cusped SnapPy census
\cite{FuterPurcellSchleimer2025}. Their procedure applies beyond
these finite data sets but is not guaranteed to distinguish every pair.
Ren proved that the conjecture is equivalent to its restriction to
hyperbolic surgeries on hyperbolic knots
\cite[Theorem~3, arXiv version~2]{Ren2025}.
The filtered instanton argument of Daemi, Lidman, and Miller Eismeier
excludes the pairs $\pm1/q$. Combined with the earlier restrictions,
their Corollary~1.4 leaves only opposite integer slopes of magnitude
two on a genus-two knot; it also imposes additional Alexander and Jones
polynomial conditions~\cite{DaemiLidmanMillerEismeier2025}.
We use only the following consequence.
\begin{proposition}[Cosmetic surgery reduction]\label{prop:introduction:reduction}
If a smooth nontrivial knot $K\subset S^3$ has distinct purely cosmetic
slopes, then $g(K)=2$ and the two slopes are $-2$ and $2$.
\end{proposition}
For completeness, the zero slope causes no additional case:
$H_1(S^3_{p/q}(K);\Z)$ is cyclic of order $|p|$ for $p\ne0$ and is
infinite cyclic for $p=0$. Primitivity makes $0/1$ the unique slope with
zero numerator. The meridional slope is covered by the sphere-filling
theorem just mentioned.
We henceforth suppose that $g(K)=2$ and that there is an oriented
diffeomorphism
\begin{equation}\label{eq:introduction:cosmetic}
\phi:Y_{-2}\longrightarrow Y_2,
\qquad Y_j=S^3_j(K),\quad Y_\infty=S^3.
\end{equation}
The three-dimensional smoothing theorem permits the original
homeomorphism to be replaced by a homotopic diffeomorphism
\cite[Theorem~2.3]{Massuyeau2011}. Homotopy preserves its degree,
so this replacement preserves orientation. We will obtain a
contradiction without imposing any peripheral condition on $\phi$.
\subsection{The argument}
There are two counts. The first is an integer instanton count
$\Omega\ne0$ on a closed parameterized cobordism. The second is an
oriented one-dimensional spinor count whose boundary would force
$2^\eta\Omega=0$. The work is to construct the two counts with compatible
metrics, incidence conditions, determinant lifts, and orientations.
The comparison follows the nonabelian-monopole cobordism strategy of
Pidstrigach--Tyurin and Okonek--Teleman
\cite{PidstrigachTyurin1995,OkonekTeleman1996}, with the analytic and
instanton-link development of Feehan--Leness
\cite{FeehanLenessPU2I,FeehanLenessPU2II,FeehanLenessLinksI}.
The doubled phase class is part of that method. Here the simultaneous
family, its relative boundary analysis, and the regularization of
mixed projections are constructed explicitly.
The ordinary count uses instanton Floer and Donaldson theory
\cite{Floer1988,Donaldson2002,DonaldsonKronheimer1990}, with
determinant-one gauge transformations.
At the rational ends with first homology of order at most two, all flat
reducers are central. At zero surgery we instead use the determinant
class evaluating oddly on the capped Seifert surface. Ordinary surgery
families give units modulo two. The negative direction requires an
interval of metrics, a degree-two sphere test, and a signed sum over two
determinant lifts. The two minimal lens-wall contributions cancel with
these signs. Symplectic caps and ordinary Donaldson invariants provide a
nonzero rational cap pairing whose probes can be moved off all variable
blocks, following the cap and nonvanishing methods of
Kronheimer--Mrowka~\cite{KronheimerMrowka2004,KronheimerMrowka1995}.
Repetition on the free integral Floer lattice modulo a power of
two then produces the nonzero integer $\Omega$.
The same repetition makes the complex coupled Dirac index $n_D$
positive while preserving nonvanishing; the index calculation and
integral congruences are in Section~\ref{sec:stack}.
The phase circle acts by multiplying the spinor. Starting from the
zero-dimensional ordinary count, adding spinors, imposing
$\eta=n_D-1$ complex phase cuts, and taking the effective phase
quotient gives expected dimension
\[
2n_D-2\eta-1=1.
\]
Near an ordinary instanton the phase link is $\CP^{n_D-1}$, and each
phase cut has class twice the hyperplane class, as in the link
calculation of Feehan--Leness~\cite[Lemma~3.28 and
Proposition~3.29, arXiv version~3]{FeehanLenessPU2II}. These links contribute
$2^\eta\Omega$ with the ordinary instanton signs.
Reducible fields and broken or bubbled limits could supply additional
ends. To control them, we construct toric metric sectors using the
Hessian description of Guillemin and the scalar-curvature formula of
Abreu~\cite{Guillemin1994,Abreu1998}. Their periods constrain every
parallel line reduction, that is, a splitting of the bundle into line
bundles preserved by the connection. Long tube estimates and
isolated-cap estimates make those constraints simultaneous, including
at genuine corners of the parameter family. The resulting quadratic
and index inequalities exclude fixed tuples and every mixed broken or
bubbled boundary except a face at which one minimal trace cap separates.
At each surviving lens face, precisely one minimal trace cap separates
from a single free main component. These limits have no particles, and
regular cylindrical gluing applies. The two determinant sectors at each face cancel using
the same local orientation comparison that defined the negative
instanton unit. The oriented boundary formula now contradicts
$\Omega\ne0$.
The analytic construction proceeds in stages. The ordinary representatives
and their compactness properties are established independently of the
nonvanishing argument. For the spinor problem, energy bounds and ideal
compactness precede fixed-type regularization and its index exclusions;
these precede the finite projected regularization and the construction of
the remaining regular ends. Gluing these ends precedes the final
interior regularization of the one-dimensional free stratum.
The generic choices are made around the
ordinary data, within a neighborhood preserving $\Omega$. The final
boundary formula is applied only after these stages are complete.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
box/.style={draw,rounded corners=2pt,align=center,text width=5.4cm,
minimum height=1.05cm,inner sep=5pt,font=\small},
arrow/.style={-{Stealth[length=2mm]},thick},
node distance=9mm and 8mm]
\node[box] (ordinary) {Ordinary units and cap pairing\\
Sections~\ref{sec:ordinary}--\ref{sec:caps}};
\node[box,right=of ordinary] (metrics) {Metric families and line estimates\\
Sections~\ref{sec:geometry}--\ref{sec:estimates}};
\node[box,below=of ordinary] (count) {Integral repetition\\
$\Omega\ne0$, $n_D>0$; Section~\ref{sec:stack}};
\node[box,below=of metrics] (bounds) {Boundary index inequalities\\
Section~\ref{sec:indices}\\
\footnotesize Hypotheses supplied in
Sections~\ref{sec:analysis}--\ref{sec:gluing}};
\node[box,below=16mm of $(count)!0.5!(bounds)$,text width=8.8cm] (analytic)
{Compatible perturbations, \mbox{compactness}, and regular end gluing\\
Sections~\ref{sec:analysis}--\ref{sec:gluing}};
\node[below=8mm of analytic,font=\small] (boundary)
{Instanton links: $2^\eta\Omega$ \qquad Lens pairs: $0$};
\draw[arrow] (ordinary)--(count);
\draw[arrow] (metrics)--(bounds);
\draw[arrow] (count.south)--(analytic.north west);
\draw[arrow] (bounds.south)--(analytic.north east);
\draw[arrow] (analytic)--(boundary);
\end{tikzpicture}
\caption{The two parts of the proof meet in one compatible boundary
count. The analytic construction supplies the regularity hypotheses used
by the dimension inequalities; those inequalities in turn identify the
allowed ends.}
\label{fig:introduction:structure}
\end{figure}
\subsection{Local constructions and organization}
Three constructions carry the main technical content. First, a
parameterized negative cobordism is made into an integral chain map by a
local comparison of two determinant lifts, while its invertibility is
detected modulo two (Theorem~\ref{thm:negative:unit} and
Proposition~\ref{prop:negative:local-sign}). Second, a family of toric periods, together with a
separate flux estimate on a long $S^2\times S^1$ tube, yields line-class
inequalities that persist across compatible separations
(Theorems~\ref{thm:geometry:family} and \ref{thm:estimates:clamp}). Third,
exact-support incidence conditions and finite equivariant branch
perturbations permit negative-dimension arguments on projections of
mixed strata (Lemmas~\ref{lem:analysis:incidence} and
\ref{lem:analysis:projection}, and Proposition~\ref{prop:analysis:finite-induction}).
These projections forget zero-spinor reducer fields while retaining
their parameters; the remaining equations are independent of the
forgotten fields. The finite construction is confined to images of
candidate tuples whose retained main fields solve the equations and
cuts, with omitted lens caps and cylinder levels recorded through discrete
nonnegative charge assignments.
Their statements retain the local hypotheses explicitly,
so that these constructions can be used apart from the final knot
surgery contradiction.
The proof distinguishes two kinds of neck throughout. The surgery
seams $Y_j$ and the auxiliary isolation collars are long but finite.
The genuine infinite faces in the spinor compactification are the
specified $S^3$ cuts, denoted $J$, and the lens boundaries of
square-$-4$ sphere neighborhoods. Their limiting metrics have positive
scalar curvature. Thus the proof does not require a general regular
spinor Floer theory on the knot surgeries, or gluing at positive particle
levels.
Sections~\ref{sec:ordinary}--\ref{sec:stack} construct the ordinary
nonzero count and its topological data.
Sections~\ref{sec:geometry}--\ref{sec:estimates} construct the metrics and
prove the line estimates with their order of choices.
Section~\ref{sec:indices} proves the boundary inequalities under precise
regularity and incidence hypotheses.
Section~\ref{sec:analysis} constructs representatives, proves compactness,
and supplies the regularity needed for the boundary exclusions and the
instanton links. Section~\ref{sec:gluing} constructs the surviving lens
collars, regularizes the remaining free interior, and completes the
boundary count proving Theorem~\ref{thm:main}.
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/01-introduction.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/02-ordinary.tex ====
\section{Ordinary instanton theory and the handle units}
\label{sec:ordinary}
We write $Y_j=S^3_j(K)$, including $Y_\infty=S^3$, and write
$A\subset Y_0$ for the capped Seifert surface. On $Y_0$ the determinant
line evaluates oddly on $A$; on $Y_{\pm1},Y_{\pm2}$ and $Y_\infty$ it is
trivializable. All gauge transformations have determinant one. In
particular, fixing a determinant means fixing its connection and its
transition identifications, not passing to the full group of projective
gauge transformations.
We establish the following ordinary input. The word ``unit'' means an
isomorphism on instanton homology over $\F_2$; it does not assert that a
particular matrix on the chain group is invertible.
\begin{theorem}[Ordinary handle units]\label{thm:ordinary:units}
Let $C(Y_j;\Z)$ be the irreducible, unframed instanton complex with the
determinants just specified. Compatible small perturbations and coherent
orientations can be chosen so that the rigid counts define integral
chain maps, and the rigid and interval counts have the following
properties over $\F_2$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Each rigid meridional handle map $Y_r\longrightarrow Y_{r+1}$,
for $r=0,1$, is a unit whenever its determinant has the specified end
restrictions. The rigid count on their composite is their composition
up to chain homotopy. Denote this composite by
$f_+:C(Y_0)\longrightarrow C(Y_2)$.
\item On the path $2\longrightarrow\infty\longrightarrow0$, let
$F_l,A$ be the capped Seifert classes of squares $-2,0$, respectively,
with $F_l\cdot A=0$, and let $S_l=A-F_l$. Choose the base determinant
$c(F_l,A)=(0,1)$ and the two determinants
$c+e\PD(S_l)$, $e=0,1$, identified off $S_l$. The sum modulo two of
the interval counts of actual index $-1$ is a unit
$g_+:C(Y_2;\F_2)\longrightarrow C(Y_0;\F_2)$.
\item The mirror construction, read backwards, gives units
$f_-:C(Y_{-2})\longrightarrow C(Y_0)$ on the rigid path through $-1$,
and $g_-:C(Y_0;\F_2)\longrightarrow C(Y_{-2};\F_2)$ on
$0\longrightarrow\infty\longrightarrow-2$. For $g_-$ the base
evaluations are $(1,0)$ and the second determinant differs by the
Poincar\'e dual of the intervening $(-2)$-sphere.
\end{enumerate}
For the summed interval maps, each summand has an integral orientation;
the assertion that the sum is a chain map and a unit is required only
after reduction modulo two. End identifications, including any tensor
automorphisms, are fixed before composing the maps.
\end{theorem}
The proof occupies this section. After fixing the complexes and index
conventions, we construct the
handle families and their determinant data. We then establish the
perturbation and matching results needed to compute every boundary
face of the interval and pentagon counts.
\subsection{Complexes and index conventions}
\begin{lemma}[Reducibles at the ordinary ends]
\label{lem:ordinary:central-ends}
At a trivializable-determinant rational end occurring here, every
reducible flat connection is central after a square-root normalization
of the determinant. Its adjoint cohomology has
$(h^0,h^1)=(3,0)$. The admissible determinant on $Y_0$ has no reducible
projectively flat connection. Small perturbations fixing the central
connections preserve these properties.
\end{lemma}
\begin{proof}
The first homology of the rational ends is either zero or $\Z/2$.
A reducible $\SU(2)$ representation is Abelian, so factors through that
first homology and takes values in $\{1,-1\}$. Its adjoint local system
is therefore the trivial rank-three system. Rational homology-sphere
cohomology gives $h^0=3$ and $h^1=0$. The two possible central lifts
at an order-two end remain distinct fixed-determinant flat connections;
their adjoint corrections coincide.
A reducible projectively flat connection on $Y_0$ would split as two
line connections with equal real first Chern class. Its determinant
would consequently pair evenly with the integral surface $A$, contrary
to the choice of determinant. At a central rational flat the Hessian
on the Coulomb slice is invertible, since $h^1=0$. The implicit function
theorem, with the residual compact stabilizer acting on that slice,
therefore preserves its isolation when the perturbation vanishes there.
If new reducible critical points appeared for perturbations tending
to zero, compactness in dimension three would give a reducible flat
limit. The preceding isolation, or the admissibility obstruction at
$Y_0$, excludes such a sequence.
\end{proof}
Thus the irreducible-only Floer construction applies to these ends.
We use the determinant-one version throughout
\cite[Section~7.1, arXiv version~2]{KronheimerMrowka2010}; the extension to
three-manifolds with two-torsion first homology is explicitly recorded
in \cite[Section~2, Theorem~2.3]{GhoshSivekZentner2023}. After a nondegenerate
perturbation there are finitely many generators. The differential
counts oriented index-one trajectories modulo translation. The
central-contact estimate proved below excludes central strata from
the compactifications defining the differential and its square.
At $Y_\infty$ choose the unperturbed flat problem, so that
\begin{equation}\label{eq:ordinary:sphere-complex}
C(Y_\infty)=0.
\end{equation}
We may forget the finite Floer grading when discussing linear
isomorphisms; whenever a moduli space is counted, its actual integer
index, rather than only its residue class, is specified.
\begin{convention}[Relative sectors and fixed-limit index]
\label{conv:ordinary:index}
The cobordism $C_j$ is the oriented knot trace from $Y_\infty$ to $Y_j$.
Paths of cobordisms are written in chronological order; their induced
operators compose from right to left. The symbol $C_j$ for a
cobordism is distinct from the complex $C(Y_j)$.
On a completed piece, $i$ denotes the unframed ASD index with the
boundary representatives fixed. Gauge transformations may approach
independent parallel stabilizers at the ends. Normal remainders have
small positive exponential weights. We retain all relative bundle
sectors compatible with the fixed determinant transitions, and select
the required actual total index. A matching across an isolated flat
adds $h^0$ to the sum of the two fixed-limit indices.
At $L=S^1\times S^2$ with trivialized determinant, harmonic flat
parameters are included before residual conjugation. At a
noncentral flat $(h^0,h^1)=(1,1)$, and at a central flat
$(h^0,h^1)=(3,3)$. Varying one fixed asymptote adds $h^1$; matching two
fixed-asymptote pieces adds $h^0+h^1$.
Choose the ordinary integral determinant-line orientation systems,
with true-end orientations, homology orientations and integral
determinant lifts fixed. These systems are orientable for the
determinant-one problem. In any fixed relative bundle sector the
space of reference connections with fixed product-flat collars is
affine, hence connected; a comparison made by determinant-line
excision therefore extends by orientation transport with a constant
relative sign. Gluing these comparisons is associative. Changing
an internal framing degree changes the relative charge sector, so
a cap prescribed to have charge $1/4$ uses that prescribed framing
degree. These orientation conventions are those of the ordinary
ASD/Floer construction \cite{DonaldsonKronheimer1990}; see in particular
\cite[Section~5.4, Proposition~5.11, and Section~5.6]{Donaldson2002}
for determinant-line gluing and determinant-one conventions.
\end{convention}
\begin{lemma}[Charge and index bookkeeping]\label{lem:ordinary:index}
Put $E=-p_1(\ad E)/4$, interpreted by Chern--Weil on the completed
connection. Increasing relative charge by one increases $i$ by eight.
If a connected piece has $b_1=0$ and only central rational ends, then
\begin{equation}\label{eq:ordinary:central-index}
i=8E-3(1+b^+).
\end{equation}
For replacements with identical boundary flats and fixed transitions,
including all matching terms, the index difference is
\begin{equation}\label{eq:ordinary:excision}
\Delta i=8\Delta E-\frac32\Delta(\chi+\sigma).
\end{equation}
The nonnegative lower charge bounds used below persist for sufficiently
small perturbations. On a product neck with a bounded Floer functional
perturbation, the required error is bounded independently of the neck
length.
\end{lemma}
\begin{proof}
The real ASD symbol contributes $-3(\chi+\sigma)/2$, and its bundle
term is $-2p_1(\ad E)=8E$. APS boundary terms for a central flat are
those of three trivial real coefficients. Under
Convention~\ref{conv:ordinary:index}, the scalar fixed-limit complex
has index $-1-b^+$; tensoring by the parallel adjoint fiber gives
\eqref{eq:ordinary:central-index}. This also gives $-3$ for a flat
central cylinder, as required by the addition of its three matching
directions. Excision cancels the identical boundary terms and proves
\eqref{eq:ordinary:excision}. These are the usual ASD index formulas
with the unframed correction retained; see \cite{DonaldsonKronheimer1990}
and \cite[Equation (10), p.~24]{DaemiLidmanMillerEismeier2025}.
For unperturbed ASD fields, the trace-free charge is a nonnegative
multiple of the curvature norm squared. In each application its
possible values lie in a specified discrete translate of $\Z$.
If a sequence of perturbations tending to zero violated the stated
lower bound, compactness and charge quantization would give an ASD
limit, with any lost charges positive, violating that same bound.
On a long product one uses the gradient identity instead of a bound
proportional to length. With the normalization of Chern--Simons
absorbed in a fixed positive constant, a gradient trajectory satisfies
\[
\int_{a}^{b}\|\dot a(t)\|^2\,dt
=\operatorname{CS}(a(a))+h(a(a))
-\operatorname{CS}(a(b))-h(a(b)).
\]
The perturbation contribution is bounded by $2\|h\|_\infty$.
On concatenated product segments the intermediate functional values
telescope. Consequently smallness can be fixed before increasing
the product length. This is also the energy estimate used when
discarding a nonconstant collar level.
\end{proof}
A regular count over a $p$-dimensional metric family, without
insertions, has $i=-p$. Its one-dimensional boundary relation has
$i=1-p$. An irreducible nonconstant trajectory has index at least one
before quotienting by translation. A \emph{particle} is an ideal
curvature-concentration point with a positive integral charge. A particle
of charge $q$ lowers
the field index by $8q$ and has at most four location parameters.
These conventions will be used even when a piece is filled only
virtually for an index comparison.
\subsection{The polygon model and its metric families}
The polygon and associahedral surgery families follow the constructions
of Culler--Daemi--Xie~\cite[Section~4.3 and Theorem~4.56,
arXiv version~1]{CullerDaemiXie2020} and
Daemi--Lidman--Miller Eismeier~\cite[Section~5]{DaemiLidmanMillerEismeier2025}.
We supply the local plumbing, determinant and boundary calculations
needed for the present ends and integral units.
\begin{lemma}[Polygon traces and plumbing cuts]\label{lem:ordinary:polygon}
A path of adjacent slopes has a smooth toric local model over a polygon.
Choose clockwise primitive slope vectors $w_0,\ldots,w_m$ with
$\det(w_i,w_{i+1})=-1$. Collapse the circle of slope $w_i$ over the
corresponding side and the whole torus at each corner, and attach the
remaining boundary to the knot exterior times an interval. An arc
from the exterior boundary to side $i$ gives the primary cut
$Z_i=Y_{w_i}$. An arc between sides $i,j$, $j-i\ge2$, gives a cut
$M_{ij}$ bounding the linear plumbing associated with the intervening
sides. Its spheres have consecutive intersections $+1$ and squares
$-d_k$, where
\begin{equation}\label{eq:ordinary:fan-relation}
w_{k-1}+w_{k+1}=d_kw_k.
\end{equation}
If the two endpoint slopes agree, the plumbing boundary is
$L=S^1\times S^2$. Its primitive radical is the image of the boundary
$S^2$. Replacing this cap by $D=S^1\times D^3$ joins the two equal
collapsed sides and leaves the exterior identity cylinder.
\end{lemma}
\begin{proof}
At a corner the two adjacent slope circles form an integral basis.
The quotient is therefore the usual smooth two-disk corner model,
rather than an orbifold. Along an interior side the change from
the preceding basis to the following basis reverses the tangent
circle parameter and has normal degree $-d_k$, proving
\eqref{eq:ordinary:fan-relation} and the self-intersection formula.
The common corner contributes intersection $+1$ to neighboring
spheres. Trivializations along the exterior side give precisely
the usual two-handle trace, since $\infty\to j$ is $C_j$,
$j\to\infty$ is its orientation-reversed reverse, and
$r\to r+1$ adds a $(-1)$-framed meridian to $C_r$. Removing a
side of square $-1$ is the corresponding corner blow-down.
The inverse image of an arc from the exterior side is the filled
three-manifold at that slope. The inverse image of an arc between
two collapsed sides is the boundary of their intervening chain,
which proves the cut descriptions. When the endpoint circles
are equal, their collapse gives a sphere, and the transverse
circle remains free; the boundary is $S^2\times S^1$.
The sphere along that arc is primitive, and its image in the
plumbing is the primitive null vector of the intersection
matrix. Filling by $S^1\times D^3$ replaces the arc by a
half-disk with one straight collapsed edge. This joins the equal
sides without changing the exterior gluing and proves the last
assertion.
\end{proof}
\begin{lemma}[Compatible associahedral metric families]
\label{lem:ordinary:metric-families}
A two-step path has an interval family with endpoint cuts
$Z_1,M_{02}$. A three-step path has a pentagon family with faces
$Z_1,Z_2,M_{02},M_{13},M_{03}$. A four-step path has the corresponding
three-dimensional associahedral family. Each face carries the product
families on its split pieces, and compatible simultaneous cuts are
exactly the disjoint noncrossing arc systems.
Two consecutive final handles may instead be retained as a fixed
segment. For the path $2,\infty,0,1,2$, this gives a pentagon with
faces
\begin{equation}\label{eq:ordinary:compressed-cycle}
Z_1,\quad Z_2,\quad M_{02},\quad M_{04},\quad M_{14}
\end{equation}
in cyclic order. The $M_{04}$ edge has endpoints $M_{02}$ and
$M_{14}$; no cut internal to the fixed segment is introduced.
\end{lemma}
\begin{proof}
Use collars of the polygon arcs of
Lemma~\ref{lem:ordinary:polygon}. At shared slope sides, separate
their endpoints slightly in noncrossing order; at the exterior
side use distinct ordered slices. Thus every compatible arc
system has disjoint collars. Face-product coordinates of an
associahedron specify independent lengths for precisely these
collars. Choose the metrics on completed pieces inductively
over faces, with fixed product metrics on the cross-sections:
round metrics on lens spaces and the product metric on $L$.
For finite lengths glue and round these collars. Once prescribed
on a neighborhood of the boundary of a parameter face, the
metrics extend over its compact interior because positive-definite
symmetric forms form a convex cone.
For two steps, the only alternatives are splitting at the
intermediate filling or isolating the intervening sphere.
For three steps, the noncrossing arc description gives the five
listed faces and their corner incidences. Keeping the last two
handles fixed simply treats their union as the third block;
the same three-block construction applies. In the original
indices its cap cuts are $M_{02},M_{14},M_{04}$, giving
\eqref{eq:ordinary:compressed-cycle}. Omitting the interior
cut of a fixed block leaves that block's data unchanged and
does not create another boundary parameter. The construction
inserts only neck lengths; it introduces no twist coordinate.
\end{proof}
\begin{lemma}[Topological and determinant data for the unit families]
\label{lem:ordinary:unit-topology}
The single steps $r\to r+1$ for $r=0,1$, and their composite
$0\to1\to2$, have $H_1=0$ and $b^+=0$. A determinant on a single
step with the prescribed end restrictions is determined modulo
twice an integral class; changing it within that parity changes
the map only by tensor identifications at the ends.
The pentagons proving the single-step units may be taken on
\begin{equation}\label{eq:ordinary:two-triples}
s\to\infty\to r\to s,
\qquad r\to s\to\infty\to r,
\qquad s=r+1.
\end{equation}
Their hard caps have chain $[-1,-1]$, primitive radical
$P_1+P_2$, and determinant evaluations $(1,-1)$ up to a common
sign. The compressed pentagon for $g_+$ has chain
$[-2,-1,-2]$, radical $S_l+2T_1+R$, and determinant evaluations
\begin{equation}\label{eq:ordinary:hard-determinants}
c_0(S_l,T_1,R)=(1,-1,1),
\qquad c_1(S_l,T_1,R)=(-1,0,1).
\end{equation}
All these full pentagon cobordisms and their interval subpieces
have $b^+\le1$.
\end{lemma}
\begin{proof}
An attached meridian normally generates the incoming surgery
group, so the single steps have trivial first homology. Precede
$r\to r+1$ by $C_r$ and blow down the meridional corner. The
result is $C_{r+1}$ blown up once, with orthogonal classes
$G,T$, $G^2=r+1$, $T^2=-1$. The incoming trace class is
$G-T$, and the image of the step's intersection lattice is its
orthogonal complement, by the relative homology exact sequence.
For $r=0$ it is generated by $G-T$ and is null; for $r=1$ it
is generated by $G-2T$, of square $-2$. In particular neither
step has positive rank. Preceding $0\to1\to2$ by $C_0$
gives the twice blown-up $C_2$, with incoming null class
$G-T_1-T_2$. Its orthogonal complement has no positive
subspace. This proves the assertions about $b^+$ and, again
by normal generation, $H_1$.
The rank-one integral cohomology of a step tests the indicated
generator, and the end restrictions fix its parity. Since
$H_1=0$ there is no torsion ambiguity in this statement. Two
determinants of that parity differ by twice a line class.
Tensoring the rank-two bundle by that line identifies their
adjoint equations and induces the corresponding fixed
identifications on the end complexes. The rank and
invertibility assertions about the map are invariant under
these end automorphisms.
For the first triple in \eqref{eq:ordinary:two-triples}, use
orthogonal classes $F_l,G,T$ with squares $-s,s,-1$ and put
\[
P_1=G-T-F_l,\qquad P_2=T,
\qquad c(F_l,G,T)=(0,0,-1).
\]
Their intersections give the chain $[-1,-1]$ and
$c(P_1,P_2)=(1,-1)$. For the second triple, blow down
$P_1=T$ to $-C_r+C_r$ with classes $F_l,G$ of squares
$-r,r$, and put
\[
P_2=G-F_l-T,\qquad
c(F_l,G,T)=(1,1,1).
\]
Again the chain determinant is $(1,-1)$. The intermediate
$-C_s$ class is $F_l+T$, with even determinant evaluation;
thus the order-two end, when present, has the required
trivializable restriction. The zero-surgery restriction is
odd in the relevant coordinates. These presentations have
at most one positive direction, as do their subpieces.
For the compressed path the full trace is $-C_2$ followed by
the twice blown-up $C_2$. In the orthogonal basis
$F_l,G,T_1,T_2$ the squares are $-2,2,-1,-1$. Set
\begin{equation}\label{eq:ordinary:three-chain}
\begin{split}
A&=G-T_1-T_2,\\
S_l&=G-T_1-T_2-F_l,\qquad R=T_2-T_1,\\
c_0(F_l,G,T_1,T_2)&=(0,0,-1,0),\qquad
c_e=c_0+e\PD(S_l).
\end{split}
\end{equation}
Direct intersection gives squares $-2,-1,-2$ and adjacent
intersections $1$. The determinant evaluations are precisely
\eqref{eq:ordinary:hard-determinants}, and both annihilate
the radical. On $-C_2+C_0$ they restrict to the data of
Theorem~\ref{thm:ordinary:units}(ii). At $Z_2$ the remaining
piece has an allowed determinant for $f_+$. The displayed
ambient diagonal form also proves the positive-rank bound.
\end{proof}
\subsection{Compatible perturbations}
The handle families require perturbations that agree on shared faces
and vary independently on separated pieces. Interior equation
perturbations provide this freedom; at true Floer ends they must be
gradients of time-independent invariant functions. We construct
these two types of perturbation separately.
\begin{lemma}[Local perturbations and fixed-flat evaluation]
\label{lem:ordinary:perturbations}
For the finite ordinary families used in this section one can choose
arbitrarily small compatible perturbations with the following
properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Irreducible equations, together with any specified finite
collection of auxiliary evaluations, are transverse, using variations
supported in the same component. On a split face, variations on
different components are independent.
\item At a specified harmonic reducible, perturbations can vanish in
value while prescribing arbitrary finite-rank complex-linear maps
from the off-diagonal kernel to the off-diagonal cokernel. Thus a
complex index-zero normal operator can be made invertible, and a
complex index-one normal operator can be made surjective.
\item Evaluation of an irreducible field at an $S^1\times S^2$ end
is transverse to a local flat slice. Fixing a central flat imposes
three real conditions in that slice.
\end{enumerate}
The equation perturbations are independent of auxiliary cut marks.
They may be taken cylindrical at designated ordinary product ends,
with exponentially decreasing tails at separated auxiliary ends.
\end{lemma}
\begin{proof}
Choose a base frame and finitely many based loops and paths in the
component. Their holonomies and transports are equivariant under
the compact frame group. At an irreducible field some finite set
has precisely the central stabilizer. To see the corresponding
infinitesimal assertion, suppose that every loop variation is a
simultaneous infinitesimal conjugation. Extend that infinitesimal
conjugating element along paths from the base point. The loop
condition makes the extension independent of the path, and the
resulting section is an infinitesimal gauge transformation. Thus
finite samples detect any prescribed finite-dimensional horizontal
tangent space.
In a slice of the sample space choose equivariant coefficient
functions, and transport arbitrary compactly supported output forms
from small balls. To test universal surjectivity at a smooth base
configuration, first use the auxiliary linearization from
$W^{1,q}_{\delta_S}$ to $L^q_{\delta_S}$, with $q>4$. When the
higher charts use a H\"older weight $\delta_H$, choose
$0<\delta_S<\delta_H$ in the same root-free interval of the
normal end operator. Field values on paths
and at points are continuous in this domain; coefficients arising
from moving paths or finite parameters are evaluated at the smooth
base. Vary auxiliary cut values independently, thereby killing the
finite-dimensional cut component of any cokernel functional. Its
equation component is then an $L^{q'}$ dual functional, where
$q'=q/(q-1)$. Smooth compactly supported outputs detect every
nonzero such functional, even when required to avoid finitely many
labeled particle locations, since they are dense in $L^q_{\delta_S}$.
No smoothness of an adjoint cokernel field is asserted; derivatives
of graph terms can have distributional adjoints. A finite selection
of outputs spans the finite-dimensional cokernel. Primal elliptic
bootstrap transfers this surjectivity to the higher-order charts:
a low-order solution of the linearized equation with higher-order
right-hand side gains the required regularity. The fixed output
forms are smooth, while transport and sample linearizations preserve
the available H\"older order with the prescribed summable bounds;
scalar cutoff functionals multiply smooth profiles. Sobolev
embedding first supplies positive H\"older regularity, the local
elliptic inverse gains one order, and iteration gives each higher
order. The end estimates transfer the solution to weight
$\delta_H$ because there are no intervening indicial roots;
no equal-weight H\"older-to-Sobolev inclusion is used.
A countable dense library and Sard--Smale then give simultaneous
transversality for the countably many relative sectors and breaking
types in question.
At a reducible, retain noncentral sample data to distinguish its
parallel torus. The off-diagonal tangent and target spaces carry
the same complex torus character. Slice-linear equivariant
functions with zero value on commuting samples realize arbitrary
complex-linear maps between finite kernel and cokernel subspaces.
Finite-rank perturbation theory then gives the asserted invertibility
or surjectivity. The value on the entire Abelian path remains zero.
Further irreducible variations are supported away from that path in
configuration space.
To vary the asymptote, extend a prescribed infinitesimal harmonic
flat variation over the end. Its equation error is compactly
supported after a cutoff and is cancelled by the same interior
universal variations. This proves transversality of the evaluation.
Near a central $S^1\times S^2$ flat the slice consists of the three
parameters $h\vartheta$, $h\in\mathfrak{su}(2)$, with the full
parallel group acting on $h$. On the irreducible exterior the
condition $h=0$ has codimension three. Quotienting the flat space
first to a conjugacy interval would obscure these three equations
and is not our convention.
One may implement the samples by thickened loops, or by graph
samples multiplied by small-ball curvature cutoffs. In the latter
description the cutoff is zero if the graph enters a ball with
curvature above a fixed small-energy threshold. Consequently a
particle disables a graph it meets; graph terms away from particles
converge in regular gauges. Countably many terms with summable
smooth operator norms suffice. At product faces extend each term
from its component and turn off incompatible terms. Product
cutoffs and a relative generic extension over parameter interiors
preserve the face data. Moving paths can be handled in successively
higher Sobolev or little-H\"older orders, so that the finite
differentiability required by Sard--Smale is available. No mark
variable enters the equation coefficients. This construction is
the ordinary, zero-spinor specialization of the more detailed
compactness-compatible construction in
Section~\ref{sec:analysis}.
\end{proof}
For actual Floer ends the perturbations in the last lemma must be
gradients of time-independent invariant functions. The next lemma
supplies this additional requirement, rather than treating arbitrary
interior curvature perturbations as Floer perturbations.
\begin{lemma}[Gradient transversality on ordinary trajectories]
\label{lem:ordinary:gradient-transversality}
Small time-independent invariant thickened-holonomy functions, fixing
the central critical points, can make the irreducible critical points
and all irreducible trajectories needed here regular. Their
gradients are bounded, and their Hessians along a compact smooth
configuration arc are bounded operators on $L^2$.
\end{lemma}
\begin{proof}
Average invariant functions of simultaneous based holonomies over
a disk of base points and submersively thickened paths. The first
variation is a bounded transported insertion along the paths;
Cauchy--Schwarz followed by integration over the thickening gives
an $L^2$ bound for the Hessian. Use weighted summable smooth norms
for a countable collection of such functions. Thin-loop values
and first variations on a compact smooth arc are limits of these
averages. The detection argument of
Lemma~\ref{lem:ordinary:perturbations} therefore supplies arbitrary
finite-dimensional differentials on an irreducible sample slice,
which proves universal transversality for the critical points.
We verify the corresponding assertion for trajectories. In temporal
gauge write $\dot a=-\grad(\operatorname{CS}+h)(a)$. Two smooth
solutions agreeing on a time slice agree everywhere. Indeed, on
a compact interval their difference satisfies
\[
\dot\xi+D\xi=B(t)\xi,
\]
where $D$ is time-independent and symmetric and $B(t)$ is bounded
on $L^2$. On a nonzero interval set
$m=\langle D\xi,\xi\rangle/\|\xi\|^2$. Differentiation and
completion of the variance square give
\[
m'\le \tfrac12\|B\|^2,
\qquad
(\log\|\xi\|^2)'=-2m+O(\|B\|).
\]
The first inequality bounds $m$ above towards the right, and the
second then prevents the norm from reaching zero at a finite
right endpoint. Apply the same argument after reversing time and
replacing $D$ by $-D$ to exclude a finite left endpoint. This proves
uniqueness in both directions.
If a spatial gauge transformation stabilizes one slice, uniqueness
implies that it stabilizes the entire trajectory. Thus every slice
of an irreducible trajectory is irreducible. Comparison with a
critical trajectory shows that the velocity never vanishes and
neither limiting critical point is attained. Comparison with a
time translate shows that repetition modulo spatial gauge would
make the positive energy of a nonconstant segment repeat
indefinitely, contradicting finite energy. Hence the trajectory
is an injectively immersed curve in the spatial quotient.
Let $\eta$ annihilate the universal linearization, including all
additional gradient perturbations supported away from the limiting
critical points. Compactly supported temporal-connection and
spatial-field variations imply
\begin{equation}\label{eq:ordinary:trajectory-adjoint}
d_a^*\eta=0,
\qquad -\dot\eta+\mathcal H_a\eta=0,
\qquad \mathcal H_a=*d_a+\operatorname{Hess}h(a).
\end{equation}
A small compact segment of the immersed trajectory can be isolated
from the rest by finitely many sample functions: injectivity,
compactness up to its limiting critical points, and
finite-dimensional tangent detection give such a slice chart.
In that chart choose a function whose restriction to the curve is
zero but whose differential in any chosen horizontal normal
direction is arbitrary. Pairing its gradient with $\eta$ forces
$\eta(t)=c(t)\dot a(t)$. Next choose arbitrary functions along the
curve, with compact support in the chart. The universal pairing
becomes $\int c(t)\frac{d}{dt}f(a(t))\,dt=0$, so $c$ is constant.
Thickened averages approximate all these tests, hence give the
same conclusion.
Differentiating the trajectory equation gives
$\ddot a=-\mathcal H_a\dot a$. Combining this with
\eqref{eq:ordinary:trajectory-adjoint} and $\eta=c\dot a$ shows that,
if $c\ne0$, both $\mathcal H_a\dot a$ and $\ddot a$ vanish. The
$L^2$ norm of the velocity would then be constant; finite energy
forces it to be zero. Thus $c=0$, proving universal surjectivity.
Local slices, elliptic regularity and simultaneous Sard--Smale
complete the construction. This argument asserts no regularity
for arbitrary reducible trajectories; those are controlled by
their index and charge instead.
\end{proof}
The same choices regularize the metric families of
Lemma~\ref{lem:ordinary:metric-families}. Relative cutoffs extend the
auxiliary data from their prescribed product faces, and relative
generic extension over each parameter interior leaves those face
restrictions unchanged.
\subsection{Ordinary gluing and harmonic necks}
At an irreducible nondegenerate match the end operator has no zero
modes. We first recall this gluing argument, then treat the harmonic
flat modes at the auxiliary cap boundaries.
\begin{lemma}[Ordinary irreducible gluing]\label{lem:caps:irreducible-gluing}
Suppose completed ordinary ASD pieces have matching isolated,
nondegenerate irreducible end connections, fixed determinant-preserving
transition identifications, and compatible cut data. At a regular
matched cut configuration, gluing gives the usual local product chart,
including the neck-length parameters when these are variable. In
particular an isolated regular matched configuration gives one nearby
solution at each sufficiently long prescribed neck. The chart exhausts
solutions converging to that configuration and carries the coherent
determinant-line gluing orientation.
\end{lemma}
\begin{proof}
At a nondegenerate irreducible critical point the gauge-fixed tangential
operator has $h^0=h^1=0$, hence a spectral gap. The fields and their
linearized variations converge exponentially on the ends. Pregluing
therefore has exponentially small equation and cut errors. Split off
the finite-dimensional tangent space of the regular matched problem;
on its complement the half-problem operators have bounded right
inverses. Their cutoff splice gives a parametrix on the joined neck.
There is no tangential zero mode to match. Increasing the cutoff
transition length makes its derivative error small, and exponential
end decay controls the remaining splice error. A Neumann series
therefore gives a right inverse on that complement, uniformly in
sufficiently long necks. The gauge Laplacian has the same property.
For an isolated matched cut configuration, index zero makes these
operators invertible after the indicated gauge fixing and cuts.
The contraction theorem corrects the pregluing uniquely; with a
nonzero matched tangent space it gives the asserted smooth local chart.
Conversely, exponential end estimates put any converging solution in
this slice and in the contraction neighborhood, giving exhaustion.
These are the gapped irreducible gluing estimates of ordinary instanton
Floer theory \cite{DonaldsonKronheimer1990,Donaldson2002}.
The center $\{\pm1\}$ acts trivially on the adjoint data and extends
over either piece, so its two representatives do not give two matches.
The only permitted matching isomorphisms preserve the fixed determinant
transition. Pairing adjacent end orientation lines in cobordism order
gives the asserted orientation. In the cap applications of
Section~\ref{sec:caps} all cut degrees are even. The asymptotic cut data have the decay required by the same
parametrix.
This argument is for irreducible matches; it does not use the
auxiliary reducible-cap hypothesis of
Lemma~\ref{lem:ordinary:regular-gluing}.
\end{proof}
The analytic input at an $S^1\times S^2$ neck differs from that at
a rational sphere. In particular its central flat is not a smooth
endpoint of a one-dimensional slice. We retain its three harmonic
coordinates until the gauge quotient is taken.
\begin{lemma}[Decay with harmonic flat modes]
\label{lem:ordinary:neck-decay}
Give $L=S^1\times S^2$ the product metric, and let $\alpha$ be a
flat connection with parallel stabilizer $H$. On a sufficiently
small-energy neck, modulo gauge, the spatial field has the form
\begin{equation}\label{eq:ordinary:harmonic-slice}
a(t)=\alpha+h(t)\vartheta+b(t),
\quad h(t)\in H^0_\alpha(L;\ad),
\quad b(t)\perp H^1_\alpha(L;\ad),
\quad d_\alpha^*b(t)=0.
\end{equation}
Here $\vartheta$ is the circle one-form and the temporal component
has zero parallel part. For $[-T,T]$ and any sufficiently small
fixed decay rate $\delta>0$, the normal part satisfies estimates
of the form
\begin{equation}\label{eq:ordinary:normal-decay}
\|b(t)\|_{k}
\le C_k\bigl(B_-e^{-\delta(t+T)}
+B_+e^{-\delta(T-t)}\bigr)
+C_k\int_{-T}^T e^{-\delta|t-s|}\rho_k(s)\,ds,
\end{equation}
where $B_\pm$ measure the end-strip normal data and $\rho_k$ is
the decreasing perturbation or background error. The derivative
of $h$ is bounded by the same normal size and error. Its total
drift is therefore bounded independently of $T$, and on middle
strips the field is exponentially close to a single, possibly
varying, middle flat.
These assertions include a central $\alpha$, in whose fixed slice
$h$ has three real coordinates. They give matching up to $H$,
exponential convergence on completed ends, and convergence of
holonomies uniformly over compact smooth families of bounded
paths on the relevant strips. At isolated lens flats the same
assertions hold with no harmonic spatial term.
\end{lemma}
\begin{proof}
For a product metric the harmonic one-forms are exactly
$h\vartheta$, with $h$ parallel. Since
$\vartheta\wedge\vartheta=0$, every $\alpha+h\vartheta$ in this
slice is an actual flat connection, including at a central
$\alpha$. The residual parallel group acts by conjugation on
$h$. Time-dependent parallel rotations remove the parallel part
of the temporal connection.
The slice equation determines the remaining temporal component
$\varphi$ by $d_\alpha^*d_a$ on the orthogonal complement of
parallel sections. This operator remains invertible for small
$h,b$, and its right-hand side vanishes when $b=0$. Its estimates
therefore have the form
$\|\varphi\|_{k+1}\le C_k\epsilon\|b\|_k+C_k\rho_k$ after
shrinking the slice. Projecting the ASD equation to the
co-closed normal space gives
\[
(\partial_t+K_\alpha)b=\mathcal R,
\qquad K_\alpha=*d_\alpha,
\qquad
\|\mathcal R\|_k\le C_k\epsilon\|b\|_k+C_k\rho_k.
\]
There is no forcing from a pure harmonic field. Projection to
harmonic forms gives the analogous bound for $|\dot h|$.
On the normal space $K_\alpha$ is self-adjoint with a spectral
gap. Propagate its positive spectral part from the left and its
negative part from the right. The resulting Green kernels are
bounded by $Ce^{-\lambda|t-s|}$ for a gap $\lambda>0$.
Convolution with these kernels and absorption of
$C\epsilon\|b\|$ in a weight $\delta<\lambda$ prove
\eqref{eq:ordinary:normal-decay}. Interior elliptic estimates
give the higher derivative versions. Integration of the
harmonic derivative gives a bound by
$C\epsilon(B_-+B_+)+C\int\rho$, with no factor $T$.
Integrating only between a middle strip and $t=0$ gives the
exponential closeness to $\alpha+h(0)\vartheta$ claimed in the
statement.
To justify using one slice along a long small-energy stretch,
start near a cluster flat on a middle strip. Local compactness
places every boundary strip of a maximal slice interval close
to some flat. The normal estimate and the length-independent
harmonic drift bound keep the solution inside a slightly larger
fixed slice until a strip leaves the small-energy regime.
Shrinking the energy threshold first excludes such an exit in
the stretches under discussion. This also avoids changing a
three-dimensional central slice into a one-dimensional
noncentral slice during the argument.
Any remaining non-small stretch produces a nonconstant level
or a particle by ordinary compactness. The collar charge on
$S^1\times S^2$ is integral because Chern--Simons is constant
on its connected flat space, so a nonflat level costs at least
one charge. After the finitely many possible losses have been
extracted, the preceding estimates apply to all remaining
stretches. Smooth convergence on strips implies the holonomy
assertion by the transport ODE and uniform bounds on the
chosen path family. For a lens flat $h^1=0$, and the same
proof uses only the gapped normal operator.
\end{proof}
\begin{lemma}[Regular matching at an auxiliary reducible end]
\label{lem:ordinary:regular-gluing}
Suppose two completed ordinary pieces meet at one of the auxiliary
flats above. Assume that one piece is a cap whose parallel stabilizer
is the whole matching group $H$; it has one matching end, and its
other end conditions, if present, are decaying. Assume that the
connection on the other piece is irreducible; prescribed unglued
true-end asymptotes are also irreducible. Frame the cap at the
matching end and leave the other piece unframed. Suppose the
matched half-problem, including all flat parameters, metric parameters
and cuts, is isolated and transverse. Then for all sufficiently long
necks each such match gives exactly one solution modulo
determinant-one gauge. Every convergent solution sufficiently close
to this broken configuration arises in this way. The regular
operators and gauge-fixing operators have inverses with at most
polynomial growth in neck length.
The assertion also holds for successive removal of pendant caps
in a tree of regular cuts, provided each removal satisfies these
hypotheses with compatible data. Orientations are the determinant-line gluing
orientations; there is no factor from ordering a splitting, from
the central subgroup $\{1,-1\}$, or from additional projective
gauge twists.
\end{lemma}
\begin{proof}
Preglue the halves with their common flat. By
Lemma~\ref{lem:ordinary:neck-decay}, the equation and cut errors are
exponentially small. The gauge-augmented linearization on each half
has bounded extended kernel fields whose limits consist of a
parallel temporal mode and a harmonic spatial variation. All normal
remainders decay exponentially.
The square operator on the finite glued manifold, with its true
ends completed, is
\[
D_T:
W^{k+1,q}_\delta(\Omega^1(\ad E))\oplus\R^{p+a}
\longrightarrow
W^{k,q}_\delta(\Omega^0(\ad E)\oplus\Omega^+(\ad E))
\oplus\R^b.
\]
Here $q>4$, positive weights occur only at the fixed true ends,
$p$ counts metric parameters, $a$ counts cut marks, and $b$ is the
cut codimension. The gauge row is $d_A^*$. The irreducible core
eliminates a global infinitesimal stabilizer. Index addition gives
\[
\ind D_T=i_N^{\rm fix}+i_V^{\rm fix}+h^0+h^1+p+a-b=0,
\]
the last equality being isolated regularity of the matched problem.
Indeed varying both flat slices adds $2h^1$, matching subtracts
$h^1$, and framing the cap adds $h^0$. At a central $L$ flat,
both $h^0$ and $h^1$ are three, so all three spatial matching
equations are retained. This index-zero assertion concerns the
full finite operator. The cap's decaying augmented operator can
still have the parallel scalar cokernel of its gauge row, even
when its physical framed problem is regular. Extended temporal
modes and the following flux calculation account for that
cokernel; framing alone does not remove it.
We first identify the zero modes in the matched kernel. On the
stabilizer cap integrate the linearized gauge-slice equation against
a globally parallel stabilizer section $\xi$. If
$a=\tau\,dt+\beta+o(1)$, with $\tau\in H^0_\alpha$ and
$\beta\in H^1_\alpha$, integration on a truncation $N_R$ gives
\[
0=\int_{N_R}\langle d_A^*a,\xi\rangle
=-\int_{L_R}\langle a(\nu),\xi\rangle+o(1).
\]
There is one possible nondecaying boundary flux; all other end
terms tend to zero. Global parallel sections restrict onto the
whole end stabilizer, so the limit forces
$\langle\tau,\xi_\infty\rangle=0$ for every $\xi_\infty$, and
hence $\tau=0$. The spatial mode $\beta$ has no normal contraction.
The same argument kills all three temporal modes at a central
flat. Although framing restricts the allowed cap gauges, the
parallel sections remain legitimate test sections in this
integration identity. Temporal matching gives the same conclusion on the other
half. The remaining spatial flat modes, field variations,
parameters and cut variations solve exactly the matched half
linearization. Its kernel is zero by the isolated transversality
hypothesis. A residual infinitesimal gauge in augmented gauge is
also zero: integration of its gauge Laplacian gives its covariant
derivative norm, with zero end terms. At true irreducible ends use
decaying gauge parameters.
Here is a useful quantitative version of this argument. Use a
uniform local norm, the supremum of the elliptic norms on unit
strips, together with the finite parameter norm and the fixed
positive outer-end weights. If no polynomial lower bound held,
one could find unit-norm fields and parameters on necks of length
$2T$ with linearized residual $o(T^{-1})$ (and, if necessary,
smaller than any fixed inverse power for higher-order norms).
Their limits on the two halves solve the homogeneous half
equations. Projection to the neck zero modes shows that their
derivatives are $o(T^{-1})$ plus the exponentially decreasing
normal tails. More explicitly, writing $z_T$ for that projection,
between distances $R$ from the two neck ends one has
\[
|z_T(T-R)-z_T(-T+R)|
\le CT\|D_Tx_T\|_{\mathrm{uloc}}+Ce^{-\delta R}.
\]
First let $T$ and then $R$ tend to infinity. Thus the two limiting
constants agree. The
previous paragraph makes both half limits and all parameters
zero. A remaining unit norm translating down the neck would
limit to a bounded homogeneous full-cylinder solution. The
spectral gap leaves only a zero mode, but its value is already
zero by the same matching estimate. This contradicts local
elliptic compactness. In particular a mode of amplitude one
spread over a neck, whose derivative is of order $T^{-1}$,
cannot invalidate the argument: it is excluded by the stated
residual bound in the uniform local norm.
The full operator has index zero, so this lower bound implies
injectivity and hence surjectivity, giving an inverse with polynomial norm.
Equivalently one constructs its right inverse from the inverses
on the matched half spaces and the finite-dimensional neck zero
modes. The gauge Laplacian has the same polynomial bound by
the long-cylinder Poincar\'e estimate and the irreducible core
on the other side. Exponential pregluing errors dominate these
polynomial losses, so the contraction theorem produces one
solution in a shrinking slice neighborhood.
For exhaustion, a convergent sequence has, on its middle strips,
an exponentially close \emph{variable} flat, by
Lemma~\ref{lem:ordinary:neck-decay}. Cut using that same flat on
both halves. In weak positive weights on the normal remainders,
and the ordinary norm on the flat parameter, the half data
converge and have exponentially small equation and matching
errors. The injective regular half linearization improves this
closeness modulo gauge to exponential closeness. Hence the
sequence lies in the uniqueness neighborhood just constructed.
At a central flat the allowed stabilizer rotation acts on the
small harmonic variation as well as on the frame.
For the stated tree, remove the pendant stabilizer caps in the
specified order and use product gluing. At each removal the flux
argument has just one nondecaying cap end; this is stronger than
merely assuming that the cut graph has no cycles.
The cap stabilizer absorbs the matching $H$-rotations. The
central gauge subgroup is already part of the gauge quotient,
and the fixed determinant transitions do not permit summing
over nonliftable $\SO(3)$ twists. A change of integral framing
degree changes the relative charge and hence the actual index;
it belongs to a different sector, not to an additional copy of
the same zero-dimensional match. Finally the usual APS/Floer
determinant-line identifications, associative under these
gluings, give the stated integral orientations.
\end{proof}
\subsection{Cap shifts and primary-face exclusions}
\begin{lemma}[The elementary cap shifts]\label{lem:ordinary:cap-shifts}
The isolated exceptional-sphere faces of the single-step pentagons
contribute at least two indices relative to a flat-ball filling.
The $M_{14}$ face of the compressed pentagon has the same lower
bound. The odd-determinant $(-2)$-disk-bundle cap at $M_{02}$
has a unique projective boundary flat class, of trace zero and
$(h^0,h^1)=(1,0)$, and its relative contribution including matching is
\begin{equation}\label{eq:ordinary:minus-two-shift}
\delta=8(E-\tfrac18),\qquad E\in\tfrac18+\Z.
\end{equation}
At $E=1/8$ its only generic framed zero-dimensional match is the
harmonic reducible, with multiplicity one. The two determinant
choices at this face have identical exterior data and cancel
modulo two.
\end{lemma}
\begin{proof}
For a split bundle write $d$ for the difference of the summand
classes. Its trace-free charge is $-d^2/4$. On an odd
exceptional sphere the least such square is $-1$, so
$E\in1/4+\Z$ and its nonnegative minimum is $1/4$.
The exceptional disk bundle and the flat ball have the same
$\chi+\sigma=1$. Equation~\eqref{eq:ordinary:excision},
including the separating $S^3$ matching, therefore gives
the shift $8E\ge2$.
The $M_{14}$ chain has matrix and inverse
\[
Q=\begin{pmatrix}-1&1\\1&-2\end{pmatrix},
\qquad Q^{-1}=\begin{pmatrix}-2&-1\\-1&-1\end{pmatrix}.
\]
Its two determinant evaluation vectors are $(-1,1)$ and
$(0,1)$; both have square $-1$. The cap is unimodular, its
link is $S^3$, and again $\chi+\sigma=1$.
Consequently $E\in1/4+\Z$ and the same shift is at least two.
For the $(-2)$ disk bundle an odd difference evaluates to one
on the section and has square $-1/2$, giving reference charge
$1/8$. Its boundary is $\RP^3$. The projective flat
representation with this determinant obstruction is represented,
after square-root normalization, by
$\diag(i,-i)$ on its generator; it is the unique such class.
Its stabilizer is a torus and $h^1=0$.
To count the charge-$1/8$ cap, we first show that its framed normal
index is zero. Compactify the resolution to the ruled surface $\mathbb F_2$ and contract its
exceptional section to obtain $\mathbb P(1,1,2)$. Let $T$
be a fiber class on $\mathbb F_2$. It has degree one on the
exceptional section and corresponds to $\mathcal O(1)$ on
the common smooth complement. The real off-diagonal ASD
symbol is complex; on a closed K\"ahler surface its complex
index is
$-\chi(\mathcal O(T))-\chi(\mathcal O(-T))$.
On $\mathbb F_2$ these fiber twists are pullbacks of
$\mathcal O(1),\mathcal O(-1)$ on the base, with no higher
direct image, giving Euler characteristics $2,0$. On
$\mathbb P(1,1,2)$ the weight-one monomials give
$h^0(\mathcal O(1))=2$. The monomial decomposition of the
three-coordinate \v{C}ech complex has no degree-one cohomology,
and its degree-two cohomology is dual to weighted polynomials
of degree $-k-4$ for $\mathcal O(k)$. For $k=1,-1$ these
spaces vanish. Thus there is no higher cohomology for
$\mathcal O(1)$ and no cohomology for $\mathcal O(-1)$.
Thus its two Euler characteristics are also $2,0$.
Excision on their common complex bundle complement has zero
normal index difference. The cone ball has framed index
zero, as follows from the flat-ball operator restricted to
the invariant fields, so the framed normal index on the
resolution cap is zero. Equivalently, the invariant flat-ball
complex has no harmonic one-forms or self-dual two-forms,
and framing removes its parallel gauge contribution.
The tangential framed operator has
index zero and no kernel or cokernel because $H^1=b^+=0$.
Unframing subtracts its one-dimensional stabilizer, and
matching adds that dimension back. This gives zero total
shift at $1/8$. All other relative sectors change charge
by an integer and index by eight, proving
\eqref{eq:ordinary:minus-two-shift}. The complement
identifications here are complex bundle identifications;
matching just real Chern forms would not suffice for this
index excision.
At the reference charge the tangential harmonic solution is
unique. Lemma~\ref{lem:ordinary:perturbations} makes the
index-zero complex normal operator invertible without changing
that solution. A framed irreducible would carry a positive
dimensional orbit of the boundary torus, contradicting a
regular zero-dimensional framed count. Additional levels or
particles would increase charge and are unavailable at the
minimum. Finally adding $\PD(S_l)$ changes the determinant
evaluation by $-2$ on this cap, hence is tensoring by a line
of degree $-1$. This tensor operation preserves the adjoint
equation, and its order-two action at the boundary preserves
the trace-zero class. Choose tensor-related cap perturbations
and the same exterior perturbations. The two minimum
contributions then agree and cancel over $\F_2$.
\end{proof}
\begin{lemma}[The primary central-group estimate]
\label{lem:ordinary:central-cost}
In a broken ordinary configuration, let a maximal consecutive group
consist of $q$ reducible pieces or levels with only central rational
contacts. Suppose its pieces have $b_1=0$, and denote by
$b^+_{\rm group}$ the sum of their positive ranks. Its index
contribution, including all internal matches and its two exterior
central matches, is at least
\begin{equation}\label{eq:ordinary:group-cost}
3(1-b^+_{\rm group}).
\end{equation}
In particular this contribution is at least three when the group
has positive rank zero. A direct central match between two
irreducible pieces adds three. The estimate remains valid when
the isolated minimum odd $(-2)$ cap is virtually filled by its
reference cap.
\end{lemma}
\begin{proof}
Apply \eqref{eq:ordinary:central-index} to every group member
and use its nonnegative charge. Their total is at least
$-3q-3b^+_{\rm group}$. There are $q-1$ internal contacts
and two exterior contacts, each contributing three. Adding
them gives \eqref{eq:ordinary:group-cost}. Direct matching
uses the same stabilizer dimension. For a group reaching a
prescribed irreducible true end, the exterior contact is
realized by a nonconstant trajectory from the central state
to that irreducible; this trajectory is not included in the
group and has index at least one. No reducer can meet the
admissible end $Y_0$. The last assertion follows from the
zero framed shift of the reference filling in
\eqref{eq:ordinary:minus-two-shift}; its removal changes
neither the exterior index convention nor the lower bound.
\end{proof}
\begin{lemma}[Compactifications in the unit dimensions]
\label{lem:ordinary:low-dimensional-compactness}
For the rigid maps, intervals and pentagons above, all moduli spaces
needed in dimensions zero and one have no particles and no unwanted
central contacts at primary cuts or true ends. Primary $S^3$ faces
are empty. Exceptional-sphere faces and the $M_{14}$ face are empty
in these dimensions. The only possible contribution at the
compressed $M_{02}$ face is the paired minimum contribution of
Lemma~\ref{lem:ordinary:cap-shifts}, away from the corners.
All other primary boundary strata are ordinary chain breakings or
the stated compositions.
\end{lemma}
\begin{proof}
The positive subspaces of split main pieces inject orthogonally
into the unsplit form, so their total positive rank is at most
one. If $p'$ parameters remain on irreducible pieces, their
total index is at least $-p'$ by regularity. We now check each
possible number of primary cuts; this avoids supposing that
a non-admissible primary face behaves like an admissible one.
For a single rigid $b^+=0$ map every forbidden central group
has the surplus three in
Lemma~\ref{lem:ordinary:central-cost}. The chain dimensions
are zero and one, so the surplus excludes it. The same
argument permits an ordinary one-parameter continuation and
stretching the middle of $f_+$: even subtracting that one
parameter leaves a surplus two.
For an interval $p=1$ the relevant indices are $-1,0$; for
a pentagon $p=2$ they are $-2,-1$. With no primary cut,
a central group of positive rank zero costs at least three,
leaving a lower total index $3-p$ and hence exceeding the
required indices. If a central group uses the available
positive rank, it must contain the whole uncut cobordism
piece: a group confined to true-end cylinder levels has
positive rank zero. It then requires two nonconstant
irreducible end trajectories. They cost at least two in
total; there is no irreducible cobordism piece from which
to subtract the metric parameter count.
At one primary cut a group using the positive rank contains
a main piece incident to at least one true end and hence
requires at least one such trajectory. At most $p-1$
parameters remain on the other pieces. Its total is
therefore at least $1-(p-1)$, which is positive for the
interval and at least zero for the pentagon. Both exceed
the needed boundary indices. Other central groups again
have surplus three. At two primary cuts of a pentagon
no metric parameters remain. Every group contribution is
nonnegative and every irreducible piece has nonnegative
index, whereas the boundary problem has index $-1$.
This excludes its central configurations as well. Since
$S^3$ has no irreducible generator, its primary faces are
empty in all these cases.
For an exceptional or $M_{14}$ cap, virtual filling leaves
the preceding nonnegative group bounds and adds at least
two indices. At most one parameter remains on the other
pieces of a pentagon face, and none on an interval face.
Thus these faces cannot occur. At $M_{02}$ the minimum
reference filling has shift zero. It can occur at a
rigid exterior parameter, as in
Lemma~\ref{lem:ordinary:cap-shifts}; its corners impose
the additional parameter loss and are excluded by the
same primary and exceptional bounds. A higher sector
there adds at least eight indices.
For completeness, particle positions do not defeat these
inequalities. A lost charge $q\ge1$ lowers field index
by $8q$ and supplies at most $4q$ position parameters.
Thus the dimension of an ideal stratum is at most the
unbroken expected dimension minus $4q$, before any
additional face loss. It is negative in dimensions at
most one. If a graph perturbation or auxiliary evaluation
is disabled by the particle, retain its position as a
parameter and use the componentwise variations of
Lemma~\ref{lem:ordinary:perturbations}; no insertion of
positive degree occurs in these unit families to restore
the lost dimension. The same argument applies to
particles on levels. Charge discreteness and the bounded
gradient energy identity prevent accumulation of further
nonconstant levels.
Consequently regular compactness leaves precisely the
ordinary irreducible chain breaks and the stated
compositions. Notice that reducible main components
joining two nondecaying primary ends have been
\emph{excluded} here by dimension. We never apply
Lemma~\ref{lem:ordinary:regular-gluing} to such a component;
all its stabilizer absorbers in the following counts are
single-ended auxiliary caps.
\end{proof}
\subsection{The hard cap and its restriction degree}
At the remaining outer face, replacing the sphere chain by
$S^1\times D^3$ leaves an identity cylinder. We will show that its
exterior contributes the continuation map and that the original cap
supplies an odd number of matches with each generic noncentral
boundary flat, summed over the two determinants in the compressed case.
\begin{lemma}[Flat replacement and the continuation exterior]
\label{lem:ordinary:flat-replacement}
At the outer cut $M_{03}$ of a single-step pentagon, or $M_{04}$
of the compressed pentagon, write $N$ for the sphere-chain cap
and $V$ for its exterior. Replace $N$ by $D=S^1\times D^3$,
keeping the determinant transitions fixed. The filled cobordism
is the appropriate identity cylinder, possibly with a fixed
change of end identification. If $i_{\rm filled}$ denotes its
index, then
\begin{equation}\label{eq:ordinary:hard-index}
i_{\rm total}=i_{\rm filled}+8E_N-3,
\qquad i_{\rm filled}=i_V^{\rm fix}+h^1,
\qquad E_N\in\tfrac14+\Z.
\end{equation}
For the pentagon boundary problem $i_{\rm total}=-1$, the only
possible exterior is regular and irreducible, has
$i_{\rm filled}=0$, and has a noncentral boundary flat.
Necessarily $E_N=1/4$. Its count with the flat replacement cap
is the continuation isomorphism.
\end{lemma}
\begin{proof}
Both chain determinants annihilate their primitive radical, so
restrict trivially to $L$ and extend over $D$. The last assertion
of Lemma~\ref{lem:ordinary:polygon} identifies the filled
cobordism. Leave the replacement cap unperturbed. A flat on
$L$ extends uniquely to $D$ with its chosen boundary
representative: both fundamental groups are generated by the
same circle. Its fixed-limit unframed index is $-h^0$.
After framing it has a regular family of flats mapping
identically to the boundary flat slice. Thus fixed-limit
addition gives
$i_V^{\rm fix}-h^0+(h^0+h^1)=i_V^{\rm fix}+h^1$.
For the two-sphere chain $\chi(N)=3$, $\sigma(N)=-1$;
for the three-sphere chain $\chi(N)=4$, $\sigma(N)=-2$.
In both cases $\chi(N)+\sigma(N)=2$, whereas
$\chi(D)+\sigma(D)=0$. Excision therefore gives the first
identity of \eqref{eq:ordinary:hard-index}. The split
reference differences in the two caps have square $-1$,
as computed explicitly in
Lemma~\ref{lem:ordinary:harmonic-caps} below. Their charge
is $1/4$. Chern--Simons is constant on the flat space of
$L$, and fixing the determinant and relative transitions
makes changes of the remaining bundle degree integral.
Hence $E_N\in1/4+\Z$, including charge on extracted hard-neck
levels. Nonnegativity gives $E_N=1/4+k$ with $k\ge0$.
At total index $-1$, \eqref{eq:ordinary:hard-index} becomes
$i_{\rm filled}=-8k$.
The exterior has no remaining metric parameter; the one
parameter on this face belongs to the cap. Its moduli space
with variable asymptote therefore has dimension
$i_V^{\rm fix}+h^1=i_{\rm filled}$. The virtual filling
has positive rank zero, so
Lemma~\ref{lem:ordinary:central-cost} excludes true central
breaks there; an admissible end also excludes reducibles.
A central $L$-restriction imposes the full three conditions
of Lemma~\ref{lem:ordinary:perturbations}. Thus generic data
leave only regular irreducibles of dimension zero, with a
noncentral restriction, and force $k=0$.
To identify their count, stretch the same $L$ neck in the
filled cylinder. A nonflat $D$-side sector or collar level
costs at least one integral charge, lowering the remaining
regular index by eight. Particles are excluded by the same
index loss. The only limiting replacement fields are
therefore the regular flat family on $D$ just described.
Lemma~\ref{lem:ordinary:regular-gluing} applies: the global
parallel stabilizer on $D$ is the entire end stabilizer,
and $D$ has one matching end. Its unique gluing identifies
this count with the usual continuation count. Continuation
is a chain homotopy equivalence over $\Z$ with coherent
orientations. A fixed tensor or collar identification can
compose it with an automorphism, which is still an
isomorphism. In the compressed pentagon the exterior and
its data are common to the two determinant choices.
\end{proof}
\begin{lemma}[Minimum-charge harmonic caps]
\label{lem:ordinary:harmonic-caps}
Let $N$ be either hard cap of
Lemma~\ref{lem:ordinary:unit-topology}, with one of its specified
determinants $c$. At charge $1/4$ the only possible unordered
split difference is $\{c,-c\}$. For each metric of its cap
interval there is a unique Abelian ASD connection modulo gauge
with the ordered difference $d=c$, allowing the boundary flat
to vary. These connections form a harmonic path.
At a fixed noncentral boundary flat the cap index is $-2$.
Its tangential unframed index is $-2$ and its complex normal
index is zero. If a noncentral exterior angle is a transverse
value of the harmonic path, the framed cap with its one metric
parameter has one regular contribution at each crossing.
There is no second contribution from exchanging its summands
or from a residual boundary framing.
\end{lemma}
\begin{proof}
For the chain $[-1,-1]$ a class annihilating the radical has
evaluations $(a,-a)$ and square $-a^2$. The determinant
parity makes $a$ odd, so charge $1/4$ forces $a=\pm1$.
For $[-2,-1,-2]$, write the difference evaluations as
$(d_l,d_m,d_r)$. The radical condition and square are
\begin{equation}\label{eq:ordinary:three-chain-charge}
d_l+2d_m+d_r=0,
\qquad -\frac{d^2}{4}=\frac{d_l^2+d_r^2}{8}.
\end{equation}
The first and last evaluations are odd. Charge $1/4$
forces $d_l,d_r\in\{1,-1\}$. When $e=0$, $d_m$ is
odd, so $d_l=d_r$ and the possibilities are
$\pm(1,-1,1)$. When $e=1$, $d_m$ is even, so
$d_l=-d_r$ and the possibilities are $\pm(-1,0,1)$.
This proves both the uniqueness assertion and $c^2=-1$.
The class $d$ restricts trivially in real cohomology to
$L$, hence lies in the image of compactly supported
cohomology. Cylindrical-end Abelian Hodge theory gives
its $L^2$ harmonic representative. The intersection form
on this image is negative definite and $b^+=0$, so that
representative is anti-self-dual. Integrating it gives
an Abelian connection with the prescribed integral
summand class; its limiting flat is determined by this
connection. The fixed determinant supplies the other
summand. There is no missing self-dual obstruction when
the harmonic boundary parameter is allowed to vary.
For uniqueness, the difference of two such curvatures
is exact, anti-self-dual, and decays. Stokes' theorem on
long truncations gives zero for its square; the flat
limiting difference contributes no boundary curvature.
An anti-self-dual exact difference of square zero
vanishes. The plumbing has $H^1(N;\R)=0$ and is simply
connected, so a remaining flat difference of line
connections is gauge. Smooth dependence of the inverse
Abelian operator on the metric produces the harmonic
path. The same argument supplies the uniqueness of the
minimum odd $(-2)$ reference used in
Lemma~\ref{lem:ordinary:cap-shifts}.
At a noncentral flat the replacement cap has index $-1$.
Its difference from $N$ is $8(1/4)-3=-1$, so the fixed
cap index is $-2$. In the diagonal scalar complex there
is one parallel stabilizer and one boundary-flux
obstruction when the flat is fixed; $H^1(N)=b^+(N)=0$
leaves no other tangential kernel or obstruction.
Thus the tangential index is $-2$. Subtracting it from
the full real index leaves zero, so the complex normal
index is zero. Framing restores one real dimension;
a transverse metric crossing of the prescribed flat
removes the remaining flux obstruction. Derivative
variations from Lemma~\ref{lem:ordinary:perturbations}
make the complex normal operator invertible while
leaving the harmonic path fixed. This gives a regular
zero-dimensional framed cap problem. A putative
irreducible cap, unframed and at this fixed angle, would
have expected dimension $-2+1=-1$ and is absent for
generic data.
An unordered splitting is counted once. Choose $d=c$
to order its two lines. At a fixed noncentral boundary
representative its actual normalized eigenvalue, not
merely its conjugacy class, prescribes which boundary
eigenline is assigned to this ordered line. Swapping
both lines describes the same unordered connection and
does not create a new boundary choice. The parallel
torus of the split cap absorbs all matching rotations.
This is exactly the single-ended stabilizer situation
of Lemma~\ref{lem:ordinary:regular-gluing}, so there is
no additional multiplicity.
\end{proof}
\begin{lemma}[Normalized eigenholonomy and the odd restriction degree]
\label{lem:ordinary:holonomy-degree}
For the $[-1,-1]$ cap, the harmonic path has odd mod-two
intersection number with every generic noncentral conjugacy
class of boundary flats. For the $[-2,-1,-2]$ cap the sum of
those intersection numbers over the two determinant choices
is odd.
\end{lemma}
\begin{proof}
Let $\beta$ generate the free circle of $L$, and choose
an integral spanning surface or chain $\Delta$ for $\beta$
in $N$. Fix the determinant trivialization $\tau$ at
$L$, compatible throughout the family. With curvature
normalized to integral periods, Abelian Stokes' theorem
for the ordered line of difference $d$ gives its
normalized eigenholonomy as
\begin{equation}\label{eq:ordinary:eigenholonomy}
u(d,c)=(-1)^{\langle c,(\Delta,\tau)\rangle}
\exp\left(\pi i\int_\Delta d_{\rm harm}\right),
\end{equation}
up to inversion. Indeed the ordered summand curvature
is $(F_{\det}+d_{\rm harm})/2$. Exponentiating its
integral over $\Delta$ gives the exponential in
\eqref{eq:ordinary:eigenholonomy}; the relative integral
of $F_{\det}$ gives the displayed half-determinant sign.
The sign cannot be discarded when determinants are
compared. Inversion changes the ordering but not the
conjugacy class or the mod-two degree.
The primitive radical is the image of the boundary
sphere by Lemma~\ref{lem:ordinary:polygon}. Boundary
duality says that a spanning chain of the free boundary
circle pairs once, up to sign, with that sphere.
Consequently
\begin{equation}\label{eq:ordinary:radical-pairing}
\Delta\cdot(P_1+P_2)=1\pmod2
\quad\hbox{or}\quad
\Delta\cdot(S_l+2T_1+R)=1\pmod2,
\end{equation}
as appropriate.
We justify the curvature localizations used at the
ends of the harmonic paths. Across a rational
plumbing link there are no real harmonic one-forms.
The linear version of the gapped estimate in
Lemma~\ref{lem:ordinary:neck-decay} gives exponential
decay for the harmonic curvature on the stretched
collar. Hodge compactness on the fixed cores and
uniqueness of the harmonic representative identify its
limits with the prescribed piecewise representatives.
Their tail integrals tend to zero. Extending compactly
supported representatives by zero therefore computes
the limiting integral over $\Delta$ by its relative
intersection pairing. This is a linear Hodge
calculation; it does not require nonlinear gluing at
a higher-energy stratum.
For $[-1,-1]$, the two ends isolate the exceptional
spheres $P_1$ and $P_2$. For the same ordered difference
$c=(1,-1)$ their limiting localized representatives are
respectively $-\PD(P_1)$ and $\PD(P_2)$. Each yields
a central boundary value, because its integral over
$\Delta$ is integral. Their quotient in
\eqref{eq:ordinary:eigenholonomy} is
$(-1)^{\Delta\cdot(P_1+P_2)}=-1$ by
\eqref{eq:ordinary:radical-pairing}. The two endpoints
are therefore the opposite central values.
For $[-2,-1,-2]$, first consider the end isolating
$S_l$, with cap $B=\nu S_l$ and rational link
$Q=\partial B=\RP^3$. Let $p$ be the integral relative
Thom class of $S_l$, represented by the line $\mathcal O(S_l)$
with its canonical nonzero section $s$ off $S_l$.
For the determinant lines choose
$D_1=D_0\otimes\mathcal O(S_l)$ and
$\tau_1=\tau_0\otimes s|_L$, as required by the fixed
outside identification. Thus their relative first Chern
classes differ by $p$, not by an unspecified lift of its
absolute class. In particular the two half-determinant
signs differ by $(-1)^k$, where
$k=\langle p,[\Delta]\rangle=\Delta\cdot S_l$.
The common remaining curvature can be seen explicitly.
In rational cohomology put $w=c_0+p/2$. Then
\[
p^2=-2,\qquad c_0\cdot p=1,\qquad
w\cdot p=0,\qquad w^2=-\tfrac12,\qquad
c_e=w+(e-\tfrac12)p.
\]
Under the $Q$ stretch, linear Hodge splitting localizes
$\pm p/2$ on $B$ and the same class $w$ on the complement.
Each part has charge $1/8$. Only the \emph{difference}
of the two harmonic curvatures localizes entirely on $B$.
Its limit there is the harmonic representative of $p$,
and its limit on the complement is zero. A compact
Thom representative and this harmonic representative
differ by an exact form whose primitive decays modulo
an exact boundary form, since $H^1(Q;\R)=0$.
Split $\Delta$ across $Q$ and apply Stokes' theorem;
the neck tails have vanishing integrals. Consequently
the integral over $\Delta$ of the harmonic difference
tends to $k$. The exponential factor in
\eqref{eq:ordinary:eigenholonomy} therefore changes by
$\exp(\pi i k)=(-1)^k$, cancelling the relative
determinant sign.
Thus the two paths have the same boundary value at
this end; that common value need not be central.
At the other end the unimodular cap on $T_1,R$ carries
the entire charge. Its two localized differences are
\begin{equation}\label{eq:ordinary:unimodular-localization}
d_0=\PD(T_1),\qquad d_1=-\PD(T_1)-\PD(R).
\end{equation}
The evaluations of these classes on $S_l,T_1,R$
are exactly \eqref{eq:ordinary:hard-determinants}.
Choose $\Delta$ to avoid this cap. This is possible
because its separating link is $S^3$: van Kampen
applied to the simply connected plumbing shows that
the complement is simply connected, so $\beta$ bounds
there. The curvature integrals in
\eqref{eq:ordinary:eigenholonomy} are now zero.
Equation~\eqref{eq:ordinary:radical-pairing} reduces
to $\Delta\cdot S_l=1\pmod2$. Hence the determinant
signs of the two paths differ by $-1$. Their values
at this end are the opposite central values.
Identify the conjugacy space of flat $\SU(2)$
connections on $L$ with the closed angle interval
$[0,\pi]$.
Figure~\ref{fig:ordinary:cap-endpoints} records the endpoint data.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[font=\small,
sphere/.style={circle,draw,inner sep=1.7pt},
path/.style={-{Stealth[length=1.6mm]},thick}]
\node at (1,1.9) {$[-1,-1]$};
\node[sphere,label=above:$-1$,label=below:$P_1$] (p1) at (0.3,1.1) {};
\node[sphere,label=above:$-1$,label=below:$P_2$] (p2) at (1.7,1.1) {};
\draw (p1)--(p2);
\node (zero) at (4.2,1.1) {$0$};
\node (pi) at (7.5,1.1) {$\pi$};
\draw[path] (zero)--(pi);
\node[align=center] at (5.85,0.4) {$P_1$ and $P_2$ isolations\\(either order)};
\node at (1,-0.6) {$[-2,-1,-2]$};
\node[sphere,label=above:$-2$,label=below:$S_l$] (sl) at (0,-1.4) {};
\node[sphere,label=above:$-1$,label=below:$T_1$] (t1) at (1,-1.4) {};
\node[sphere,label=above:$-2$,label=below:$R$] (r) at (2,-1.4) {};
\draw (sl)--(t1)--(r);
\node (common) at (4.2,-1.4) {$\theta_*$};
\node (z0) at (7.5,-0.95) {$0$};
\node (zpi) at (7.5,-1.85) {$\pi$};
\draw[path] (common)--(z0);
\draw[path] (common)--(zpi);
\node[align=center] at (4.2,-2.55) {$S_l$ isolation};
\node[align=center] at (7.5,-2.55) {$T_1,R$ isolation};
\end{tikzpicture}
\caption{Cap chains and their harmonic-path endpoint values in the
flat conjugacy interval. The two compressed paths share a value
$\theta_*\in[0,\pi]$ and end at opposite central values. The branches
are not assigned determinant labels. Only the endpoint data are shown;
no monotonicity of the paths is asserted.}
\label{fig:ordinary:cap-endpoints}
\end{figure}
The preimage parity of a generic interior
angle under any path is determined by which side of
that angle contains each endpoint. For the first
cap its two endpoints are $0,\pi$, so the parity is
one. For the second cap the contributions of the
common endpoint cancel between the two paths, while
the other endpoints are $0,\pi$; their total parity
is again one. Choose the exterior angles away from
the finitely many relevant limiting values and
transverse to these paths. Such a choice is allowed
by Lemma~\ref{lem:ordinary:perturbations}. Small
normal regularizations preserving the Abelian paths
do not change the parity.
\end{proof}
\begin{lemma}[No additional ends of the minimum cap matches]
\label{lem:ordinary:cap-end-compactness}
For a generic noncentral exterior angle, the cap matches counted
in Lemmas~\ref{lem:ordinary:harmonic-caps} and
\ref{lem:ordinary:holonomy-degree} form a compact zero-dimensional
set. The two types of cap-interval endpoint give no extra
contribution. Gluing them to the continuation exterior gives
exactly its count modulo two, summed over both determinants
in the compressed case.
\end{lemma}
\begin{proof}
At exceptional-sphere or unimodular-cap isolation, the isolated
part already carries charge $1/4$. No charge remains on
its complement. The limiting boundary angle is therefore
one of the localized harmonic endpoint values computed in
Lemma~\ref{lem:ordinary:holonomy-degree}, and a generic
exterior angle avoids it.
At the $S_l$ isolation in the compressed case, the
trace-zero $(-2)$ reference cap has its minimum charge.
After filling that piece, the other part has fixed
index $-2$ and no parameter. It has no regular
irreducible solution. A split solution extends
through the traceless link by the unique minimum
split filling and again has unordered difference
$\{c_e,-c_e\}$ by
Lemma~\ref{lem:ordinary:harmonic-caps}. It therefore
has the already computed common endpoint angle,
also avoided by the exterior. Additional charge
losses are impossible at total charge $1/4$;
particles have integral positive charge, and collar
levels between the same prescribed flat classes
also have positive integral charge.
All remaining matches occur at transverse interior
angles. Their tangential and normal regularity and
multiplicity one were proved in
Lemma~\ref{lem:ordinary:harmonic-caps}. The cap has
one matching end and absorbs its full parallel
stabilizer. Lemma~\ref{lem:ordinary:regular-gluing}
therefore both constructs all nearby solutions and
exhausts every convergent sequence. The odd degree
of Lemma~\ref{lem:ordinary:holonomy-degree} supplies
one copy modulo two of each continuation exterior.
No gluing at any excluded higher-energy cap or at
a reducible main connector has been invoked.
\end{proof}
\subsection{Boundary identities and the bridge}
\begin{proof}[Proof of Theorem~\ref{thm:ordinary:units}]
First form the rigid maps on the two increasing steps. Their
chain identities and the identification of the rigid count
on $0\to1\to2$ with their composition follow from the
$b^+=0$ case of
Lemma~\ref{lem:ordinary:low-dimensional-compactness}, the irreducible
gluing of Lemma~\ref{lem:caps:irreducible-gluing}, and determinant-line
orientations. All collars used in this comparison are
fixed. Denote one single-step map by $u_r$.
For its inverse comparison use both triples in
\eqref{eq:ordinary:two-triples}. The complementary
interval on $s\to\infty\to r$ has no non-chain
boundary: its $S^3$ endpoint is empty and its exceptional
cap endpoint has shift at least two. Thus its index
$-1$ count is a chain map. The five faces of the two
pentagons are as follows; the same labels refer to
arcs in their respective ordered paths.
\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Face & $s,\infty,r,s$ & $r,s,\infty,r$\\
\midrule
$Z_1$ & empty $S^3$ split & composition across $Y_s$\\
$Z_2$ & composition across $Y_r$ & empty $S^3$ split\\
$M_{02}$ & exceptional cap, empty & exceptional cap, empty\\
$M_{13}$ & exceptional cap, empty & exceptional cap, empty\\
$M_{03}$ & continuation on $Y_s$ & continuation on $Y_r$\\
\bottomrule
\end{tabular}
\end{center}
Here the empty statements are precisely
Lemma~\ref{lem:ordinary:low-dimensional-compactness};
the last row is
Lemmas~\ref{lem:ordinary:flat-replacement}--\ref{lem:ordinary:cap-end-compactness}.
No other face exists by
Lemma~\ref{lem:ordinary:metric-families}.
The compactified one-dimensional pentagon moduli
spaces, of actual index $-1$, consequently give,
over $\F_2$, chain homotopies of the form
\begin{equation}\label{eq:ordinary:single-step-identities}
u_r h_1\simeq J_s,
\qquad h_2\widetilde u_r\simeq J_r,
\end{equation}
where $J_s,J_r$ are continuation isomorphisms. The
map $\widetilde u_r$ has the determinant induced by
the second pentagon; it is tensor-equivalent to
$u_r$, by Lemma~\ref{lem:ordinary:unit-topology}.
The first identity makes $u_r$ surjective on homology.
The second makes $\widetilde u_r$ injective, and hence
imposes the opposite inequality on the dimensions of
the two finite-dimensional homology groups. They
have equal dimension. Therefore $u_r$ is an
isomorphism. This argument does not require the
two complementary interval maps or their determinant
transitions to be identical. Tensor equivalence
also proves the assertion for every allowed
single-step determinant. It follows that
$f_+=u_1u_0$ is a unit.
Now use the paired interval on $2\to\infty\to0$.
Its $S^3$ endpoint is empty. At its other endpoint
the two minimum trace-zero cap contributions agree
and cancel modulo two by
Lemma~\ref{lem:ordinary:cap-shifts}; every higher
sector is excluded. The remaining ends are exactly
the differential terms. Thus the paired index
$-1$ count $g_+$ is a chain map modulo two.
The compressed pentagon on $2,\infty,0,1,2$ has
exactly the faces in
\eqref{eq:ordinary:compressed-cycle}. We record
each term separately:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Face & Contribution to the paired count\\
\midrule
$Z_1$ & empty primary $S^3$ split\\
$Z_2$ & $f_+g_+$\\
$M_{02}$ & equal trace-zero terms, cancelling modulo two\\
$M_{14}$ & positive cap shift, empty\\
$M_{04}$ & continuation $J_2$ by combined odd degree\\
\bottomrule
\end{tabular}
\end{center}
At the $M_{04}$ endpoints the only possible additional
cap degenerations are $M_{02}$ and $M_{14}$, and
Lemma~\ref{lem:ordinary:cap-end-compactness} has
already excluded them at a generic exterior angle.
The compact one-dimensional paired pentagon count
therefore gives
\begin{equation}\label{eq:ordinary:two-step-identity}
f_+g_+\simeq J_2\qquad\text{over }\F_2.
\end{equation}
Since $f_+$ and $J_2$ are isomorphisms, so is $g_+$.
Apply the same constructions to the mirror knot
and reverse the order of the ends, identifying the
oppositely oriented three-manifolds. The ordinary
counts dualize. Negating a determinant, or changing
the sign of the flip sphere, changes the two choices
by tensoring with lines, with the latter line
canonically trivial off the sphere. Their prescribed
restrictions are the ones in part (iii), and the
mod-two unit argument is unchanged. This proves
the claims for $f_-$ and $g_-$.
\end{proof}
\begin{corollary}[The ordinary bridge]\label{cor:ordinary:bridge}
Fix an orientation-preserving diffeomorphism
$\phi:Y_{-2}\longrightarrow Y_2$, with fixed bundle and collar
identifications. The stacked cobordism and its rigid count
\begin{equation}\label{eq:ordinary:bridge}
T=f_-\,\phi^{-1}\,f_+:Y_0\longrightarrow Y_0
\end{equation}
define an integral chain map inducing a unit over $\F_2$ and
an isomorphism over $\Q$. The cobordism has
\begin{equation}\label{eq:ordinary:bridge-topology}
H_1(T;\Z)=0,\qquad \chi(T)=4,\qquad \sigma(T)=-2.
\end{equation}
Its two increasing-handle halves are simply connected.
\end{corollary}
\begin{proof}
The maps $f_\pm$ have coherent integral orientations,
and the fixed diffeomorphism induces an integral
chain isomorphism. Stretching their rational seams
introduces only $b^+=0$ primary pieces. The surplus
three of Lemma~\ref{lem:ordinary:central-cost}
excludes central contacts in the chain dimensions,
so the rigid count equals the displayed composition
up to integral chain homotopy. Each factor is a
unit modulo two by Theorem~\ref{thm:ordinary:units}.
To obtain the rational assertion, take the mapping
cone of this map of finite free integral complexes.
Its homology modulo two vanishes. The universal
coefficient exact sequence implies that its
integral homology has no free summand, since any
such summand would survive tensoring with $\F_2$.
Thus the cone is acyclic over $\Q$, proving the
rational isomorphism. No integral invertibility
of $T$ on torsion is required.
Each meridional handle kills a normally generating
meridian in the incoming surgery group. Hence
both handle halves are simply connected and their
gluing has $H_1=0$. There are four two-handles,
giving $\chi=4$. Each half has one negative
direction and no positive direction: its remaining
intersection direction comes from the zero-surgery
boundary and is null, as in
Lemma~\ref{lem:ordinary:unit-topology}. Signature
additivity across the rational seam gives
$\sigma=-1-1=-2$.
\end{proof}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/02-ordinary.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/03-negative.tex ====
\section{The negative interval unit}\label{sec:negative}
The ordinary interval units have an additional negative-direction analogue
with a sphere incidence condition. This condition is essential: it is a
degree-two insertion, obtained by allowing the loop in a degree-three
holonomy condition to move through a one-parameter sweep of a sphere.
Throughout this section the determinant transitions on the true ends are
fixed, and all gauge transformations have determinant one. Sweep
homotopies preserve the chosen sphere orientation.
The proof has four stages: the minimal cap calculation, the comparison
of its two determinant lifts, the chain-map identity, and the
four-step family proving that the map is a unit modulo two. The local
comparison fixes the weights that will also cancel the spinor lens ends
in Section~\ref{sec:gluing}.
Set
\[
W'=(-C_2)+C_{-2}\colon Y_2\longrightarrow Y_{-2}.
\]
The capped Seifert classes $F_l,F_r$ are orthogonal, have square $-2$, and
give integral coordinates on $H_2(W';\Z)$. The two capping disks give an
embedded sphere
\begin{equation}\label{eq:negative:sphere}
S=F_l-F_r,\qquad S^2=-4.
\end{equation}
The interval of metrics has one endpoint at the primary splitting
$J=Y_\infty=S^3$ and the other at the splitting off
$N'=\nu S$, whose oriented boundary is the corresponding lens space of
order four. These are the two arc cuts in the polygon construction of
Lemma~\ref{lem:ordinary:polygon}. Write $t$ for the interval parameter.
Choose an orientation of $S$ and a sweep $\gamma_s$, $0\le s\le1$, from a
constant loop to a constant loop, of degree one onto $S$. Reparametrize
it to be constant near $s=0,1$. On the lens collar keep the entire sweep
inside $N'$. For each $e\in\{0,1\}$ use the determinant
\begin{equation}\label{eq:negative:lifts}
c_e=e\PD(S),
\end{equation}
with the two determinant bundles identified off $N'$. Both determinants
have square roots on $W'$: their evaluations on the integral basis
$(F_l,F_r)$ are $(0,0)$ or $(-2,2)$, and $H_1(W';\Z)=0$.
Divide holonomy by the holonomy of that root. Equivalently, for a sphere
sweep the determinant square root is the one continued from its
collapsed starting loop. The condition denoted by $x(S)$ is
\begin{equation}\label{eq:negative:cut}
\Hol_0(a,\gamma_s)=-\id\in\SU(2).
\end{equation}
The target is central, so the condition is independent of the frame
used to record holonomy. At the collapsed loops the normalized holonomy
is $+\id$, and the condition is absent there, also after a sufficiently
small regularizing perturbation supported away from those endpoints.
The index $i$ is the fixed-limit unframed ASD index of
Section~\ref{sec:ordinary}. Thus a zero-dimensional count defining the
interval map has
\begin{equation}\label{eq:negative:dimension}
i+\underbrace{1}_{t}+\underbrace{1}_{s}
-\underbrace{3}_{\text{holonomy}}=0,
\qquad i=1.
\end{equation}
Let $B_e$ be this count between the prescribed irreducible generators.
We will specify the relative orientation of its two summands below and
write $B=B_0+B_1$ with those orientations understood.
\begin{theorem}[Negative interval unit]\label{thm:negative:unit}
The counts defining $B$ can be chosen coherently so that
\[
B\colon C(Y_2;\Z)\longrightarrow C(Y_{-2};\Z)
\]
is an integral chain map. Its reduction modulo two induces an
isomorphism on Floer homology. Its integral chain-homotopy class is
unchanged by compatible changes of the interval data and sphere sweep.
The relative sign of the two determinant lifts is local at $N'$,
is independent of the exterior solution, and is preserved under
composition with other cobordisms.
\end{theorem}
We first establish the cap calculation and the signs. We then compute
the map modulo two by a four-step family. The analytic inputs are the
ordinary compactness, perturbation, and regular-gluing statements of
Section~\ref{sec:ordinary}, together with the compatible library of
Proposition~\ref{prop:analysis:ordinary-library}. In the application
below these inputs are used with fixed relative bundle sectors,
product-compatible parameter collars, perturbations independent of
incidence marks, and fixed true-end data. At a harmonic reducer the
perturbation value is zero and its normal derivative may be varied
equivariantly. We only glue transverse, isolated matches with no
particle or positive-charge neck loss. The following dimension and
charge arguments verify those last hypotheses before gluing is used.
\subsection{The minimal cap and relative orientations}
\begin{lemma}[The order-four cap]\label{lem:negative:cap}
Use a determinant square-root normalization on $N'$. For a fixed
boundary flat $\alpha$, let $h^0(\alpha)$ be the dimension of its parallel
adjoint sections. The framed cap index and its charge satisfy
\begin{equation}\label{eq:negative:cap-index}
\delta:=i_{N'}^{\mathrm{fix}}+h^0(\alpha)=8E,
\qquad
\begin{cases}
E\in\Z,&\alpha\text{ central},\\
E\in\frac14+\Z,&\alpha\text{ trace zero}.
\end{cases}
\end{equation}
A flat cap misses $x(S)$. The minimum cap which meets the cut has
$E=1/4$, trace-zero boundary flat, and framed real index two. With
arbitrarily small compatible equivariant perturbations, its framed
cut-down solutions are harmonic reducible crossings, with signed total
$\pm1$. Each crossing gives one match with a regular unframed exterior;
there is no additional ordering or framing multiplicity.
\end{lemma}
\begin{proof}
Let $x\in H^2(N';\Z)$ evaluate as one on $S$. Rationally $x^2=-1/4$.
The normalized boundary representations of its cyclic fundamental group
have generator holonomy $\pm\id$ or a conjugate of
$\diag(i,-i)$. A reference split connection with summand difference
$d=2x$ has
\[
E=-\frac{d^2}{4}=\frac14.
\]
With a fixed boundary representative, changing the relative bundle
degree changes $E$ by an integer and the index by eight. Both central
boundary values have the trivial adjoint boundary correction; the
flat $+\id$ reference has framed index zero. The trace-zero reference has
framed tangent index zero, since $H^1(N';\R)=0$ and $b^+(N')=0$,
and framed normal complex index one. Here is the relative calculation
of the latter correction. Contract the negative section in
$\mathbb F_4$ to obtain $\mathbb P(1,1,4)$. On their identified smooth
complements the two normal twists agree as complex bundles. For the
fiber twists of degrees $2,-2$ their Euler characteristics are $3,-1$
on the resolution and $3,0$ on the weighted plane. Indeed the fiber
line on $\mathbb F_4$ is pulled back from $\mathcal O_{\CP^1}(1)$,
so the two Euler characteristics are $\chi(\mathcal O(2))=3$ and
$\chi(\mathcal O(-2))=-1$. On the weighted plane with weights
$1,1,4$, the degree-two part has three monomials; the degree-minus-two
part has no sections, no intermediate cohomology, and no top
cohomology, the latter also following from the canonical degree $-6$.
The ASD normal
symbol contributes minus the sum of these characteristics; hence its
relative complex index is
\[
[-3-(-1)]-[-3-0]=1.
\]
The corresponding flat cone reference has framed index zero.
Excision gives the asserted normal index, and hence
\eqref{eq:negative:cap-index}. The minimal unframed dimension one is
also the $m=1$ case of the order-four calculation in
\cite[Lemmas~5.2--5.3]{FintushelStern1997}.
The cap is simply connected. A flat connection after determinant
normalization therefore has trivial holonomy on every loop of the
sweep. Compactness makes this avoidance persist for sufficiently
small perturbations. Nonnegative charges in
\eqref{eq:negative:cap-index}, including charges of any neck levels,
then show that the first possible cut cap has $E=1/4$.
At this charge a reducible has unordered difference $\{2x,-2x\}$.
Order the summands by requiring $d(S)=2$. Their difference restricts
to the degree-two character modulo four, and the ordered normalized
eigenvalue specifies its eigenline at the fixed boundary representative.
Thus ordering does not double the number of connections. Abelian
Hodge theory supplies a unique ASD extension with these data: existence
uses the vanishing of the self-dual obstruction, while the difference
of two such solutions has exact curvature, zero $L^2$ curvature norm
by integration by parts, and then is gauge trivial because $H^1(N')=0$.
The normalized chosen eigenline has degree $d(S)/2=1$. If $D_s$ is
the part of the swept sphere preceding $\gamma_s$, its holonomy is
\[
\lambda(s)=\exp\left(2\pi i\int_{D_s}F_{\mathrm{eig}}/(2\pi i)\right).
\]
The continuous real phase changes by one from one collapsed loop to
the other. Consequently $\lambda$ has winding one, up to reversal of
the sphere orientation. Its transverse crossings of $-1$ have signed
sum $\pm1$. Both eigenvalues equal $-1$ at such a crossing; they
describe a single instance of \eqref{eq:negative:cut}.
For completeness, regularity here concerns the \emph{framed} cap.
Its tangent operator is already regular of real index zero. The
normal operator is complex linear of index one. Holonomy derivatives
detect its kernel modulo infinitesimal gauge: if every based-loop
variation is a simultaneous infinitesimal conjugation, transport from
the base point constructs that gauge; its off-diagonal part decays
because the trace-zero boundary flat has no off-diagonal parallel
sections. A finite collection detects the finite-dimensional kernel.
In a slice fixing the boundary torus, torus-equivariant linear outputs
in these detected directions prescribe kernel-to-cokernel variations.
These outputs vanish on commuting data. They therefore make the
complex operator surjective without moving the harmonic solution.
After this step its normal kernel has complex dimension one.
The holonomy target tangent at $-\id$ is the direct sum of a real toral
line and an off-diagonal complex line. Vary the cut so that the mark
derivative is nonzero in the toral line at every crossing, and its
derivative on the normal kernel is a complex isomorphism onto the
off-diagonal line. These variations are allowed by the same sample
slice construction. The resulting dimension is
\begin{equation}\label{eq:negative:framed-cut}
\underbrace{2}_{\text{framed cap index}}
+\underbrace{1}_{\text{mark}}-3=0.
\end{equation}
The off-diagonal isomorphism is complex oriented, so the remaining
crossing sign is precisely the toral winding sign. This proves the
signed count assertion.
An irreducible cap satisfying the cut would have a positive-dimensional
orbit under the residual framing torus $H=\U(1)$. Equivariant
transversality would make the framed cut locus zero-dimensional, so
such a solution is absent. A reducible crossing is fixed by $H$:
its parallel stabilizer absorbs that action. Accordingly one can
glue the framed cap to the unframed exterior, or equivalently divide
two framed matching spaces by the diagonal $H$; either convention
gives exactly one match per crossing. At the minimum there is no
positive cap or neck charge available, and a flat cap misses the cut,
so there is no further end of this local count. These regularity
choices are required only at the finitely many transverse exterior
parameter values; they can be localized, extended over parameter
collars, and copied by tensoring to the second lift.
\end{proof}
\begin{proposition}[Local relative signs]\label{prop:negative:local-sign}
There is a relative orientation convention for the two lifts in
\eqref{eq:negative:lifts} under which their minimal lens ends cancel
over $\Z$. The comparison uses identical exterior data, including
relative bundle degree, and is independent of the exterior field.
It commutes with gluing on other seams and is multiplicative for
disjoint sphere caps. Adding a coupled complex Dirac determinant
preserves this convention.
\end{proposition}
\begin{proof}
Fix coherent Floer orientations for the unflipped sector. Fix also a
representative $\alpha$ of the trace-zero lens flat, and realize the
two framed relative cap sectors at charge $1/4$. On the cap,
$\PD(S)=-4x$, so tensoring by a line of first Chern class $a=-2x$
changes the determinant by $2a=\PD(S)$. Starting with $c(S)=0$,
the relative Chern numbers change by
\begin{equation}\label{eq:negative:tensor-charge}
\Delta c^2=-4,\qquad \Delta c_2=-1,\qquad
\Delta\left(c_2-\frac{c^2}{4}\right)=0.
\end{equation}
Here $\alpha$ is recorded in the fixed common exterior determinant
normalization. The twisting line has the order-two flat character
on the boundary in that convention.
It carries $\diag(i,-i)$ to $\diag(-i,i)$, which is conjugate back
to $\alpha$. Choose a determinant-preserving parallel boundary
identification making that conjugation. Its remaining ambiguity is
the connected centralizer, and does not reverse an orientation.
Equality of the trace-free charges with this boundary identification
fixes the relative $c_2$ change in \eqref{eq:negative:tensor-charge}.
Thus gluing this tensor cap to the unchanged exterior gives precisely
the flipped bundle, with unchanged relative degree outside the cap.
Here is the determinant-line comparison, including its relation to any
previous choice of orientations for the individual sectors. Write
\[
K_e=\det D_{N',e}^{\mathrm{fr}}\otimes\det\R_s
\otimes\bigl(\det\mathfrak{su}(2)\bigr)^*
\]
for the framed cap line with its mark and cut factors, and let
$\lambda_V$ include the common unframed exterior and its remaining
parameter and cut factors. With their fixed ordering, determinant
gluing gives maps $\Gamma_e:\lambda_V\otimes K_e\longrightarrow
\lambda_e$ to the full constrained orientation lines. Tensoring and
the chosen boundary conjugation compare the full cap deformation and
cut complexes and give $\tau_{\mathrm{cap}}:K_0\longrightarrow K_1$.
Set
\begin{equation}\label{eq:negative:orientation-comparison}
\Phi=\Gamma_1\circ(\id\otimes\tau_{\mathrm{cap}})
\circ\Gamma_0^{-1}.
\end{equation}
For conventional sector orientations $o_e$, define $s\in\{1,-1\}$
by $\Phi(o_0)=s\,o_1$. Choose weights according to
\begin{equation}\label{eq:negative:lift-weights}
\varepsilon_1=-s\varepsilon_0,
\qquad
\Phi(\varepsilon_0o_0)=-\varepsilon_1o_1.
\end{equation}
This is exactly the opposite count orientation required for paired
crossings. It neither assumes nor needs $s=1$. In particular a Weyl
boundary conjugation can reverse an ordered toral generator and
conjugate an ordered off-diagonal complex coordinate; all such signs,
including the framing and target factors, are part of
$\tau_{\mathrm{cap}}$.
Take tensor-related equation and cut data and tensor-related roots:
a root after tensoring is the old root tensored with the twisting
line. These are the intrinsic cap roots continued from the collapsed
loop; their boundary identifications need not coincide with the fixed
common exterior normalization used to record $\alpha$. Thus the
boundary value changes from $\alpha$ to $-\alpha$ in the latter
convention, whereas normalized sweep holonomy is unchanged before the boundary
conjugation. Conjugation preserves the central target $-\id$ and its
orientation in $\SU(2)$. Thus the matched cut points correspond and
\eqref{eq:negative:lift-weights} makes their contributions opposite.
The weights may equivalently be absorbed into the orientations of
$B_e$. In later compositions they give a weighted sum of the ordinary
sector counts. The canonically oriented cap pairing used for
nonvanishing is left unchanged: the subsequent repetition argument
uses this signed map $B$, whose reduction modulo two is independent
of the weights. No equality with an unweighted sum of Donaldson
invariants on the assembled manifold is required.
We explain why the convention extends consistently away from the wall.
The integral-lift ASD determinant line with chosen true-end orientations
is orientable on the \emph{determinant-one gauge quotient}, as fixed in
Convention~\ref{conv:ordinary:index}. Its orientation double cover
therefore has trivial monodromy, including around a path closed by a
gauge transformation. The reference connection spaces with the
specified collar representative and relative degree are affine and
connected, so their quotients are connected. Connectedness permits
transport from the chosen reference; quotient orientability makes that
transport independent of the path. This extends the comparison
\eqref{eq:negative:orientation-comparison} as a comparison of
orientation torsors with one constant relative sign.
In choosing $\alpha$, fix its internal framing degree so that the cap
sector has charge $1/4$. An integral gauge degree changes this relative
charge by the same integer. Hence a boundary gauge relating two actual
minimum-charge walls in this sector has degree zero. In the fixed
boundary trivialization its homotopy class lies in
$[\partial N',\SU(2)]\cong H^3(\partial N';\Z)\cong\Z$, classified
by that degree. A degree-zero gauge is nullhomotopic; use a nullhomotopy
on the collar and the identity beyond it to extend it over the cap.
Nonzero degree would change the prescribed relative sector. The
remaining parallel ambiguity in the boundary conjugation lies in the
connected torus stabilizer and extends as a parallel cap gauge. There
is consequently no additional boundary-gauge sign in the comparison.
Interpolations here take place in connection space; no path of
solutions is asserted.
Associativity of determinant-line gluing now proves naturality under
adding an exterior cylinder or cobordism: in either gluing order the
same cap reference and the identity on the same exterior are used.
The two raw framed cap indices are the same even number, namely two,
and the two constrained indices in \eqref{eq:negative:framed-cut} are
zero. With one fixed ordering of cuts and parameters there is no
exterior-dependent interchange sign. Comparisons on disjoint caps
therefore multiply. Finally a coupled Dirac determinant has its
complex orientation, and tensoring and the boundary conjugation act
complex linearly on that factor. It introduces no new relative sign.
\end{proof}
The square roots used to normalize holonomy do not identify the two
global relative bundles with their fixed exterior and true-end data.
In particular a root of their determinant difference need not be
trivial on that exterior. The comparison just proved is the localized
cap replacement with its boundary conjugation. The same comparison
will be used for the coupled regular cap in
Proposition~\ref{prop:gluing:regular-cap}.
\subsection{The chain map and its invariance}
\begin{proposition}[Chain property and invariance]\label{prop:negative:invariance}
With the signs of Proposition~\ref{prop:negative:local-sign}, $B$ is
an integral chain map. Compatible homotopies of equation data and
incidence representatives give integral chain homotopies. The allowed
representative homotopies include smooth conjugation-equivariant
homotopies of the normalized based holonomy paths, relative to their
collapsed endpoints, under the following requirements. Their exact
compact supports remain in the specified parts, they persist under
ideal convergence away from a support hit, and the equation data do
not depend on the incidence mark. At the lens endpoint the test is
supported in $N'$, its two lifts are tensor-paired with the same
exterior, and its flat-cap and collapsed-endpoint avoidance holds
throughout. The data admit regularization preserving these
requirements. The homotopy of the incidence representative need not
be small. Near the primary $J$ endpoint the unconstrained equation
exclusion requires no additional behavior of the representative.
\end{proposition}
\begin{proof}
The boundary relation for the count \eqref{eq:negative:dimension} uses
$i=2$. At $J$, with no remaining interval parameter, a central contact
costs at least three by Lemma~\ref{lem:ordinary:central-cost} and
$b^+(W')=0$. This excludes the face even if the incidence condition
is discarded. At a true central end without the $J$ cut the same
cost is accompanied by a nonconstant irreducible trajectory towards
the prescribed generator. That trajectory costs at least one more
index, giving $4-1>2$ even after the interval parameter is allowed.
A bubble costs eight index before its four position parameters;
even discarding the degree-two insertion leaves too much codimension
in these zero- and one-dimensional families. Equivalently, project
to the remaining equation solution before introducing the incidence
mark: its index has decreased by eight and it has at most one
equation parameter. It is excluded by ordinary regularity and the
same reducible-group estimate.
At the lens endpoint a flat cap misses the cut. Thus
Lemma~\ref{lem:negative:cap} gives $\delta\ge2$, with neck charge
included. The exterior is measured after a reference filling by the
cap; it has $b^+=0$ and no equation parameters. For a central outer
limit the reducible estimate remains
$8E_{\mathrm{ext}}-3(1+b^+_{\mathrm{filled}})$: excision subtracts
exactly the cap contribution $\delta$, so no boundary correction is
lost. Reducible groups and forbidden central contacts are consequently
excluded as above. Regular irreducible exteriors have nonnegative
index. The only lens boundary with $i\le2$ is therefore
\[
i=2,\qquad \delta=2,\qquad i_{\mathrm{ext}}=0.
\]
It consists of an isolated regular irreducible exterior and a minimal
cut cap, without further breaking. Lemma~\ref{lem:ordinary:regular-gluing}
applies: the framed cap is regular and absorbs the boundary stabilizer,
and all matching and cut derivatives are transverse. It gives exactly
the end counted by Lemma~\ref{lem:negative:cap}. The two lifts cancel
by Proposition~\ref{prop:negative:local-sign}.
All remaining ends are the ordinary irreducible trajectory breaks at
the true ends. Their oriented boundary formula is precisely the
chain-map equation for $B$ (with the usual degree convention).
For a homotopy allow one additional common family parameter, so that
at most two parameters enter the equation data, independently of the
incidence mark. Its chain-homotopy count has index zero and its boundary
relation index one. At the
primary face the estimate is $3-1>1$; at a true central contact it is
$4-2>1$. Lens boundaries, when present, are the same paired minimal
caps with their common exterior. Bubbling is again excluded after
dropping the insertion. These are all faces of the parameterized
compactification. The oriented boundary formula supplies the desired
chain homotopy. These arguments use smallness of the equation
perturbations for the energy exclusions, but no smallness of the
incidence-representative homotopy: its relative endpoint condition and
flat avoidance exclude mark endpoints and zero-charge lens caps,
its support and persistence hypotheses identify the ideal limits, and
its tensor pairing gives the same lens cancellation. In particular
sweeps may be transported along collars or changed near $J$, where
the exclusion did not use the test.
To check explicitly that such a homotopy preserves the insertion,
work first in a framed configuration slice. Its evaluation maps
are a homotopy
\[
F_h:\mathcal B\times([0,1],\{0,1\})
\longrightarrow(\SU(2),\{\id\}),\qquad 0\le h\le1.
\]
Pulling back the oriented degree-three class of the central incidence
condition and integrating over the mark gives its degree-two
transgression. A homotopy relative to the mark endpoints preserves
this class. The construction descends from framed slices because
the homotopy is conjugation-equivariant and conjugation preserves the
target orientation. Its regular parameterized incidence cobordism
is exactly the one used in the boundary argument above.
\end{proof}
We turn to the mod-two computation. Consider the four-step path
\begin{equation}\label{eq:negative:four-step}
2\longrightarrow\infty\longrightarrow0
\longrightarrow\infty\longrightarrow-2
\end{equation}
on a cobordism $W$. Its three-dimensional associahedral family has
nine facets: the three primary cuts $Z_1,Z_2,Z_3$ and the six cap cuts
$M_{02},M_{03},M_{04},M_{13},M_{14},M_{24}$. A facet carries the
product of the remaining parameter families, as in
Lemma~\ref{lem:ordinary:metric-families}.
\subsection{The four-step family and its boundary faces}
\begin{lemma}[Topology and determinant sectors of the four-step family]
\label{lem:negative:four-topology}
The group $H_2(W;\Z)$ has integral coordinates $F_l,F_r,A,U$, with
$F_l^2=F_r^2=-2$, an orthogonal hyperbolic pair
$A^2=U^2=0$, $A\cdot U=1$, and $F_l,F_r$ orthogonal to that pair.
Moreover $H_1(W;\Z)=0$ and $b^+(W)=1$. The interior side spheres are
\[
S_l=A-F_l,\qquad U,\qquad S_r=A-F_r,
\]
and form the chain with weights $[-2,0,-2]$. For
\begin{equation}\label{eq:negative:four-determinants}
c=\PD(U)+e\PD(S_l)+e'\PD(S_r),\qquad e,e'\in\{0,1\},
\end{equation}
the face at $Z_2$ is the product of the two ordinary interval units
of Theorem~\ref{thm:ordinary:units}.
\end{lemma}
\begin{proof}
The two outside traces split off across $S^3$ and give $F_l,F_r$.
The doubled zero-trace portion is represented by a zero-framed knot
handle and a linked meridian handle. The latter gives the square-zero
sphere $U$ and the capped Seifert surface gives its primitive dual
$A$. Their linking number is one; the two zero framings give their
self-intersections zero. This is an integral handle basis, and the
meridian normally generates the relevant fundamental group, so the
displayed homology and positive rank follow. The common capped
surface can also be represented on $Z_2=Y_0$ and pushed to either
side of the middle cut. Subtracting its capping disk on the outside
trace gives $S_l,S_r$, with intersections exactly as stated.
On the left half of $Z_2$, $\PD(U)$ evaluates as $(0,1)$ on
$(F_l,A)$; on the right it evaluates as $(1,0)$ on $(A,F_r)$.
The two additions in \eqref{eq:negative:four-determinants} are supported
in their respective halves. Thus the determinant transition at $Z_2$
can be chosen common to all four sectors. The resulting restrictions
are those of $g_+:Y_2\to Y_0$ and its negative-side counterpart
$g_-:Y_0\to Y_{-2}$. Each has index $-1$ with one parameter;
their product face has index $-2$ with two parameters and induces a
mod-two isomorphism. A fixed tensor or endpoint identification in
this description only composes it with a Floer automorphism.
\end{proof}
\begin{lemma}[The other associahedron facets]\label{lem:negative:faces}
In the boundary relation at index $i=-2$ for the family
\eqref{eq:negative:four-step}, the $M_{04},M_{03},M_{14},Z_1,Z_3$
facets and all their degenerations contribute nothing. The
$M_{02},M_{24}$ contributions cancel modulo two by their respective
determinant flips. The remaining terms are the product at $Z_2$,
the face $M_{13}$, and ordinary true-end breaks.
\end{lemma}
\begin{proof}
At $M_{04}$ the enclosed chain has intersection matrix
\[
Q=\begin{pmatrix}-2&1&0\\1&0&1\\0&1&-2\end{pmatrix},
\qquad
Q^{-1}=\begin{pmatrix}-\tfrac14&\tfrac12&\tfrac14\\
\tfrac12&1&\tfrac12\\\tfrac14&\tfrac12&-\tfrac14\end{pmatrix}.
\]
Replacing its middle zero-sphere cap by $D=S^1\times D^3$ joins the
two $\infty$ sides, leaving exactly $N'$. This identifies their
oriented links. The determinant evaluation vector is
$Q(e,1,e')^t$, so its restriction to that link is trivializable.
Every boundary eigenline extends over the chain, because the
restriction from its integral cohomology onto the lens cohomology is
surjective. A split reference with difference evaluations
$(d_l,b,d_r)$ has
\begin{equation}\label{eq:negative:chain-square}
d^2=-\frac{(d_l-d_r)^2}{4}+b(d_l+d_r)+b^2.
\end{equation}
Here $d_l,d_r$ are odd, since
$c(S_l)=1-2e$ and $c(S_r)=1-2e'$. Hence the square is integral and
$E\in\frac14\Z$. Charge zero is impossible, even at a corner:
nonnegative charges on all cap pieces and levels would make all of
them flat; their matching would give a flat adjoint cap, whereas its
restriction to an outside sphere has nonzero $w_2$. Thus $E\ge1/4$.
Compared with $N'$, this cap has Euler characteristic greater by two
and the same signature. The excision formula therefore gives its
framed contribution, including the matching correction, as
\begin{equation}\label{eq:negative:large-shift}
8E-3\ge-1.
\end{equation}
The exterior has no parameters and has $b^+=0$ after virtual $N'$
filling. The central-group estimate, including any true-end
trajectories, and irreducible regularity give exterior index at least
zero. The sum is greater than $-2$, excluding $M_{04}$.
For $M_{03}$ the chain is $[-2,0]$, and the reversed chain occurs at
$M_{14}$. The inverse intersection matrix gives
\[
d^2=2b(d+b).
\]
Its outside evaluation $d$ is odd; $b$ and $d+b$ have opposite
parities, so $d^2$ is divisible by four. The cap charge is a positive
integer. Positivity follows by the same odd-sphere flat obstruction,
including at broken corners. Comparison with a ball gives shift
$8E-3\ge5$. The virtually filled exterior has $b^+=0$.
Even allowing all remaining parameters on its irreducible components
cannot reduce this contribution to index $-2$. These facets are
therefore absent as well.
The small caps $M_{02},M_{24}$ are the odd-determinant $-2$ disk
bundles of Section~\ref{sec:ordinary}. Their trace minimum has charge
$1/8$ and framed shift zero relative to its reference filling.
Nonminimum charge has strictly larger shift. At a minimum a single
harmonic reducer gives the framed regular match, and the flip supported
on that sphere gives a second copy with the same exterior. The two
copies cancel modulo two. To justify applying this local description,
we check that no exceptional exterior or corner can intervene.
Away from the larger cap facets already excluded, fill back any small
odd cap by its reference. The sum of positive ranks of the primary
pieces is at most one. A maximal reducible group contributes at least
$3(1-b^+_{\mathrm{group}})$, including its two exterior matches, by
Lemma~\ref{lem:ordinary:central-cost}. If there is no primary cut,
a group using the positive rank must contain the whole main cobordism
and needs two nonconstant trajectories to the prescribed irreducible
ends. If there is one primary cut, such a group touches an end and
needs at least one trajectory; with at most $p=2$ remaining parameters
its lower bound is $1-p\ge-1>-2$. At two or more primary cuts at most
one parameter remains, and the group bound alone is greater than
$-2$. A central contact with no such group costs at least three
before parameter subtraction. Nonminimum small caps and bubbles
increase these bounds. Consequently the only small-cap matches are
minimum caps with exteriors that are irreducible and rigid in their
remaining parameters, away from the corners.
The primary cuts $Z_1,Z_3$ are on $S^3$, which has no irreducible
Floer generators. Any prospective contact there is one of the
forbidden central configurations just excluded. The only surviving
primary facet is $Z_2$, the product in
Lemma~\ref{lem:negative:four-topology}. The nine-facet list is now
exhausted except for $M_{13}$. Ordinary irreducible breaking at the
two true ends supplies the differential terms in the family homotopy
relation.
\end{proof}
The remaining facet $P=M_{13}$ is a pentagon. It separates
$N_U=\nu U\cong S^2\times D^2$ from $V=W\setminus N_U^\circ$ and
has link $L=S^1\times S^2$. Its five edges are
\begin{equation}\label{eq:negative:pentagon-edges}
Z_1,\ Z_3,\ M_{03},\ M_{14},\ M_{04}.
\end{equation}
In particular it meets neither small flip facet.
\subsection{The selected center and the interval count}
The remaining task is to turn the pentagon count into the interval count
that defines $B$. We first identify the flat on $L$ selected by a flat
$N_U$. Replacing $N_U$ by $D=S^1\times D^3$ will then express this
matching as a loop-holonomy condition. The final step contracts the loop
through two disks whose union represents $\pm S$, producing the
degree-two sphere test.
\begin{lemma}[The root mismatch and the selected center]
\label{lem:negative:root}
Only the sectors $e=e'$ have a flat limit at $L$. After a tensor
identification fixed on the true ends, their determinants are
\begin{equation}\label{eq:negative:surviving-determinant}
\PD(U)+e\PD(F_l-F_r),\qquad e=0,1.
\end{equation}
Replacing $N_U$ by $D=S^1\times D^3$ gives $W'$ with determinant
$e\PD(S)$. If the exterior is normalized using a square root from
$W'$, a flat $N_U$ matches it exactly when the normalized holonomy
of the generator of $\pi_1(L)$ is $-\id$.
\end{lemma}
\begin{proof}
The determinant evaluates on $U$ as $e+e'$. A projectively flat
connection on $S^1\times S^2$ has trivial $w_2$ on its $S^2$, since
its representation factors through the circle. Thus $e\ne e'$
is impossible. In the $e=e'=1$ sector tensor by the line of class
$-\PD(S_l)$, canonically trivial off that sphere. This changes the
determinant by $-2\PD(S_l)$ and gives
\[
\PD(U)-\PD(S_l)+\PD(S_r)
=\PD(U)+\PD(F_l-F_r).
\]
It proves \eqref{eq:negative:surviving-determinant}; the $e=0$ sector
requires no tensoring. The class $\PD(U)$ can be represented inside
$N_U$, while the second term survives the replacement. The polygon
replacement joins the identical $\infty$ sides and yields $W'$,
including its fixed true-end identifications.
There are square roots of the determinant separately on $N_U$ and on
$V$: use the root on $W'$ for $V$. Their root structures on $L$
differ by a flat sign line $Q\in H^1(L;\Z/2)$. The Mayer--Vietoris
connecting homomorphism sends $Q$ to the obstruction $c\bmod2$ to
a square root on $W$. This obstruction is nonzero, since
\[
c(A)=U\cdot A+e(F_l-F_r)\cdot A=1.
\]
Equivalently, if the root structures matched they would glue to a
global root, contradicting this odd evaluation. As
$H^1(L;\Z/2)=\Z/2$, $Q$ is its nontrivial character and is $-1$
on the circle generator. A normalized flat connection on the simply
connected cap $N_U$ has generator holonomy $+\id$. Changing from
its root to the exterior root multiplies that holonomy by $Q$, giving
$-\id$. Conversely this sign change makes any exterior limit at
$-\id$ agree with the flat cap. This establishes the selected center
and not merely its image in the projective flat space.
\end{proof}
\begin{lemma}[The surviving pentagon count]\label{lem:negative:pentagon}
At index $i_{\mathrm{total}}=-2$ the face $P$ consists of isolated
regular exterior matches at the selected center with a flat $N_U$.
Their count induces a mod-two isomorphism. It is also the count on
$W'$ over $P$ at index $i'=1$, with the degree-three condition
\begin{equation}\label{eq:negative:beta-cut}
\Hol_0(a,\beta)=-\id,
\qquad \beta=S^1\times\{0\}\subset D,
\end{equation}
when the $D$--exterior neck is sufficiently long and finite.
\end{lemma}
\begin{proof}
Leave $N_U$ unperturbed at this cut. Relative to a virtual flat
$D$ filling, index excision gives
\begin{equation}\label{eq:negative:zero-replacement-index}
i_{\mathrm{filled}}
=i_V^{\mathrm{fix}}+h^1
=i_{\mathrm{total}}+3-8E_{N_U}\le1,
\qquad E_{N_U}\in\Z_{\ge0}.
\end{equation}
Any nonflat hard-neck level is included in $E_{N_U}$. The charge is
integral because the Chern--Simons value is constant on this flat
space and the cap has a zero-charge reference. The filled exterior
has $b^+=0$ and at most two equation parameters. If $E_{N_U}\ge1$,
then $i_{\mathrm{filled}}\le-7$, which is incompatible with
irreducible regularity or the central-group lower bound. Positive
neck or cap charge is therefore impossible. An additional primary
$S^3$ cut leaves at most one parameter, and costs three. A forbidden
true central contact also needs an extra irreducible trajectory and
costs at least $4-p>1$ for $p\le2$. The larger cap corners have
already been excluded in Lemma~\ref{lem:negative:faces}.
At the selected center $h^0=h^1=3$. Varying the flat in a framed
slice uses three real parameters; imposing the center removes all
three. It is not a codimension-one endpoint condition on the
conjugacy interval. The dimension of a prospective exterior match is
\[
i_{\mathrm{filled}}+p-3\le1+2-3=0.
\]
It can be nonnegative only at $E_{N_U}=0$, $p=2$, and
$i_{\mathrm{total}}=-2$, and is then zero. Generic compatible
exterior data give transverse matches.
The flat $N_U$ is regular when the boundary flat is allowed to vary
before matching. Its framed fixed-limit infinitesimal kernel is
zero, since $H^1(N_U)=0$; the three boundary flux obstructions in
the self-dual equation are killed by the three flat parameters,
componentwise by the Abelian relative Hodge exact sequence. Its full
parallel group absorbs the matching rotations. Thus this is an
isolated regular matched problem of exactly the type in
Lemma~\ref{lem:ordinary:regular-gluing}, with coefficient one.
The associahedron boundary relation, using
Lemma~\ref{lem:negative:faces}, identifies its map on homology with
the unit at $Z_2$. This proves the first assertion.
For the second assertion replace $N_U$ by $D$ and retain the same
limiting exterior. The index becomes $i'=1$. The framed flat
family on $D$ is locally the three-dimensional group of generator
holonomies near $-\id$; its evaluation on $\beta$ is the identity.
Consequently \eqref{eq:negative:beta-cut} regularly fixes that flat,
and $i'+2-3=0$. A nonflat $D$ component or hard-neck loss costs
a positive integer charge and is excluded by the same filled-exterior
estimate. Regular gluing therefore gives exactly the previous
center matches at every sufficiently long finite neck.
Here is the required uniform boundary check. Keep the exterior of
$M_{04}$ fixed and common for the two sectors, independent of its
cap parameters; both flip spheres are inside that cap. After
replacement denote the corresponding edges by primes. The
$Z_1,Z_3$ edges still cut on $S^3$ and are excluded with $p\le1$
even without \eqref{eq:negative:beta-cut}. The cap at $M_{04}'$
is $N'$; a nonflat contribution has framed shift at least two and
the exterior has no parameters, so it cannot occur at $i'=1$.
At $M_{03}',M_{14}'$ the caps are balls and a nonflat contribution
has shift at least eight. On these three edges the zero-charge
caps in the filled manifold are simply connected and contain $\beta$;
their normalized flat holonomy is $+\id$, so they miss the cut.
The same assertion holds if the long $D$ neck breaks simultaneously:
its matching zero-charge fields glue to a flat on the same simply
connected filled cap. The normal-gap estimate and controlled
harmonic drift of Lemma~\ref{lem:ordinary:neck-decay} show that a
loop in a middle slice sees that same flat orbit. This rules out
escape down the long neck and proves exhaustion, including the
compatible corners. The true-end determinant identifications remain
fixed throughout the tensor and replacement operation.
\end{proof}
\begin{lemma}[Contraction of the pentagon to the interval]
\label{lem:negative:sweeps}
The map on homology counted in Lemma~\ref{lem:negative:pentagon}
agrees modulo two with $B$.
\end{lemma}
\begin{proof}
We describe the contraction disks, since their location determines
which boundary avoidances remain valid. In the polygon after the
$U$ replacement, let $E=[a,b]$ be the merged $\infty$ side. Its
preimage is
\[
\bigl([a,b]\times S^1\bigr)
/\bigl(\{a\}\times S^1\text{ and }\{b\}\times S^1
\text{ collapsed separately}\bigr),
\]
an embedded sphere. Let $[u,v]\subset(a,b)$ be the inserted $D$
portion and choose $u<t_0<v$. The surviving circle over $t_0$
is $\beta$. The preimages of
$[a,t_0]$ and $[t_0,b]$ are disks $\Delta_l,\Delta_r$ with common
boundary $\beta$. Their union is the merged side sphere. Before
replacement its class is, up to orientation, $S_l-S_r$; hence after
replacement it is $\pm(F_l-F_r)=\pm S$.
The $M_{04}'$ arc encloses the entire merged side and contains both
disks. The $M_{03}'$ arc can end on the merged side at a point
$t_+>v$, so its one-corner cap contains $\Delta_l$ and all of $D$;
the $M_{14}'$ arc can end at $t_-<u$, and its cap contains
$\Delta_r$ and all of $D$. These choices come from nesting the
original $N_U$ cut inside the respective larger cap. The two-disk
chart at the unique basis corner identifies each one-corner cap
with a ball. These two ball cuts are incompatible facets and are
never imposed simultaneously. At a corner with $M_{04}'$, the
appropriate one-corner cap is nested in that outer cap. Slight
collar displacements put the disks in the required interiors without
altering their boundary or their union class.
Choose an embedded interval $I\subset P$ from the interior of
$M_{04}'$ to a primary $Z$ edge, separating $M_{03}'$ from
$M_{14}'$, as in Figure~\ref{fig:negative:contraction}.
It divides $P$ into two regions. On the first region
contract $\beta$ through $\Delta_l$, and on the second contract it
through $\Delta_r$. Denote the contraction parameter by $s$.
Transport the contractions over the parameter collars so that, at
each cap edge, the loops remain in the disk just specified. The
metrics and ASD perturbations have the two parameters of $P$ and
are independent of $s$. At $s=0$ we recover the original
degree-three condition; at $s=1$ the loop is collapsed and misses
$-\id$. Each contraction is a one-dimensional constrained family:
\[
i'+\underbrace{2}_{P}+\underbrace{1}_{s}-3=1.
\]
\begin{figure}[htbp]
\centering
\input{figures/negative-contraction}
\caption{The two contractions of the pentagon loop test.
Left: the interval $I$ separates the incompatible ball edges
$M_{03}'$ and $M_{14}'$; each region uses the disk lying in its ball cap.
Right: the inverse image of the merged $\infty$ side $[a,b]$ is a
sphere. The circle over $t_0$ is $\beta$, and the preimages of
$[a,t_0]$ and $[t_0,b]$ are $\Delta_l$ and $\Delta_r$.
The shaded annulus is the part in the inserted $D$ piece.
Concatenating the disk contractions gives a sweep of $\pm S$.
The drawing is schematic, and the $D$ neck remains finite.}
\label{fig:negative:contraction}
\end{figure}
We check its compactification uniformly before using its boundary
formula. At a primary edge, the unconstrained equation projection
has at most one parameter and is excluded by the central cost.
The contraction mark cannot remedy an empty equation projection.
At a true central end the extra irreducible trajectory gives cost
at least $4-p>1$ for $p\le2$ \emph{equation} parameters, again
independent of the mark. Positive cap or neck loss and bubbles obey
the same inequalities as in Lemma~\ref{lem:negative:pentagon}.
At $M_{04}'$, $M_{03}'$, or $M_{14}'$ a zero-charge cap misses the
test because the chosen contraction lies in that simply connected
cap. The collapsed end is also uniformly disjoint from the cut.
The latter avoidance is valid even while the hidden $D$ neck becomes
arbitrarily long. The sphere meets $D=S^1\times D^3$ in $S^1$
times a diameter of $D^3$. In each stretched collar its intersection
is consequently $S^1$ times a pole of $S^2$ times the neck interval.
The contractions can therefore be taken along the merged side, using
at each mark a free-circle loop in one slice of the neck or in one
of the two bodies, with uniformly bounded spatial length and
derivatives. Central holonomy requires no comparison
of frames transported down the neck. Suppose uniform avoidance
failed for arbitrarily small part perturbations. Ideal compactness
and the charge exclusions just proved would give matching flat
data on a zero-charge cap and its neck pieces. These glue to a
flat on the whole simply connected cap, and the root from $W'$
normalizes it absolutely. A mark remaining on a body then has
holonomy tending to $+\id$. For a mark escaping along the middle,
Lemma~\ref{lem:ordinary:neck-decay} gives the same conclusion:
normal modes decay and the harmonic slice remains in the matching
flat orbit. Both conclusions contradict the value $-\id$.
The argument works also at nested compatible corners, using the
innermost required cap. Thus first choose the sufficiently small
limiting part perturbations with this avoidance, and only then the
large finite $D$ neck. This order also permits the small generic
cut perturbations needed for regularity.
Apply the one-dimensional mod-two boundary formula on the two
regions. Their original pentagon contributions add to the original
count; the collapsed ends and the exterior parameter edges contribute
nothing. True irreducible end breaks give a chain homotopy. The
remaining boundary lies on $I$, where the two disk contractions
concatenate to a sweep of their union sphere. Its dimension is
\[
i'+\underbrace{1}_{I}+\underbrace{1}_{s}-3=0.
\]
This is exactly the interval count with $x(S)$, up to reversal of
the sweep and the fixed tensor identifications. Reversal is
irrelevant modulo two.
Finally this induced interval data may be made identical to the
chosen data for $B$. Keep the exterior at $M_{04}'$ fixed and common
for the two lifts. At index one there are no solutions near that
endpoint: a nonflat cap costs at least two with no exterior
parameters, and a flat cap misses the sweep. Hence tensor-paired
cap regularizations and cut data may be inserted there, stationary
at the endpoint, without changing the count. Subsequent comparisons
are precisely the compatible homotopies in
Proposition~\ref{prop:negative:invariance}. This proves the lemma.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:negative:unit}]
Propositions~\ref{prop:negative:local-sign} and
\ref{prop:negative:invariance} give the integral chain map, its
homotopy invariance, and its coherent local sign convention.
Lemma~\ref{lem:negative:four-topology} identifies the $Z_2$ face
with a product of ordinary mod-two units.
Lemma~\ref{lem:negative:faces} and the associahedron boundary formula
identify that product on homology with the surviving pentagon count.
Lemmas~\ref{lem:negative:pentagon} and \ref{lem:negative:sweeps}
identify the latter with $B$ modulo two. Thus $B$ induces an
isomorphism, as asserted.
\end{proof}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/03-negative.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/04-caps.tex ====
\section{Caps and a nonzero ordinary pairing}\label{sec:caps}
This section constructs a nonzero ordinary pairing and puts every
Donaldson insertion in two fixed end caps. These caps supply the fixed
states paired with the repeated cobordism maps in Section~\ref{sec:stack}.
The intervening cobordisms will carry no moving surface insertions.
We retain the determinant-one gauge group and the fixed collar
identifications of Section~\ref{sec:ordinary}. Write $x_0$ for the point
class and use the normalization
\begin{equation}\label{eq:caps:normalization}
\mu(\xi)=-\tfrac14 p_1(\ad E)/\xi .
\end{equation}
A \emph{tight surface} is an embedded oriented surface $B$ with
$B^2=2g(B)-2>0$. The capped minimal Seifert surface $A\subset Y_0$ has
genus two and square zero in the closures below.
\begin{theorem}[Fixed caps with a nonzero pairing]\label{thm:caps:pairing}
There are two fixed compact caps $C_-$ and $C_+$ with common rational
boundary $Y_2$, fixed determinant data trivialized at that boundary, and
finite collections of integral point and surface probes in their interiors
with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Each cap has $H_1(C_\pm;\Z)=0$ and $b^+(C_\pm)>0$. Each contains
a square-zero copy of $A$ on which the determinant class evaluates as $1$,
and a tight genus-two surface of square two.
\item Each cap has one additional, separate exceptional summand. The
determinant evaluates as zero on its exceptional sphere, and every probe
is supported in the old part of the cap.
\item For suitable generic cap data and small ordinary Floer data at the
rational boundary, the cut counts give a rational cycle $x$ and a rational
cocycle $y$ on the irreducible determinant-one Floer complex, with
\begin{equation}\label{eq:caps:nonzero-pairing}
yx\ne0.
\end{equation}
Fixed integral multiples of these are an integral cycle and cocycle.
\item The total real degree $z$ of all cap probes is fixed. Reducible
cap limits in any of the bounded charge ranges used below are excluded by
the cap period choices. The period choices admit the finite isolation
extensions of Proposition~\ref{prop:estimates:cap}.
\end{enumerate}
The cycle and cocycle can be recomputed with the final cap data used in
Section~\ref{sec:stack}; their pairing is the same nonzero closed
Donaldson count, up to the fixed normalization and orientation sign.
\end{theorem}
We first construct a closure, prove that it has a nonzero Donaldson
invariant, and move its insertions away from a rational seam. The exact
sections used to represent the insertions are supplied by
Proposition~\ref{prop:analysis:ordinary-library}; the present section proves
the labelled identities, the compactness budgets, and the comparison with
closed Donaldson invariants. This use of the ordinary analytic library
does not require the spinor compactification or the nonzero count to be
constructed later.
\subsection{The closure and its tight surfaces}
The foliation, contact, and symplectic-cap route to a closed
nonvanishing problem is assembled in
Kronheimer--Mrowka~\cite[Theorem~8, Lemmas~9--14, and
Proposition~15]{KronheimerMrowka2004}. The construction below establishes
simple connectivity and the tight surfaces used in the nonvanishing
argument. It also gives the stated geography bounds.
\begin{lemma}[Enhanced symplectic closure]\label{lem:caps:closure}
There is a symplectic closure $X_0$ of $Y_0$ such that its two halves are
simply connected, $A$ is symplectic, and each half contains a tight
genus-two surface $B$. The closure can be chosen to satisfy
\begin{equation}\label{eq:caps:geography}
q(X_0):=\frac{\chi(X_0)+\sigma(X_0)}4\ge3,
\qquad s(X_0):=2\chi(X_0)+3\sigma(X_0)\ge q(X_0)-2.
\end{equation}
Moreover, $[A]$ is primitive and there is $w\in H^2(X_0;\Z)$ with
$\langle w,A\rangle=1$.
\end{lemma}
\begin{proof}
Gabai's taut-foliation construction for zero surgery on a nontrivial knot
allows the capped norm-minimizing Seifert surface to be a leaf
\cite[Corollary~8.10 and Remark~8.13]{Gabai1987}.
The contact approximation and weak symplectic product
construction preserve positivity on this leaf
\cite{EliashbergThurston1998}. Thus a product neighborhood of $Y_0$
embeds in a symplectic cobordism, with $A$ symplectic and outer boundaries
weakly compatible with their contact structures. We describe the additional
cap choices; they strengthen
the homological conclusions alone of the usual cap construction
\cite[Theorem~8 and Lemmas~9--12]{KronheimerMrowka2004}.
Choose a supporting open book with connected binding at each end
\cite[Theorems~3--4]{Giroux2002}.
Positive stabilizations can increase the page genus while restoring
connected binding in pairs of stabilizations. We may therefore take its
genus to be
\[
G=\frac{(d-1)(d-2)}2
\]
for an arbitrarily large $d\ge6$. Attach Eliashberg's symplectic binding
handle \cite[Theorem~1.1]{Eliashberg2004}. The new boundary is a bundle of
closed genus-$G$ surfaces over the circle, with a prescribed closed
boundary two-form positive on the fibers. Viewed from this new boundary,
the handle cobordism has only a relative two-handle. In particular its
fundamental group is generated by the image of that boundary.
Factor the required inverse boundary monodromy into positive twists on
nonseparating curves~\cite[Lemma~2, arXiv version~2]{AkbulutOzbagci2002}.
Such a factorization exists for the closed fiber: nonseparating twists
generate the closed surface mapping class group, and a positive chain
relation expressing the identity contains each of the standard generators.
Solving this relation for a chosen inverse expresses that inverse as a
positive word; conjugation does the same for any nonseparating twist.
Append as many positive identity factorizations as necessary from generic
degree-$d$ plane pencils. These pencils have only irreducible nodal
singular fibers. Their blown-up total spaces are
$\CP^2\#d^2\overline{\CP}^{\,2}$ and have sections. Removing a regular
fiber leaves a simply connected disk fibration: its fundamental group is
normally generated by the meridian of the removed fiber, and a punctured
section is a disk bounding this meridian. Equivalently, the vanishing
cycles of one plane-pencil identity block normally generate the fiber
fundamental group. Appending one such block consequently kills the
fundamental group of the disk fibration with the originally prescribed
monodromy. Let $Z$ denote the resulting simply connected disk filling.
There is no additional boundary-flux hypothesis in this construction.
Indeed, Poincar\'e--Lefschetz duality gives
\[
H^3(Z,\partial Z;\R)\cong H_1(Z;\R)=0,
\]
so the class of the prescribed boundary two-form extends over $Z$.
It is positive on the regular fiber and on each singular fiber component,
since every singular fiber is irreducible. Here is the relative form
construction with this particular boundary condition. On a boundary
collar extend the prescribed two-form, and choose representatives of the
extended class positive on the vertical tangent planes in fiber
neighborhoods. At a critical point use the usual local Lefschetz model
and a form taming that model. Representatives in overlapping base
neighborhoods differ by exact forms after their fiber integrals have been
matched. Patching their primitives with a partition of unity on the base
therefore preserves both closedness and vertical positivity; the correction
terms contain a base differential and vanish on vertical pairs. Carry
out this patching relative to the boundary collar. Adding a sufficiently
large pullback of an area form on the disk gives a symplectic form,
including on the finitely many critical neighborhoods. That added form
restricts to zero on the tangent bundle of the boundary. Thus its
tangential boundary restriction is exactly the prescribed one. Equal
restrictions give compatible collars by the symplectic hypersurface
neighborhood construction. This also agrees with the boundary-form and
flux arrangements in
\cite[Lemmas~3.4--3.5 and Remark~3.6]{Eliashberg2004}.
Gluing $Z$ kills the group of the binding-handle cobordism, because its
new boundary surjects on that group. It follows that the resulting half
of the closure is simply connected. Do this at both ends. Van Kampen's
theorem then gives $\pi_1(X_0)=0$.
For completeness, two adjacent plane-pencil identity blocks give the
required tight surface inside either cap. Conjugate their fiber
identifications so that a chosen nonseparating vanishing cycle agrees in
the two blocks, and put these chosen critical points next to the
intervening product region. Join their thimbles across that region. The
Lefschetz framings give an embedded sphere $S$ of square $-2$. A curve in
the fiber dual to the vanishing cycle gives a rim torus $R$ in the product
region, with $R^2=0$ and $S\cdot R=1$. Take disjoint parallel copies
$R_1,R_2$ and resolve their two intersections with $S$. The resulting
connected embedded oriented surface satisfies
\[
\chi(B)=2+0+0-4=-2,
\qquad B^2=(S+R_1+R_2)^2=-2+4=2.
\]
Thus $g(B)=2$ and $B$ is tight. This is the sphere-and-two-tori
construction of \cite[Corollary~8.5]{KronheimerMrowka1995}, in the
matching-thimble form used in
\cite[Lemma~4.6, arXiv version~2]{Sivek2015}. It requires neither a new
cap theorem nor a genus-two nonvanishing theorem.
One additional plane-pencil block has Euler characteristic $3+d^2$
and signature $1-d^2$. Fiber sum along genus $G$ adds another $4G-4$
to the Euler characteristic. Consequently its increments are
\begin{equation}\label{eq:caps:block-increments}
\Delta q=G,
\qquad \Delta s=1-d^2+8G.
\end{equation}
For $d\ge6$, $\Delta s-\Delta q=1-d^2+7G>0$. Sufficiently many
blocks give \eqref{eq:caps:geography}. In particular
$b^+(X_0)=2q(X_0)-1\ge5$.
Finally symplectic adjunction gives $K_\omega\cdot A=2$. The second
homology of the simply connected $X_0$ is torsion free. If $[A]=ka$
with $k>1$, this equality forces $k=2$ and $K_\omega\cdot a=1$.
But $a^2=0$, contradicting the fact that $K_\omega$ is characteristic.
Hence $[A]$ is primitive. Unimodularity of the intersection form supplies
an integral $w$ with $\langle w,A\rangle=1$.
\end{proof}
\subsection{An extremal Floer line and Donaldson nonvanishing}
\begin{proposition}[Donaldson nonvanishing for the closure]\label{prop:caps:nonvanishing}
For the integral determinant $w$ of Lemma~\ref{lem:caps:closure},
there are $m,a\ge0$ and $h\in H_2(X_0;\Z)$ such that
\begin{equation}\label{eq:caps:initial-invariant}
h\cdot A=1,
\qquad D^w_{X_0}(x_0^m h^a)\ne0.
\end{equation}
\end{proposition}
We prove this by comparing positive Lefschetz words. A pencil on a
blowup of $X_0$ supplies the word to be studied; an elliptic surface
supplies a model word with a nonzero Donaldson invariant. On a
one-dimensional generalized Floer summand the model acts by a nonzero
scalar. Conjugation and Hurwitz moves extract the pencil word from a
product of model words, forcing its action on that same line to be
nonzero. Simple type and the blowup formula then recover the prescribed
determinant $w$ on $X_0$.
The argument uses ordinary irreducible gluing in
Lemma~\ref{lem:caps:irreducible-gluing}, the Donaldson structure theorem,
and the calculation for elliptic surfaces. It does not identify the
Donaldson series of an arbitrary symplectic manifold with its
Seiberg--Witten series. We begin with the generalized Floer summand
on which the comparison will take place.
\begin{lemma}[The product extremal generalized spaces]\label{lem:caps:extreme}
Let $V_g$ be the full determinant-one instanton homology over $\C$ of
$\Sigma_g\times S^1$, with determinant dual to $\{p\}\times S^1$,
where $g\ge2$. Set $S=\mu(\Sigma_g)$ and $P=\mu(x_0)$.
Every joint eigenvalue $(\lambda,\nu)$ has $|\lambda|\le2g-2$.
Those with equality are exactly
\begin{equation}\label{eq:caps:extreme-spectrum}
(\lambda,\nu)=\bigl(i^r(2g-2),(-1)^r2\bigr),
\qquad r=0,1,2,3.
\end{equation}
Each corresponding simultaneous generalized eigenspace has dimension one.
\end{lemma}
\begin{proof}
The gauge convention matters here. The full group is the determinant-one
group of \cite[Section~7.1]{KronheimerMrowka2010}; Mu\~noz's ring
calculation uses the additional involution quotient. The spectrum and the
conversion between these conventions are described in
\cite[Proposition~7.1 and Section~7.2]{KronheimerMrowka2010}.
Proposition~7.4 there states an ordinary eigenspace assertion. We give the
extra local-algebra argument needed for a generalized eigenspace,
using the localization framework of
Mu\~noz~\cite[Section~5, equations~(6)--(7)]{Munoz1999Ring}.
In the quotient ring use Mu\~noz's coordinates
$\alpha=2S$, $\beta=-4P$, and $\gamma$. Its invariant summand is
$\C[\alpha,\beta,\gamma]/J_g$, where
$J_g=(R_g^1,R_g^2,R_g^3)$. With
$(R_0^1,R_0^2,R_0^3)=(1,0,0)$, the recurrences of
\cite[Theorem~16]{Munoz1999Ring} are
\begin{align}
R_g^1&=\alpha R_{g-1}^1+(g-1)^2R_{g-1}^2,
\label{eq:caps:munoz-first}\\
R_g^2&=(\beta+(-1)^g8)R_{g-1}^1
+\frac{2(g-1)}gR_{g-1}^3,
\label{eq:caps:munoz-second}\\
R_g^3&=\gamma R_{g-1}^1.\label{eq:caps:munoz-third}
\end{align}
The extreme points of this quotient are
\[
(a_0,b_0,0)=
\bigl(\pm4(g-1)i^g,\,(-1)^{g-1}8,\,0\bigr).
\]
Here is an explicit check of the two polynomial facts required for
localization. Put $\zeta_r=R_r^1$. Eliminating the other two
polynomials from the recurrence gives
\[
\zeta_{r+1}=\alpha\zeta_r
+r^2(\beta+(-1)^r8)\zeta_{r-1}
+2r(r-1)\gamma\zeta_{r-2}.
\]
For $f_r(\alpha)=\zeta_r(\alpha,b_0,0)$, induction gives
\begin{align*}
b_0=-8:\quad
f_{2m}(\alpha)&=\prod_{j=1}^{m}
\bigl(\alpha^2-16(2j-1)^2\bigr),&
f_{2m+1}(\alpha)&=\alpha\prod_{j=1}^{m}
\bigl(\alpha^2-16(2j-1)^2\bigr),\\
b_0=8:\quad
f_{2m}(\alpha)&=\alpha^2\prod_{j=1}^{m-1}
(\alpha^2+64j^2),&
f_{2m+1}(\alpha)&=\alpha\prod_{j=1}^{m}
(\alpha^2+64j^2).
\end{align*}
The second even formula is for $m\ge1$, with $f_0=1$ separately.
For even $g$ use the first row, and for odd $g$ the second.
In either case $f_{g-1}(a_0)\ne0$, whereas $a_0$ is a simple root
of $f_g$. These are the extreme factors in
\cite[Proposition~20]{Munoz1999Ring}.
Thus $R_{g-1}^1$ is a unit in the localized algebra at this point.
Equation~\eqref{eq:caps:munoz-third} first forces $\gamma=0$.
Since $R_{g-1}^3$ is divisible by $\gamma$,
Equation~\eqref{eq:caps:munoz-second} then forces $\beta=b_0$.
The remaining relation $R_g^1(\alpha,b_0,0)=0$ has a simple root at
$a_0$ by the displayed factors.
After localization it therefore forces $\alpha=a_0$. The local algebra
is $\C$, not a nonreduced extension of $\C$.
More precisely, the full quotient ring decomposition is
\[
\bigoplus_{k=0}^{g-1}\Lambda_0^kH^3\otimes
\C[\alpha,\beta,\gamma]/J_{g-k}.
\]
The terms with $k>0$ are primitive exterior-power summands tensored
with the analogous quotient for a smaller genus.
Their surface eigenvalues have strictly smaller modulus. They contribute
nothing to an extreme local algebra. This proves the one-dimensional
\emph{generalized} assertion in the quotient theory.
The full determinant-one group splits into the two eigenspaces of the
degree-four involution $\tau$, its positive eigenspace being the
quotient theory above. The grading character $U$ associated to an
eighth root of unity satisfies
\[
SU=iUS,\qquad PU=-UP,\qquad U\tau=-\tau U.
\]
For operators of cohomological degrees two and four one may take
$U|_{V_j}=e^{-\pi i j/4}$; the opposite grading convention reverses
the character. It exchanges the two sectors, multiplies the surface
eigenvalue by $i$, and reverses the point eigenvalue. It carries generalized
spaces isomorphically to generalized spaces. Applying it to the two
quotient extremes gives all four points in
\eqref{eq:caps:extreme-spectrum}, with the normalization
\eqref{eq:caps:normalization}. The same ring presentation gives the
stated upper bound for every remaining eigenvalue.
\end{proof}
\begin{lemma}[A nonseparating pencil]\label{lem:caps:pencil}
After blowups at points disjoint from $A$, the tight surfaces, and $Y_0$,
the closure $X_0$ admits a genus-$g$ Lefschetz fibration with a section,
$g\ge2$, all of whose vanishing cycles are nonseparating. Its positive
monodromy word is the identity in the mapping class group of a surface
with one marked point.
\end{lemma}
\begin{proof}
Rationally approximate and scale the symplectic form, and take a
sufficiently high-degree Donaldson pencil \cite{Donaldson1999}.
Let $F_0$ be its fiber and $b=F_0^2$ its positive number of base points.
Its genus $g_0$ is large. Since $b^+(X_0)>1$, Taubes's canonical-class
inequality $K_\omega\cdot[\omega]\ge0$
\cite[Theorems~1--2]{Taubes1995} and symplectic adjunction give
\[
2g_0-2=F_0^2+K_\omega\cdot F_0\ge b.
\]
Use \emph{full} degree doubling: every base point is used in joining the
two copies of the punctured old fiber. In the doubling monodromy formula,
the new vanishing cycles are nonseparating, and an old separating cycle
remains separating only if one of its sides contains no used base point
\cite[Theorem~4]{AurouxKatzarkov2008}. If $C$ is a component of a
reducible pencil fiber, then
\[
F_0\cdot C=k\int_C\omega>0.
\]
A distinct fiber can meet $C$ only at base points. Both sides of every
old separating cycle therefore contain used base points, so none remains
separating after full doubling. This is also the mechanism in
\cite[Theorem~3.10]{Smith2001}.
The finite base-point set can be chosen disjoint from the specified
surfaces and hypersurface. Blow it up. Any exceptional section marks a
point in every regular fiber; the critical handles are disjoint from this
section. The monodromy around the sphere is consequently the identity
in the pointed mapping class group. The resulting fibration is denoted
$\widetilde X_0\to S^2$.
\end{proof}
\begin{lemma}[A model word on an elliptic surface]\label{lem:caps:model}
For every $g\ge2$ there is a genus-$g$ Lefschetz fibration with a section
on a manifold deformation equivalent to $E(g+1)$, all vanishing cycles
nonseparating, whose Donaldson series for the section determinant has a
nonzero coefficient at $e^{(2g-2)t}$ on the fiber line.
\end{lemma}
\begin{proof}
Put $n=g+1$. Consider the double cover of
$\mathbb P^1_x\times\mathbb P^1_y$ branched in a smooth curve of
bidegree $(4,2n)$, and project to the $x$-line. A regular fiber is a
double cover of the $y$-line branched at $2n$ points, hence has genus
$n-1=g$. Fix $y_0$ and impose on the branch polynomial $b$ the condition
\[
b(x,y_0)=s(x)^2,
\]
where $s$ is a quadratic section with two simple zeros. The equations
$z=\pm s(x)$ above $y=y_0$ give sections. At a zero of $s$, require
$\partial b/\partial y\ne0$. The total branch curve is then smooth
there, its projection to $x$ is unramified, and each displayed section is
smooth. The fact that the two sections meet at these points causes no
failure of either section to be a section.
Here is the required genericity away from that slice. Variations
vanishing on $y=y_0$ are divisible by its defining linear form; their
remaining bidegree is $(4,2n-1)$. They supply first point jets, second
vertical jets, and the first vertical jets at two distinct points of any
other fixed fiber. Thus singular branch points impose three independent
conditions on a two-dimensional point space; triple vertical roots
likewise impose three; and two double roots on one fiber impose four
conditions on a three-dimensional space of a fiber and two points.
A generic member of the imposed affine system has none of these
phenomena. The same argument in the projective coordinate charts covers
points at infinity. All singular fibers therefore arise from a single
simple branch collision. The normalization of such a double cover has
$2n-2>0$ remaining branch points and is connected. Its node is
nonseparating.
The projection to the $y$-line has elliptic regular fiber $F$.
To use the elliptic-surface Donaldson series, we must identify the
surface together with this fiber class. We first identify a smooth double cover of bidegree
$(4,2)$ with the cubic-pencil $E(1)$, then use cyclic base change and
deformation to obtain the required model of $E(n)$.
Choose a generic pencil
\(y_0A(x,z)+y_1B(x,z)=0\) of curves of bidegree \((2,2)\) on \(\mathbb
P^1_x\times\mathbb P^1_z\). Its eight basepoints are distinct and transverse.
Its incidence surface is the blowup at these basepoints. Projection of the
incidence surface to \(\mathbb P^1_x\times\mathbb P^1_y\) is a finite double
cover. Indeed, at a fixed \(x\) the two quadratic coefficient vectors are
freely variable, and their being proportional is the rank-at-most-one
condition on a \(3\times2\) matrix, of codimension two. Allowing \(x\) to vary
adds only one dimension, so a generic pair has no proportional coefficient
vectors at any \(x\). The three coefficients of the quadratic in \(z\)
therefore have no common zero on the \((x,y)\)-base. Its discriminant has
bidegree \((4,2)\). The finite map is flat, since its smooth surface source is
Cohen–Macaulay and its target is a smooth surface. Locally a flat double cover
is \(w^2=b\); its smooth total space therefore implies that its discriminant
is smooth.
Write \(H_x,H_z\) for the ruling classes and \(E_1,\ldots,E_8\) for the
exceptional classes. The system \(|L|\), where \(L=H_x+H_z-E_1\), is the
resolution of projection of the Segre quadric from the first basepoint to
\(\mathbb P^2\). It contracts the two ruling transforms \(A_1=H_x-E_1\),
\(A_2=H_z-E_1\) and the seven curves \(E_2,\ldots,E_8\). These are nine
disjoint exceptional curves; genericity ensures that none of the other seven
basepoints lies on those two rulings. The elliptic pencil has fiber \[
F=2H_x+2H_z-\sum_{i=1}^{8}E_i =3L-A_1-A_2-\sum_{i=2}^{8}E_i. \] Thus the
blowdown sends it to a plane cubic pencil with nine simple basepoints. This is
the usual cubic-pencil construction of \(E(1)\), and each blown-up basepoint
gives a section.
Choose a disk in the \(y\)-sphere containing no critical value and choose two
distinct interior points. The connected cyclic degree-\(n\) cover of the
sphere branched at these two points pulls the discriminant back to a smooth
curve of bidegree \((4,2n)\): over the branch values the elliptic fibers are
regular, so the discriminant is transverse to those ruling fibers. Over the
complementary disk the base cover consists of \(n\) disjoint disks. The
inverse image of the chosen disk is a sphere with \(n\) boundary circles,
since its Euler characteristic is \(n-2(n-1)=2-n\). The elliptic fibration
over the original disk is a product. Its pullback is therefore the same
elliptic fiber times this punctured sphere, attached to the \(n\) copies over
the complementary disk with the product identifications. This is the untwisted
iterated fiber sum of \(n\) copies of the cubic-pencil \(E(1)\), namely
\(E(n)\). The base-changed section remains a section, so there are no multiple
fibers.
Finally choose one \(y\)-ruling fiber transverse to both this pullback
discriminant and the smooth discriminant constructed above with the required
genus-\(g\) sections. Smooth divisors of bidegree \((4,2n)\) transverse to
this fixed ruling fiber form a nonempty Zariski-open subset of the projective
linear system, hence are path connected. Over the corresponding connected
open subset of nonzero sections in
$H^0(\mathbb P^1\times\mathbb P^1,\mathcal O(4,2n))$, the equations
$z^2=b$ in the fixed line bundle $\mathcal O(2,n)$ form a proper holomorphic
family of smooth double covers. Thus our cover is complex deformation
equivalent to the cyclic-base-change cubic-pencil model of $E(n)$.
Along a smooth path within this open set the double covers form a proper
smooth family; the inverse images of the
chosen ruling fiber form a smooth subfamily. A lift of the parameter vector
field tangent to this subfamily integrates to an orientation-preserving
diffeomorphism carrying the elliptic fiber to the elliptic fiber. Thus our
model is diffeomorphic to \(E(n)\) with its required class \(F\) preserved.
If $G_f$ denotes the genus-$g$ fiber, then $F\cdot G_f=2$ and
$G_f^2=0$. The elliptic-surface Donaldson calculation has extreme basic
classes $\pm(n-2)F$ with nonzero coefficients, all other basic classes
having smaller evaluations on $G_f$
\cite{KronheimerMrowka1995}; the elliptic series is computed in
\cite[Section~6, Theorem~6.7, arXiv version~1]{FintushelStern1997},
with its parameters $p=q=1$. In the convention with
determinant even on $F$, its series has the factor
$\sinh(F)^{n-2}$, times a nonzero constant and $\exp(Q/2)$.
Either genus-$g$ section $z=\pm s(x)$ above $y=y_0$ lies in an
elliptic fiber, so its dual determinant has zero evaluation on a
generic $F$; other determinant
conventions change the nonzero basic coefficients only by signs.
Therefore
\begin{equation}\label{eq:caps:model-series}
\mathcal D^{\rm section}_{E(n)}(tG_f)
:=D^{\rm section}_{E(n)}\bigl((1+x_0/2)e^{tG_f}\bigr)
\end{equation}
has nonzero extreme exponential coefficients at
$\pm2(n-2)=\pm(2g-2)$, as claimed.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:caps:nonvanishing}]
Let $g$ be the genus of the pencil on $\widetilde X_0$ supplied by
Lemma~\ref{lem:caps:pencil}. Remove products over two regular disks
from the genus-$g$ model fibration of
Lemma~\ref{lem:caps:model}. Its remaining Lefschetz word is an annular
cobordism with both ends identified with the marked product
$\Sigma_g\times S^1$. Use the determinant dual to its actual section,
including its induced collar identifications. Let $M$ be its ordinary
map on $V_g$, and let $u_g,v_g$ be the state and costate of the two marked
product-disk pieces. The annular map commutes with $S$ and $P$:
represent a fiber or point insertion on either product collar and move
it across the annulus. The even degrees introduce no insertion-order
sign. Ordinary admissible compactness, the support budget in
Lemma~\ref{lem:caps:labelled-compactness} below, and fixed-transition
gluing of Lemma~\ref{lem:caps:irreducible-gluing} justify this operation.
In particular no simple-type assertion about a relative piece is being
used.
By primary decomposition for two commuting endomorphisms of a
finite-dimensional vector space, their simultaneous generalized spectral
projectors are polynomials in $S,P$. The nonzero extreme exponential
coefficient in \eqref{eq:caps:model-series}, computed by capping $M$,
therefore gives an extreme line $V_*$ on which
\begin{equation}\label{eq:caps:three-factors}
v_g\Pi_*M\Pi_*u_g\ne0.
\end{equation}
Lemma~\ref{lem:caps:extreme} says that this space has dimension one.
Consequently the projected disk state, projected disk costate, and
restriction of $M$ are all nonzero; the last is a nonzero scalar.
The conclusion does not depend on arranging a common sign for all
monomial gluing conventions in advance. Indeed, consider the two
sequences with $j$ fiber insertions and respectively zero and one point
insertion. The extreme exponentials in \eqref{eq:caps:model-series}
imply that at least one sequence has upper exponential growth rate at
least $2g-2$. Multiplying its terms by orientation signs preserves this
rate. If every extreme projection of the disk--annulus--disk composition
vanished, its terms would instead be bounded by a polynomial in $j$
times $\rho^j$ for some $\rho<2g-2$, by the Jordan decomposition.
This is impossible. Thus some extreme line always satisfies
\eqref{eq:caps:three-factors}. The normalization of $S,P$ is fixed by
\eqref{eq:caps:normalization}; the ordinary gluing has no additional
factor accumulating with each insertion.
Let $a_1\cdots a_N$ be the pointed positive word of
$\widetilde X_0$. Every $a_i$ is a twist on a nonseparating curve away
from the mark. These curves form one orbit of the pointed mapping class
group. For each $i$ conjugate a copy of the model word so that a selected
first twist is $a_i$. A marked oriented fiber identification, a compatible
determinant identification, or continuation induces an isomorphism on
Floer homology commuting with $S,P$. Its restriction to $V_*$ is
nonzero. Each conjugated model word thus acts by a nonzero scalar on the
same $V_*$, and so does their product.
Extract the selected twists to the front in their prescribed order.
The elementary Hurwitz exchange for a preceding positive word $R$ is
\begin{equation}\label{eq:caps:hurwitz}
Ra=a(a^{-1}Ra).
\end{equation}
It leaves the selected $a$ unchanged and replaces $R$ by a positive
conjugated word. Repeated use gives the factorization
$(a_1\cdots a_N)R'$ of the product of the model words. The whole word
and its prefix are pointed identity words; therefore so is $R'$.
Cutting between them gives marked product boundaries, and the composite
of their maps is nonzero on $V_*$. Both factors, in particular the
desired prefix, are therefore nonzero there.
We spell out why this word operation preserves the data needed by the
gauge theory. A pointed Lefschetz cobordism is built from a marked product
by attaching the ordered Lefschetz two-handles away from the section
track. Hurwitz changes are changes of vanishing paths for those handles;
conjugation is a change of marked fiber identification. The section
track remains straight in the attaching coordinates. The prefix is
therefore the cobordism of $\widetilde X_0$, with its section, up to its
boundary identifications. There is no additional loop-of-identifications
invariant: contractibility of $\operatorname{Diff}_0(\Sigma_g)$ and
the point-evaluation fibration, with its fiber based at the identity, give
$\pi_1(\operatorname{Diff}_0(\Sigma_g,p))=\pi_2(\Sigma_g)=0$
\cite[Theorem~1 and Section~1.1, Proposition~1]{Gramain1973}.
Capping the marked section is allowed to have nonzero normal Euler
number. At every step the determinant is dual to this tracked section,
with its fixed transition functions; we never replace it by a sum over
possible seam lifts.
One can track the normal Euler number explicitly by replacing the marked
point with a small disk. Each twist away from the mark has its canonical
lift fixing the disk boundary. A pointed identity word has product
$t_\partial^e$ in this boundary-fixing mapping class group, where
$t_\partial$ is the central boundary twist and $e$ records the section's
normal Euler number (with the convention $e=-[\text{section}]^2$).
The Hurwitz exchange \eqref{eq:caps:hurwitz} holds for these canonical
lifts. Lifts of a pointed conjugator differ by a central boundary
twist, which cancels in conjugation. The extracted prefix therefore has
exactly the canonical lifted word, and hence the section Euler number,
of the original pencil. The complement carries the sum of the model
exponents minus the prefix exponent; no factor is required
to have exponent zero. Its actual section determinant and boundary
identifications are transported throughout. A change of marked product
identification transports the disk states by an isomorphism commuting
with $S,P$, so their nonzero projections to $V_*$ remain nonzero.
Cap the prefix with the two nonzero projected disk states. Since
$V_*$ is a line, the resulting projected composition is nonzero.
The projector is a polynomial in $S,P$, so some monomial of fiber and
point insertions gives a nonzero ordinary Donaldson invariant on
$\widetilde X_0$. Insertions may be placed on separate product collars.
Their exact geometric representatives and closed interpretation are
justified in Lemmas~\ref{lem:caps:labelled-compactness}
and~\ref{lem:caps:closed-comparison} below; those lemmas apply to these
admissible product boundaries as well as to $Y_0$.
We have obtained a nonzero invariant on $\widetilde X_0$ for its
section determinant. It remains to pass to the prescribed determinant
$w$ and remove the pencil blowups. The tight surface $B$ survives these
blowups, so the tight-surface simple-type theorem
\cite[Theorem~8.1]{KronheimerMrowka1995} applies to $\widetilde X_0$, which is simply connected with odd $b^+>1$.
A nonzero invariant then implies a nonzero $(1+x_0/2)$-series.
To see that no cancellation is hidden in this assertion, simple type
reduces every point power using $x_0^2=4$, while, for fixed surface degree
$a$, the terms $D(h^a)$ and $D(x_0h^a)$ have degrees differing by four.
Only one can satisfy the fixed determinant index congruence modulo eight.
The basic-class structure theorem expresses this series as
\begin{equation}\label{eq:caps:structure-series}
\mathcal D^{w'}(h)=e^{Q(h)/2}
\sum_j(-1)^{((w')^2+K_j\cdot w')/2}\beta_j e^{K_j\cdot h},
\end{equation}
with distinct $K_j$ and nonzero $\beta_j$ independent of $w'$
\cite[Theorem~1.7]{KronheimerMrowka1995}.
Changing to the pullback of $w$ changes coefficients only by signs and
cannot annihilate this sum of distinct exponentials. The pullback
blowup formula~\cite[Proposition~1.9]{KronheimerMrowka1995} has a
nonzero exceptional factor, so nonvanishing of
that pullback series implies nonvanishing of the series on $X_0$.
Thus some homogeneous polynomial $h\mapsto D^w_{X_0}(x_0^m h^a)$
is nonzero.
The affine integral lattice $\{h:h\cdot A=1\}$ is nonempty because
$A$ is primitive, and is Zariski dense in its affine hyperplane. A
nonzero homogeneous polynomial cannot vanish on that whole hyperplane:
rescaling it would make it vanish on the open set $h\cdot A\ne0$.
There is therefore an integral point in the hyperplane at which the
polynomial is nonzero. For $a=0$ any integral point of that hyperplane
will do. This proves \eqref{eq:caps:initial-invariant}.
\end{proof}
\subsection{Labelled probes and their exact representatives}
The closure is cut along $Y_0$, while the fixed caps must meet along the
rational homology sphere $Y_2$. Inserting a positive number of copies of
the bridge $T=f_-\phi^{-1}f_+$ from
Corollary~\ref{cor:ordinary:bridge} creates such a seam. To retain a
nonzero invariant we must also carry the surfaces across these copies;
the ordinary map $T$ alone does not record their insertions. We keep
the occurrences separately labelled and construct a finite deformation
of the Floer complex that records them. Its algebra will show that some
positive power of the deformed bridge still has a nonzero pairing.
Represent the $a$ occurrences of $h$ in
\eqref{eq:caps:initial-invariant} by separately labelled oriented
surfaces. Their intersections with $Y_0$ all represent the meridian
class in $H_1(Y_0;\Z)$, since $h\cdot A=1$. A surface cobordism
between homologous one-cycles in $Y_0$, inserted in a collar together
with its reverse, changes each intersection to a prescribed meridian
$\gamma_i$. Choose the $\gamma_i$ to be disjoint parallel loops.
General position and tubing give pairwise transverse representatives
with no triple intersections. Put the $m$ point representatives in one
cap, disjoint from all of them.
The two handle halves of $T$ are simply connected, so the meridian
loops bound relative surfaces on the two halves separately. Fix these
surfaces, joining the corresponding labels at the two ends of $T$.
Choose its determinant with evaluation one on both end copies of $A$ and the prescribed trivialization at its internal
rational seam. The increasing-handle coordinates in
Section~\ref{sec:ordinary} give precisely these restrictions. Glue it
to $w$ and repeat the same determinant transitions for every copy of
$T$. The closed manifold obtained by inserting $T^k$ in $X_0$ at
$Y_0$ is denoted $X'_k$, with $X'_0=X_0$.
\begin{lemma}[Exact cuts and the low-dimensional support budget]
\label{lem:caps:labelled-compactness}
For the labelled surfaces just described, and also for the separated
fiber and point insertions used above, the exact adjoint tests can be
chosen coherently on cylinders, caps, and cobordisms. In cut dimensions
zero and one, their compactifications have only the ordinary irreducible
breaking strata. No curvature particle contributes. These statements
hold for the parameterized comparisons of exact tests as well.
\end{lemma}
\begin{proof}
At a fixed or moving point, use the dependence locus of two sections of
the universal complexified adjoint bundle of rank three. Its principal
stratum has complex codimension two, hence real codimension four, and
represents
\[
c_2(\ad E\otimes\C)=-p_1(\ad E).
\]
The zero-rank stratum has larger codimension and is avoided in the
dimensions at issue. A moving point on an oriented surface contributes
two position parameters, so its net cut degree is two. A fixed point
has degree four.
The section construction of Proposition~\ref{prop:analysis:ordinary-library}
uses holonomy and frame-equivariant sampled transports from the marked
point to finite systems of loops detecting irreducibility. The center
acts trivially on the adjoint output. Irreducibility gives full value
variation at the output, hence transverse choices of the two sections.
On a cylinder the choices are translation invariant relative to the
mark; on cobordisms they approach those choices at the ends. The
sampling terms carry small-ball energy cutoffs along their paths.
For thin paths one uses arbitrarily fine path libraries, with sufficiently
rapidly decreasing coefficients. The cutoff limits include curvature
atoms in the limiting energy measure. Thus a term either has its
prescribed convergent off-atom transport or is cut off when its sampling
neighborhood acquires an atom. Loop systems avoiding a given finite
set of atoms still detect irreducibility and still give independent
value variations. The resulting exact cut condition persists unless
its marked point itself hits a curvature atom. On a particle stratum
the retained equation is allowed to depend on its retained atom
positions. Indeed a smooth sampling cutoff replaces its energy
argument by
\[
\int\rho\,|F_a^0|^2+8\pi^2\sum_j\ell_j\rho(p_j),
\]
which is smooth in the positions $p_j$ on that stratum. Thus persistence
means convergence to a section on the residual-field-and-particle
stratum, not necessarily to the ordinary section on the residual field
alone. The latter equality is not needed for the dimension budget.
This is the
support-exact feature for which the construction, rather than a
representative depending indiscriminately on a whole surface
neighborhood, is needed here.
Choose parameters transverse simultaneously on all lower-charge strata,
including remaining marks and the positions of particles. This remains
possible for the particle-dependent sections just described: at any
such configuration a detecting graph and sufficiently small sampling
balls can avoid the finite atom set, and the associated surviving term
has unrestricted adjoint output variation. The particle positions are
counted once among the stratum parameters, not again as new cut data.
Ends carrying
the odd determinant on $A$ are admissible. Every nonconstant cylinder
level and every cobordism piece considered in this labelled construction
is consequently irreducible. A constant critical cylinder cannot carry
a generic mark: modulo translation its mark has only the one loop
coordinate, against a codimension-four condition. Floer compactness
therefore assigns any escaping mark to a nonconstant trajectory level,
unless that mark hits an escaping particle.
For a compact-part particle, the precise budget is as follows. The
``Released degree'' column records the net degree of discarded cuts, after their
mark parameters are included.
\begin{center}
\begin{tabular}{lrrr}
\toprule
Position of the particle & Released degree & Position dimension & Sum\\
\midrule
Outside the supports & $0$ & $4$ & $4$\\
On one surface & $2$ & $2$ & $4$\\
At two transverse surfaces & $4$ & $0$ & $4$\\
At a fixed point & $4$ & $0$ & $4$\\
\bottomrule
\end{tabular}
\end{center}
There are no triple intersections or point--surface incidences. If the
lost charge is $\ell$ and it is carried by $r$ distinct particles, then
$r\le\ell$, while the ASD index drops by $8\ell$. The residual cut
stratum, with particle positions retained, therefore has dimension at
most
\begin{equation}\label{eq:caps:bubble-budget}
d-8\ell+4r\le d-4\ell,
\end{equation}
where $d$ is the original cut dimension, including any comparison
parameter. This is negative for $d\le1$ and $\ell>0$.
The same reasoning applies on extracted nonconstant cylinder levels.
If an atom escapes along a constant-flat region with no nonconstant
level retained there, it can discard at most one label, since the
parallel cylindrical supports $\R\times\gamma_i$ are disjoint.
All equations on retained levels persist, so this case is at least as
strongly excluded. This is the incidence estimate of
Lemma~\ref{lem:analysis:incidence} in the present setting.
At the remaining broken configurations, exponential convergence to
nondegenerate critical points and
Lemma~\ref{lem:caps:irreducible-gluing} give exactly
the products of the cut factors. The increasingly distant sampling
offsets may be truncated on long gluing collars with faster than
exponential tails; their errors then lie within the ordinary gluing
estimates. Section choices and determinant-line orientations are
compatible with these products. Thus the only ends of an oriented
one-dimensional cut moduli space are its ordinary irreducible breaks
and its prescribed comparison endpoints.
\end{proof}
\begin{lemma}[Comparison with closed Donaldson insertions]
\label{lem:caps:closed-comparison}
The closed counts obtained from these exact tests and the fixed
transition functions are the Donaldson invariants for that determinant,
with each surface and point class replaced by its multiple $4\mu$.
They have no additional sum over determinant lifts. The assertion
applies to $X'_k$ and to the fiber gluings used in
Proposition~\ref{prop:caps:nonvanishing}.
\end{lemma}
\begin{proof}
All the $X'_k$ are simply connected: the original two halves and the
handle halves of $T$ are simply connected, and Van Kampen applies at
each connected seam. They retain a tight surface and $b^+>1$.
Their determinants remain odd on the retained copies of $A$.
The same properties needed here hold for the simply connected model and
pencil manifolds. We first explain comparison from long admissible
collars, where the Floer perturbation was chosen before the gluing
length.
A reducible would have integral line difference $d$ with odd evaluation
on every odd-class slice. On a fixed slice metric its tangential
curvature therefore has a uniform positive $L^2$ lower bound, by pairing
with a fixed dual form and Cauchy--Schwarz. Integrated along a long
product collar this lower bound grows linearly with its length. The
near-ASD equation, with gradient-type Floer perturbation and sufficiently
small remaining error, converts it to a growing lower bound for the
charge on that product. Other admissible collars have the same
property, and their gradient energy identities have endpoint terms
that telescope. In a fixed total charge range this rules out a
reducible while the Floer perturbations are turned off on collars that
are still sufficiently long. It is therefore legitimate to perform
this first step with the earlier fixed small Floer data; no reversal of
the order of smallness and gluing length is required. Thereafter choose
a generic path to an unperturbed closed metric, with perturbations as
small as needed. Generic periods exclude the finitely relevant
reducible classes throughout this path; $b^+>1$ permits a generic
one-parameter path. The bubble budget
\eqref{eq:caps:bubble-budget} applies to these comparison homotopies.
Localize the exact section data near the surface representatives and in
small disjoint point balls at the final generic unperturbed metric.
An irreducible ASD connection on one of these simply connected
manifolds is locally irreducible: a local parallel line extends by ASD
unique continuation, and simple connectivity removes monodromy of this
extension. An irreducible flat exception cannot occur. Thus the
localized choices retain the needed independent transversality.
For a fixed point the adjoint rank-drop locus is already the usual
geometric representative of $-p_1$ on its ball.
For a surface compare one test at a time. Omit that degree-two test;
the remaining partially cut ASD moduli space has dimension two.
It is compact. Indeed, take the chosen small neighborhoods so that
at most two surface neighborhoods meet at any point and no point ball
meets a surface neighborhood. If a particle occurs, discard the tests
whose neighborhoods contain it. At most degree four is released per
particle. For this argument project to the residual ASD solution and
ignore particle-position parameters: every sampling graph, output,
and cutoff for a retained test lies in its atom-free neighborhood.
Consequently these retained equations are exactly the ordinary
lower-level local equations and are independent of particles outside
their neighborhoods. This localization, unlike the earlier general
particle-stratum argument, is what permits forgetting the positions.
The residual problem has dimension at most
$2-8\ell+4\ell<0$. Hence it has no solutions. This establishes the
compactness needed for the comparison, rather than assuming compactness
of an arbitrary partially cut moduli space.
On this compact two-dimensional space the moving-point intersection
number is exactly the slant product of $c_2(\ad E\otimes\C)$ with
the surface. By \eqref{eq:caps:normalization} it is $4\mu$ of that
surface. Replacing the surface tests one at a time identifies the full
count with the ordinary Donaldson intersection number, and the point
tests contribute $4\mu(x_0)$ in the same normalization. Generic
homotopies of these exact tests have no extra ends by
Lemma~\ref{lem:caps:labelled-compactness}.
Finally, the glued transition functions are fixed actual
determinant-preserving bundle identifications, not arbitrary projective
identifications with a prescribed characteristic class. A projective
seam automorphism has an obstruction in $H^1(Y;\Z/2)$ to lifting to
the determinant-one gauge group. A nonzero obstruction excludes that
automorphism from the matching group at the outset, even if it would
preserve $c_1$ and $c_2$. The full determinant-one Floer complex likewise
keeps distinct the orbits related by such a nonliftable twist.
For the remaining, liftable automorphisms, obstruction theory on the
three-manifold identifies components of the determinant-one gauge group
with $H^3(Y;\Z)$: its fiber $\SU(2)$ is two-connected, and conjugation
acts trivially on $\pi_3(\SU(2))=\Z$. Thus a degree-zero allowed
automorphism is homotopic to the identity. A nonzero degree changes
$c_2$ and changes the ASD index by eight times that degree. The actual
zero-dimensional index fixes this remaining integer. In the pointed
word construction the actual section and its dual determinant transport
these identifications through every conjugation and Hurwitz move.
Therefore ordinary gluing here counts the required fixed determinant
problem; neither a hidden $H^1(Y;\Z/2)$ twist nor a new sum of lifts is
introduced.
This restriction on matching isomorphisms does not omit closed
connections. Fix the actual closed rank-two bundle $E$ with the
prescribed determinant and second Chern class, and restrict it to the
pieces using the transition induced by $E$. Every closed connection on
$E$ has literal restrictions with equal full determinant-one boundary
orbits. Compactness and gluing are performed in that gauge convention.
A projective change of gauge with nonzero lifting obstruction can move
one full boundary orbit to another; the composition sums over all full
orbits and does not quotient by this action. Thus such boundary states
are retained, rather than their corresponding closed solutions being
discarded. Ordinary maps also retain all compatible relative bundle
sectors. Their integer degree bookkeeping sums to $c_2(E)$ at the
selected total index. Conversely a full-state match with these fixed
determinant transitions and this total degree glues a rank-two bundle
with $c_1(E),c_2(E)$, hence the prescribed closed bundle, since these
classes classify rank-two complex bundles over a four-dimensional
complex. The usual determinant-one gauge-matching quotient identifies
its connections with the closed gauge orbits, and
Lemma~\ref{lem:caps:irreducible-gluing} gives multiplicity one. This
applies in particular to the annulus--disk compositions above.
For comparison, the published
Fukaya--Floer gluing formula in \cite[Theorem~3.1]{Munoz1999FukayaFloer}
uses a convention permitting an additional nonliftable seam twist and
has a corresponding sum of two determinant invariants. We have instead
constructed and glued the determinant-one counts with fixed collars;
that published sum is not being invoked as a fixed-lift formula.
\end{proof}
\subsection{The finite deformation and a positive power of the bridge}
Let $C$ be the ordinary integral admissible Floer complex of $Y_0$,
with its fixed determinant. Tensor with $\Q$ when needed. Keep the
labels distinct and set
\begin{equation}\label{eq:caps:label-ring}
L_a=\Q[t_1,\ldots,t_a]/(t_1^2,\ldots,t_a^2),
\qquad J=(t_1,\ldots,t_a).
\end{equation}
The variables have even parity (one may give each degree $-2$).
For a subset $P\subset\{1,\ldots,a\}$ write $t_P=\prod_{i\in P}t_i$.
For $a=0$, use $L_0=\Q$.
\begin{proposition}[Finite labelled Floer construction]\label{prop:caps:labelled}
There are a differential $d_L$ on $C_L=C\otimes L_a$, a cycle
$u\in C_L$, a cocycle $v\in\Hom_{L_a}(C_L,L_a)$, and a chain map
$T_L:C_L\to C_L$ such that $T_L\bmod J=T$. The all-label coefficient
of $vT_L^k u$ is, up to a fixed nonzero normalization and orientation
sign, the closed Donaldson invariant on $X'_k$ with the indicated
labelled closed surfaces and fixed point insertions. In particular,
\begin{equation}\label{eq:caps:coefficient-zero}
[t_1\cdots t_a](vu)\ne0.
\end{equation}
\end{proposition}
\begin{proof}
For each subset $P$, count cylinder trajectories with one moving point
on $\R\times\gamma_i$ and its codimension-four adjoint test for every
$i\in P$. A rigid such trajectory has unparametrized index
$2|P|$, or parametrized trajectory index $1+2|P|$.
Let its count be $d_P$, and set
\[
d_L=\sum_Pt_Pd_P.
\]
Do not retain a mark or a test for a label outside $P$. Define the cap
coefficients $u_P,v_P$ by the corresponding relative surfaces in the
caps and their fixed point insertions. Define $(T_L)_P$ by the
relative surfaces in $T$. All these counts use the exact tests of
Lemma~\ref{lem:caps:labelled-compactness}.
The oriented boundary of a one-dimensional cylinder space is a broken
pair with each label assigned to exactly one factor. Its algebraic
identity is
\[
\sum_{P_1\sqcup P_2=P}d_{P_1}d_{P_2}=0.
\]
This is exactly the coefficient of $t_P$ in $d_L^2$, since repeated
labels vanish in $L_a$. The same boundary calculation on a cap or a
cobordism gives $d_Lu=0$, $vd_L=0$, and
$d_LT_L=T_Ld_L$, with the prescribed common grading sign if that
convention is used. End-generator orientation lines are paired by
ordinary determinant-one composition. All insertion degrees are even,
so distributing labels introduces no additional Koszul sign. There
are no factorials: these are distinct labels, each used at most once.
Stretch the $Y_0$ seams in the closed manifold. Its ends are admissible.
Lemma~\ref{lem:caps:labelled-compactness} excludes particles and unwanted
constant levels, and Lemma~\ref{lem:caps:irreducible-gluing} gives the product of the
cut factors, with labels distributed among them. This is precisely
the coefficient expansion of $vT_L^ku$. The fixed transition data and
Lemma~\ref{lem:caps:closed-comparison} identify each coefficient with
its actual fixed-determinant Donaldson invariant. At $k=0$, the
all-label coefficient is the nonzero invariant
\eqref{eq:caps:initial-invariant}, multiplied by $4^{a+m}$ and the
fixed orientation sign. This proves \eqref{eq:caps:coefficient-zero}.
\end{proof}
\begin{lemma}[Nilpotent deformation and positive powers]\label{lem:caps:positive-power}
There is $k\ge1$ for which
\begin{equation}\label{eq:caps:positive-coefficient}
[t_1\cdots t_a](vT_L^ku)\ne0.
\end{equation}
\end{lemma}
\begin{proof}
By Corollary~\ref{cor:ordinary:bridge}, $T$ is a rational
quasi-isomorphism. Filter the mapping cone of $T_L$ by powers of $J$.
Since $J^{a+1}=0$, this is a finite filtration. Its associated graded
complex is a direct sum of copies of the rational mapping cone of $T$,
tensored with the finite-dimensional spaces $J^r/J^{r+1}$, and is
acyclic. Induction through the short exact sequences of filtration
steps shows that the cone of $T_L$ is acyclic. Therefore the induced
endomorphism $F$ of $H_*(C_L)$ is an automorphism. This homology is
finite-dimensional over $\Q$; no freeness over $L_a$ has been asserted
or needed.
Let $N=\dim_\Q H_*(C_L)$ and write the characteristic polynomial of
$F$ as $a_0+a_1s+\cdots+a_Ns^N$. Its constant coefficient is nonzero
because $F$ is invertible. Cayley--Hamilton gives
\begin{equation}\label{eq:caps:cayley-hamilton}
\id=-a_0^{-1}\sum_{j=1}^Na_jF^j.
\end{equation}
Apply the functional $[t_1\cdots t_a]\,v(\,\cdot\,)u$ to this
identity on homology. Its value on the identity is nonzero by
\eqref{eq:caps:coefficient-zero}. At least one positive-power value
must therefore be nonzero, proving \eqref{eq:caps:positive-coefficient}.
\end{proof}
\subsection{Moving all probes into the final caps}
Fix a positive $k$ supplied by Lemma~\ref{lem:caps:positive-power}.
The associated nonzero closed count on $X'_k$ has only even-degree
insertions. The dimension congruence consequently forces $b^+(X'_k)$
to be odd; $b_1=0$ and the ASD dimension is
$8\kappa-3(1+b^+)$. Split $X'_k$ at an internal rational $Y_2$ seam.
Since $H_1(Y_2;\Q)=H_2(Y_2;\Q)=0$, Mayer--Vietoris gives
\begin{equation}\label{eq:caps:homology-split}
H_2(X'_k;\Q)=H_2(C_-^0;\Q)\oplus H_2(C_+^0;\Q).
\end{equation}
Expand each probe class with this direct sum and use multilinearity of
the closed invariant. The resulting finite sum is nonzero, so at least
one allocation of all the probes to the two cap interiors has nonzero
value. Clear its rational denominators. Integral surface classes in a
cap have embedded oriented representatives in its interior, after
resolving double points if necessary. Take separate representatives for
separate labels, arrange pairwise transversality without triple points,
and keep them disjoint from the point probes.
Each $C_\pm^0$ is simply connected by the same handle-half and Van Kampen
argument used above. It contains its old tight surface $B$, so it has
$b^+>0$, and contains a copy of $A$ on which its determinant evaluates
as one. Its determinant is trivialized at the rational boundary.
Blow up one further point in the interior of each cap, away from all
these surfaces and probes. Extend the determinant with zero exceptional
evaluation. The pullback blowup formula
\cite[Proposition~1.9]{KronheimerMrowka1995}, restricted to the old homology,
preserves the invariant, because its exceptional factor has constant
term one. Denote the resulting caps by $C_-,C_+$.
\begin{proposition}[Cap periods and the ordinary pairing]\label{prop:caps:periods}
Generic period choices on the old cap parts exclude reducible ASD cap
solutions in the bounded charge ranges needed for the cap cycles,
comparison homotopies, and the closed pairing. After sufficiently long
finite isolation of the separate exceptional summands, and sufficiently
small ordinary Floer data, stretching the common $Y_2$ seam computes
the nonzero allocated invariant as the pairing of a cycle and a cocycle
on the irreducible complex.
\end{proposition}
\begin{proof}
For a reducible with determinant $c$, its integral line difference is
$d=2l-c$ for some integral $l$. In particular
\[
d(A)\equiv c(A)=1\pmod2,
\]
so $d$ is nonzero in rational cohomology. A reducible ASD solution with
rational flat end would require $d^+=0$. Since $b^+(C_\pm)>0$,
generic cap periods avoid this condition. The precise cap version of
the period argument, including compactly supported metric variation,
small perturbations, bounded charge, and the long isolated exceptional
summand, is Proposition~\ref{prop:estimates:cap}. Its metric choices
apply here because the old cap contains both $A$ and $B$ and the probes
are supported away from the isolated summand.
One must use its bounded-charge conclusion, rather than claiming that
arbitrary cap data give the same cap states. Here are the charge bounds
that supply its hypothesis. Let $z_W$ be the local probe degree on one
cap. Its cycle entries have actual index $i=z_W$, and the
one-dimensional spaces giving the cycle identity have $i=z_W+1$.
For a comparison with $t\in\{0,1\}$ parameters the corresponding
condition is $i+t-z_W\in\{0,1\}$. Only these finitely many integer
indices occur. With fixed reference lifts for the finitely many end
generators $\alpha$, the APS formula takes the form
\[
i=8\kappa+b_W(\alpha).
\]
The boundary correction $b_W(\alpha)$ is uniformly bounded as the
ordinary perturbations become sufficiently small. Indeed the flat
moduli space is compact; elliptic spectral estimates bound uniformly
the number of eigenvalues near zero, so only uniformly finitely many
such eigenvalues can contribute integer spectral-flow jumps under a
small perturbation. The smooth boundary and transgression terms are
also bounded on this compact family near the flat moduli space.
At a central flat the correction
is explicitly $-3(1+b^+(W))$ by
Equation~\eqref{eq:ordinary:central-index}. Thus all these cap problems
have a bounded charge range, including reducible candidates before
period exclusion. The closed pairing has the fixed total charge
\[
8\kappa_{\rm closed}
=z+3(1+b^+(X)).
\]
These bounds do not deteriorate with a Floer collar length. On a
gradient collar the perturbed Chern--Simons identity bounds the
topological charge below by the difference of the bounded perturbation
at its two endpoints, with a fixed error from the permitted integrable
non-gradient terms. Summing through successive segments or broken
levels telescopes those endpoint terms. Thus the charge of an entire
outgoing chain has a lower bound independent of its length and of the
number of levels. Subtracting this bound from the fixed total charge
bounds the charge on the compact cap part from above. The same argument
applies to either cap in the closed pairing, using the fixed local
geometry on the other compact part. Chern--Weil then bounds the
curvature on compact parts. On a prefix of length $N$, the remaining
small self-dual error contributes at most a fixed constant times
$\delta^2N$, so the bound has the form
\[
\int_{\text{cap and prefix through }N}|F_a^0|^2
\le E_{\mathcal P}+C_W\delta^2N.
\]
Here $E_{\mathcal P}$ can depend on the fixed finite problem, while
the coefficient of $N$ is controlled by the fixed collar data. This
is the prefix bound required for the exponentially decreasing harmonic
cutoff tails in Proposition~\ref{prop:estimates:cap}. It also covers
central outgoing chains; their later dimension exclusion is not used
to obtain this energy bound. A formal infinite tail may be asymptotic
to a perturbed nonflat connection, so no finite unperturbed curvature
norm on that entire tail is asserted. Isolation depths and smallness
thresholds can now be chosen for all the fixed index ranges just listed.
Count regular irreducible cap solutions with their local cuts, obtaining
$x$ and $y$. Bubbling is absent by the exact support budget of
Lemma~\ref{lem:caps:labelled-compactness}. Reducible cap pieces are
absent by the period choice. At a central seam the three-dimensional
central stabilizer contributes three matching indices, as in
Lemma~\ref{lem:ordinary:central-cost}. Each of the two irreducible cap
pieces has nonnegative residual dimension after its local cuts. Such
a central matching therefore cannot occur in a zero-dimensional closed
count. More generally a maximal central cylinder group includes its
exterior matchings and has the same surplus of at least three. At a
central end in a cap-cycle identity it must also have the requisite
nonconstant outgoing trajectory. It cannot be an end of a
one-dimensional cut cap moduli space. The only remaining boundary
points are ordinary irreducible Floer breaks. Their signed sum gives
$\partial x=0$ and $y\partial=0$.
At a sufficiently long finite common neck, every rigid closed solution
is therefore an irreducible match of two rigid cap solutions. Conversely
each such match glues by Lemma~\ref{lem:caps:irreducible-gluing}.
Fixed determinant transitions identify the glued bundle, and cancelling
adjacent orientation lines gives the closed count up to its fixed
comparison sign. There is no extra factor of two from the common
central subgroup $\{\pm1\}$ of an irreducible stabilizer: it is already
included in the determinant-one gluing convention. We obtain the
pairing $yx$, which equals the nonzero allocated invariant up to the
nonzero $4\mu$ normalization.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:caps:pairing}]
Use the caps and probes just constructed and
Proposition~\ref{prop:caps:periods}. Their topology gives (i) and (ii),
and their ordinary pairing gives (iii). All exact point and surface
classes are integral multiples of the Donaldson classes. The remaining
rational factors come from the fixed denominators in
\eqref{eq:caps:homology-split} and, if used, rational perturbation
weights. Clearing these fixed denominators yields integral cycles and
cocycles in the finite integral complex, while multiplying $yx$ by a
nonzero integer. Let $z$ be the sum of the real degrees of the resulting
fixed probes. This proves (iv), with the extension of the cap period
choices supplied by Proposition~\ref{prop:estimates:cap}.
For later metric choices, recompute the pairing by stretching the same
seam in the old closed, blown-up manifold. Its fixed-determinant closed
Donaldson invariant is unchanged because $b^+>1$; the exact-test
comparison is Lemma~\ref{lem:caps:closed-comparison}. The cap period
exclusion and central matching budget apply again at those choices.
This proves the last assertion without requiring separate
metric-independence of either individual cap state.
\end{proof}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/04-caps.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/05-stack.tex ====
\section{The stack, its lifts, and the nonzero instanton count}
\label{sec:stack}
We now arrange the ordinary maps and the fixed cap pairing into a closed
family with a nonzero integer count. The cap pairing is nonzero over
\(\Q\), while the maps are units modulo two. Repetition on the free
integral Floer lattice will reconcile these two inputs and make the
coupled Dirac index positive. Before choosing that repetition, we compute
the stack's lift data and identify its actual count in the metrics used
for the spinor argument. That identification requires a separate
exclusion of central limits at rational seams; invariance of the
individual maps alone does not supply it.
Throughout this section, the diffeomorphism
\(\phi:Y_{-2}\longrightarrow Y_2\) is the arbitrary oriented cosmetic
identification fixed in Section~\ref{sec:ordinary}. No compatibility with
the knot exterior is imposed. We use the caps of
Theorem~\ref{thm:caps:pairing}, their fixed compact probes of total real
degree \(z\), and the signed negative interval map
\(B:C(Y_2)\longrightarrow C(Y_{-2})\) of
Theorem~\ref{thm:negative:unit}. Put
\[
H=f_+f_-:C(Y_{-2})\longrightarrow C(Y_2).
\]
Thus \(H\) is the ordinary rigid map along
\(-2\to-1\to0\to1\to2\), with the odd determinant at \(Y_0\).
All compositions below use fixed determinant transitions and
determinant-one gauges.
\subsection{The arrangement and its cell lattices}
Insert \(n\) negative intervals \(W'_1,\ldots,W'_n\) between the caps.
After an interval use \(\phi\) to return to the \(Y_2\) convention,
except at \(m\) selected bridges, where the positive path defining
\(H\) is inserted instead. The return after the last interval meets
the final cap in the \(Y_2\) convention. Selected bridges are at least
four intervals apart; the repeated pattern used below has exact spacing
four. Long initial and terminal pads contain no positive bridge.
A positive bridge together with its two
adjacent negative intervals is a \emph{macro}. Macros are disjoint and
have ordinary single intervals between them. Write \(J_i\) for the
potential primary \(S^3\)-cut in \(W'_i\), and write
\[
S_i=F_{l,i}-F_{r,i},\qquad S_i^2=-4,
\]
for its test sphere. Cutting at all the \(J_i\)'s and filling the new
spherical boundaries by balls gives two outside pieces and one cell
between each pair of successive \(J_i\)'s. This is a topological
decomposition used for intersection-form computations; it does not
assert that any metric neck has been stretched.
\begin{figure}[ht]
\centering
\input{figures/stack-pattern}
\caption{One repeated four-interval pattern between the caps. The
first two intervals and the positive path form a macro; the positive
cell lies between the primary cuts inside those intervals. The cut
marks are schematic, and the cell lattice is computed after filling
its spherical boundaries. Initial and terminal negative pads are
suppressed. All ordinary \(Y\)-seams have finite length; each
\(B\) carries its own primary \(J\)-face and lens face, which are
the genuine ends of its parameter interval.}
\label{fig:stack:pattern}
\end{figure}
\begin{proposition}[Cell lattices]
\label{prop:stack:lattice}
All these filled pieces have \(H_1=0\), and their integral intersection
lattices are unimodular. A negative cell has orthogonal trace classes
\(F_r,F'_l\), both of square \(-2\), and its lattice has the basis
\begin{equation}
a=\frac{F_r+F'_l}{2},\qquad b=\frac{F_r-F'_l}{2},\qquad
a^2=b^2=-1,\quad ab=0.
\label{eq:stack:negative-lattice}
\end{equation}
A positive cell \(P\) has an integral basis
\(G,U,T_1,T_2,T_3,T_4\) with
\begin{equation}
Q_P=\begin{pmatrix}2&1\\1&0\end{pmatrix}\oplus(-I_4),\qquad
F_r=G-\sum_{b=1}^4T_b,\qquad F'_l=G-2U.
\label{eq:stack:positive-lattice}
\end{equation}
Here \(U\) is represented by a sphere of square zero, and
\(UF_r=UF'_l=1\). The middle \(0\)-trace class is
\begin{equation}
F_0=G-T_3-T_4.
\label{eq:stack:middle-class}
\end{equation}
The macro contains an ordinary plumbing of the three spheres
\begin{equation}
X_1=-S_j,\quad U,\quad X_2=S_{j+1},\qquad
X_1^2=X_2^2=-4,\quad UX_1=UX_2=1,\quad X_1X_2=0.
\label{eq:stack:macro-plumbing}
\end{equation}
In particular \(b^+(P)=1\) and \(\sigma(P)=-4\).
\end{proposition}
\begin{proof}
The trace halves have zero first homology. Gluing two such halves along
a rational homology sphere preserves this property by the
Mayer--Vietoris sequence. The same argument applies to the caps and to
the positive handle path. Filling spherical boundary components by
balls does not change \(H_1\). Poincar\'e duality and the universal
coefficient theorem therefore give a unimodular intersection form on
the integral second homology.
In a negative cell the two trace classes have intersection matrix
\(\operatorname{diag}(-2,-2)\), whose determinant is four. They span
an index-two sublattice. Its discriminant group is
\((\Z/2)^2\). Of its three nonzero elements, the classes of
\(F_r/2\) and \(F'_l/2\) have square \(-1/2\), and so cannot be
adjoined to an integral lattice. The remaining element,
\((F_r+F'_l)/2\), has integral square \(-1\). It gives the unique
integral index-two extension, namely \eqref{eq:stack:negative-lattice}.
This argument uses no information about \(\phi\) beyond its being a
diffeomorphism of the filled manifolds. In particular, its action on
\(H_1(Y_{\pm2})=\Z/2\) is necessarily the identity.
For the positive cell use the polygon model from
Section~\ref{sec:ordinary}. Between its two spherical boundaries the
consecutive rays are
\[
(-1,0),\ (-2,1),\ (-1,1),\ (0,1),\ (1,1),\ (2,1),\ (1,0).
\]
The five internal side spheres form the chain
\([-1,-2,-2,-2,-1]\). The sum of these spheres has zero intersection
with each member of the chain. Smoothing its four transverse
intersections gives a sphere: the dual graph is a tree, and each
smoothing joins two spheres. Its square is
\(-1-2-2-2-1+2\cdot4=0\). This is \(U\).
Its two endpoint incidences give the stated intersections with the
neighboring test spheres.
Blow down the four corner subdivisions introduced by the four
increasing steps. The remaining doubled \(2\)-trace has a trace class
\(G\) with \(G^2=2\). Doubling its cocore disk across the
\(Y_2\) seam gives the sphere \(U\), with \(U^2=0\) and
\(GU=1\). Under restoration of the corner subdivisions this is
the smoothed side-chain sphere described above; the blowup points
are disjoint from its representative. This rank-two form is unimodular.
Restore the subdivisions in their chronological order from slope
\(-2\) to slope \(2\), and use their total exceptional transforms
as \(T_1,\ldots,T_4\). These have intersection form \(-I_4\) and
are orthogonal to the pullbacks of \(G,U\). The incoming trace passes
through all four blown-up corners, whereas the middle trace passes
through the last two. Their proper transforms are therefore
\(G-\sum T_b\) and \(G-T_3-T_4\). The reverse outgoing trace is
orthogonal to the incoming one, meets \(U\) once, and has square
\(-2\); in the doubled-trace coordinates it is \(G-2U\).
This proves \eqref{eq:stack:positive-lattice} and
\eqref{eq:stack:middle-class}. The local corner charts give transverse
positive plumbing intersections, so the smoothing and the two adjacent
\(-4\)-spheres give \eqref{eq:stack:macro-plumbing} inside the macro.
There is no use of a negative bridge identification in this construction.
The signature and positive rank follow from the displayed matrix.
\end{proof}
An evaluation vector \(v=(x,u;t_1,t_2,t_3,t_4)\) in the positive
basis consequently has square
\begin{equation}
v^2=2xu-2u^2-\sum_{b=1}^4t_b^2.
\label{eq:stack:positive-dual-square}
\end{equation}
On a negative cell the corresponding formula is
\begin{equation}
v^2=-\frac{(vF_r)^2+(vF'_l)^2}{2}.
\label{eq:stack:negative-dual-square}
\end{equation}
These formulas will also be used on rational orthogonal summands of
pieces cut at only some of the \(J_i\)'s.
\subsection{Characteristic lifts and the index telescope}
The spin-\(c\) structure used for the coupled Dirac operator will have
characteristic determinant class \(l\). For a rank-two bundle with
determinant class \(c\), write \(\Lambda=l+c\). We choose \(c_0\)
and \(\Lambda_0\) so that \(l=\Lambda_0-c_0\) is characteristic
and remains fixed when the determinant lift changes.
On every negative cell set
\[
c_0=0,\qquad (\Lambda_0F_r,\Lambda_0F'_l)=(-2,0).
\]
On a positive cell specify their evaluations by
\begin{equation}
c_0=(2,1;1,0,1,0),\qquad
\Lambda_0=(-2,-1;0,1,0,-1).
\label{eq:stack:base-lifts}
\end{equation}
On each outside piece use the fixed cap determinant, extending it with
zero evaluation on the bordering negative trace. Choose
\(\Lambda_0-c_0\) characteristic, with
\(\Lambda_0F_{l,1}=0\) on the first outside trace and
\(\Lambda_0F_{r,n}=-2\) on the last. On the separate extra exceptional
class \(E_{\rm exc}\) of each cap choose
\(\Lambda_0E_{\rm exc}\) to be an odd integer; its absolute value
will later be made large. The old cap data are fixed before this choice.
\begin{lemma}[Existence and restrictions of the lifts]
\label{lem:stack:lifts}
These choices define integral classes on the closed stack, and
\(l=\Lambda_0-c_0\) is characteristic. For every bit vector
\(e=(e_1,\ldots,e_n)\in\{0,1\}^n\), set
\begin{equation}
c_e=c_0+\sum_{i=1}^ne_i\PD(S_i),\qquad
\Lambda_e=\Lambda_0+\sum_{i=1}^ne_i\PD(S_i).
\label{eq:stack:bit-lifts}
\end{equation}
Then \(\Lambda_e-c_e=l\) is independent of \(e\). The determinant
has the required odd restriction at every positive \(Y_0\) seam and
is trivializable at all rational primary seams. Moreover
\begin{equation}
c_0S_i=0,\qquad \Lambda_0S_i=2,\qquad
S_iS_j=0\quad(i\ne j).
\label{eq:stack:sphere-pairings}
\end{equation}
\end{lemma}
\begin{proof}
In the basis \(a,b\) of a negative cell, \(\Lambda_0\) has
evaluations \((-1,-1)\), so its difference from \(c_0=0\) is
characteristic. On a positive cell this difference has evaluations
\[
(-4,-2;-1,1,-1,-1).
\]
The first two are even, as the diagonal entries of the rank-two form
are even, and the other four are odd. This is exactly the
characteristic congruence on the integral basis in
Proposition~\ref{prop:stack:lattice}. Direct evaluation gives, for
either type of cell,
\[
(c_0F_r,c_0F'_l)=(0,0),\qquad
(\Lambda_0F_r,\Lambda_0F'_l)=(-2,0).
\]
A square-\(-2\) integral class in an integral lattice is primitive:
if it were \(dF\), \(d>1\), its square would be divisible by
\(d^2\). In a unimodular lattice it consequently has a dual class
of pairing one. Starting with any characteristic class on an outside
piece, add an even multiple of that dual to give the prescribed even
evaluation on the bordering trace. The extra exceptional summand is
orthogonal and has determinant evaluation zero; its characteristic
evaluation can be any odd integer. Gluing over \(S^3\)'s identifies
these integral classes, since the first and second cohomology of the
seams vanish. This proves existence of the closed classes and the
characteristic assertion.
On a positive cell \(c_0F_0=1\), and
\(F_0F_r=F_0F'_l=0\). Thus every adjacent bit change preserves the
odd evaluation at the middle seam. At a \(\pm2\) rational seam the
restriction of a line class is its trace evaluation modulo two.
Those evaluations are even, including after a bit change. Hence the
determinant is trivializable there. The remaining rational seams have
first homology zero. Finally the two halves of each \(S_i\) have the
base evaluations just computed, including at the outside pieces.
The spheres lie in distinct intervals, which proves
\eqref{eq:stack:sphere-pairings}.
\end{proof}
Write
\(\kappa=c_2(E)-c_e^2/4\) and
\(\Theta=(\Lambda_e^2-\sigma(X))/4\). To keep \(\kappa\) fixed
when one bit changes from zero to one, the second Chern number must
\emph{decrease} by one. Indeed
\begin{equation}
c_e^2=c_0^2-4\sum_i e_i,\qquad
\Lambda_e^2=\Lambda_0^2.
\label{eq:stack:global-bit-squares}
\end{equation}
There is a useful more local version of the second equality.
\begin{lemma}[The cell telescope]
\label{lem:stack:telescope}
The contribution of the cell between \(J_j\) and \(J_{j+1}\) to
\(\Theta\) is
\begin{equation}
m_{\rm cell}+\frac{e_j-e_{j+1}}2,
\qquad
m_{\rm cell}=\begin{cases}0&\text{negative cell},\\1&\text{positive cell}.
\end{cases}
\label{eq:stack:cell-theta}
\end{equation}
Consequently there are constants \(b_0,\Theta_0\), depending only
on the outside choices, such that
\begin{equation}
b^+(X)=m+b_0,\qquad \Theta=m+\Theta_0.
\label{eq:stack:total-topology}
\end{equation}
For an interior filled component consisting of \(L\) cells between
actual \(J\)-cuts, with \(m_I\) positive cells and endpoint bits
\(e_a,e_b\),
\begin{equation}
b_I^+=m_I,\qquad \Theta_I=m_I+\frac{e_a-e_b}{2},\qquad p_I=L-1.
\label{eq:stack:interior-topology}
\end{equation}
Here \(p_I\) counts interval parameters with separated lens parameters
virtually restored. Endpoint differences telescope when contiguous
components are joined.
\end{lemma}
\begin{proof}
The restriction of a bit change to a cell is
\(-e_j\PD(F_r)+e_{j+1}\PD(F'_l)\). Orthogonality of the two
trace classes and their base evaluations give
\[
\Lambda_e^2-\Lambda_0^2
=4e_j-2e_j^2-2e_{j+1}^2=2(e_j-e_{j+1}).
\]
The base square on a negative cell is \(-2\), equal to its
signature. On a positive cell both base squares in
\eqref{eq:stack:base-lifts} are zero by
\eqref{eq:stack:positive-dual-square}, whereas its signature is
\(-4\). This proves \eqref{eq:stack:cell-theta}.
Summing the internal cells leaves \((e_1-e_n)/2\).
The first outside trace contributes \(-e_1/2\), and the last
contributes \(+e_n/2\). All bit dependence cancels. Positive
subspaces add across spherical cuts, proving
\eqref{eq:stack:total-topology}. The same sum on an interior segment
proves \eqref{eq:stack:interior-topology}; there is one retained
interval between each consecutive pair of its \(L\) cells.
\end{proof}
\begin{corollary}[Spacing]
\label{cor:stack:interior}
An interior segment with \(m_I\ge1\) positive cells has
\(L\ge4m_I-3\). More precisely, if \(a,b\) count negative cells
before its first and after its last positive cell, then
\(L\ge4m_I-3+a+b\). Equality holds when it contains no enlarged
gap between positive cells.
\end{corollary}
\begin{proof}
The positive cell positions are at least four apart. The distance
from the first to the last is therefore at least \(4(m_I-1)\).
Include those endpoint cells and the \(a+b\) additional cells.
\end{proof}
The dimension of the ordinary closed family with its \(n\) sphere
tests and cap probes is \(D_I+n-2n-z\). Thus the zero-dimensional
condition is
\begin{equation}
D_I=8\kappa-3(1+b^+)=n+z.
\label{eq:stack:ordinary-dimension}
\end{equation}
The coupled Dirac index is \(n_D=\Theta-\kappa\), as computed in
Lemma~\ref{lem:indices:closed}. Consequently
\begin{equation}
n_D=\Theta-\kappa
=\frac{5m-n}{8}+\Theta_0-\frac{z+3+3b_0}{8}.
\label{eq:stack:dirac-growth}
\end{equation}
These are topological index formulas; none presupposes the existence
of a solution. For the four-interval repeated pattern with fixed pads,
\(n=4m+d\) for a fixed integer \(d\), so the last index grows as
\(m/8\). We still have to identify the actual ordinary count with
the product and choose repetitions that preserve the cap pairing and
the bundle-degree congruence.
\subsection{Tests and the relative macro map}
Use a product of the negative interval parameters. The genuine faces
are the primary \(J\)-faces and the \(-4\)-lens faces of those
intervals. All other seams, including the outer \(Y\)-seams of
macros and single intervals, have long \emph{finite} collars.
The metric construction of Theorem~\ref{thm:geometry:family} keeps
these genuine faces and provides the local metrics later needed for
the Abelian estimate. In the present ordinary argument we use only
the topology, the face-product structure, and the small-perturbation
properties stated next; no Abelian spinor inequality is used.
\begin{lemma}[Compatible sphere representatives]
\label{lem:stack:tests}
The sphere tests can be chosen with the following properties, preserving
the integral map \(B\) on homology and its mod-two unit property.
For a commuting connection whose ordered line difference is \(v\),
a whole test avoids \(-1\) if \(vS_i=0\). Toward the primary
face it becomes two consecutive sweeps representing
\(F_{l,i}\) and \(-F_{r,i}\); each sweep then avoids \(-1\)
when its half evaluation is zero. During the change of prescription,
avoidance is retained if both half evaluations vanish. At an actual
\(J_i\)-split each surviving test is assigned compactly to exactly
one side. At a lens face it is supported on the lens cap and its data
at the two determinant lifts are tensor-paired as in
Proposition~\ref{prop:negative:local-sign}.
\end{lemma}
\begin{proof}
Apply the smooth conjugation-equivariant operation of
Lemma~\ref{lem:analysis:zero-winding} to the normalized holonomy path
of each sphere sweep. At a commuting connection its winding is half
the line-difference evaluation. The operation fixes nonzero-winding
commuting paths and moves every zero-winding commuting path into the
closed positive-real hemisphere, giving a uniform gap from \(-1\).
It is defined on path space before evaluating on connections, so one
smooth prescription applies across all commuting configurations.
At a primary split the simply connected traces supply sphere maps for
\(F_{l,i}\) and \(-F_{r,i}\), and their pinch represents \(S_i\).
Lemma~\ref{lem:analysis:segmentation} first homotopes the whole sweep
to this concatenation and then interpolates to the separate modifications
of the two half sweeps. When both half evaluations vanish, the whole
and half logarithms agree on their respective segments, so the uniform
avoidance gap persists throughout this interpolation. At the actual
face each event is defined in a frame intrinsic to its half; a zero
half evaluation therefore gives avoidance on that side. These changes
take place within the interval, before its true length collars and away
from the internal \(Y\)-necks used below. The metric construction
stops using the whole prescription on a side before that side begins
to switch.
Use product data near primary faces and the original tensor-paired
data near lens faces. The two representative lemmas belong to the
independent ordinary library in
Proposition~\ref{prop:analysis:ordinary-library}; they do not use the
nonzero count proved below. Generic equivariant cut variations smaller
than half the uniform gap retain the avoidances and give regular cuts
on the remaining irreducible locus.
To verify invariance of \(B\), use the homotopy to the path operation
provided by Lemma~\ref{lem:analysis:zero-winding}. It is relative to
the mark endpoints and fixes the constant path and every nonzero-winding
commuting path, including the minimum lens-cap paths of
Lemma~\ref{lem:negative:cap}. Thus flat avoidance and tensor
compatibility at the lens are retained. The operation uses only the
original sweep; its exact-support and ideal-persistence properties are
those of Lemma~\ref{lem:analysis:incidence}, and equation data remain
independent of the incidence mark. This initial homotopy need not avoid
\(-1\) at every intermediate winding-zero path, nor be a small change
of sweep: Proposition~\ref{prop:negative:invariance} imposes neither
condition. The later whole-to-halves interpolation has the stronger
both-zero-half avoidance just proved. At a primary face the
unconstrained index exclusion applies even without the test, and at a
lens face the two lift contributions cancel by
Proposition~\ref{prop:negative:local-sign}. The invariance proposition
therefore preserves \(B\) on homology throughout these changes,
including for standalone intervals.
\end{proof}
In estimates involving ideal limits, failure of an avoidance can occur
only when a particle meets the exact support of that test. Distinct
tests have disjoint supports in their intervals. One lost unit of
charge cannot be used to explain two such failures. The ordinary
transversality and support-incidence estimates are those of
Lemma~\ref{lem:ordinary:perturbations}.
\begin{lemma}[Local charge costs]
\label{lem:stack:charge-costs}
For the main pieces of a single interval or macro, virtually restore
all separated lens caps with the split references
of differences \(0,\pm4,\pm2\), respectively at the extendibly
central, other central, and trace-zero states. Include their charges,
particles, and other positive losses in the piece charge. For a
reducible main piece with \(s\) assigned tests and no positive cell,
\begin{equation}
8\kappa_{\rm piece}\ge4s.
\label{eq:stack:negative-test-cost}
\end{equation}
For a reducible main piece containing the positive cell of a macro,
\begin{equation}
\kappa_{\rm piece}\in\tfrac12\Z,\qquad
\kappa_{\rm piece}\ge\tfrac12.
\label{eq:stack:positive-charge}
\end{equation}
The statements hold for sufficiently small perturbations, throughout
the finite metric templates and the long-neck comparisons used below.
\end{lemma}
\begin{proof}
For an ordered splitting the line difference satisfies \(v=c_e\)
modulo two. All negative trace evaluations are therefore even.
After virtual filling its charge has the form
\[
\kappa_{\rm piece}=-v^2/4+\ell,\qquad
\ell\in\Z_{\ge0}.
\]
The stated reference fillings have charges \(0,1,1/4\); their
boundary eigenlines extend the given splitting. Excess charge in a
cap or its trajectories, and charge of particles, contributes to
\(\ell\). On a negative piece a nonzero even half evaluation
contributes at least two to \(-v^2\), by
\eqref{eq:stack:negative-dual-square}. Every assigned test either
charges such a half or, by Lemma~\ref{lem:stack:tests}, charges a
unit of \(\ell\). In a whole test a nonzero difference entails a
nonzero half; during switching the same assertion follows unless
both halves vanish; at a primary face only its assigned half is used.
For a separated lens, vanishing of both halves would give the flat
reference, which misses the test. Different tests use different
halves or different units of loss. Thus each costs at least four in
\(8\kappa=-2v^2+8\ell\), proving
\eqref{eq:stack:negative-test-cost}.
On the positive cell, \(vG\) is even and precisely two of the four
\(vT_b\)'s are odd; bit changes preserve these two properties. Equation~
\eqref{eq:stack:positive-dual-square} shows that \(v^2\) is even.
Added negative trace halves also have even square contribution, so
\(\kappa\in\frac12\Z\). The small-perturbation energy bound
first gives \(\kappa\ge0\) in this discrete set. Charge zero is
impossible. In fact, the sphere between the first two increasing
steps represents \(\pm(T_2-T_1)\), and the determinant evaluates
oddly on it. It lies in the first half of the positive path, away
from the internal \(Y_{-2},Y_0,Y_2\) collars and from the genuine
cut collars. For fixed artificial metric sizes its neighborhood has
compactly controlled geometry. Were charge zero possible for
perturbations tending to zero, the energy identity would force the
trace-free curvature to tend to zero on this neighborhood. A flat
projective limit over this sphere has trivial \(w_2\), contradicting
the odd determinant evaluation.
This argument remains uniform in the long-neck comparisons. On a
long Floer collar retain a gradient perturbation; the perturbed
Chern--Simons identity bounds its charge below by an error tending
to zero with the perturbation size, independently of collar length.
The endpoint perturbation terms telescope over successive levels.
On genuine separating collars use unperturbed deep regions, and
make other errors arbitrarily small in integrated square norm.
The compact odd sphere is untouched. Thus the same contradiction
excludes zero in true split limits and in the comparison to long
internal product collars. Discreteness now gives
\eqref{eq:stack:positive-charge}.
\end{proof}
\begin{proposition}[The relative macro]
\label{prop:stack:macro}
The two-parameter, two-test family on the chronological cobordism
\(B\,H\,B\), with the above data and irreducible outer limits,
defines an integral map on Floer homology at instanton index \(i=2\).
It is independent of the metric interpolation preserving the genuine
faces, and agrees, up to the common orientation convention, with
\(BHB\). In particular it induces a unit modulo two.
\end{proposition}
\begin{proof}
We give the low-dimensional exclusions needed for the chain and
homotopy identities. Let \(t=0\) for the family and \(t=1\) for
a homotopy. It suffices to consider \(i\le3-t\). If \(k\) of
the two primary \(J\)'s are cut, \(k\in\{0,1,2\}\), let \(p\)
and \(s\) on a main piece denote its remaining interval parameters
and its assigned tests, with lens parameters virtually restored.
Then
\[
\sum p=2-k,\qquad \sum s=2.
\]
Equation and cut data on a piece depend only on its retained
parameters. A regular irreducible piece costs at least \(2s-p\).
At most one additional parameter is subtracted from the irreducible
aggregate in a homotopy.
A separated lens loses a parameter and removes its degree-two test
from the main problem, but has framed index shift
\(\delta=8E\ge2\). Its net additional cost, measured against
\(2s-p\), is at least one. A unit particle loses eight indices;
its position and any released cut degrees recover at most four by
the exact-support incidence convention. These losses are therefore
strict. A purely negative reducible piece has index at least
\(4s-3\), by Lemma~\ref{lem:stack:charge-costs}. Matching at each
central contact adds three.
Consider first an unwanted centered contact or a maximal consecutive
reducible group not containing a positive-cell reducer. A group of
\(r\) negative reducers contributes at least \(4s_R-3r\), and
its \(r-1\) internal and two exterior central matchings contribute
\(3(r+1)\). Its aggregate cost is at least \(4s_R+3\).
A centered contact without reducers has the same extra cost three.
Adding regular irreducible costs, and allowing the one homotopy
parameter, gives the lower bound
\[
4s_R+3+2(2-s_R)-(2-k)-t
\ge5+k-t>3-t.
\]
Unused parameters on reducible pieces only increase this bound.
Now suppose the unique positive cell lies on a reducible piece.
Its index is at least \(4-6=-2\). A maximal reducible group
containing it, including all internal and both exterior central
matchings, costs at least four; negative members add their
nonnegative test costs. At \(k=0\), the whole main cobordism is
reducible and must be joined to each prescribed irreducible outer
limit by a genuine nonconstant irreducible trajectory. These two
trajectories give the bound six. At \(k=1\), let \(u\in\{0,1\}\) record whether the split
test is assigned to the positive piece. That piece reaches an outer
end, so a true-end trajectory is required. The other piece has no
retained parameter and has \(1-u\) tests. If it is irreducible,
the bound is \(4+1+2(1-u)-t\ge5-t\). If it is reducible,
both outer ends require trajectories and the bound is
\(4+4(1-u)+2\ge6\). At \(k=2\) there are no retained
interval parameters. Let \(u_i\) record assignment of each split
test to the positive piece. An irreducible outer piece contributes
at least \(2(1-u_i)\); a reducible outer piece, together with its
forced true-end trajectory, contributes at least \(4(1-u_i)+1\).
Including the positive group and subtracting the homotopy parameter
at most once gives \(4-t\). Each bound is strictly greater than
\(3-t\). These are all placements and assignment choices for
the positive reducer.
It follows that only regular irreducible pieces occur in the relevant
boundary strata. Their strict loss estimates exclude particles and
all lens possibilities except a single trace-minimum lens cap in the
boundary index. Its two lift contributions cancel by
Proposition~\ref{prop:negative:local-sign}. Ordinary Floer breaking
and regular gluing consequently give the integral chain identity and
the invariance on homology.
It remains to identify this operation with \(BHB\). Homotope the
local toric metrics to product
metrics through families with the same genuine cuts. The space of
metrics on the remaining compact cores permits the interpolation;
there is no need to retain a spinor clamp during this homotopy.
Keep the tests of Lemma~\ref{lem:stack:tests} and the paired lens
data. Finite partial stretches may still retain the positive
rank of the unsplit macro; they are covered by the preceding macro
homotopy bounds and Lemma~\ref{lem:stack:charge-costs}. For the
final comparison send all three internal lengths to infinity so
that their minimum tends to infinity. Every limit then contains
the three cuts \(Y_{-2},Y_0,Y_2\), and its main factors in path
order are
\[
B_L,\qquad H_-:\ -2\to-1\to0,\qquad
H_+:\ 0\to1\to2,\qquad B_R.
\]
Both \(B\) factors are negative definite. The two increasing
paths also have \(b^+=0\). Explicitly, preceding \(H_-\) by
the \(-2\)-trace gives \(C_0\#2\overline{\CP}^{2}\),
with basis \(A^2=0,T_1^2=T_2^2=-1\). Its incoming class is
\(A-T_1-T_2\); the orthogonal image of the handle path has
\(t_1+t_2=0\) and square \(-2t_1^2\). Preceding \(H_+\)
by the \(0\)-trace gives \(C_2\#2\overline{\CP}^{2}\),
with \(G^2=2,T_3^2=T_4^2=-1\) and incoming class
\(G-T_3-T_4\). A vector \(aG+t_3T_3+t_4T_4\) in its
orthogonal satisfies \(2a+t_3+t_4=0\) and has square \(-(t_3-t_4)^2/2\).
The relative Mayer--Vietoris sequence identifies these orthogonal
images with the handle-path homology; the null direction is the
\(Y_0\) surface direction. Thus neither path contributes a
positive subspace.
At a compatible genuine \(J\)-face, either \(B\) splits into
negative trace halves; at a lens face, virtual restoration of the
negative lens cap recovers the same nonpositive form. The two
increasing paths remain unchanged. A reducer cannot meet the
admissible \(Y_0\) seam, so all reducer groups lie in negative
interval pieces, trace halves, or rational cylinder levels. They
therefore have the same cost \(4s_R+3\), including their internal
and two exterior central matchings. Regular increasing-path
factors cost nonnegative index and carry no interval parameters or
tests. Consequently an unwanted centered contact again costs at
least \(5+k\), while a lens, particle, or nonconstant irreducible
trajectory adds strictly to the regular index bound. At \(i=2\)
only the ordinary rigid irreducible matches remain, with no true
interval faces or positive losses. The regular gluing theorem therefore identifies the map with the product
for sufficiently long finite internal lengths. The coherent
orientations are the compositions of the ordinary orientations, with
the same local flip comparisons. The mod-two unit assertion follows
from Theorems~\ref{thm:ordinary:units} and
\ref{thm:negative:unit}.
\end{proof}
\subsection{The closed gluing budget}
We next stretch only the external negative \(Y\)-seams separating
caps, macros, and single intervals. The relative macro proposition
fixed irreducible outer limits; a limit of this closed stretching may
instead have central flats on the seams. We therefore need a local
index bound allowing central outer limits, followed by a global bound
including their matching contributions.
\begin{lemma}[Local excess, including central outer limits]
\label{lem:stack:local-excess}
For a single interval or macro let \(b=1\) or \(2\), respectively,
and let \(i_{\rm loc}\) be its index at its actual outer limits,
with seam trajectories treated separately. A single interval has
\(i_{\rm loc}-1\ge0\). A macro with \(a\) central outer limits
has
\begin{equation}
i_{\rm loc}-2\ge-2a.
\label{eq:stack:macro-excess}
\end{equation}
If both outer limits are irreducible, equality for either kind of
factor requires an ordinary stable rigid solution: there are no
primary or lens faces, particles, or positive trajectory losses.
\end{lemma}
\begin{proof}
Suppose \(k\) primary cuts occur in the factor. Restore the lens
caps virtually, assigning any primary-neck charges to the pieces.
Index addition and the parameter and test totals give
\begin{equation}
E:=i_{\rm loc}-b
=\sum_{\rm pieces}(i+p-2s)+4k,
\qquad \sum p=b-k,\quad\sum s=b.
\label{eq:stack:excess-identity}
\end{equation}
Indeed primary matching adds \(3k\), while the lost parameters
supply the remaining \(k\). Call the summands in parentheses
\emph{scores}. Regular irreducibles have score at least zero,
strictly positive for the losses described in the preceding proof.
Negative reducers have score at least
\(-3+2s+p\). A reducer containing the positive cell has score
at least \(-2+p-2s\).
For a single interval with \(k=0\), an irreducible has nonnegative
score, and a reducer has \(s=p=1\), also giving score at least zero.
For \(k=1\), both parameters disappear and the two test assignments
are \(u,1-u\), with \(u\in\{0,1\}\). If both pieces are
irreducible, \(E\ge4\). If just one is reducible, assigned
\(s_R\) tests, then \(E\ge1+2s_R\ge1\). If both are
reducible, their scores plus four give \(E\ge0\); both outer
limits are then central. This proves all assertions for a single
interval.
For a macro first suppose the positive-cell piece is irreducible.
At \(k=0\) its score is nonnegative. At \(k=1\), the only
possible reducer is the negative outer piece; with \(u\) the
assignment of the split test to the positive piece, its score is
at least \(-1-2u\), so \(E\ge3-2u\ge1\). If it is
irreducible as well, the stronger bound is four. At \(k=2\),
let \(u_i\) indicate assignment of the \(i\)-th split test to
the positive piece and \(r_i\) indicate reducibility of the
corresponding outer piece. Then
\[
E\ge8-\sum_{i=1}^2r_i(1+2u_i)\ge2.
\]
Thus all primary faces are strict in this case.
It remains to consider a positive-cell reducer. At \(k=0\) the
whole macro is reducible, with \(p=s=2\), giving \(E\ge-4\)
and \(a=2\). At \(k=1\) its parameter count is one and its
test count is \(1+u\), where \(u\in\{0,1\}\) is the split
test assignment. The other piece has no parameter and has
\(1-u\) tests. Write \(r=1\) if that piece is reducible and
\(r=0\) otherwise. Equation~\eqref{eq:stack:excess-identity}
becomes
\[
E\ge1-2u-r(1+2u).
\]
For \((u,r)=(0,0),(1,0),(0,1),(1,1)\), the respective lower
bounds are \(1,-1,0,-4\). The positive reducer forces one central
outer limit, and the other reducer, if present, forces the second.
Hence \(a\ge1+r\), and each of these four bounds implies
\eqref{eq:stack:macro-excess}.
At \(k=2\) the positive reducer has no parameter and has
\(u_1+u_2\) tests. Each outer piece has no parameter and has
\(1-u_i\) tests. Thus
\[
E\ge6-2(u_1+u_2)-\sum_{i=1}^2r_i(1+2u_i).
\]
For \(r_1+r_2=0,1,2\), its minimum over the assignments is,
respectively, \(2,-1,-4\). Since \(a\ge r_1+r_2\), all
three minima imply \eqref{eq:stack:macro-excess}. Notice that
an irreducible outer piece can itself have a central outer limit;
only this inequality, not equality, is needed. When \(a=0\)
the bound is at least two.
Finally, with irreducible outer limits a nonsplit whole reducer is
impossible. The strict primary-face bounds just proved, and the
strict lens and particle costs on remaining irreducibles, show that
zero excess then occurs only in the ordinary stable rigid count.
\end{proof}
\begin{proposition}[Closed composition in the final metrics]
\label{prop:stack:composition}
Choose the old cap metrics with generic periods and take sufficiently
long finite isolation collars for their separate extra exceptional
summands. For any fixed finite stack, the sphere tests and sufficiently
small ordinary data can be chosen so that, for all sufficiently long
finite external negative \(Y\)-necks, the oriented closed count
\(\Omega\), summed over the determinant bits, equals the ordinary
composition of its cap states, single intervals, and macro maps, up
to one overall sign. The same conclusion holds after sufficiently
small further generic changes preserving the genuine-face
prescriptions. These data can also satisfy
Theorem~\ref{thm:estimates:clamp}.
\end{proposition}
\begin{proof}
The fixed indices and total charge, together with uniformly bounded
end corrections, give bounded cap charge ranges for this finite stack.
Include also the cap-cycle, comparison-homotopy, and old closed-pairing
ranges. With the old-cap periods fixed, Proposition~\ref{prop:caps:periods},
in the uniform isolation form of Proposition~\ref{prop:estimates:cap},
permits exceptional isolation depths and perturbation tolerances to be
chosen for all these ranges at once. It excludes cap reducers also in broken
relative limits as the external \(Y\)-necks stretch. The charge
bounds used there remain uniform in those lengths: the gradient
Chern--Simons terms telescope along any chain of end levels, and the
resulting cap and finite-prefix bounds suffice for the period test.
They make no assertion of finite unperturbed curvature norm on an
infinite tail approaching a perturbed nonflat.
Each regular cap irreducible now has nonnegative excess over its fixed
probe degree. The last assertion of Theorem~\ref{thm:caps:pairing}
allows the cap pairing to be recomputed with these final data by
stretching the old closure's rational seam. Its closed invariant is
unchanged, since \(b^+>1\) and the probes represent the same classes.
Thus the pairing stays nonzero without requiring separate metric
independence of the two cap states.
Now consider a limit as all external seams become long. A seam
contributes nonnegative additional index from trajectories and
matching. If any central state occurs in it, its total contribution
is at least three: group contiguous reducible cylinder levels and
include their two exterior matching terms, exactly as in
Lemma~\ref{lem:ordinary:central-cost}. Write \(S\) for the number
of seams containing a central state, and \(A\) for the sum of the
numbers of central outer limits over the macros. Since macros are
separated by single intervals and separated from the caps by pads,
a seam is adjacent to at most one macro. Hence \(A\le S\).
Lemma~\ref{lem:stack:local-excess} and cap regularity give total
excess at least
\[
3S-2A\ge S.
\]
But the total excess is zero by
\eqref{eq:stack:ordinary-dimension}. Thus \(S=0\).
With all external limits irreducible, every factor has nonnegative
excess. Equality and the last assertion of
Lemma~\ref{lem:stack:local-excess} exclude all true faces and
positive losses. Nonconstant irreducible seam trajectories cost
at least one and are excluded as well. Precisely the rigid
irreducible factor matches remain.
Choose regular ordinary data independently on the pieces, with
product compatibility at all external seams and genuine cuts.
Ordinary compactness, the preceding exclusions, and regular Floer
gluing now identify the closed count with the product for
sufficiently long finite external lengths. This proves also that
no broken or ideal cut tuple occurs at a genuine face for these
choices. Otherwise a sequence at arbitrarily large external
lengths would have a limiting tuple excluded by the same budget.
At fixed suitable lengths the absence is open by compactness, and
the regular zero-dimensional count is unchanged by sufficiently
small further data changes.
For orientations, order the closed ASD determinant line in
cobordism order, pairing an intermediate end line with its dual.
The parameter, mark, and test lines are then reordered into the
factor order. Their dimensions and the indices of the rigid
factors are fixed, so the reordering sign is independent of the
matched solutions. Equivalently, the fixed cap degree determines
the first Floer parity, and the fixed degrees of the successive
maps determine every subsequent parity. This gives a single
comparison sign for the whole product. The local two-lift
comparison commutes with this operation: excise its framed lens
cap before or after the other gluings, using the same transition
and trace reference. The compared cap indices are equal and even.
Thus all bit comparisons are precisely the ones used to define the
maps in Section~\ref{sec:negative}.
It remains to explain compatibility with the later metric choices.
The metric existence assertion in
Theorem~\ref{thm:geometry:family} and the finite-choice assertion in
Corollary~\ref{cor:estimates:choices} apply to the fixed topology,
lifts, and index bounds, independently of the value of \(\Omega\).
They supply finite artificial isolation sizes and a positive
allowable size for fixed gradient perturbations on the external
\(Y\)-collars before those collars are sent into their gluing
range. The ordinary argument above works for arbitrarily small
such gradients. Choose them below that tolerance, then take the
external lengths sufficiently large and finite. Any additional
non-gradient errors are chosen with arbitrarily small total squared
norm for those already chosen lengths. The final open generic
adjustments preserve the count. Thus no error estimate is required
to hold for a non-gradient perturbation accumulated over an
uncontrolled \(Y\)-length.
\end{proof}
Changing the fixed determinant connection, or the background connection
used in the coupled Dirac equation, does not change this ordinary
calculation. In the ordinary projective equations its effect is
absorbed by the scalar half-difference of the two determinant
connections. The normalized holonomy tests use the same determinant
lifts. Thus the independent background choices needed later do not
alter \(\Omega\).
\subsection{Repetition on the integral free quotient}
The cap pairing is rationally nonzero, whereas the maps are known to
be units modulo two. The passage between these assertions uses the
free quotient of integral Floer homology, rather than discarding
torsion inside a mod-two homology group.
\begin{lemma}[The finite-lattice argument]
\label{lem:stack:finite-lattice}
Let \(C,C'\) be finite complexes of finitely generated free abelian
groups, and let \(f:C\to C'\) be a chain map whose reduction modulo
two is a quasi-isomorphism. On the free quotients
\[
L=H(C;\Z)/\operatorname{Tor},\qquad
L'=H(C';\Z)/\operatorname{Tor},
\]
its induced map is an isomorphism after localization at two. In
particular \(L/2^NL\to L'/2^NL'\) is an isomorphism for every
\(N\ge1\).
If \(L=L'\) has rank \(r\), all these maps modulo \(2^N\)
have orders dividing a single integer depending only on \(r,N\).
\end{lemma}
\begin{proof}
The mapping cone of \(f\) is a finite free complex whose mod-two
homology is zero. Apply the universal coefficient sequence over
\(\Z_{(2)}\). In each degree the finitely generated localized
cone homology has zero quotient by its maximal ideal \((2)\),
so it vanishes by Nakayama's lemma. Thus \(f\) induces an
isomorphism on integral homology after localization. An isomorphism
preserves torsion, hence also induces an isomorphism on the free
quotients after localization. In integral bases its square matrix
therefore has odd determinant. The adjugate formula makes it
invertible modulo every \(2^N\). The last assertion follows by
taking the exponent of the finite group
\(\GL_r(\Z/2^N\Z)\). It does not require integral homology to
be torsion-free.
\end{proof}
\begin{theorem}[A nonzero family with positive coupled index]
\label{thm:stack:nonzero}
Fix the old caps and their compact probes. Prescribe any lower bounds
on the two outside pad lengths, and prescribe any odd evaluations of
\(\Lambda\) on their separate extra exceptional classes. There
exist arbitrarily large finite stacks with positive bridges at
spacing four, satisfying those pad bounds, for which the following
hold simultaneously:
\begin{enumerate}
\item the lifts are those of Lemma~\ref{lem:stack:lifts}, with a
common allowed charge \(\kappa\) and
\(D_I=n+z\);
\item the actual oriented ordinary family count satisfies
\(\Omega\ne0\);
\item the coupled Dirac index \(n_D=\Theta-\kappa\) is positive;
\item all auxiliary collars are finite, and the metric and small-data
choices can be made to satisfy Theorem~\ref{thm:geometry:family} and
Theorem~\ref{thm:estimates:clamp}.
\end{enumerate}
The integers specifying the stack and all topological lift data are
chosen before the artificial isolation sizes, perturbation bounds,
and external Floer gluing lengths.
\end{theorem}
\begin{proof}
Clear the fixed cap denominators once. Thus choose integral cap
cycles and an integral cap functional whose pairing is a nonzero
integer \(q\); the corresponding exact surface insertions are the
fixed integral multiples of the rational probes used in
Theorem~\ref{thm:caps:pairing}. The functional kills torsion, so
it factors through the free integral Floer lattice \(L\) of
\(Y_2\). Choose \(N\) such that \(q\not\equiv0\pmod{2^N}\).
The rational pairing shows that \(r=\rank L\) is positive.
Let \(\mathcal A=\phi_*B\) be a negative step returned to the \(Y_2\)
convention, and let \(\mathcal D=HB\) be a positive-return step.
By Theorems~\ref{thm:ordinary:units} and
\ref{thm:negative:unit}, and Lemma~\ref{lem:stack:finite-lattice},
both operators are automorphisms of \(L/2^NL\). The chronological
four-interval pattern is one positive-return step followed by three
negative steps, so its operator is \(\mathcal A^3\mathcal D\). Geometrically its
first two intervals and intervening positive bridge form the macro;
its last two intervals are single factors. Proposition~
\ref{prop:stack:macro} proves that its directly counted clamped
macro has the same integral homology operation, with the already
fixed common sign convention. For the finite-order argument it
would suffice that this direct operation is itself a mod-two unit.
Choose a common exponent \(E\) of
\(\GL_r(\Z/2^N\Z)\), enlarged to be divisible by eight.
Choose initial and terminal pad lengths \(p_-,p_+\) that are
multiples of \(E\) and meet the prescribed lower bounds. Let
\(q_0\) be any sufficiently large positive multiple of \(E\)
and repeat the four-interval pattern \(q_0\) times. Then
\[
m=q_0,\qquad n=p_-+4q_0+p_+,
\]
and each pad and the whole repeated middle operation are the identity
on \(L/2^NL\). Here \(q_0\), the repetition number, is distinct
from the fixed nonzero pairing \(q\).
Use literally periodic local block, orientation, and determinant
transition data within each set of repetitions. Thus each operation
really is a power of one automorphism. The data may depend on this
finite stack: the common exponent depends only on \(r,N\), so the
argument is unchanged if the automorphism changes when smaller
perturbations are required. Continuation identifies the free
lattices, and the integral cap pairing remains \(q\), up to a
fixed sign, by the closed comparison in the proof of
Proposition~\ref{prop:stack:composition}. A common sign of a map
causes no difficulty, since it too is an element of the same finite
group.
We check the degree congruence before making any metric choices.
The old nonzero cap pairing has an allowed bundle degree. Rational
Mayer--Vietoris at its seam identifies its positive rank with
\(b^+(C_-)+b^+(C_+)=b_0\) and its determinant square with the sum
of the two cap squares. Adjoining the outside trace halves adds only
negative directions with zero determinant evaluation. Internal
negative cells also have \(c_0=0\), and a positive cell has
\(c_0^2=0\). Thus the base determinant square of the closed stack
equals that for the old pairing, while its positive rank increases
by \(m\). Equation~
\eqref{eq:stack:ordinary-dimension} therefore changes the required
quantity \(8\kappa\) by \(n+3m\). Our choices make both
\(n\) and \(m\) multiples of eight, so this change is divisible
by eight. The required \(c_2=\kappa+c_0^2/4\) is integral.
Every bit sector is also integral, by
\eqref{eq:stack:global-bit-squares}. This is the residue restriction
on the allowed degrees; it has not been inferred from a formal
nonzero count.
The pad lengths and exceptional evaluations are now fixed.
Equation~\eqref{eq:stack:dirac-growth} becomes
\[
n_D=\frac{q_0}{8}
+\Theta_0-\frac{p_-+p_++z+3+3b_0}{8}.
\]
It is positive for all sufficiently large allowed \(q_0\).
Increasing the repetition number by \(E\) preserves every
congruence and increases \(n_D\) by \(E/8\).
For each such fixed finite problem choose the metric isolations and
small data as in Proposition~\ref{prop:stack:composition}, preserving
periodicity during the factor computation. That proposition
identifies the actual count with the product. Modulo \(2^N\)
this product pairs the final cap states exactly as the identity
operation, and hence
\[
\Omega\equiv\pm q\not\equiv0\pmod{2^N}.
\]
It follows that \(\Omega\ne0\). Subsequent sufficiently small
generic data for the one-component closed problem need no
periodicity and preserve this integer. The same proposition gives
the compatibility with the metric tests and completes all assertions.
\end{proof}
With the stack integers and lifts fixed, the displayed indices bound
all charge ranges that will occur, including the auxiliary relative
cap calculations. Table~\ref{tab:stack:choices} records the order of
the finite choices; its length and perturbation quantifiers are those
of Corollary~\ref{cor:estimates:choices}.
One deformation used in comparing the old cap closure with unperturbed
closed data deserves separate mention. Turn off a gradient perturbation
while its seam remains long, preserving its gradient form throughout.
This retains the uniform charge bound used in the period exclusion.
\begin{table}[htbp]
\centering
\small
\begin{tabular}{@{}p{.04\textwidth}p{.32\textwidth}p{.56\textwidth}@{}}
\toprule
& Choice & Dependence and use \\
\midrule
1 & Old caps, probes, and periods & The constants $C_W$ depend only on
the old cap data (Proposition~\ref{prop:estimates:cap}). \\
2 & Outside delays and odd exceptional evaluations & Chosen using the
old-cap bounds, before middle repetitions
(Lemma~\ref{lem:indices:end-exclusion}). \\
3 & Pads and repetition integers & Congruences, nonvanishing, and
$n_D>0$ determine a fixed finite problem
(Theorem~\ref{thm:stack:nonzero}). \\
4 & Tube scale $\epsilon$ & Chosen for this fixed finite problem. \\
5 & Regularizer $d$, finite rational isolations, and a positive gradient
tolerance & Isolation thresholds may depend on the whole finite
problem; the gradient tolerance precedes the external seam lengths. \\
6 & Periodic ordinary factor data, then finite $Y$ lengths & Data lie
below the chosen tolerances. The $Y$ lengths may then be any sufficiently
large finite gluing lengths (Proposition~\ref{prop:stack:composition}). \\
7 & Further nongradient errors & Their allowed total squared cost may
depend on the selected finite $Y$ lengths. \\
8 & Small closed generic changes & These preserve the computed integer
$\Omega$ and need not preserve periodicity. \\
\bottomrule
\end{tabular}
\caption{Order of the finite choices. Only the old-cap constants are
independent of later exterior choices. The genuine $J$ and lens lengths,
distinct from the finite lengths listed here, may subsequently diverge
at parameter faces.}
\label{tab:stack:choices}
\end{table}
\subsection{The coupled spinor data}
The next sections couple the fixed spin-\(c\) determinant
\(l=\Lambda-c\) to the rank-two bundle of determinant \(c\).
Let $W^\pm$ be the rank-two spinor bundles of this spin-$c$ structure,
and let $E$ be the rank-two Hermitian bundle with $c_1(E)=c$.
A configuration consists of a unitary connection $a$ on $E$, with
its determinant connection fixed, and a coupled spinor
$\Phi\in\Gamma(W^+\otimes E)$. The coupled Dirac operator is
$D_a:\Gamma(W^+\otimes E)\longrightarrow\Gamma(W^-\otimes E)$.
In the equations of Section~\ref{sec:indices}, $P_D$ is a section of
$W^-\otimes E$, while $P_+$ has the same target as
$\rho(F_a^{0,+})$. These denote the Dirac and curvature perturbations.
The phase circle multiplies $\Phi$; on the determinant-one gauge
quotient its ineffective subgroup at a free orbit is $\{1,-1\}$.
Definition~\ref{def:indices:types} gives the full field-type
classification. Table~\ref{tab:gluing:orientations} records how the
ordinary determinant, framing, and sign conventions continue into
the final spinor count.
At a phase-fixed configuration the spinor is either zero, or lies
in one line of a parallel splitting. If \(v\) is the ordered
summand difference, with the active line first in the latter case,
its spin-\(c\) line class is \(K=\Lambda+v\). For virtual
\(-4\)-cap fillings use the split references above, choosing the
order at the nonextendible center so that \(|K S_i|\le4\);
this is possible because \(|\Lambda S_i|=2\), and is automatic
at the trace-zero state. These conventions fix the interface with
Section~\ref{sec:indices}. The symbol \(P\) for a positive bridge
cell in this section is distinct from the later component type
\(P\), which denotes a field not fixed by phase.
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/05-stack.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/06-geometry.tex ====
\section{A simultaneous family of metrics}
\label{sec:geometry}
The positive bridges require a range of self-dual periods. We construct
that range inside the interval cube, while keeping precisely the genuine
cuts prescribed in Section~\ref{sec:stack}. Two distinctions matter in this
construction. First, a vanishing coefficient in a period is an auxiliary
plumbing degeneration; it is not the middle three-sphere cut of an
interval. Second, the narrow-period region is controlled by a long tube
with spherical fibers. We retain that tube while replacing the toric
metric near the corner where both coefficients vanish.
Fix a finite stack, its determinant and spin data, and its finitely many
bit choices. The number of intervals, all exceptional evaluations, and
the topological charge ranges are fixed in every limiting assertion below.
Constants for compact pieces may depend on this finite problem. The
constant attached only to an old cap will be identified separately in
Section~\ref{sec:estimates}. A \emph{genuine length} is a length tending to
infinity at a face of the interval cube. The only such cuts are the
middle spheres $J_i\cong S^3$ and the boundaries of the $(-4)$ disk bundles
$N_i$. All other lengths used here, including the external surgery seams
$Y_j$, will ultimately be chosen finite. The $Y$ collars varied below
are the external rational seams joining the ordinary factors: macros,
single intervals, and caps.
In a macro containing positive bridge $j$, orient the adjacent spheres as
\begin{equation}
X_1=-S_j,\qquad X_2=S_{j+1}.
\label{eq:geometry:oriented-spheres}
\end{equation}
Write $U$ for the sphere in Proposition~\ref{prop:stack:lattice}. Thus
$X_i^2=-4$, $U^2=0$, $U X_i=1$, and $X_1X_2=0$. The regular neighborhoods
needed below are
\begin{equation}
C_2=\nu(X_1\cup U\cup X_2),\qquad
C_{1,i}=\nu(U\cup X_{3-i}),\qquad C_0=\nu U,\qquad N_i=\nu X_i.
\label{eq:geometry:neighborhoods}
\end{equation}
Here neighborhoods are rounded, and their separating hypersurfaces are
specified up to isotopy. In particular $C_0$ is a neighborhood supporting
a tube, not an additional infinite cut.
\begin{theorem}[The metric family]
\label{thm:geometry:family}
For the fixed finite stack, there are families with the following
properties. They are defined for all sufficiently small $\epsilon>0$,
all sufficiently small regularizers $d>0$, and arbitrarily long finite
auxiliary rational collars. The auxiliary bounds may depend on
$\epsilon$ and the fixed problem. The family parameter is the original
cube of interval parameters, with no added boundary faces.
\begin{enumerate}[label=\textup{(\roman*)}]
\item At its genuine faces the family has independent product length
collars at $J_i$ and $\partial N_i$, compatible at intersections of
faces. The $J_i$ cross-sections are round. On a component separated by
$J_i$, the limiting data depend only on the interval parameters retained
on that component. The $N_i$ metric is complete with positive scalar
curvature and a positive-scalar-curvature product end. Its link and its
scale on the stretching collar are independent of the tangential cube
parameters. The link is joined to the round lens metric through
positive-scalar-curvature metrics. These metric choices are common to
the two bit terms compared at that lens face.
\item On the positive-width toric region of a macro constructed below,
as $d\to0$ the retained main piece has a complete
metric conformal to a toric K\"ahler metric with positive cohomology ray represented by
\begin{equation}
H=U+aX_1+b_0X_2,\qquad
0\le a,b_0\le\tfrac14,\qquad a+b_0>0.
\label{eq:geometry:period}
\end{equation}
It has nonnegative scalar curvature, with strict positivity on an open
set. A zero coefficient removes the corresponding outer part by a
$C_{1,i}$ separation; a coefficient $1/4$ separates $N_i$. The class
$H$ is supported rationally on the retained main piece in all these
cases. Its positive rank is one.
\item Every macro in the small-width replacement region contains an
$S^2\times S^1\times I$ tube, with fiber $U$, of length comparable to
$\epsilon^{-1}$. On its bulk the metric is, or approaches smoothly,
\begin{equation}
g_{\mathrm{long}}
=g_{S^2(1/\sqrt2)}+\tfrac12\,d\xi^2+2\,d\theta_x^2.
\label{eq:geometry:long-model}
\end{equation}
The same description holds for every toric macro whose normalized
length $L=1/(a+b_0)$ tends to infinity. All such tube lengths are $O(\epsilon^{-1})$.
The tubes lie in $C_0$ and are disjoint from both middle spheres $J_i$.
\item Along any sequence of these parameters, every additional
unbounded region is one of the following: an exact rational product
collar, an annulus approaching a positive-scalar-curvature product, or
one of the tubes in \textup{(iii)}. The remaining pieces, after the
specified satellite rescalings, have uniformly bounded geometry on
fixed truncations. There are only finitely many such pieces for the
fixed stack. Away from fixed-length tube ends, the error from
\eqref{eq:geometry:long-model}, with any fixed number of spatial
derivatives, tends to zero as the distance from both ends tends to
infinity. In particular the integrated errors used in
Section~\ref{sec:estimates} are $o(\epsilon^{-1})$, uniformly under
deeper regularization and longer auxiliary rational collars.
For scalar-curvature costs, the middles of the $Y_j$ collars
are omitted, retaining prefixes of length $O(\epsilon^{-1/2})$;
no positivity is required on a $Y_j$ cross-section.
\item For each fixed real cohomology class $\Lambda$ there are closed
comparison forms with its prescribed vertical flux on these tubes,
zero curvature on the long parts of rational collars, and
$o(\epsilon^{-1})$ energy away from the tubes. More precisely, the
uniform comparison and error assertions are those of
Lemma~\ref{lem:geometry:comparison}.
\end{enumerate}
The construction is smooth at finite parameter values and has the
ordinary product description on every genuine length collar. Only
spatial derivative estimates are asserted uniformly as lengths diverge;
no bound on derivatives in a compactifying length coordinate is needed.
\end{theorem}
The proof occupies the rest of this section. The theorem supplies the
geometry for the simultaneous Abelian tests, whose statement and proof
are Theorem~\ref{thm:estimates:clamp}. Harmonic background connections
and their exact product convention at genuine faces are constructed in
Section~\ref{sec:estimates}; this will also ensure that the complementary
data agree under the lens flip.
\subsection{Embedded plumbings and the period polygon}
\begin{lemma}[Placement of the interfaces]
\label{lem:geometry:plumbing}
The neighborhoods in \eqref{eq:geometry:neighborhoods} can be placed
strictly inside the macro so that $U$ is disjoint from $J_1\cup J_2$,
$C_{1,i}$ is disjoint from $J_i$, $N_{3-i}\subset C_{1,i}$, and
$C_0\subset C_{1,1}\cap C_{1,2}$. The simultaneous separating
interfaces required below can be chosen disjoint or nested. The two
interfaces $\partial C_{1,1}$ and $\partial C_{1,2}$ are never required
simultaneously, and neither is a $C_0$ separation required together
with an $N_i$ separation.
\end{lemma}
\begin{proof}
The four increases of framing in the positive path replace the
$(-2)$ trace by a $2$ trace blown up at four distinct points of its
core. This is the corner-blowdown description in
Proposition~\ref{prop:stack:lattice}. Its cocore disk can be chosen
away from those four points. On adjoining the reverse $2$ trace, the
two cocores join to a sphere. The two normal disk trivializations
agree along their common boundary with opposite boundary orientations,
so the normal Euler number of this double is zero. The double meets
each neighboring end core once. With the orientations
\eqref{eq:geometry:oriented-spheres}, these intersections are positive.
The cocore double lies between the two middle cuts, whereas $X_i$
passes through its own middle cut $J_i$. Consequently the double is
disjoint from both $J_i$.
At each intersection choose oriented complex disk coordinates in
which the two spheres are the coordinate axes. Plumbing their normal
disk bundles by exchanging base and fiber disks gives a neighborhood
of the embedded configuration; its Euler numbers are $-4,0,-4$.
Taking sufficiently small disk fibers gives $C_2$ inside the macro.
The subconfiguration $U\cup X_{3-i}$ misses $J_i$, so its smaller
regular neighborhood $C_{1,i}$ also misses $J_i$. Shrink $N_{3-i}$
inside it and choose a still smaller $C_0$ common to both subchain
neighborhoods. This proves the asserted geometric placements.
The restrictions on simultaneous interfaces follow from the parameter
regions used below: a low-side separation has the other coefficient
bounded away from zero in normalized units, and thus cannot coexist
with the other low-side separation. An opposite high-side $N_{3-i}$
separation is inside $C_{1,i}$. The region using only $C_0$ has both
coefficients small, so no high-side separation occurs there. We
therefore need only nested subchain neighborhoods and disjoint disk
bundle neighborhoods; small isotopies make their collars disjoint.
\end{proof}
For $0<a,b_0<1/4$, consider the polygon
\begin{equation}
\mathcal P(a,b_0)=
\{(x,y):4|y|\le x\le1,\ -a\le y\le b_0\}.
\label{eq:geometry:polygon}
\end{equation}
The angle lattice is $2\pi\Z^2$. In cyclic order the inward normals
are
\begin{equation}
n_p=(1,4),\quad n_1=(0,1),\quad n_o=(-1,0),\quad
n_2=(0,-1),\quad n_q=(1,-4).
\label{eq:geometry:normals}
\end{equation}
The vertex $(0,0)$ has determinant eight. We always remove its cone
point and attach the resulting outer lens boundary to the macro
exterior. Figure~\ref{fig:geometry:polygon} shows the compact divisors
that remain.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=3.2,>=Latex]
\fill[blue!4] (0,0)--(.48,-.12)--(1,-.12)--(1,.17)--(.68,.17)--cycle;
\draw[thick] (.08,-.02)--(.48,-.12)--(1,-.12)--(1,.17)--(.68,.17)--(.08,.02);
\draw[dashed] (.08,-.02) arc[start angle=-90,end angle=90,x radius=.018,y radius=.02];
\node[left] at (.04,0) {$\partial C_2$};
\node[below] at (.76,-.12) {$X_1$};
\node[right] at (1,.025) {$U$};
\node[above] at (.84,.17) {$X_2$};
\node[below left] at (.29,-.075) {$x+4y=0$};
\node[above left] at (.39,.10) {$x-4y=0$};
\draw[->,gray] (1.24,-.12)--(1.24,.17);
\node[right] at (1.24,.025) {$a+b_0$};
\begin{scope}[shift={(2.03,.025)},xscale=.36,yscale=.36]
\draw[thick] (-.75,0)--(.75,0);
\foreach \x/\lab/\wt in {-.75/X_1/-4,0/U/0,.75/X_2/-4}{
\fill[white] (\x,0) circle (.075);
\draw[thick] (\x,0) circle (.075);
\node[above] at (\x,.09) {$\lab$};
\node[below] at (\x,-.09) {$\wt$};
}
\end{scope}
\end{tikzpicture}
\caption{The punctured polygon and its compact divisor chain. The
three affine edge lengths are $1-4a$, $a+b_0$, and $1-4b_0$. Shrinking
a horizontal edge separates its $(-4)$ disk bundle. Sending $a$ or
$b_0$ to zero instead moves the outer cone point to a subchain
interface; the true middle-sphere cut is introduced outside that
interface.}
\label{fig:geometry:polygon}
\end{figure}
\begin{lemma}[The toric model and its periods]
\label{lem:geometry:toric}
The toric space associated to the punctured polygon is $C_2$. With
$w=a+b_0$ and normalized coordinates
\begin{equation}
\xi=x/w,\quad y'=y/w,\quad L=1/w,\quad
r_1=a/w,\quad r_2=b_0/w,
\qquad r_1+r_2=1,
\label{eq:geometry:normalization}
\end{equation}
it has a toric K\"ahler metric
\begin{equation}
\begin{split}
g&=\tfrac12 M_{ij}\,d\mu_i\,d\mu_j
+2(M^{-1})_{ij}\,d\theta_i\,d\theta_j,
\qquad (\mu_1,\mu_2)=(\xi,y'),\\
M&=C+\sum_{\nu}\frac{n_\nu n_\nu^t}{l_\nu},
\qquad C=\begin{pmatrix}1&0\\0&0\end{pmatrix},\\
(l_p,l_1,l_o,l_2,l_q)
&=(\xi+4y',r_1+y',L-\xi,r_2-y',\xi-4y').
\end{split}
\label{eq:geometry:hessian}
\end{equation}
Its positive K\"ahler ray is \eqref{eq:geometry:period}, and
\begin{equation}
HX_1=1-4a,\quad HU=a+b_0,\quad HX_2=1-4b_0,
\qquad H^2=2(a+b_0)-4(a^2+b_0^2)>0.
\label{eq:geometry:period-pairings}
\end{equation}
At $a=0$ or $b_0=0$, with $w>0$, the retained piece is the indicated
punctured subplumbing $C_{1,i}$. At a coefficient $1/4$, the retained
piece is punctured at the collapsed $X_i$ divisor. The affine term
for a collapsed facet is retained in the limiting potential. These
limits may occur together.
\end{lemma}
\begin{proof}
Every successive normal pair except $(n_q,n_p)$ has determinant one.
The corresponding torus corner chart is a smooth bidisk: the two
circle subgroups normal to its sides collapse on its coordinate axes.
Gluing these charts along an edge identifies the normal disk fibers
with the framing determined by the third normal. The relations
\[
n_p+n_o=4n_1,\qquad n_1+n_2=0,\qquad n_o+n_q=4n_2
\]
therefore give Euler numbers $-4,0,-4$, respectively. At the remaining
corner $|\det(n_q,n_p)|=8$, so its removed neighborhood is a cone on
a lens space of order eight. These charts construct exactly the
plumbing in Lemma~\ref{lem:geometry:plumbing}, without changing its
underlying smooth four-manifold.
The matrix $M/2$ is the Hessian of
\[
\tfrac14\xi^2+\tfrac12\sum_\nu l_\nu\log l_\nu.
\]
It is positive definite in the polygon interior since its normals
span $\R^2$. The coefficient $\tfrac12$ of each primitive boundary
term gives the smooth polar-coordinate extension along a simple
facet. At a determinant-one corner this holds in both disk
coordinates; at the removed corner it holds on the finite quotient
chart. This is the toric boundary convention of
\cite{Guillemin1994}, also verified directly by setting the normal
moment variable equal to half the square of a radial coordinate.
For the unnormalized polygon the affine lengths of the compact
facets are $1-4a$, $w$, and $1-4b_0$; symplectic areas are $2\pi$
times these lengths. The intersection matrix in the ordered basis
$(X_1,U,X_2)$ is
\begin{equation}
\begin{pmatrix}-4&1&0\\1&0&1\\0&1&-4\end{pmatrix}.
\label{eq:geometry:plumbing-matrix}
\end{equation}
It is nonsingular with determinant eight and signature $(1,2)$.
Solving for the class with these three periods gives
$U+aX_1+b_0X_2$. Normalizing the metric divides its K\"ahler form by
$w$ and hence does not change this ray. Direct multiplication gives
\eqref{eq:geometry:period-pairings}; positivity follows also from
$a^2+b_0^2\le(a+b_0)/4$.
If $a=0$, the $X_1$ coefficient is absent, so the class is carried by
the subchain $U,X_2$. The new vertex is removed, while the affine
term belonging to the collapsed outer edge remains as a redundant
term in the Hessian. The other low limit is identical with the
indices exchanged. If $a=1/4$, then $HX_1=0$. The rational
orthogonal decomposition at $\partial N_1$ consequently puts the
whole class on the complement of $N_1$. The same argument applies
to $X_2$. The subchain matrix has determinant $-1$; each isolated
disk bundle has determinant $-4$. Thus all these separating
boundaries are rational homology spheres, so the decompositions over
$\R$ are nondegenerate. Removing only negative directions, or
retaining the subchain of signature $(1,1)$, leaves positive rank one.
\end{proof}
\subsection{Scalar curvature of the Hessian metrics}
We give the curvature calculation because its quantitative strictness
is needed when the tips are changed conformally. It applies to both
the main polygon and the satellite metrics.
\begin{lemma}[Curvature and dependence bounds]
\label{lem:geometry:scalar}
Let $l_\nu$ be positive affine functions with gradients $n_\nu$
spanning $\R^2$, and let $C\ge0$ be constant. For the metric
\eqref{eq:geometry:hessian}, put
\[
B=M^{-1},\quad
A_\nu=B^{1/2}n_\nu/\sqrt{l_\nu},\quad
P_{\nu\mu}=A_\nu^tA_\mu,\quad
z_\nu=l_\nu^{-1/2},\quad q_\nu=z_\nu P_{\nu\nu}.
\]
Then $0\le P\le I$, $\Scal(g)=2\mathcal S$, and
\begin{align}
\mathcal S
&=2\sum_\nu z_\nu^2P_{\nu\nu}^2
-q^tPq-z^tP^{\circ3}z \notag\\
&=q^t(I-P)q+
z^t\bigl((I-P)\circ(P\circ P)\bigr)z\ge0.
\label{eq:geometry:scalar-identity}
\end{align}
Here $\circ$ is entrywise product. For every nonzero relation
$\sum_\nu\gamma_\nu n_\nu=0$, its two nonnegative terms satisfy
\begin{align}
q^t(I-P)q
&\ge \frac{(\sum_\nu\gamma_\nu P_{\nu\nu})^2}
{\sum_\nu\gamma_\nu^2l_\nu},
\label{eq:geometry:first-bound}\\
z^t\bigl((I-P)\circ(P\circ P)\bigr)z
&\ge\frac{\left\|\sum_\nu\gamma_\nu A_\nu A_\nu^t\right\|_{\mathrm{HS}}^2}
{\sum_\nu\gamma_\nu^2l_\nu}.
\label{eq:geometry:second-bound}
\end{align}
The main metrics have strictly positive scalar curvature on compact
sets away from punctures, including their smooth toric axes. This
strictness remains uniform on fixed compact truncations of their
$L\to\infty$ tips and of the three-direction satellites below.
\end{lemma}
\begin{proof}
Let $A$ be the matrix with columns $A_\nu$. Then
\[
AA^t=B^{1/2}(M-C)B^{1/2}
=I-B^{1/2}CB^{1/2}\le I.
\]
Hence $P=A^tA$ is a positive contraction. For completeness set
$G_{\nu\mu}=n_\nu^tBn_\mu$. Differentiating
$B M=I$ gives
\[
\partial_kB
=\sum_\nu \frac{n_{\nu k}}{l_\nu^2}
Bn_\nu n_\nu^tB.
\]
Differentiating once more, contracting the two derivative indices,
and collecting the derivative of $l_\nu^{-2}$ separately from the
two derivatives of $B$ yields
\[
-\partial_i\partial_jB_{ij}
=2\sum_\nu\frac{G_{\nu\nu}^2}{l_\nu^3}
-\sum_{\nu,\mu}
\frac{G_{\nu\nu}G_{\mu\mu}G_{\nu\mu}+G_{\nu\mu}^3}
{l_\nu^2l_\mu^2}.
\]
The real Riemannian scalar curvature of a toric metric is minus the
contracted second derivative of its inverse potential Hessian
\cite[Theorem~4.1(i), arXiv version~1]{Abreu1998}; it is twice the K\"ahler scalar
curvature denoted by $R$ in that reference.
That inverse Hessian is $2B$ here, proving
\eqref{eq:geometry:scalar-identity}, including its factor two. The
second equality uses
$\operatorname{diag}(P_{\nu\nu}^2)-P^{\circ3}
=(I-P)\circ(P\circ P)$. Both summands are nonnegative by the
Schur product theorem.
Let $k_\nu=\gamma_\nu\sqrt{l_\nu}$. The relation on the normals
gives $Ak=0$, hence $Pk=0$ and
\[
I-P\ge \frac{kk^t}{\|k\|^2}.
\]
Pairing with $q$ proves \eqref{eq:geometry:first-bound}.
Schur-multiplying the same inequality by $P\circ P$ and pairing
with $z$ proves \eqref{eq:geometry:second-bound}, because
\[
\sum_{\nu,\mu}\gamma_\nu\gamma_\mu
(A_\nu^tA_\mu)^2
=\left\|\sum_\nu\gamma_\nu A_\nu A_\nu^t\right\|_{\mathrm{HS}}^2.
\]
For the full polygon use the positive dependence
$\gamma=(1,1,2,1,1)$ in the order $(p,1,o,2,q)$.
Its first numerator is strictly positive in the interior. At a
simple facet its active column $A_\nu$ tends to a unit vector, so
$P_{\nu\nu}\to1$ and this numerator remains positive. At a smooth
vertex the two active columns tend to orthonormal vectors, with the
same conclusion. These assertions follow either by inverting $M$
in active-normal coordinates or by taking the Schur complement of
its singular normal term. They also prove a positive lower bound
on any compact smooth truncation.
For the tip with $\xi=O(1)$ and $L\to\infty$, discard the vanishing
$n_o$ term and use $n_1+n_2=0$. Its denominator is
$l_1+l_2=1$, and its first numerator is
$(P_{11}+P_{22})^2$. In the interior this is positive. On an
active horizontal edge one diagonal term tends to one. On an
active sloping edge, the projections of $n_1,n_2$ to the remaining
tangent direction are nonzero; inversion on that direction again
gives a positive limit. At a smooth vertex one horizontal edge is
active. Thus strictness extends to all smooth boundary points of
each fixed truncated tip.
After satellite rescaling the constant $C$ disappears and exactly
three distinct normal directions remain. Their unique relation
has three nonzero coefficients. The three rank-one symmetric
matrices $n_\nu n_\nu^t$ are linearly independent: in coordinates
where two directions are the axes, the third has a nonzero
off-diagonal entry. Conjugation by $B^{1/2}$ and multiplication by
positive $l_\nu^{-1}$ preserve independence. Thus the numerator
of \eqref{eq:geometry:second-bound} is nonzero in the interior.
At a simple edge its active unit column is orthogonal in the limit
to the other columns, so the corresponding nonzero coefficient
prevents cancellation. At a smooth vertex the two active columns
are orthonormal and again prevent cancellation. The same bound
is positive on each compact truncation, including the smooth axes.
\end{proof}
\begin{lemma}[Cone links]
\label{lem:geometry:cones}
At every puncture, the homogeneous Hessian formed from precisely the
vanishing linear terms is a metric cone. Its smooth link has scalar
curvature at least six. The link at a collapsed $X_i$ admits a path
through positive-scalar-curvature metrics to the round lens metric.
\end{lemma}
\begin{proof}
Write $E=\mu_i\partial_{\mu_i}$ for the moment Euler field at the
vertex. For homogeneous $l_\nu$ and $C=0$,
\[
g(E,\cdot)=\tfrac12 d\sum_\nu l_\nu,\qquad
R^2=2\sum_\nu l_\nu.
\]
The dilation of moment variables by $s^2$ multiplies $g$ by $s^2$,
and $2E=R\partial_R$ has squared length $R^2$. Thus
$g=dR^2+R^2h$. Its scalar curvature is
$R^{-2}(\Scal(h)-6)$, so Lemma~\ref{lem:geometry:scalar} gives
$\Scal(h)\ge6$. Each bounding side has primitive normal. The
torus action therefore has no finite isotropy along its collapsed
circle, and the link is smooth, even when the cone vertex itself
is a finite quotient singularity.
At $N_1$ the adjacent linear forms are $p$ and $v$ below, and the
redundant form is $(p+v)/4$. Multiply its Hessian contribution by
$s\in[0,1]$. For $s>0$ this is the contribution of the positive
linear form $s(p+v)/4$ itself, since
$\nabla(sl)\nabla(sl)^t/(sl)=s\nabla l\nabla l^t/l$.
The nonnegative scalar calculation applies for every $s$, including
the two-form limit $s=0$. Identifying the radius-one links by
radial dilation gives a smooth path with scalar curvature at least
six. At $s=0$ the two independent primitive-side terms give the
flat quotient of $\C^2$, whose link is the round lens space. The
other side is obtained by reflection.
\end{proof}
\subsection{Shrinking edges, satellites, and cylindrical completions}
In the normalized polygon the parameters measuring an edge shrink
are
\begin{equation}
r_i,\qquad h_i=L/4-r_i.
\label{eq:geometry:shrink-parameters}
\end{equation}
Only one $r_i$ can be small, because $r_1+r_2=1$. A small $h_i$
occurs near the corresponding right-hand corner and separates
$N_i$. A small $r_i$ occurs at a different corner and separates
$C_{1,i}$. These zones can be chosen disjoint. The sole nesting
is the original determinant-eight puncture inside an $r_i$
satellite. We retain this distinction in choosing the conformal
factors.
\begin{lemma}[Uniform collar and satellite models]
\label{lem:geometry:collars}
For every fixed derivative order $k$, the metric near an edge with
shrink parameter $\delta=r_i$ or $h_i$ has the following description.
There is a homogeneous radius $R$ and a cone metric $g_{\mathrm c}$
such that, in scale-invariant $C^k$ norms on every dyadic annulus,
\begin{equation}
\|g-g_{\mathrm c}\|_{C^k(g_{\mathrm c})}
\le C_k\bigl(\delta/R^2+R^2\bigr),
\qquad C\sqrt\delta\le R\le R_*.
\label{eq:geometry:cone-error}
\end{equation}
The estimates include smooth toric axes and are uniform on the
$L\to\infty$ tips. In affine moment coordinates $\mu_{\mathrm{loc}}$
centered at the relevant homogeneous cone vertex, the scale
$\mu_{\mathrm{loc}}=\delta\widehat\mu_{\mathrm{loc}}$ gives
$g=\delta(g_{\mathrm{sat}}+o(1))$ on compact sets away from any
remaining cone point. The metric $g_{\mathrm{sat}}$ is a
three-direction Hessian metric with strictly positive scalar
curvature on those compact sets.
There are conformal changes making all the necessary annuli
asymptotically cylindrical, with fixed positive limiting link
sizes. They have nonnegative scalar curvature and preserve
strict positivity on smooth compact regions. Arbitrarily deep
in those annuli one may interpolate to the exact limiting product
over a bounded logarithmic width, with $C^k$ errors tending to
zero. In particular the separated $N_i$ cap can be made positive
everywhere, including its exact product end.
\end{lemma}
\begin{proof}
The two local calculations are explicit. At a small $h_1$, set
\[
p=\xi+4y',\qquad v=L-\xi.
\]
The three relevant forms are $p,v,(p+v)/4-h_1$. Their homogeneous
limits are $p,v,(p+v)/4$. At a small $r_1$, set
\[
q=\xi-4y',\qquad l=r_1+y'.
\]
The relevant forms are $q,l,q+8l-8r_1$, with homogeneous limits
$q,l,q+8l$. Reflection gives side two. In each case use the sum
of the three homogeneous forms to define $R$ as in
Lemma~\ref{lem:geometry:cones}.
Only the redundant homogeneous form is shifted. It is uniformly
comparable to $R^2$: in the high model $(p+v)/4=R^2/10$, and in
the low model $R^2/4\le q+8l\le4R^2/9$. Replacing that
denominator by its shift of size $O(\delta)$ therefore gives a
relative error $O(\delta/R^2)$. The bounding side forms are
unchanged, including where one vanishes on an axis. The constant
matrix and the remaining regular
facet terms give a relative error $O(R^2)$. Differentiating after
rescaling the annulus to unit size gives the same bound at every
fixed derivative order. Near a simple axis, use the square root
of its vanishing affine form as the radial disk coordinate. The
singular normal term is then precisely the smooth polar disk
metric, and its coefficients and those of the inverse Hessian
obey the same estimates. This proves
\eqref{eq:geometry:cone-error} through the axis. At the left tip
as $L\to\infty$, the omitted $l_o^{-1}$ term is bounded with all
scaled derivatives and tends to zero; all other omitted
denominators stay bounded away from zero on the chosen zone.
Thus the constants are uniform there.
Use $\mu_{\mathrm{loc}}=(p,v)$ at a high-side shrink and
$\mu_{\mathrm{loc}}=(q,l)$ at a low-side shrink. In the variables
$\widehat\mu_{\mathrm{loc}}=\mu_{\mathrm{loc}}/\delta$, the three
singular Hessian terms have factor $\delta^{-1}$ while
$d\mu_{\mathrm{loc}}$ has factor $\delta$.
The moment and angle terms of the metric both have
factor $\delta$. Removing that factor leaves exactly the three
affine-direction metric. The constant and regular terms tend
to zero. Lemma~\ref{lem:geometry:scalar} proves its asserted
strict positivity.
Choose a nondecreasing smooth function $\chi_\infty$ which is zero
below a large fixed number and one above a larger fixed number,
and put
\begin{equation}
F(t)=\int_t^\infty\chi_\infty(s)s^{-2}\,ds,
\qquad
f=1+c\,\chi_0(R)\delta^{-1/2}F(R/\sqrt\delta).
\label{eq:geometry:conformal-factor}
\end{equation}
Here $\chi_0$ is one at small fixed radius and cuts off at a
slightly larger fixed radius. Choose those radii inside the
zone above. The new metric is $f^2g$. In dimension four its
scalar curvature is $f^{-3}(-6\Delta_g+\Scal(g))f$, where
$\Delta_g=\operatorname{div}\grad$.
For $q(R)=\delta^{-1/2}F(R/\sqrt\delta)$ a direct calculation gives
\begin{equation}
\Delta_g q
=-\frac{\chi_\infty'(R/\sqrt\delta)|dR|^2}
{\sqrt\delta\,R^2}
+\frac{\chi_\infty(R/\sqrt\delta)}{R^3}
\bigl(2|dR|^2-R\Delta_gR\bigr).
\label{eq:geometry:radial-laplacian}
\end{equation}
For the exact cone, $R\Delta R=3|dR|^2$. By
\eqref{eq:geometry:cone-error}, choosing the initial ramp point
large and $R_*$ small gives
$R\Delta_gR>(2+\sigma)|dR|^2$ for one fixed $\sigma>0$
where $q$ starts decreasing. Thus $-\Delta_gq$ is positive
there. Where $q$ is constant, strict positivity of the base
scalar curvature suffices. On the outer cutoff annulus, the
main or rescaled-tip metrics have a common positive scalar
lower bound; choosing $c>0$ sufficiently small absorbs all
derivatives of $\chi_0$. We obtain
$(-6\Delta_g+\Scal(g))f>0$ on the smooth part. The coefficient
$c$ may be tapered smoothly to zero when $\delta$ leaves its
small-parameter zone. Choose the taper only where the same
strict compact bounds hold. The limiting factor is $1+c/R$
on the cylindrical annulus, with fixed $c>0$ near $\delta=0$.
At the original puncture the homogeneous radius is
$R_8=\sqrt{4\xi}$. Add a cutoff multiple $c_8/R_8$. If no
$r_i$ is small, this is the preceding construction without an
inner smoothing scale. If $r_i\to0$, place its cutoff at a
fixed sufficiently small radius in the rescaled satellite,
that is, at $R_8$ comparable to a small fixed multiple of
$\sqrt{r_i}$. It then lies where the first factor in
\eqref{eq:geometry:conformal-factor} is constant of order
$r_i^{-1/2}$. Write $R_8=\sqrt{r_i}\rho$ and cancel this scale
against $g=r_i g_{\mathrm{sat}}+o(r_i)$. The added factor is
a fixed multiple of $c_8/\rho$ on a fixed-scale satellite.
Its cutoff error is therefore absorbed by the satellite's
strict compact positivity after choosing $c_8$ small. This
also shows that the nested link does not shrink to zero.
At most one such nesting occurs. The other zones are
disjoint, so their choices do not interfere.
On a cylinder use $s=-\log R$. The cone metric multiplied by
$(c/R)^2$ is $c^2(ds^2+h)$. The difference between $1+c/R$
and $c/R$ has relative size $O(R/c)$, in addition to the error
\eqref{eq:geometry:cone-error}. Interpolating to
$c^2(ds^2+h)$ over a fixed $s$-width deep in the annulus has
errors tending to zero with all fixed spatial derivatives.
Because the product scalar curvature is bounded positively
away from zero, a sufficiently deep interpolation preserves
positivity. The same argument applies at the nested puncture
after rescaling. On a separated high-side satellite, the
compact core and the entire outward conformal annulus are
positive; the exact product continuation is positive as well.
This is the required complete positive metric on $N_i$.
\end{proof}
\subsection{The interval cube and its true ends}
Use coordinates $t_i\in[-3,3]$ on the two intervals of a macro.
The true $J_i$ face is $t_i=-3$, and the true $N_i$ face is
$t_i=3$. Choose a smooth nondecreasing function $\alpha_i(t_i)$
which is zero for $t_i\le0$, equals $1/4$ for $t_i\ge1$, and
increases between these values on $(0,1)$. It may be taken
stationary to all orders at the endpoints. A sphere test keeps
its whole-loop rule for $t_i\ge0$. Its conversion to the two
segment rules of Section~\ref{sec:stack} takes place only at
$t_i<0$, after that side is no longer used as a whole test.
Set $w_0=\alpha_1+\alpha_2$. For $w_0\ge3\epsilon$ the
target coefficients are $(\alpha_1,\alpha_2)$. For a finite
smooth metric replace them by
\begin{equation}
a_i=\alpha_i+d w_0(1-8\alpha_i),\qquad
(a_1,a_2)=(a,b_0),\qquad
w=w_0\bigl(1+d(2-8w_0)\bigr).
\label{eq:geometry:regularization}
\end{equation}
For a common sufficiently small $d>0$, both coefficients lie
strictly between zero and $1/4$. At low saturation
$\alpha_i=0$, and at high saturation $\alpha_i=1/4$, respectively,
the shrink parameter is
\begin{equation}
r_i=\frac{d}{1+d(2-8w_0)},\qquad
h_i=\frac{d}{1+d(2-8w_0)}.
\label{eq:geometry:saturation-scale}
\end{equation}
Thus there is one uniform small scale for productizing at either
saturation.
\begin{proposition}[Assembly in the positive-width region]
\label{prop:geometry:assembly}
For $w_0\ge3\epsilon$, the toric models and their collars form
a smooth family on the fixed macro. The low and high saturation
regions extend to the prescribed true faces. At fixed $\epsilon$,
as $d\to0$, all retained positive-width main pieces converge
on compact subsets to the complete conformal metrics of
Theorem~\ref{thm:geometry:family}\textup{(ii)}, uniformly under
further genuine length insertions.
\end{proposition}
\begin{proof}
Always remove the original outer cone point, complete it by
Lemma~\ref{lem:geometry:collars}, and attach its product collar
to the exterior of $C_2$. The collar is an artificial finite
collar. Its productization may be moved arbitrarily far down
the puncture as $d\to0$, so on every compact subset the
retained metric is exactly conformal to
\eqref{eq:geometry:hessian} in the limit.
Near a saturation, productize at $R\asymp d^{1/4}$ whenever
the corresponding $\delta$ is at most $2d$, and turn off this
productization by $\delta=3d$. Equations
\eqref{eq:geometry:cone-error} and
\eqref{eq:geometry:saturation-scale} show that its Hessian error
is $O_k(\sqrt d)$, while its conformal product error is
$O_k(d^{1/4})$. A fixed-width cutoff in $\log R$ therefore
has vanishing errors. Use the adjacent affine forms to fix
the angular coordinates of this product. They determine its
link independently of all tangential parameters.
For $t_i\ge1$, this gives an exact collar separating $N_i$.
Attach any additional length inside the exact product, with
length zero initially and tending to infinity only as
$t_i\to3$. Keep the link and the coefficient $c$ fixed on
that collar. The satellite on the $N_i$ side is the positive
cap of Lemma~\ref{lem:geometry:collars}. Its metric, like
every retained sector, is unaffected by length inserted
in another disjoint product collar. The metric on the
complementary sector is unchanged by the bit supported
on $X_i$.
For $t_i\le0$, the collar instead separates $C_{1,i}$. It
must be retained while the exterior is changed to introduce
$J_i$. Since $w_0\ge3\epsilon$, the other coefficient is
positive; there is no competing second low separation.
First extend the outer-puncture productization back to a
fixed sufficiently far-out portion of the rescaled satellite,
where it is already close to a positive product. Shorten
that obsolete outer collar there. Similarly extend the
product structure a bounded distance toward the compact
part on the $C_{1,i}$ side, while retaining the auxiliary
$\partial C_{1,i}$ collar, whose length tends to infinity as
$d\to0$. For fixed auxiliary choices this collar remains finite
as $t_i\to-3$. All of these extensions
take place in positive-product regions, so they have bounded
geometry and preserve positive scalar curvature along their
long parts.
The exterior of this retained collar now consists of bounded
pieces. On those pieces interpolate to a metric with a
round product collar at $J_i$, and fixed data on its far
side. This is possible because $C_{1,i}$ misses $J_i$ by
Lemma~\ref{lem:geometry:plumbing}. Keep the inner collar
fixed during the interpolation. Only after reaching the
round product data insert the genuine $J_i$ length, tending
to infinity as $t_i\to-3$. The far side was made independent
of the other interval parameter before this insertion.
Consequently the true split has the required local parameter
dependence. The retained main metric and its long
$\partial C_{1,i}$ collar have not been changed.
Use successive closed subintervals of the low or high
saturation range for these operations, with each operation
stationary near the joints. If another parameter has no
active operation it simply waits. Disjoint genuine collars
then permit independent insertion lengths. In a low-high
corner the $N_{3-i}$ collar is inside $C_{1,i}$ and its
length insertion commutes with the exterior low-side
operation. This gives the asserted compatibility.
It remains to justify that these varying polygons and
interfaces are placed on one fixed manifold. Start from a
fixed punctured plumbing polygon, with marked cross curves
for the interfaces actually in use. A cross curve near a
shrinking $X_i$ edge bounds that divisor's disk bundle. A
cross curve at small $r_i$ joins the $X_i$ edge to the far
sloping side; its inner side contains exactly the subchain
$U,X_{3-i}$. A cross curve joining the two horizontal sides
in the long strip bounds the chosen $U$ neighborhood. These
are exactly the neighborhood types in
Lemma~\ref{lem:geometry:plumbing}.
On each side of the polygon, place the endpoints of these
curves by increasing coordinate changes. Their order does
not change, since only compatible disjoint or nested curves
are marked together. Choose the curves as segments or
graphs in rectangular side charts and extend their motion
by vector fields tangent to the polygon faces, supported
away from unmarked corners. On the remaining disk interiors
extend the motion with a cutoff. This constructs smooth
face-preserving diffeomorphisms carrying the marked collars
to the fixed interfaces. Near a face they are smooth
positive rescalings of its normal variable, so their lifts
in the square-root disk coordinates are smooth. Carry the
angle torus along with these maps; its lattice and all
transition functions are fixed. The two low placements use
disjoint parameter zones and fix the common smaller
$C_0$ whenever it is marked. Thus they impose no crossing
condition on one another.
For any exterior interpolation first shorten unretained
long products, trivialize the resulting bounded pieces by
these maps, and interpolate the positive-definite tensors
there with their product collars fixed. Reinsert retained
products afterward. This is a convex interpolation on a
compact family of bounded pieces and hence gives uniform
spatial geometry; it does not require a uniform parameter
derivative in a stretching coordinate. The construction
is stationary on smaller genuine face collars except for
the prescribed lengths and retained tangential parameters.
It proves both smooth assembly and the stated convergence.
\end{proof}
\subsection{Small width and the long spherical tube}
\begin{lemma}[The tube and the corner replacement]
\label{lem:geometry:tube}
The family of Proposition~\ref{prop:geometry:assembly} extends
across $w_0\le3\epsilon$ to the low-low corner, with exactly
the true cuts $J_1,J_2$. Every macro in this region contains
a tube of fiber class $U$ and length comparable to
$\epsilon^{-1}$. The extension satisfies the uniform
description in Theorem~\ref{thm:geometry:family}\textup{(iii)--(iv)}.
\end{lemma}
\begin{proof}
First suppose $w_0\in[\epsilon,3\epsilon]$, so
$L\asymp\epsilon^{-1}$ and no high coefficient is saturated.
Put $z=y'+r_1$. On the strip between the two horizontal
faces, $0\le z\le1$. The two horizontal Hessian terms add
to $1/[z(1-z)]$ in the $y'$ direction. If both $\xi$ and
$L-\xi$ tend to infinity, the other facet terms tend to zero.
Thus the limiting metric is
\begin{equation}
\frac{dz^2}{2z(1-z)}+2z(1-z)d\theta_y^2
+\tfrac12d\xi^2+2d\theta_x^2.
\label{eq:geometry:explicit-tube}
\end{equation}
Under $z=(1+\cos\varphi)/2$ its first two terms are
$\tfrac12(d\varphi^2+\sin^2\varphi\,d\theta_y^2)$.
This proves the radius and normalization in
\eqref{eq:geometry:long-model}. The class of the spherical
fiber is $U$, since it is the toric sphere transverse to
the longitudinal strip between the two horizontal facets.
For every fixed derivative order, on
$T\le\xi\le L-T$ the error is bounded by a number
$\eta_k(T)\to0$ as $T\to\infty$, uniformly in $r_i\in[0,1]$.
This follows by differentiating the remaining reciprocal
affine terms; the denominators are at least a fixed
multiple of the distance to a strip end when $T$ is large.
Using $\sqrt z$ and $\sqrt{1-z}$ near the two sphere poles
gives the same conclusion through the axes. All conformal
tip changes were supported in bounded end regions and
do not meet this bulk strip.
Choose a large fixed $T$. First interpolate on the strip
to make the metric exactly \eqref{eq:geometry:long-model}
between $2T$ and $L-2T$, using cutoffs in fixed-width
transition strips. The interpolated errors obey the same
bulk bounds, in particular with two derivatives. Within the parameter
region $\epsilon\le w_0\le3\epsilon$, deform the toric model from regularizer $d$
to one fixed small positive value. The obsolete
artificial collar lengths then become bounded. Throughout
these stages retain the low-saturation exterior procedure
of Proposition~\ref{prop:geometry:assembly} if either
parameter is low. It changes only an exterior of the
retained subchain and does not reach the tube.
Now bring the exact tube to a fixed length comparable to
$\epsilon^{-1}$. The remaining tips, in normalized
coordinates, have bounded size, and their effective
$r_i$ are bounded away from zero because the regularizer
has been fixed positively. Interpolate these bounded
pieces to template metrics with product collars at both
$J_i$, retaining the exact tube. Its placement inside
$C_0$ is disjoint from both $J_i$. If a $J_i$ insertion
has already started, perform the interpolation on the
underlying shortened metric with its product collar
fixed, then reinsert precisely the same length. Arrange
the original low-side stages to supply this product
before the corresponding insertion starts. The far
side stays fixed, so no dependence on the other interval
parameter is introduced at a true cut.
These successive operations can be assigned disjoint
subranges of $w_0/\epsilon$, with stationary overlaps.
Below $w_0=\epsilon$ retain only the fixed long tube,
the bounded templates, and the ordinary low-end
prescriptions. This includes $w_0=0$, where neither a
normalized polygon nor a positive-width period is used.
Both genuine $J_i$ lengths are still inserted in their
own collars. Thus the corner has no extra face or
extra multiplicity.
For the uniform assertion, take any sequence of parameters.
The identities $r_1+r_2=1$ and $L\ge2$ imply that the two
small-$r_i$ parameter regimes are mutually exclusive.
Also $h_i\ge1/2-r_i$ excludes simultaneous small $r_i,h_i$
on the same side. A low degeneration and the opposite
high degeneration, or the two high degenerations, may
occur simultaneously; their spatial neighborhoods are
disjoint. The only nested spatial degeneration is the
already described outer puncture inside a low satellite.
Lemma~\ref{lem:geometry:collars} lists
the smooth main scale, cone annulus, and rescaled satellite
for each shrinking edge. The only remaining possible
unbounded toric direction is $L\to\infty$, just analyzed
by \eqref{eq:geometry:explicit-tube}. A deactivated
rational collar in a low homotopy was shortened inside
a positive product. Every free exterior interpolation
was on bounded pieces. Hence this list exhausts all
unbounded regions after passing to a subsequence.
For a toric macro retained above $3\epsilon$,
$L\le C/\epsilon$ by \eqref{eq:geometry:regularization};
in the replacement region its chosen length has the
same bound. Fixing $T$ leaves a bounded number of
truncated tips, whose geometric costs are bounded by
a constant $C_T$, independent of deeper regularizations
and longer rational products. A bounded local error
tending to zero on the tube bulk has integrated cost
at most
\begin{equation}
C_T+\eta(T)\,C\epsilon^{-1},\qquad \eta(T)\longrightarrow0.
\label{eq:geometry:integrated-error}
\end{equation}
Multiplying by $\epsilon$, letting $\epsilon\to0$ first,
and then $T\to\infty$ proves the stated
$o(\epsilon^{-1})$ assertion. The rational product collars inside a macro
have positive scalar curvature and their own exact
models, so their arbitrary lengths contribute no error
of this type. Their nonexact annular ends have the
same uniform bounded-truncation description. This
establishes the full sequential assertion.
\end{proof}
Intervals outside positive macros may use any smooth interval
path with the same true cuts, taking the positive complete
disk-bundle metric of Lemma~\ref{lem:geometry:collars} at
the lens end and the round middle cut at the other end.
The positive-definite tensors with fixed collars form a
convex space on the shortened compact piece, so such paths
exist with stationary endpoints. Different macro procedures
are products on disjoint regions. The external rational
$Y_j$ collars and the old-cap exceptional-sphere isolation
collars are disjoint from all these modifications. They
can therefore be inserted with arbitrary finite lengths.
This proves the metric and true-face parts of
Theorem~\ref{thm:geometry:family}.
\subsection{Cohomology support and uniform comparison forms}
We finish with the two topological interfaces needed in the
Abelian estimates. They prevent a period comparison from
using a class on a piece which has disappeared, and prevent
long rational collars from adding an uncontrolled background
energy cost.
\begin{lemma}[Support at period vertices]
\label{lem:geometry:period-support}
Let $H$ be a positive-width limiting period in
\eqref{eq:geometry:period}. Express it as a convex
combination of the four vertex classes
\[
U,\qquad U+X_1/4,\qquad U+X_2/4,
\qquad U+(X_1+X_2)/4.
\]
Every vertex with nonzero weight is carried rationally by
the retained smooth main piece. It omits every side whose
coefficient is zero, and is orthogonal to every divisor
whose coefficient is $1/4$. Its pairings may consequently
be computed with compactly supported Poincar\'e forms on
that piece, independently of any reference filling at a
true lens cut. A side occurring in such a nonempty vertex
still has its whole-sphere test in the approximating
positive-width family.
\end{lemma}
\begin{proof}
Set $p=4a$, $q=4b_0$. The four weights are
$(1-p)(1-q)$, $p(1-q)$, $(1-p)q$, and $pq$.
If $p=0$, all nonzero-weight vertices omit $X_1$ and lie
in the subchain $U,X_2$. If $p=1$, all such vertices
contain $X_1/4$, and their intersection with $X_1$ is
$1+X_1^2/4=0$. The same statements hold for $q$ and
$X_2$. At a rational boundary, Mayer--Vietoris over
$\R$ splits second cohomology, and the intersection
form splits orthogonally. Thus absence of a coefficient
gives support on the retained subchain, and orthogonality
to an isolated negative divisor gives support on its
complement. The relative-to-absolute cohomology map is
surjective in degree two because the boundary has
$H^1=H^2=0$ over $\R$. Each supported class therefore
has a compactly supported representative on the completed
main piece.
A positive limiting coefficient has $t_i>0$ unless it
is on the high saturated side, where $t_i\ge1$.
For all sufficiently late approximating parameters
that side is in the whole-loop range. If its $N_i$
has been cut off it is also, by convention, a whole
test. No side in the low switching range is used.
\end{proof}
\begin{lemma}[Comparison forms, uniformly in the auxiliary lengths]
\label{lem:geometry:comparison}
Fix a finite set of real degree-two classes on the fixed
stack, or on its genuine-cut components with specified
bounded reference extensions. On every family in
Theorem~\ref{thm:geometry:family} each such class $\Lambda$
has a closed imaginary-valued representative $b_*$, with
$b_*/(2\pi i)$ representing $\Lambda$, such that the
following hold. On each long tube of fiber $U$ it is
\begin{equation}
b_*=i\lambda_U\,\operatorname{vol}_{S^2(1/\sqrt2)},
\qquad \lambda_U=\Lambda U,
\label{eq:geometry:comparison-flux}
\end{equation}
apart from bounded end transitions. It vanishes on the
long middle of every rational collar. For every fixed
truncation size $T$, its energy off the bulk tubes is
bounded by a constant $C_T$ on all compact and rescaled
pieces, independently of deeper regularizations and
longer rational collars. On the tube bulk its norm and
volume differ from their product values by an error of
the form \eqref{eq:geometry:integrated-error}. Consequently
\begin{equation}
\int |b_*|^2=O(\epsilon^{-1}),\qquad
\int_{\mathrm{off\ tubes}}|b_*|^2=o(\epsilon^{-1}),
\label{eq:geometry:comparison-energy}
\end{equation}
where bounded end regions may be included in either
term. The same uniform $o(\epsilon^{-1})$ assertion
holds for the squared negative scalar-curvature costs
on the region obtained by omitting the $Y_j$ middles
and retaining prefixes of length $O(\epsilon^{-1/2})$,
and for bounded local defects from the bulk model.
There is no scalar-curvature assertion on the omitted
$Y_j$ middles. Constants may depend on
the fixed classes and finite problem. These assertions
do not assert uniform bounds as the integral class
itself ranges over an unbounded set.
\end{lemma}
\begin{proof}
First shorten every long rational collar and every
long tube to bounded length, keeping its product
identification and the corresponding collapse map.
For a shrinking edge also cut in the cone annulus,
retain fixed compact truncations of its main and
rescaled satellite pieces, and shorten the intervening
product. Do this inside the satellite for its nested
outer puncture if present. The proof of
Lemma~\ref{lem:geometry:tube} shows that there are only
finitely many resulting decomposition types. To check
compactness also when $L$ is unbounded, observe that then
both $h_i=L/4-r_i$ tend to infinity. Off the bulk strip
there remain only the tip $\xi\le2T$, the tip
$L-\xi\le2T$, and possibly one low-side satellite.
At the latter, the rescaling and nested puncture
truncation of Lemma~\ref{lem:geometry:collars} give
fixed compact charts. At the opposite tip use
$(L-\xi,z)$, where $z=y'+r_1$; the two sloping-facet
terms tend to zero, and the other terms converge
smoothly on the retained charts. When $L$ is bounded,
ordinary parameter compactness together with the
finitely many edge rescalings gives the same conclusion.
The low-side exterior homotopies were performed on
bounded pieces. Thus within each decomposition type
the truncated metrics and their collar identifications
vary in compact smooth families.
The moment sides and angle lattice are fixed; no
unbounded change of homology basis occurs in passing
to these truncations. Uniform equivalence of these
metrics on each fixed truncation is enough for the
following de Rham construction.
Choose closed representatives of a basis for the
cohomology of each shortened manifold. On a rational
product collar, a degree-two class restricts to zero
over $\R$. Its representative there is therefore
$d\eta$, and subtracting $d(\chi\eta)$ sets it to
zero on a smaller middle collar. On a tube slice,
$H^2(S^2\times S^1;\R)$ is generated by the spherical
area form. Since that sphere has area $2\pi$,
\eqref{eq:geometry:comparison-flux} has exactly the
prescribed integral. The difference between it and
the original representative is exact on a middle
slab. The same cutoff construction replaces the
representative by the prescribed vertical form
there. Use disjoint smaller collars for these
operations. Local primitives can be chosen by a
fixed bounded right inverse on the shortened
collars, so their norms and the resulting energy
are bounded uniformly on each compact family.
Restore every long product using the same tangential
closed form on its whole middle: zero on a rational
collar and \eqref{eq:geometry:comparison-flux} on a
tube. Closure is preserved, and the collapse
identification preserves the original real class.
The cutoff transitions remain on bounded pieces.
For a varying decomposition choose finitely many
overlapping parameter neighborhoods carrying these
constructions. Affine combinations with a partition
of unity in the parameters remain closed on each
four-manifold, represent the same class, and have
the same prescribed middle values. A finite
combination retains the uniform norm bounds. This
also handles the stationary overlaps in the low-side
and small-width interpolations.
These are curvature forms, and the argument does
not trivialize a torsion line bundle on a rational
boundary. Such a restriction admits a flat unitary
connection; using that flat reference and splicing
a real primitive changes its curvature by the
exact forms just constructed. Fixed torsion
holonomy and determinant transition maps are
therefore retained throughout.
Four-dimensional two-form energy is invariant under
conformal change and constant metric rescaling.
The large conformal factors at the satellite and
the rescaling by its shrinking parameter thus
introduce no new energy factor. Their fixed
truncations contribute at most $C_T$. The long
rational middles contribute zero. On a tube the
metric converges with the uniform bulk error
$\eta(T)$ of Lemma~\ref{lem:geometry:tube}; the fixed
vertical form has bounded product norm, so its
integral differs from the product integral by at
most $\eta(T)$ times the tube length. Since the
total tube length is $O(\epsilon^{-1})$, the
two-stage limit in \eqref{eq:geometry:integrated-error}
proves \eqref{eq:geometry:comparison-energy}.
For scalar curvature, compact truncated pieces
have a uniform scalar bound and bounded volume.
The cylindrical annuli and exact rational
products other than the $Y_j$ products have nonnegative
scalar curvature by construction; free interpolations
occur on the bounded pieces. Each retained $Y_j$
prefix has fixed bounded local geometry and length
$O(\epsilon^{-1/2})$, so the integral of its squared
negative scalar part is $O(\epsilon^{-1/2})
=o(\epsilon^{-1})$. The omitted middle may be
arbitrarily long and is not included in this assertion.
On a tube, two spatial derivative
bounds control its scalar defect from the product
value four. Its bounded defect tending to zero
in the bulk has the same estimate
\eqref{eq:geometry:integrated-error}, also after
squaring. This proves the last assertion.
\end{proof}
In the application $\lambda_U=-1-e_j+e_{j+1}$, so
$|\lambda_U|\le2$. The fiber has area $2\pi$, scalar
curvature four, and the circle and longitudinal
normalizations are exactly those in
\eqref{eq:geometry:long-model}. These fixed constants
are what make the Dirac estimate compatible with
the prescribed flux. Lemma~\ref{lem:geometry:comparison}
completes the proof of Theorem~\ref{thm:geometry:family};
the next section chooses the background connections
and then chooses finite auxiliary sizes so that all
the required tests hold simultaneously.
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/06-geometry.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/07-estimates.tex ====
\section[Curvature estimates and the Abelian clamp]{Uniform curvature estimates and the Abelian clamp}
\label{sec:estimates}
We now turn the metric periods and tube estimates of
Section~\ref{sec:geometry} into simultaneous bounds on the parallel line
classes that enter the index calculation of Section~\ref{sec:indices}.
The metric choices in Section~\ref{sec:geometry} will be made for a
\emph{fixed finite problem}. This means that the number of intervals, the
bits, the bundles, all exceptional evaluations, the charge bound, and the
finite set of reference fillings have been fixed. Denote this collection
by $\mathcal P$. Constants with a subscript $\mathcal P$ are allowed to
depend on all of it. The constants attached to an old cap $W$ below have
a different, explicitly stated, dependence.
We use real two-forms $\beta,D$, with
\[
b=i\beta,\qquad [\beta/(2\pi)]=\Lambda,
\qquad [D/(2\pi)]=v.
\]
Thus $iD$ is the difference of the two line curvatures at a reduction,
and the active spinor line has determinant class $K=\Lambda+v$.
Type $R$ means an Abelian connection with zero spinor, with an ordering
of its summands; type $S$ means a parallel splitting with a nonzero
spinor in the active summand. These component-type letters do not
denote the capped Seifert surface $A$; the full field-type classification
is Definition~\ref{def:indices:types}. Our curvature norm is normalized
so that, on a region with its inherited relative charge,
\begin{equation}
\|F_a^0\|_2^2
=2\|F_a^{0,+}\|_2^2+8\pi^2\kappa.
\label{eq:estimates:energy-charge}
\end{equation}
The identity is additive under cutting, whether or not the individual
charges are positive.
For a positive bridge put $X_1=-S_j$, $X_2=S_{j+1}$, as in
Section~\ref{sec:geometry}, and define
\begin{equation}
\begin{gathered}
\lambda=\Lambda U=-1-e_j+e_{j+1},\qquad u=vU,\\
H_I=U+\frac14\sum_{i\in I}X_i,\qquad
z_I=vH_I,\qquad \lambda_I=\Lambda H_I
\quad(I\subset\{1,2\}).
\end{gathered}
\label{eq:estimates:vertices}
\end{equation}
In particular $H_\varnothing=U$. Direct substitution gives
\begin{equation}
\lambda_1=-\frac32+e_{j+1},\qquad
\lambda_2=-\frac12-e_j,\qquad
\lambda_{12}=-1,
\qquad u\equiv\lambda\pmod2.
\label{eq:estimates:vertex-values}
\end{equation}
A whole side is a side on which the sphere test is still the whole
sphere test. A separated $N_i$ with its specified Abelian reference
filling counts as whole; a side in the transition to the two-piece
$J_i$ rule need not be used.
\begin{theorem}[The Abelian clamp]
\label{thm:estimates:clamp}
For the finite problem $\mathcal P$ and a fixed upper bound on total
topological charge, the metrics and fixed line connections of
Theorem~\ref{thm:geometry:family} can be chosen with all artificial
isolation lengths finite so that the following assertions hold
simultaneously, including at every genuine face.
\begin{enumerate}[label=\textup{(\roman*)}]
\item On an $R$ or $S$ component containing a positive bridge, either
$u=0$, or there is a nonempty subset $I$ of its adjacent whole sides
such that
\begin{equation}
\operatorname{sign}(u)z_I\leq0\quad(R),
\qquad
\operatorname{sign}(u)z_I<|\lambda_I|\quad(S).
\label{eq:estimates:clamp}
\end{equation}
\item No $R$ component contains an outside cap. For each fixed old
cap $W$, cut at its rational seams and before its extra blow-up,
there is a constant $C_W$ such that an $S$ component satisfies
\begin{equation}
v_W^2\leq C_W,\qquad (\Lambda_W+v_W)^2\leq C_W.
\label{eq:estimates:cap-squares}
\end{equation}
The constant depends only on $W$, its fixed line data, metric and
boundary collars. It is independent of the number of added
intervals and of the new exceptional evaluations.
\item On an actually separated $N_i$ the fixed background is harmonic
and anti-self-dual, and the metric has positive scalar curvature.
The complementary metrics and backgrounds agree for the two bit
choices differing by the class supported on $N_i$.
\end{enumerate}
One may first choose a positive bound for the ordinary Floer-gradient
perturbations on the finite $Y$ seams and then take those seams to any
sufficiently large finite gluing lengths. Once those lengths have
been chosen, further compatible perturbations may be made sufficiently
small, including in total squared $L^2$ cost on the seams. Precise
quantifiers are given in Corollary~\ref{cor:estimates:choices}.
The charge assumption concerns the sum over all retained fields and
nonnegative ideal losses. No sign assumption is made on the charge of
an arbitrary $\PU(2)$ component.
\end{theorem}
We prove the theorem in several steps. Only the collar, bulk-product,
and comparison-representative conclusions of
Theorem~\ref{thm:geometry:family} are used as geometric input.
\subsection{Harmonic backgrounds and local energy}
\begin{lemma}[Harmonic backgrounds and rational collars]
\label{lem:estimates:hodge}
On a completed piece with rational-homology-sphere ends, a real
cohomology class has a unique $L^2$ harmonic representative. If pieces
are joined by increasingly long product collars with rational
cross-sections, their harmonic representatives converge on compact
subsets to the representatives of the restricted classes, provided
their total norms are bounded. The convergence and the cutoff errors
are exponential in distance from the ends of a long collar.
Background connections can consequently be chosen to be flat on the
middle of each sufficiently long finite $Y$ collar and to obey the
product rule at all genuine faces. On an isolated positive toric
piece their self-dual curvature is proportional to its K\"ahler form;
on a separated $N_i$ it is zero.
\end{lemma}
\begin{proof}
For clarity, ``a class'' on a completed piece means its image from
compactly supported cohomology. The map from relative to absolute
degree-two cohomology is an isomorphism over $\R$ here: the boundary
has zero first and second real cohomology. Minimize the two-form norm
in the affine space obtained by adding the $L^2$ closure of compactly
supported exact forms to a compactly supported representative.
Orthogonal projection gives a closed and coclosed form, which is smooth
by local elliptic regularity. Compactly supported dual pairings
preserve its class. For uniqueness, the difference of two such forms
is exact. On each end it has a decreasing primitive, by the spectral
description below; a global primitive can be made equal to these
primitives outside a compact set, since $H^1$ of the cross-section
vanishes over $\R$. Integration by parts against the difference gives
zero for its squared norm. This is also the relative-cohomology
description of cylindrical Hodge theory
\cite[Proposition~4.9]{AtiyahPatodiSingerI}.
Here is the required quantitative collar fact. Let $Z$ be a fixed
rational homology sphere. Write a harmonic two-form on $[0,T]\times Z$
as $dt\wedge\alpha(t)+\gamma(t)$. The tangential one- and two-form
Laplacians have a common positive lower spectral bound $\mu_Z$.
Each eigenmode satisfies
\[
-f''+\mu f=0,\qquad \mu\geq\mu_Z.
\]
Its two exponential solutions and elementary integration give, for
$1\leq r\leq T/2$, and then by interior elliptic estimates,
\begin{equation}
\|b\|_{L^2([r,T-r]\times Z)}
\leq C_Ze^{-\sigma_Zr}\|b\|_{L^2([0,T]\times Z)},
\qquad
\|b\|_{C^k([r,r+1]\times Z)}
\leq C_{Z,k}e^{-\sigma_Z\min(r,T-r-1)}\|b\|_2,
\label{eq:estimates:rational-decay}
\end{equation}
after decreasing $\sigma_Z>0$ if necessary. The constants can be
chosen uniformly when the link metrics range over the compact
families used in Section~\ref{sec:geometry}. The same mode expansion
gives a primitive on an interior collar with the same bounds. There
are no harmonic tangential modes in this argument. In particular,
it is not an assertion about an $S^2\times S^1$ tube.
Local elliptic compactness, followed by compactly supported dual
pairings, now identifies every bounded-norm limit with the harmonic
representative of the restricted class. To see the stronger gluing
assertion, cut off the decreasing tails of the piecewise harmonic
forms and glue their primitives on the long products. This produces
a same-class closed representative $h_T$. Conversely, applying the
same cutoff to the harmonic representative $b_T$ produces
representatives on the separated pieces. Equation
\eqref{eq:estimates:rational-decay} bounds the energy change in either
direction by $Ce^{-\sigma T}$ times a bound for the total energy.
The projection identity
\begin{equation}
\|h_T-b_T\|_2^2=\|h_T\|_2^2-\|b_T\|_2^2
\label{eq:estimates:minimizing-remainder}
\end{equation}
therefore proves exponential closeness, with a possibly smaller
exponent. Interior estimates give the corresponding smooth
closeness away from the splice strips. The argument applies to
several disjoint collars at once.
For each subset of the currently long genuine collars, preglue the
harmonic data obtained by cutting just those collars. Average these
closed representatives with the product weights formed from
$\chi(T)$ and $1-\chi(T)$, where $\chi$ is zero below one large
threshold and one above a larger threshold. These weights depend
only on the metric parameters, so the average is closed on the
four-manifold and represents the same class. The preceding estimate
makes its difference from harmonic data as small as prescribed.
Taking thresholds uniformly in the other disjoint collars gives the
product rule at every multiple face. At a $J$ face the data on a
piece depend only on that piece's parameters. At an $N_i$ face the
piece is already separated, so its datum remains exactly harmonic.
The thresholds may be increased as the auxiliary sizes vary; thus
the errors can have any prescribed smaller order in the estimates
below.
A line bundle on a rational cross-section admits the flat connection
specified by its torsion class; no trivialization of that line bundle
is asserted. The small primitive in
\eqref{eq:estimates:rational-decay} cuts a connection off to this flat
one. Connections with the prescribed curvatures exist because their
periods are the prescribed integral periods. Fixed collar gauges and
the fixed line identifications give the asserted compatibility.
Finally, in four dimensions the star on two-forms and their $L^2$ norm
are conformally invariant. The toric K\"ahler form has finite squared
norm before completion, since the moment polygon has finite area.
It therefore remains an $L^2$ self-dual harmonic form after conformal
cylindrical completion. The main piece has $b^+=1$, so this form
spans its self-dual harmonic space. Since $b^+(N_i)=0$, its harmonic
datum is anti-self-dual. A class supported on $N_i$ restricts to zero
on the complement; the product construction accordingly gives the
same exterior datum for the two bit choices.
\end{proof}
We next record the local estimate with the error norms that will
actually be used. The equations and Clifford convention are those of
Section~\ref{sec:indices}; equivalently,
\[
D_a\Phi=P_D,\qquad
\rho(F_a^{0,+})=(\Phi\Phi^*)_{00}+P_+.
\]
These are the usual trace-free monopole equations
\cite{FeehanLenessPU2I}. If $\Phi$ is identified with a two-by-two
matrix with singular values $s,t$, then
\[
\bigl\langle(\Phi\Phi^*)_{00}\Phi,\Phi\bigr\rangle
=\frac14(s^4+t^4)+\frac52s^2t^2
\geq\frac14|\Phi|^4.
\]
We write $d_*=1/4$ for this lower bound; changing the common quadratic
normalization merely changes this fixed positive constant.
\begin{lemma}[Local integrated estimate]
\label{lem:estimates:local}
Set
\[
V=\frac{\Scal}{4}\id+\frac12\rho(b),\qquad
h=\max\{0,-\lambda_{\min}(V)\}.
\]
If $\zeta=f^2$ is a compactly supported real cutoff with $0\leq f\leq1$,
then
\begin{equation}
\begin{split}
\frac12\int\zeta^2|\nabla_a\Phi|^2
+\frac{d_*}{4}\int\zeta^2|\Phi|^4
\leq{}&2\int\zeta^2|P_D|^2
+\frac1{d_*}\int\zeta^2(h^2+|P_+|^2)\\
&+\frac9{d_*}\int\frac{|d\zeta|^4}{\zeta^2}.
\end{split}
\label{eq:estimates:cutoff}
\end{equation}
The last integrand is defined continuously as $16|df|^4$ where
$f=0$. Enlarging the numerical constants in
\eqref{eq:estimates:cutoff}, if the curvature perturbation is expressed
as a two-form instead of an endomorphism, accounts for its fixed
Clifford norm.
In particular, uniform local geometry, bounded $h$, and bounded
local squared error costs give uniform bounds for
$\int(|\nabla_a\Phi|^2+|\Phi|^4)$ on interior unit strips. No derivative
of $P_D$ occurs.
\end{lemma}
\begin{proof}
Pair the weak identity $D_a^*D_a\Phi=D_a^*P_D$ with
$\zeta^2\Phi$, and use the positive-spinor Weitzenb\"ock formula.
The right side is
\[
\langle P_D,D_a(\zeta^2\Phi)\rangle
=\zeta^2|P_D|^2+
2\zeta\langle P_D,\rho(d\zeta)\Phi\rangle.
\]
Thus no integration by parts differentiates the prescribed error.
The covariant derivative cross term is at most
$\frac12\zeta^2|\nabla_a\Phi|^2+2|d\zeta|^2|\Phi|^2$;
the displayed Dirac cross term is at most
$\zeta^2|P_D|^2+|d\zeta|^2|\Phi|^2$.
The negative zeroth-order terms are bounded by
$\zeta^2(h+|P_+|)|\Phi|^2$. Apply Young's inequality separately to
these two terms, using a total of at most $d_*/2$ of the quartic
coefficient, and to $3|d\zeta|^2|\Phi|^2$, using at most $d_*/4$.
The resulting remainders are bounded by the right side of
\eqref{eq:estimates:cutoff}. Finally
$|d(f^2)|^4/(f^2)^2=16|df|^4$. A cutoff supported in a fixed larger
strip and equal to one on the strip in question proves the last
assertion.
\end{proof}
\begin{lemma}[Charge on a finite $Y$ middle]
\label{lem:estimates:cylinder}
Consider a flat-background product cylinder with fixed cross-section
$Y$. On its translation-invariant middle, assume that the ordinary
connection perturbation is the gradient of a functional
$\mathcal W$ with
$\|\mathcal W\|_\infty\leq M$ and uniformly bounded gradient.
Let $r,e$ denote any additional connection and Dirac errors there.
Suppose their total squared $L^2$ norms are bounded by $\nu$.
There is a constant $C_Y$, depending only on the fixed cross-section
and the functional and gradient bounds, independent of both $\nu$
and cylinder length, such that one can
choose boundary slices in prescribed interior unit strips near its
two ends for which
\begin{equation}
\kappa_{\rm middle}\geq-C_Y-C_Y\nu.
\label{eq:estimates:cylinder-charge}
\end{equation}
The complementary bounded strips may be included in the retained
region of a cutoff estimate.
\end{lemma}
\begin{proof}
Use temporal gauge and the convention
$\rho_3(\xi)=-\rho_4(dt\wedge\xi)$ on positive spinors. Write the
equations as
\[
\dot\Phi+D_a\Phi=e,\qquad
\dot a+*F_a=q(\Phi)+\sigma\nabla\mathcal W(a)+r,
\qquad \sigma\in\{1,-1\}.
\]
With the stated Clifford and trace conventions $c_*=1/2$, and
$\langle q(\Phi),v\rangle=-c_*\langle D'_a(v)\Phi,\Phi\rangle$.
All pairings in this proof are integrated over $Y$ and are real
pairings. For $H(t)=\langle D_a\Phi,\Phi\rangle$, differentiation
and the Dirac equation give the exact identity
\begin{equation}
\begin{split}
-\langle *F_a,\dot a\rangle
={}&\|\dot a\|_2^2+c_*H'(t)+2c_*\|\dot\Phi\|_2^2
-\sigma\frac{d}{dt}\mathcal W(a)\\
&-\langle r,\dot a\rangle-2c_*\langle e,\dot\Phi\rangle.
\end{split}
\label{eq:estimates:cylinder-identity}
\end{equation}
In particular an ordinary Floer perturbation contributes endpoint
values, rather than a length times its pointwise size. Young's
inequality absorbs the last two terms into half the positive energies
at a cost of $\frac12\|r\|_2^2+c_*\|e\|_2^2$.
It remains to bound $H$ on suitable slices. Apply
Lemma~\ref{lem:estimates:local} on fixed larger strips around the two
prescribed unit strips. Their geometry and background are fixed, and
the gradient has a uniform pointwise bound, so
\[
\int_{J\times Y}(|\nabla_a\Phi|^2+|\Phi|^4)\leq A_Y(1+\nu).
\]
Since $|D_a\Phi|\leq\sqrt3|\nabla_a\Phi|$ for the tangential Dirac
operator,
\[
\int_J|H(t)|\,dt
\leq\sqrt3\,\Vol(J\times Y)^{1/4}
\{A_Y(1+\nu)\}^{3/4}.
\]
There is a slice in each unit strip with
$|H|\leq B_Y(1+\nu)^{3/4}$, where $B_Y$ is independent of $\nu$.
Integrating
\eqref{eq:estimates:cylinder-identity} between the chosen slices and
using
\[
\kappa_{[u,v]\times Y}
=-\frac1{4\pi^2}\int_u^v\langle *F_a,\dot a\rangle\,dt
\]
gives the explicit lower bound
\[
-\frac{2c_*B_Y(1+\nu)^{3/4}+2M+
\frac12\|r\|_2^2+c_*\|e\|_2^2}{4\pi^2}.
\]
Since $(1+\nu)^{3/4}\leq1+\nu$, this proves
\eqref{eq:estimates:cylinder-charge} for every $\nu\geq0$ with the
asserted constant. The same local
estimate controls the adjoining cutoff strips. Notice that the
proof does not require pointwise differentiability bounds for $e$.
\end{proof}
\subsection{The long tube and the small-width test}
\begin{proposition}[Tube comparison and retained-region energy]
\label{prop:estimates:energy}
Fix $\mathcal P$. Let $\epsilon\downarrow0$ through the family in
Section~\ref{sec:geometry}, allowing all admissible deeper
regularizations and rational isolations. Let $\mathcal R_\epsilon$
be the union of the retained regions, obtained by removing the long
translation-invariant $Y$ middles. Keep initial $Y$ prefixes of
length $R_0=C\epsilon^{-1/2}$, as well as the bounded cutoff strips.
For harmonic backgrounds, and also for the product-rule backgrounds
of Lemma~\ref{lem:estimates:hodge} chosen with smaller errors,
\begin{equation}
\sum_{\mathcal T}\|\beta-\lambda_{\mathcal T}\omega_{S^2}\|_{L^2(\mathcal T)}^2
+\|\beta\|_{L^2(\mathrm{rest})}^2=o_{\mathcal P}(\epsilon^{-1}),
\qquad \|\beta\|_2^2=O_{\mathcal P}(\epsilon^{-1}).
\label{eq:estimates:tube-background}
\end{equation}
Here $\mathcal T$ runs over the bulk tubes, bounded end pieces may be
transferred to the rest, and $\omega_{S^2}$ is the area form of
$S^2(1/\sqrt2)$. The subscript on the little-$o$ means uniformity
over those admissible choices and all parameter faces for the fixed
problem.
For solutions whose total charge is bounded above, bounded ordinary
$Y$ gradient perturbations and additional errors of squared cost
$o(\epsilon^{-1})$ give
\begin{equation}
\int_{\mathcal R_\epsilon}
(|\Phi|^4+|F_a^{0,+}|^2+|F_a^0|^2)
=o_{\mathcal P}(\epsilon^{-1}).
\label{eq:estimates:small-energy}
\end{equation}
For each fixed $\epsilon>0$, the same quantities have a finite bound
$E_{\mathcal P,\epsilon}$ independent of all subsequently lengthened
rational collars and of the omitted $Y$ lengths, with the error
costs bounded. The assertions also hold for the retained main fields
of ideal tuples with nonnegative omitted losses.
\end{proposition}
\begin{proof}
The geometric input is Lemma~\ref{lem:geometry:comparison}. After
collapsing long products to bounded products, prescribe the vertical
form $\lambda_{\mathcal T}\omega_{S^2}$ on their middle slices and zero
real curvature on rational middle slices. The restriction classes
agree with these prescriptions. Relative de Rham representatives on
the bounded truncations, followed by reinsertion of the products,
give a same-class comparison form $h$. Uniform bounded truncations
at the smooth pieces, conical annuli and rescaled satellites give
\begin{equation}
\|h\|_2^2
=\sum_{\mathcal T}\lambda_{\mathcal T}^2\Vol(\mathcal T)
+o_{\mathcal P}(\epsilon^{-1}).
\label{eq:estimates:comparison-upper}
\end{equation}
In this formula the volumes are those of the product models. Metric
errors change its two sides by $o_{\mathcal P}(\epsilon^{-1})$.
Additional rational cylinders have zero comparison curvature on
their long parts, so their arbitrary lengths add no leading term.
For the converse estimate, on each exact product fiber
$\Sigma=S^2(1/\sqrt2)$ one has
\[
\int_\Sigma\beta=2\pi\lambda_{\mathcal T},\qquad
\operatorname{area}(\Sigma)=2\pi.
\]
Fiberwise Cauchy--Schwarz and integration over the other two
coordinates therefore bound the energy from below by the summand
in \eqref{eq:estimates:comparison-upper}. More precisely, the cross
term against the model vertical form is fixed by this same flux, so
on the exact product
\[
\|\beta-\lambda_{\mathcal T}\omega_{S^2}\|_2^2
=\|\beta\|_2^2-
\lambda_{\mathcal T}^2\Vol(\mathcal T).
\]
First discard bounded tip ranges and an arbitrarily small end
fraction of each long tube. On the remainder the metric and its
needed derivatives tend uniformly to the product model. The errors
in the last identity are bounded by this metric error times the
energy, which is $O_{\mathcal P}(\epsilon^{-1})$ by minimization and
\eqref{eq:estimates:comparison-upper}. Then let the discarded
fraction tend to zero. Nonnegativity of all the remaining energies
proves \eqref{eq:estimates:tube-background}.
The background is flattened on each $Y$ middle only after the
prefix of length $R_0$. Equation
\eqref{eq:estimates:rational-decay} bounds the added squared error
by $C_{\mathcal P}e^{-2\sigma R_0}\epsilon^{-1}$, which is negligible.
The product-rule alterations at genuine faces can have smaller
cost by the threshold choice in Lemma~\ref{lem:estimates:hodge}.
On a bulk tube, $\Scal=4$ in the model and
\[
\frac{\Scal}{4}=1\geq\frac{|\lambda_{\mathcal T}|}{2}
=\frac12\|\rho(i\lambda_{\mathcal T}\omega_{S^2})\|,
\qquad |\lambda_{\mathcal T}|\leq2.
\]
Consequently the squared negative part $h^2$ from
Lemma~\ref{lem:estimates:local} has integral
$o_{\mathcal P}(\epsilon^{-1})$ there, by
\eqref{eq:estimates:tube-background} and the integrated metric-error
bounds. The same assertion holds on the rest: its background
energy is lower order, its negative scalar-curvature squared cost
is lower order by the geometric decomposition, and the retained
$Y$ prefixes have bounded local scalar cost times
$O(R_0)=o(\epsilon^{-1})$. A bounded $Y$ gradient perturbation has
the same lower-order squared cost on those prefixes. All other
errors have the stipulated total cost. Thus, for a cutoff equal
to one on the retained part away from the bounded transition strips,
\[
\int_{\mathcal R_\epsilon}(h^2+|P_+|^2+|P_D|^2)
=o_{\mathcal P}(\epsilon^{-1}).
\]
Apply \eqref{eq:estimates:cutoff}; use the local strip estimate at its
finitely many transitions. At a genuine infinite end use cutoffs
and finite-energy tails. This gives the $|\Phi|^4$ assertion.
The quadratic curvature equation gives the assertion for
$|F_a^{0,+}|^2$.
For the full curvature, sum charges over all retained components.
If $Q$ is the fixed upper charge bound and there are $q_Y$ omitted
$Y$ middles, Lemma~\ref{lem:estimates:cylinder} gives
\begin{equation}
\kappa(\mathcal R_\epsilon)
\leq Q+q_YC_Y(1+\nu).
\label{eq:estimates:retained-charge}
\end{equation}
Nonnegative particles and other omitted nonnegative losses only
decrease the left side. Substitution into
\eqref{eq:estimates:energy-charge} proves the full-curvature
assertion. This sums over the arbitrary $\PU(2)$ fields as well as
the reduced fields, and never discards a component on an assumed
charge sign.
Finally fix $\epsilon$. All bounded-truncation contributions in the
same argument are then finite uniform constants; lengthening a
positive rational cylinder adds neither negative scalar cost nor
a zero-mode background. The omitted-middle bound is independent
of length. Repeating the estimates with constants instead of
little-$o$ proves the stated $E_{\mathcal P,\epsilon}$ bound.
\end{proof}
\begin{corollary}[The small-width test]
\label{cor:estimates:tube}
For $\epsilon$ sufficiently small, every $R$ or $S$ reduction on a
small-width transition or replacement region has $vU=0$.
This choice is uniform under the subsequent deeper regularizations
and rational isolations allowed by the metric construction.
\end{corollary}
\begin{proof}
Such a region contains a bulk tube of length at least
$c_{\mathcal P}/\epsilon$, whose other circle factor has length
bounded below. For a reduction, $F_a^0$ has diagonal entries
$iD/2,-iD/2$, so $|F_a^0|^2=|D|^2/2$. Each fiber has
$\int_{S^2}D=2\pi u$. Cauchy--Schwarz therefore gives, with a
constant independent of the auxiliary lengths,
\[
\int_{\mathcal T}|F_a^0|^2
\geq c'_{\mathcal P}u^2/\epsilon.
\]
If the integer $u$ is nonzero this contradicts
\eqref{eq:estimates:small-energy} along a sequence
$\epsilon\downarrow0$. The estimates are uniform in precisely the
other parameters permitted in such a sequence. The stationary
overlap of the small-width and toric prescriptions is included.
\end{proof}
\subsection{Period inequalities and finite persistence}
\begin{lemma}[The period inequality on an isolated positive piece]
\label{lem:estimates:projection}
Let $M$ be a positive-width toric main limit, with period class
$H=U+aX_1+b_0X_2$, $a,b_0\in[0,1/4]$, $a+b_0>0$.
Let the fixed background be harmonic. Every finite-energy solution
of the unperturbed reduced equations satisfies
\begin{equation}
vH=0\quad\hbox{or}\quad
(vH)((\Lambda+v)H)<0,
\label{eq:estimates:period}
\end{equation}
with $vH=0$ when the spinor vanishes and the strict alternative when
it is nonzero.
\end{lemma}
\begin{proof}
At a reduction the trace-free curvature is
$\diag(iD/2,-iD/2)$. The two bundle-diagonal blocks of the doubly
trace-free quadratic term are likewise one half of the active
spinor quadratic term with opposite signs. The reduced curvature
equation is therefore
\[
\rho(iD^+)=(\psi\psi^*)_0,
\]
and the active Dirac operator has fixed-plus-variable curvature
$i(\beta+D)$. This checks that $v$ and $\Lambda+v$ in
\eqref{eq:estimates:period} are full line classes.
For the Clifford convention in use,
\[
\tr\bigl(\rho(i\eta^+)\rho(i\xi^+)\bigr)
=4\langle\eta^+,\xi^+\rangle,
\qquad |\psi|^4=8|D^+|^2.
\]
The integrated Dirac identity consequently reads
\begin{equation}
0=\int_M\left(|\nabla\psi|^2+\frac{\Scal}{4}|\psi|^2\right)
+2\int_M\left(\langle\beta^+,D^+\rangle+|D^+|^2\right).
\label{eq:estimates:line-Weitzenbock}
\end{equation}
To justify integration on the completion, insert cutoffs changing
on bounded-width end strips. The derivative error there is
bounded by a fixed constant times
$\Vol(\mathrm{strip})^{1/2}
(\int_{\mathrm{strip}}|\psi|^4)^{1/2}$ and tends to zero.
The other terms follow by the finite curvature energy and the
cutoff estimate. In particular no global $L^2$ assumption on
$\psi$ is needed.
Scalar curvature is nonnegative and positive on a nonempty interior
open set. The first integral in
\eqref{eq:estimates:line-Weitzenbock} is therefore nonnegative.
If it vanished, then $\nabla\psi=0$ and
$\Scal|\psi|^2=0$ everywhere. The latter equality makes $\psi$
vanish on that open set, and a parallel section which vanishes at
a point vanishes on the connected manifold $M$. Thus the first
integral is strictly positive whenever $\psi\ne0$. Write
$D^+=D_H+D_\perp$, where $D_H$ is its self-dual harmonic projection.
By Lemma~\ref{lem:estimates:hodge}, $\beta^+$ belongs to the same
one-dimensional harmonic space. With $h_H$ the harmonic form of
class $H$,
\[
\beta^+=2\pi\frac{\Lambda H}{H^2}h_H,
\qquad D_H=2\pi\frac{vH}{H^2}h_H.
\]
The second integral in \eqref{eq:estimates:line-Weitzenbock} is
therefore
\[
(2\pi)^2\frac{(vH)((\Lambda+v)H)}{H^2}
+\|D_\perp\|_2^2.
\]
It is negative if $\psi\ne0$, which proves the strict alternative.
If $\psi=0$, the curvature equation gives $D^+=0$ and hence $vH=0$.
\end{proof}
\begin{lemma}[Vertex test and finite persistence]
\label{lem:estimates:vertices}
After $\epsilon$ has been fixed as in
Corollary~\ref{cor:estimates:tube}, sufficiently small regularization,
sufficiently long finite rational isolation, and sufficiently small
perturbations give \eqref{eq:estimates:clamp} on every positive bridge.
The assertion is uniform near genuine faces and over all sufficiently
long finite $Y$ seams.
\end{lemma}
\begin{proof}
\emph{The isolated vertex test.}
First consider an isolated positive-width limit. Put
$p=4a$, $q=4b_0$. Its period is the convex combination
\begin{equation}
H=(1-p)(1-q)H_\varnothing+p(1-q)H_1
+(1-p)qH_2+pqH_{12}.
\label{eq:estimates:convex}
\end{equation}
All weights are nonnegative, and at least one nonempty vertex has
positive weight. All nonempty vertices have negative
$\Lambda$-evaluation by \eqref{eq:estimates:vertex-values}; the empty
one has $\lambda\leq0$. Consequently $\Lambda H<0$.
If $u\ne0$, its parity gives
\begin{equation}
|u|\geq|\lambda|.
\label{eq:estimates:parity-size}
\end{equation}
Indeed this is immediate for $\lambda=0,-1$, and for $\lambda=-2$
a nonzero even integer has absolute value at least two.
For type $R$, if all available nonempty vertices violated the
asserted weak inequality, multiplication of
\eqref{eq:estimates:convex} by $\operatorname{sign}(u)v$ would give
$\operatorname{sign}(u)vH>0$, contradicting
Lemma~\ref{lem:estimates:projection}. For type $S$, failure at all
such vertices, together with \eqref{eq:estimates:parity-size}, gives
\[
\operatorname{sign}(u)vH
\geq\sum_I\theta_I|\lambda_I|
=|\Lambda H|>0.
\]
Here $\theta_I$ are the four weights. A number of magnitude at
least $|\Lambda H|$ cannot satisfy either alternative of
\eqref{eq:estimates:period}. This proves the limiting test.
\emph{Persistence at finite sizes.}
Fix $\mathcal P$ and $\epsilon$, and suppose the conclusion failed
along a sequence
with regularization tending to zero, isolations tending to infinity,
and perturbations tending to zero. Pass to a fixed sign of $u$ and
to limiting period parameters. The small-width alternative is already
excluded by Corollary~\ref{cor:estimates:tube}.
By Proposition~\ref{prop:estimates:energy} the retained energy is
uniformly bounded. We will pass to an unperturbed reduction on the
retained smooth main piece. Only the pairings of vertices with positive
limiting weight must pass to this limit; those vertices remain supported
on that piece even when other classes collapse.
On a ball, Abelian Coulomb gauge gives a
connection one-form bounded in $W^{1,2}$ by its curvature norm.
Lemma~\ref{lem:estimates:local}, followed by
$d\psi=\nabla_a\psi-a\psi$ and $W^{1,2}\subset L^4$, bounds the
spinor in $W^{1,2}$ as well. To exclude an atom of its fourth-power
measure at a point, take a radial cutoff $f_R$ equal to one on
$B_{R^2}$ and zero outside $B_R$, with logarithmic transition.
In four dimensions
\[
\int|df_R|^4\leq C|\log R|^{-3}.
\]
Use $\zeta=f_R^2$ in \eqref{eq:estimates:cutoff}, first pass to the
sequence limit and then let $R\downarrow0$. The bounded background
cost is $O(R^4)$ and the error costs tend to zero, so the atom is
zero. The critical Sobolev concentration argument now gives strong
local $L^4$ convergence of the spinors: its defect measure is
supported at atoms, as follows by applying the Sobolev inequality
to a cutoff times the weakly convergent difference and partitioning
sets on which the limiting gradient-energy measure is small.
The reduced curvature equation and the interior elliptic estimate
for $d^*\oplus d^+$ then give strong $W^{1,2}$ convergence of the
Coulomb connection forms on smaller balls.
For completeness, bootstrap on balls on which their $L^4$ norms
are small. In the Dirac equation multiplication by this one-form
is a small operator from $W^{1,p}$ to $L^p$, for $2<p<4$, and can
be absorbed in the local Dirac estimate. The cutoff inhomogeneity
is already in $L^p$; choosing $p>8/3$ gives spinor integrability
greater than eight. The quadratic curvature equation then gives
connection coefficients in $W^{1,q}$ for some $q>4$. Repeated
Dirac and $d^*\oplus d^+$ estimates give smooth convergence when
the errors tend to zero in the corresponding local Sobolev norms.
This proves convergence to an unperturbed reduction without
differentiating the errors in the initial energy estimate or using
transversality. Ideal markings elsewhere and filled true lens ends
do not change its compact pairings.
Only vertices of positive limiting
weight in \eqref{eq:estimates:convex} matter. By
Lemma~\ref{lem:geometry:period-support}, each such vertex is
represented rationally in the smooth retained main part: it omits
a side whose low coefficient has vanished, and includes a collapsed
high divisor with coefficient $1/4$, making it orthogonal to that
divisor. Compactly supported dual representatives therefore bound
its evaluation by the retained $L^2$ energy. Those evaluations lie
in a fixed discrete lattice, so a subsequence makes them constant.
All of their nonempty sides are whole for late terms of the
sequence. There is no need to bound an evaluation on a vertex of
zero limiting weight, nor to bound $u$ if the empty weight is zero.
For $R$, strict violation on a nonempty vertex persists as a
strictly positive discrete value; the empty vertex, if relevant,
has $\operatorname{sign}(u)u>0$. For $S$, violation is the closed
inequality $\operatorname{sign}(u)z_I\geq|\lambda_I|$, and persists
as well. The limiting calculation just made is a contradiction,
including if an $S$ spinor limits to zero. The argument allowed
arbitrary genuine faces and arbitrary further $Y$ lengths, proving
the stated uniformity.
\end{proof}
\subsection{Fixed caps and the order of choices}
\begin{proposition}[Cap constants before exterior choices]
\label{prop:estimates:cap}
For each old cap $W$ there is a metric, unchanged on its rational
end collars, and a number $C_W<\infty$ such that
\[
\begin{gathered}
\forall\,\mathcal P\ \exists\,L_{\mathcal P},\delta_{\mathcal P}>0:\\
\text{isolation at least }L_{\mathcal P},\quad
\text{admissible errors at most }\delta_{\mathcal P}\\
\Longrightarrow\quad
\text{no cap-containing }R\text{ and \eqref{eq:estimates:cap-squares}.}
\end{gathered}
\]
Here $\mathcal P$ is any fixed finite exterior problem whose
restriction to $W$ is the fixed cap datum. Neither $C_W$ nor that
metric depends on the new exceptional evaluation or the number of
intervals. The thresholds and intermediate full-curvature bounds
may depend on both.
\end{proposition}
\begin{proof}
The cap contains a nonzero class $A$ with $A^2=0$ and $cA$ odd.
Its real intersection form is nondegenerate, since its boundaries
are rational homology spheres. A nondegenerate negative-definite
form has no nonzero isotropic vector, so $b^+(W)>0$.
Moreover every allowed restriction $v_W\equiv c_W\pmod2$ has odd
evaluation on $A$, and is nonzero in real cohomology.
Choose the metric so that none of these countably many integral
classes is anti-self-dual. Here is the period argument with the
permitted support. Suppose a nonzero harmonic representative
$\alpha$ of one such class is anti-self-dual. Choose a nonzero
self-dual harmonic form $\omega$. Unique continuation for harmonic
forms~\cite[Remark~3]{Aronszajn1957} implies
that the nonzero sets of $\alpha$ and $\omega$ meet in any prescribed
open region for metric variation. Infinitesimal changes of the
metric there give all homomorphisms between the anti-self-dual and
self-dual two-form spaces. Thus one may choose a compactly
supported variation of the star for which
\[
\int_W\langle\dot *\alpha,\omega\rangle\ne0.
\]
Differentiating the harmonic projection shows that this is, up to
a nonzero fixed factor, its derivative in the $\omega$ direction.
The wall for this class is therefore closed with empty interior
among metrics fixed on the end collars. Baire's theorem avoids
all the walls at once. This is the usual local period variation
argument; see also \cite{DonaldsonKronheimer1990}. It uses the
nonzero real class, so torsion presents no exception.
Fix this metric and its harmonic datum $\beta_W$. Let
$\omega_1,\ldots,\omega_b$ be an $L^2$ orthonormal basis of its
self-dual harmonic forms. They and their end primitives decrease
exponentially by \eqref{eq:estimates:rational-decay}. Cover $W$ by
finitely many interior sets and bounded-overlap unit end strips
$V_k$, $k\geq0$. On a completed-cap limit, the local estimate gives
\begin{equation}
\|D^+\|_{L^2(V_k)}\leq A_W
\quad\hbox{for every }k,
\label{eq:estimates:cap-local}
\end{equation}
where $A_W$ depends only on the fixed cap geometry and background.
Indeed the scalar curvature and background are bounded on all
these sets, the enlarged sets have uniformly bounded volume and
cutoffs, and the limiting equations are unperturbed. No incoming
spinor boundary value occurs in Lemma~\ref{lem:estimates:local}.
For $R$ the left side is zero. Hence the convergent sums
\begin{equation}
B_{W,a}=\frac{A_W}{2\pi}\sum_k\|\omega_a\|_{L^2(V_k)},
\qquad
B_W=\left(\sum_{a=1}^bB_{W,a}^2\right)^{1/2}
\label{eq:estimates:cap-constant}
\end{equation}
bound the coefficients, and then the norm, of the limiting positive
projection of $v_W$. These numbers contain no exterior data.
We must identify that projection with the topological restriction,
without imposing a cap-only bound on the anti-self-dual curvature.
Fix $\mathcal P$ and $\epsilon$ first. Compactly supported dual
pairings in $W$ and Proposition~\ref{prop:estimates:energy} bound all
integral evaluations of $v_W$; thus only finitely many restriction
classes occur for this problem. In a hypothetical sequence with
isolation tending to infinity, pass to a constant one of these
classes. Local convergence gives a field on the completed cap.
For a harmonic $\omega_a$, let $\alpha_a$ be its exponentially
decreasing primitive on each end. If $\chi_N$ is one through
distance $N$ and zero beyond $N+1$, then
\[
\widehat\omega_{a,N}
=\chi_N\omega_a+d\chi_N\wedge\alpha_a
\]
is a compactly supported closed representative of its relative
class. The sign follows from $d\alpha_a=\omega_a$; equivalently it
is $\omega_a-d((1-\chi_N)\alpha_a)$ on the ends.
Its pairing with $D/(2\pi)$ is the topological pairing of $v_W$
with $[\omega_a]$.
The full-curvature bound needed in these end strips can depend on
$\mathcal P$. Retain a prefix of length $N+2$ of a neighboring $Y$
collar and choose its omitted-middle slice beyond the prefix.
Lemma~\ref{lem:estimates:cylinder} still gives the same lower
middle-charge bound. Lemma~\ref{lem:estimates:local} bounds the
self-dual cost of each additional cap-prefix strip by a fixed
constant. Equation~\eqref{eq:estimates:energy-charge} then gives
\begin{equation}
\int_{\text{cap and prefixes through }N+2}|D|^2
\leq E_{\mathcal P,\epsilon}+C'_W(N+2).
\label{eq:estimates:cap-prefix}
\end{equation}
The same bound applies at a positive rational isolation collar.
For each fixed $N$ it follows either before passing to the cap
limit, once that prefix is isolated, or afterward. Its intercept
may be arbitrarily large as exterior data vary.
The correction pairing is consequently bounded by
\begin{equation}
C_W' e^{-\sigma N}
\sqrt{E_{\mathcal P,\epsilon}+C'_W(N+2)}
\longrightarrow0
\quad(N\longrightarrow\infty),
\label{eq:estimates:cap-tail}
\end{equation}
with $\mathcal P$ fixed. The remaining integral converges by
\eqref{eq:estimates:cap-local} and exponential decay of $\omega_a$.
It equals $(2\pi)^{-1}\int_W\langle D^+,\omega_a\rangle$.
This proves the required identification and the bound
$\|v_W^+\|\leq B_W$.
An $R$ limit would have $v_W^+=0$, contrary to the chosen period
condition. For an $S$ limit, possibly with vanishing limiting
spinor,
\[
v_W^2\leq\|v_W^+\|^2\leq B_W^2,\qquad
(\Lambda_W+v_W)^2
\leq(\|\Lambda_W^+\|+B_W)^2.
\]
Choose once and for all
\begin{equation}
C_W=1+\max\{B_W^2,(\|\Lambda_W^+\|+B_W)^2\}.
\label{eq:estimates:CW}
\end{equation}
For the fixed problem, a sequence violating either finite square
bound has a constant restriction class and would contradict this
strict cushion in the limit. A sequence of cap-containing $R$
fields similarly contradicts the period condition. Thus finite
thresholds exist. The exceptional negative summand is outside
$W$ under the rational orthogonal splitting, and has not been
included in either cap constant.
\end{proof}
\begin{corollary}[Order of finite choices]
\label{cor:estimates:choices}
The constants $C_W$ can be fixed before choosing the delays,
exceptional evaluations and repetition count in the stack. After
those choices have specified a finite problem $\mathcal P$, there
are $\epsilon>0$, admissible finite auxiliary sizes, numbers
$\delta>0$ and $T_0<\infty$, with the following property. For every
collection of finite $Y$ lengths $T_j\geq T_0$ and ordinary
Floer-gradient perturbations of size at most $\delta$, there is a
positive allowed size for the further non-gradient perturbations
such that all conclusions of Theorem~\ref{thm:estimates:clamp} hold.
The further size may depend on the chosen finite $T_j$; it includes
their total squared error costs. At genuine ends the perturbations
and backgrounds obey the product rule and the specified decreasing
tail bounds.
\end{corollary}
\begin{proof}
First fix the cap metrics and \eqref{eq:estimates:CW}. Next fix
$\mathcal P$, including any arbitrarily large exceptional
evaluations. Choose $\epsilon$ by
Corollary~\ref{cor:estimates:tube}. Bounded $Y$ gradients only cost
$O(\epsilon^{-1/2})$ on its retained prefixes and contribute bounded
endpoint terms on the omitted middles, so this first choice remains
valid under the later lengthenings. Choose the smaller
regularization parameter $d$, the rational isolation lengths and
the perturbation bounds for Lemma~\ref{lem:estimates:vertices} and
Proposition~\ref{prop:estimates:cap}, with this $\epsilon$ fixed.
Here is the uniformity argument for the order of the last choices.
If no such finite auxiliary choices, positive $\delta$, and $T_0$
worked for all $T_j\geq T_0$, choose failures successively with
$d\to0$, each required isolation length tending to infinity,
gradient sizes tending to zero, and $\min T_j\to\infty$.
For each chosen finite collection of lengths, let the additional
errors tend to zero in the indicated local and total norms before
selecting a failure. The off-middle energy bound is uniform in
all these lengths. After passing to subsequences, the small-width
test, the positive-width vertex argument or the cap argument
therefore supplies exactly the contradiction already proved.
This is a neighborhood assertion in all the auxiliary variables:
it supplies a fixed positive gradient tolerance before the finite
$Y$ gluing lengths are selected. It is stronger than merely
finding one successful diagonal sequence.
Finally fix those finite lengths. Further errors can be decreased
in norms depending on all the sizes now chosen. Compactly
supported errors have finite cost; errors on genuine collars can
have prescribed square-integrable decreasing tails. For an error
on a finite $Y$ middle this permits, for example, a pointwise bound
whose square times the chosen length tends to zero. Alternatively
that middle may remain exactly a gradient equation. The argument
never requires a fixed non-gradient pointwise tolerance on
arbitrarily long uncontrolled middles. On a separated positive
$N_i$ allow only the small compact instanton perturbations that
preserve its spinor-zero conclusion. This completes the choice
of all data and the proof of Theorem~\ref{thm:estimates:clamp}.
\end{proof}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/07-estimates.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/08-indices.tex ====
\section{Spinor indices and exclusion of unwanted strata}\label{sec:indices}
The purpose of this section is to determine which strata can meet the
cut-down spinor problem. We separate this numerical assertion from the
regularization needed to apply it: Definition~\ref{def:indices:analytic-hypotheses}
states the precise analytic properties used here.
In particular, none of the dimension arguments below presupposes compactness
of the moduli spaces it is intended to exclude.
\subsection{Conventions and closed indices}
Fix the integral determinant lift $c=c_1(E)$ and its connection, and use only
determinant-one gauge transformations with the prescribed transition maps.
The spin-$c$ determinant is $l=\Lambda-c$. Write
\begin{equation}\label{eq:indices:charge}
\kappa=c_2(E)-\frac{c^2}{4},\qquad
\Theta=\frac{\Lambda^2-\sigma}{4}.
\end{equation}
Here and below a square on a piece with rational homology sphere boundary is
the relative rational square. On positive spinors the Clifford convention is
$\rho(e^i\wedge e^j)=\rho(e^i)\rho(e^j)$ for orthogonal unit covectors;
positive chirality sees self-dual curvature. The equations are
\begin{equation}\label{eq:indices:equations}
D_a\Phi=P_D,\qquad
\rho(F_a^{0,+})=(\Phi\Phi^*)_{00}+P_+.
\end{equation}
The subscript $00$ means trace-free in both rank-two factors. The unperturbed
quadratic term has positive pairing with the curvature term in the Dirac
Weitzenb\"ock formula, bounded below by a positive constant times $|\Phi|^4$.
Indeed, represent a spinor by a $2\times2$ complex matrix, diagonalize it by
unitary changes of basis, and project its rank-one Hermitian square off the
two scalar factors. That projection is nonzero for every nonzero spinor;
its squared norm has a positive minimum on the unit sphere. This is the
quartic term used in Section~\ref{sec:estimates}.
\begin{definition}\label{def:indices:types}
The phase circle $\mathbb T$ acts by multiplying $\Phi$. On the quotient by
determinant-one gauge, its kernel at a free orbit is $\{1,-1\}$. A main
component has type $A$ when $\Phi=0$ and the connection is irreducible, type
$R$ when $\Phi=0$ and the connection has continuous stabilizer, type $S$ when
$\Phi\ne0$ lies in one summand of a parallel splitting, and type $P$
otherwise. A \emph{fixed-type tuple} has no component of type $P$.
The letters here denote field types; in particular $A$ is not the capped
Seifert surface, and $P$ is not the positive bridge cell.
For a split connection write $E=L_1\oplus L_2$ and
$v=c_1(L_1)-c_1(L_2)$, placing the active spinor line first in type $S$.
Then
\begin{equation}\label{eq:indices:split-classes}
v\equiv c\pmod2,\qquad K=l+2c_1(L_1)=\Lambda+v.
\end{equation}
At type $R$ choose either ordering, including an ordering at a central
connection. Actual separated lens caps are not counted as main components
when assigning these types.
\end{definition}
The description of the phase-fixed locus follows directly from stabilizers.
A gauge transformation compensating a phase outside $\{1,-1\}$ is a
parallel unitary automorphism of $E$ with distinct eigenvalues. If the
spinor is nonzero, it belongs to its corresponding parallel eigenline;
determinant one gives the complementary eigenline. Conversely,
a parallel splitting carrying the spinor in one summand compensates every
phase by the diagonal determinant-one gauge action. Thus the above list
also describes the fixed locus of the diagonal phase on a product of main
components.
\begin{lemma}\label{lem:indices:closed}
On a closed filled piece with $b_1=0$, the real instanton index, complex
coupled Dirac index, and real spinor index before phase quotient are
\begin{equation}\label{eq:indices:closed-indices}
D_I=8\kappa-3(1+b^+),\qquad
n_D=\Theta-\kappa,\qquad D_{\rm sp}=D_I+2n_D.
\end{equation}
The pure-line tangential index at type $S$ is
\begin{equation}\label{eq:indices:tangent-index}
d_s=\frac{K^2-(2\chi+3\sigma)}4.
\end{equation}
\end{lemma}
\begin{proof}
The ASD index is the index of $d_a^*\oplus d_a^+$ with
$p_1(\ad E)=-4\kappa$. For the Dirac operator the degree-four part of
$\operatorname{ch}(E)e^{l/2}\widehat A$ is
\[
\frac{c^2}{2}-c_2(E)+\frac{cl}{2}+\frac{l^2}{4}
-\frac{\sigma}{4}
=\frac{(l+c)^2-\sigma}{4}-\kappa.
\]
Here the signature theorem gives the integrated $\widehat A$ term.
This proves the first two formulas by the index theorem
\cite{AtiyahSingerIII}; the complex Dirac summand contributes twice its
index to the real deformation index. For a line spinor with determinant
$K$, its complex Dirac index is $(K^2-\sigma)/8$. Adding twice this to the
scalar gauge/self-duality index $-(1-b_1+b^+)$ gives
\eqref{eq:indices:tangent-index}, since
$2\chi+3\sigma=4(1-b_1+b^+)+\sigma$.
These conventions also agree with
\cite[Equations~(2.51) and~(2.63)]{FeehanLenessLinksI}.
\end{proof}
On cylindrical pieces we use fixed-limit Fredholm operators with a small
positive exponential weight. The gauge group permits independent parallel
limiting constants in the stabilizers; this is the \emph{unframed}
convention. Framing an end adds $h^0$, the dimension of its flat
stabilizer. The true cuts are $J=S^3$ and the lens boundary of a
$(-4)$ disk bundle. For each flat on these links, $H^1$ of the adjoint local
system is zero: this is immediate on $S^3$, and on its finite lens quotient
it follows by averaging, or by taking invariants in the cohomology of the
finite cover. The flat Dirac has no kernel by positive scalar curvature.
Thus the weights can be chosen below a common nonzero spectral gap. Filling
a $J$-end by a flat ball changes no unframed index after stabilizer matching;
the framed ball index is zero. Across $k$ such cuts the filled ASD indices
sum to $D_I^{\rm tot}-3k$, whereas Dirac indices add without a correction.
These statements follow from the APS index formula with the specified
framing convention \cite{AtiyahPatodiSingerI}.
\subsection{Lens fillings and their indices}
Let $N$ be the disk bundle with zero section $S$, $S^2=-4$. Normalize the
boundary holonomy by a square root of the determinant on $N$. The three
states are the central states $+$ and $-$, and the trace-zero state.
This normalization is local to $N$: after a determinant bit is flipped,
the two normalizations need not agree when viewed from the exterior.
An effective cap charge $\kappa_N$ includes particles in the cap and all
trajectory charge in this end, with the boundary state specified as seen
from the main component.
\begin{proposition}\label{prop:indices:lens-index}
The framed ASD and spinor corrections supplied by this end are
\begin{equation}\label{eq:indices:lens-table}
\begin{array}{c|c|c|c}
\text{state}&\text{charge congruence}&\delta_I&\delta_{\rm sp}\\ \hline
+,-&\kappa_N\in\Z&8\kappa_N&6\kappa_N\\
\text{trace}&\kappa_N\in\frac14+\Z
&2+8(\kappa_N-\frac14)&2+6(\kappa_N-\frac14)
\end{array}
\end{equation}
For a split main field there are reference fillings with
\begin{equation}\label{eq:indices:reference-fillings}
\begin{array}{c|c|c|c}
\text{state}&d=v(S)&\kappa_{\rm ref}&\text{choice for an active line}\\ \hline
+&0&0&K(S)=\Lambda S\\
-&\pm4&1&|\Lambda S+d|\le4\\
\text{trace}&\pm2&\frac14&\text{the prescribed boundary eigenline}
\end{array}
\end{equation}
Here $|\Lambda S|=2$. With these choices the pure-line tangential correction,
including stabilizer matching, is zero.
\end{proposition}
\begin{proof}
A cohomology class evaluating as $r$ on $S$ has square $-r^2/4$.
Consequently a splitting with difference $d$ has charge $d^2/16$.
The normalized eigenline has degree $d/2$ on $S$ and boundary character
$\exp(\pi i d/4)$, up to inversion of the chosen generator. Thus
$d=0,\pm4,\pm2$ give the three states in
\eqref{eq:indices:reference-fillings}. Comparing any filling with a
reference by gluing changes the charge by an integer, proving the
congruences.
We compute the local Dirac correction explicitly. In this calculation
$k(S)$ is the \emph{spin-$c$ determinant evaluation}, not the evaluation of
a twisting line. On the Hirzebruch surface $\mathbb F_4$, let $T$ be a
fiber divisor. The canonical class is $-2S-6T$, so the canonical spin-$c$
determinant evaluates as $-2$ on $S$. Twisting by $\mathcal O(aT)$ gives
$k(S)=-2+2a$. Projection to $\CP^1$ gives
\[
\chi(\mathbb F_4,\mathcal O(aT))=a+1.
\]
Contracting $S$ gives $\mathbb P(1,1,4)$, with $\mathcal O(aT)$ corresponding
to $\mathcal O(a)$ on the common smooth complement. Set
\[
H(j)=
\begin{cases}
\displaystyle\sum_{b=0}^{\lfloor j/4\rfloor}(j-4b+1),&j\ge0,\\
0,&j<0.
\end{cases}
\]
Weighted monomials compute $H^0$, intermediate cohomology vanishes, and
Serre duality uses the canonical sheaf $\mathcal O(-6)$. Hence
\[
\chi(\mathbb P(1,1,4),\mathcal O(a))=H(a)+H(-a-6);
\]
see \cite[Sections~1.2.3 and~1.4.1]{Dolgachev1982}.
Regard $X=\mathbb P(1,1,4)$ as the complex orbifold with its standard
cyclic quotient chart at the contracted point. Its orbifold line
$\mathcal L_a=\mathcal O(a)$ has invariant holomorphic-section sheaf
equal to the coherent sheaf denoted $\mathcal O(a)$ above. On each
finite uniformizing chart, averaging the Dolbeault resolution over
the finite group preserves exactness; invariant partitions of unity
make it a fine resolution. Thus the orbifold Dolbeault index of
$\mathcal L_a$ equals the coherent Euler characteristic just computed.
The canonical spin-$c$ Dirac operator twisted by $\mathcal L_a$ has
the same symbol and hence the same index
\cite[pp.~155--156]{Kawasaki1979}.
The contraction identifies the complements of $S\subset\mathbb F_4$
and the quotient point of $X$ holomorphically, identifying
$\mathcal O(aT)$ with $\mathcal L_a$. Since
$K_{\mathbb F_4}=-2S-6T$, this identification also identifies the
canonical spin-$c$ structures on the complement and their boundary
restrictions. We use this identification, including its boundary
character, for the twisted spin-$c$ structures. In the quotient
chart, write $\zeta(u,v)=(\zeta u,\zeta v)$ for $\zeta\in\mu_4$,
and choose the character convention
$\mathcal L_a\leftrightarrow\zeta^a$. The canonical determinant
character is $\zeta^2$, so twisting gives $\zeta^{2+2a}$, consistent
with $k(S)=-2+2a$ modulo four. The common-complement identification
fixes the full boundary spin-$c$ structure, not merely this
determinant residue.
Choose identical product operators on a separating smooth lens-space
collar. Cutting with complementary boundary spectral projections
gives the same exterior contribution in the two index problems.
This cutting identity is local on the smooth collar: its parametrix
and matching maps commute with finite-group averaging on the
quotient-ball side. Consequently only the indices of the two
fillings differ; no characteristic-number formula for an orbifold
with boundary is being assumed.
The quotient-ball filling is $B^4/\mu_4$, with the corresponding
equivariant flat line on $B^4$. Choose an invariant positive-scalar
cylindrical completion and flat twisting. The Weitzenb\"ock identity
kills the $L^2$ kernels in both chiralities upstairs, hence in each
required finite-group invariant subspace. The boundary Dirac
operator is invertible, so a sufficiently small decay weight crosses
no root. The ball Dirac index is therefore zero. The common-exterior
comparison gives
\[
\ind_{\C}D_N^+
=\chi(\mathbb F_4,\mathcal O(aT))-\chi(X,\mathcal L_a),
\]
which yields the following table:
\begin{equation}\label{eq:indices:dirac-table}
\begin{array}{r|r|r|r|r}
k(S)&a&\chi(\mathbb F_4,\mathcal O(aT))
&\chi(\mathbb P(1,1,4),\mathcal O(a))&\ind_{\C}D_N^+\\ \hline
-6&-2&-1&0&-1\\
-4&-1&0&0&0\\
-2&0&1&1&0\\
0&1&2&2&0\\
2&2&3&3&0\\
4&3&4&4&0\\
6&4&5&6&-1
\end{array}
\end{equation}
For the ASD operator, the framed real tangential line index on $N$ is zero,
since $H^1(N)=0$ and $b^+(N)=0$. The complex off-diagonal index is obtained
by comparing
\[
-\chi(\mathbb F_4,\mathcal O(dT))
-\chi(\mathbb F_4,\mathcal O(-dT))
\]
with the corresponding expression on the weighted plane. Indeed,
the K\"ahler symbol is the direct sum of the reversed Dolbeault symbols
for the difference line and its inverse, so its complex index is
the negative sum of these two Euler characteristics. The same
description holds for the corresponding orbifold lines, and the
common-collar comparison applies to these symbols as well.
The flat quotient ball has no deformation cohomology in degrees
one or two. Its sole unframed contribution is the negative of the
space of parallel gauge generators. Adding the boundary-framing
factor cancels this contribution, including when the boundary
character is trivial. Thus its framed index is zero. The differences
are $0,1,4$ complex dimensions for $d=0,2,4$ respectively. For example, at
$d=2$ the two resolution Euler characteristics are $3,-1$, and the
orbifold values are $3,0$, giving difference $+1$. Doubling yields framed
real ASD indices $0,2,8$.
For $\Lambda S=2$, the two Dirac determinant evaluations are $2+d$ and
$2-d$. Their index sums from \eqref{eq:indices:dirac-table} are $0,0,-1$
at $d=0,2,4$. Changing the sign of $\Lambda S$ gives the same sums.
Increasing relative charge by one with the boundary data fixed increases
the ASD index by eight and decreases the complex Dirac index by one.
Thus the coupled Dirac corrections are $-\kappa_N$ at a central state and
$\frac14-\kappa_N$ at trace. Adding twice these to the ASD corrections
proves \eqref{eq:indices:lens-table}.
Finally choose the active-line reference at $-$ with sign opposite to
$\Lambda S$. The two choices $d=\pm4$ restrict to the same allowable
boundary eigenline: their individual line degrees differ by four.
More explicitly, with the determinant root fixed, the active extensions
differ by a line $M$ of degree four up to sign on $N$, and the complementary
extensions differ by $M^{-1}$. The restriction
$H^2(N;\Z)\longrightarrow H^2(\partial N;\Z)$ is reduction modulo four.
The reference connection on $M$ therefore has trivial flat boundary
restriction; choose a parallel trivialization there and its inverse on
the complement. This gives a determinant-one identification preserving
the active boundary line and the fixed transition, and identifies the
spinors themselves, not merely their determinant classes.
Boundary trivializations introduce no $\U(1)$ degree because
$H^1(\partial N;\Z)=0$.
At $+$ and trace the prescribed choices
already give $|K(S)|\le4$. Equation~\eqref{eq:indices:dirac-table} gives
zero active-line Dirac index in all these cases. Together with the zero
framed scalar ASD/gauge index, and matching of its parallel $\U(1)$, this
gives zero tangential correction. Choosing the other reference at $-$
would instead give $|K(S)|=6$ and real tangential correction $-2$; that is
why the choice in \eqref{eq:indices:reference-fillings} is necessary.
The calculation was made with round links. The positive-scalar link
homotopies in Theorem~\ref{thm:geometry:family} preserve the boundary Dirac
gap; flat adjoint cohomology is unchanged. Index invariance and excision
with any intervening cylinders therefore give the same answers for the
chosen ends.
\end{proof}
\begin{lemma}\label{lem:indices:lens-charge}
On an actual separated $N$ and its infinite lens cylinders the spinor is
zero. If its sphere cut is retained, its effective charge satisfies
\begin{equation}\label{eq:indices:lens-charge}
\kappa_N\in\Z_{\ge1}\quad\text{at }+,-,
\qquad
\kappa_N\in\tfrac14+\Z_{\ge0}\quad\text{at trace}.
\end{equation}
At trace charge $1/4$, there are no particles or nonconstant cylinder
levels, and the cap itself has charge $1/4$.
\end{lemma}
\begin{proof}
On $N$ the fixed-line curvature is harmonic ASD and the metric has positive
scalar curvature; the same applies to its exact limiting cylinders with
flat background. Integrating the Dirac identity with finite-energy
cutoffs therefore annihilates the spinor. The sufficiently small compact
ASD perturbations allowed on $N$ preserve this conclusion by absorption
in the positive scalar term, with no Dirac perturbation there.
On an unperturbed cylindrical level a nonconstant ASD trajectory has
strictly positive charge, whereas zero charge means a constant flat.
On a cap with a compact curvature perturbation and zero spinor,
Chern--Weil gives
\[
\kappa_{\rm cap}\ge-C\|F_a^{0,+}\|_{L^2}^2
\ge-C'\|P_+\|_{L^2}^2>-\tfrac14
\]
after decreasing the permitted perturbation size. Its discrete charge
congruence therefore makes its charge nonnegative, and at least $1/4$
when its own end is trace. Particles carry positive integer charges.
The congruences in
Proposition~\ref{prop:indices:lens-index} now imply the claimed ranges,
except possibly central charge zero.
At $+$, the zero-charge unperturbed cap is flat and its sphere sweep
misses $-1$. At $-$, a flat extension is impossible because $N$ is simply
connected. This excludes charge zero with the cut. These exclusions
persist for the permitted small compact cap perturbations: a sequence
violating them as the perturbation tends to zero has no energy available
for bubbling or nonconstant levels, so converges to the flat case; the
modified sphere path has a uniform avoidance gap there. The local
compactness and no-loss assertion used in this last sentence is proved
independently of exclusion in Lemmas~\ref{lem:analysis:compactness}
and~\ref{lem:analysis:neck}.
For effective trace charge $1/4$ an integer particle is impossible. A
nonflat cap with the cut already has charge at least $1/4$ if its own end
is trace, and at least one if it is central. A zero-charge flat cap at
$+$ misses the cut. There is consequently no charge left for a
nonconstant cylinder level, proving the final assertion.
\end{proof}
\begin{corollary}\label{cor:indices:trace-minimum}
At the minimal trace reducer, the framed tangential ASD index is zero,
the normal ASD index is one complex dimension, and the coupled Dirac
operator is invertible. Its normalized eigenline sphere sweep has winding
of absolute value one.
\end{corollary}
\begin{proof}
Order the summands with $d=2$. The normalized eigenline has degree one.
The harmonic Abelian ASD extension is unique for this ordered boundary
line, since $H^1(N)=0$ and $b^+(N)=0$; any parallel boundary identification
is absorbed by the cap's parallel stabilizer. The ASD indices were
computed in Proposition~\ref{prop:indices:lens-index}. The two Dirac
determinant evaluations are $0$ and $\pm4$, so their total index is zero.
Positive scalar curvature and ASD curvature annihilate the positive
Dirac kernel; index zero then annihilates the cokernel. The degree-one
eigenline gives the asserted winding. Normal ASD surjectivity and
regularity of the cap cut require the additional choices made in
Section~\ref{sec:gluing}; they are not consequences of this index statement.
\end{proof}
\subsection{Virtual pieces and the analytic hypotheses}
The calculation below applies to a finite stack with nonzero ordinary
count and $n_D^{\rm tot}>0$. Lemma~\ref{lem:indices:end-exclusion}
specifies the required outside data; with those data fixed,
Theorem~\ref{thm:stack:nonzero} supplies the repetitions.
The ordinary dimension condition and the number of phase sections are
\begin{equation}\label{eq:indices:phase-count}
D_I^{\rm tot}+n-2n-z=0,\qquad
\eta=n_D^{\rm tot}-1\ge0.
\end{equation}
There are $n$ interval parameters, $n$ real-degree-two sphere tests, cap
insertions of total real degree $z$, and $\eta$ complex sections of phase
weight two. The free spinor dimension after these cuts and phase quotient
is therefore
\[
(D_I^{\rm tot}+n-2n-z)+2n_D^{\rm tot}-2\eta-1=1.
\]
Keep $k$ true $J$-cuts in a limiting tuple, and virtually fill all lens
cuts and all $J$-ends. The resulting $k+1$ main components are ordered
from left to right. Let $p_i$ count the remaining interval parameters on
component $i$, restoring a parameter at each of its lens ends. Let $s_i$
count its assigned sphere tests, and $z_i$ its cap degree. At a $J$-cut the
one sphere event is assigned to the side on which it occurs. Allocate
particle charges and integer $J$-trajectory charges to pieces, and include
all effective lens charges in the corresponding virtual filled indices.
Set
\begin{equation}\label{eq:indices:q-and-i}
q_i=D_{I,i}+p_i-2s_i-z_i,\qquad i_i=q_i+2n_{D,i}.
\end{equation}
\begin{lemma}\label{lem:indices:virtual-sums}
These virtual numbers satisfy
\begin{equation}\label{eq:indices:virtual-sums}
\sum_i(q_i+4)=4,\qquad
\sum_i i_i-2\eta=2-4k.
\end{equation}
For a component of type $R$ or $S$, the split reference fillings of
Proposition~\ref{prop:indices:lens-index} give
\begin{equation}\label{eq:indices:ell}
\kappa_i=-\frac{v_i^2}{4}+\ell_i,\qquad \ell_i\in\Z_{\ge0}.
\end{equation}
\end{lemma}
\begin{proof}
The $J$-gluing formulas and allocation of charges give
\[
\sum_iD_{I,i}=D_I^{\rm tot}-3k,\quad
\sum_i n_{D,i}=n_D^{\rm tot},\quad
\sum_i p_i=n-k,\quad \sum_i s_i=n,\quad \sum_i z_i=z.
\]
Thus $\sum_iq_i=-4k$, proving both sums. The fourth lost dimension per
$J$ is the removed interval parameter, in addition to the three ASD
stabilizer dimensions. For a split field,
$c_2(L_1\oplus L_2)-c^2/4=-v^2/4$. Actual particles and allocated
$J$-trajectory charges add nonnegative integers. At a lens end the actual
charge minus the chosen reference charge is also a nonnegative integer
by Lemma~\ref{lem:indices:lens-charge} and
\eqref{eq:indices:reference-fillings}. Their sum is $\ell_i$.
\end{proof}
\begin{definition}\label{def:indices:analytic-hypotheses}
Call a choice of equations and representatives \emph{admissible for the
exclusion calculation} if it has the following properties, for all ideal
main-part tuples in the fixed total charge budget, including the enlarged
candidate class described in Section~\ref{subsec:analysis:candidates}.
In this class the completed main fields satisfy their equations and
retained nonphase cuts, together with the phase cuts when imposed.
Matching across true cuts and the actual lens-cap and connector fields are
omitted. The assigned lens-cap and connector charges obey their permitted
nonnegative ranges and the total charge constraint; the omitted fields
need not be realizable.
\begin{enumerate}[label=(\roman*)]
\item On a fixed-type tuple, the restricted ASD equations with retained
nonphase cuts on type $A$ components, and the pure tangential equations
on type $S$ components, are transverse. In particular a type $A$
component has $q_i\ge0$, with
strict inequality if it has any lens end, main particle, or allocated
nonconstant $J$-trajectory loss. A type $S$ component satisfies
\begin{equation}\label{eq:indices:tangential-bound}
d_s+p_i+4\ell_i\ge0.
\end{equation}
These are componentwise assertions using only parameters retained on the
component.
\item For types $R,S$, an assigned whole sphere test with $v(S_j)=0$
requires a unit of $\ell_i$. This includes a separated lens test. During
switching to the $J$ prescription the unit is required if both half
values vanish; at a true $J$ the assigned test requires the unit if its
own half value vanishes. Different sphere tests so charged on the same
component require distinct units. Type $S$ is needed in this assertion
only for a tuple with no $P$.
\item For a tuple containing $P$, project away all type $R$ fields,
positions, and tests, retaining their parameters. The equations and
retained cuts on the other components are independent of the discarded
data. Before diagonal phase quotient and phase cuts, the necessary
projected problem has dimension at most
\begin{equation}\label{eq:indices:projection-input}
\sum_{i\notin R}
\left(i_i-\sum_{\text{lens ends of }i}(\delta_{\rm sp}-1)-2w_i\right)
+\sum_{i\in R}p_i,
\end{equation}
where $w_i$ is the total main-particle weight and other allocated integer
loss outside lens caps on that component. The $\eta$ phase sections
subtract $2\eta$, and the phase quotient subtracts one. Regularity is
required only near the projections of candidate tuples in the stated
overcount, with all $R$ fields still present before projection. At those
candidate projections, a negative dimension excludes a zero. No
assertion is made about other zeros of the enlarged equations after the
$R$ fields are discarded. This permits finite rational multisections
on free strata without changing the integral instanton maps at zero
spinor.
\end{enumerate}
\end{definition}
Here is the finite-dimensional loss calculation underlying these
requirements. It also specifies which incidence bounds the analytic
construction must preserve.
\begin{lemma}\label{lem:indices:loss-budget}
For exact sphere, cap surface, and cap point representatives as in
Section~\ref{sec:analysis}, a particle of integer weight $a\ge1$ can
recover at most $4a$ dimensions through its position and any cuts dropped
at that position. A main particle consequently has net penalty at least
four in ASD dimension and two in spinor dimension. A separated lens has
net penalty $\delta_I-1$ or $\delta_{\rm sp}-1$ in the respective
cut-down main problem.
\end{lemma}
\begin{proof}
An unconstrained particle position contributes four and releases no cut.
A particle hitting the exact support of a sphere sweep has at most two
position dimensions; dropping the entire sphere test releases two more.
The same budget holds for a cap surface cut, whose position varies in a
two-dimensional domain and whose real degree is two. At a fixed cap point
there are zero position dimensions and the released degree is four.
Two transverse cap surfaces can be hit at one point: the position is
then zero-dimensional and the released degree is four. General position
excludes triple surface hits and intersections with the fixed cap points.
Sphere supports are mutually disjoint and disjoint from cap tests.
These cases exhaust the possibilities, also for sphere maps with
self-intersections, because their exact supports are parameterized by
their two-dimensional domains. Thus the bound is four per distinct
particle position, and at most $4a$ per particle weight.
Dropping the integer charge $a$ lowers $D_I$ by $8a$ and $D_{\rm sp}$ by
$6a$, giving the stated net penalties. Ignoring matching and using the
same conservative recovery allowance for allocated integer trajectory
units only enlarges the necessary problem. At a lens end, let $\delta$
be its framed index correction. Relative to the virtual filled problem,
the main index falls by $\delta$, one parameter is removed, and its
sphere cut of degree two is omitted. The net change is $-\delta-1+2
=-(\delta-1)$.
\end{proof}
By \eqref{eq:indices:lens-table} and \eqref{eq:indices:lens-charge}, a
lens therefore costs at least one in either calculation. Its ASD penalty
is at least seven at a central state; its spinor penalty is at least five
there. At trace the spinor penalty is exactly $1+6h$ when
$\kappa_N=\frac14+h$. Type $A$ regularity therefore implies (i), with
strictness for every loss. For type $S$, fill by the sharp tangential
references and ignore all retained cuts; allowing four position
dimensions per unit of $\ell_i$ gives
\eqref{eq:indices:tangential-bound}. The remaining issue is to obtain
these regularity and independence assertions simultaneously.
Section~\ref{sec:analysis} proves them, rather than assuming regularity of
a split-component Dirac operator with negative index.
\subsection{The square estimate on a positive cell}
We now use the lattice of Proposition~\ref{prop:stack:lattice} and the
metric tests of Theorem~\ref{thm:estimates:clamp}. In a positive cell put
\[
x=vG,\qquad u=vU,\qquad t_b=vT_b\quad(1\le b\le4).
\]
The cell contributes
\begin{equation}\label{eq:indices:positive-square}
Q_0=2xu-2u^2-\sum_{b=1}^4t_b^2
\end{equation}
to $v^2$. Exactly two $t_b$ are odd. The two neighboring negative half
values will be denoted $h_1,h_2$, and the whole side evaluations are
\begin{equation}\label{eq:indices:d-values}
d_1=x-\sum_bt_b-h_1,\qquad d_2=x-2u-h_2.
\end{equation}
All $x,h_i,d_i$ are even. In particular, for a selected nonempty subset
$I$ of whole sides,
\begin{equation}\label{eq:indices:z-local}
z_I=u+\frac14\sum_{i\in I}d_i\in\tfrac12\Z.
\end{equation}
We retain the subscript here to distinguish $z_I$ from the cap insertion
degree $z$.
For endpoint bits $e,f\in\{0,1\}$, the background values are
\begin{equation}\label{eq:indices:lambda-values}
\lambda=-1-e+f,\quad
\lambda_{\{1\}}=-\tfrac32+f,\quad
\lambda_{\{2\}}=-\tfrac12-e,\quad
\lambda_{\{1,2\}}=-1.
\end{equation}
If $u\ne0$, the clamp selects a nonempty subset of available \emph{whole}
sides with
\begin{equation}\label{eq:indices:clamp-input}
\operatorname{sign}(u)z_I\le0\quad(R),\qquad
\operatorname{sign}(u)z_I<|\lambda_I|\quad(S).
\end{equation}
A side whose test has switched to the separate-half rule is not needed
in this subset. A sphere in a separated lens cap still counts as whole
for this assertion and for its reference filling.
\begin{lemma}\label{lem:indices:positive-square}
For a cell of type $R$ or $S$, combine its square with the neighboring
negative half contributions $-\frac12\sum_{i\in I}h_i^2$ selected by the
clamp. The resulting square $Q$ is at most $-2$, except possibly in type
$S$ when $I$ is a singleton and $u=z_I=1$ or $u=z_I=-1$.
In that exception $Q\le2$ and the selected whole side has $d_i=0$.
If $u=0$, the cell square alone is at most $-2$.
\end{lemma}
\begin{proof}
For $u=0$, Equation~\eqref{eq:indices:positive-square} and the two odd
$t_b$ give $Q_0\le-2$. Suppose $u\ne0$. Direct substitution from
\eqref{eq:indices:d-values}--\eqref{eq:indices:z-local} gives
\begin{align}
I=\{1\}:\quad Q
&=8z_Iu-4u^2-\sum_b(t_b-u)^2-\frac{(h_1-2u)^2}{2},
\label{eq:indices:square-one}\\
I=\{2\}:\quad Q
&=8z_Iu-4u^2-\sum_bt_b^2-\frac{(h_2-2u)^2}{2},
\label{eq:indices:square-two}\\
I=\{1,2\}:\quad Q
&=4z_Iu-2u^2-\sum_b(t_b-u/2)^2
-\sum_{i=1}^2\frac{(h_i-u)^2}{2}.
\label{eq:indices:square-both}
\end{align}
Let $a=|u|\ge1$ and $w=\operatorname{sign}(u)z_I$.
For a singleton the exceptional-square sum is at least two, also after
shifting each $t_b$ by the integer $u$: exactly two of the shifted values
remain odd. Type $R$ has $w\le0$ and therefore $Q\le-2$.
For type $S$, the singleton threshold is $1/2$ or $3/2$.
Since $w\in\frac12\Z$, the former implies $w\le0$ and the latter
$w\le1$. Thus
\[
Q\le8aw-4a^2-2\le8a-4a^2-2.
\]
For $a\ge2$ the last expression is at most $-2$. For $a=1$, every
$w\le1/2$ still gives $Q\le-2$. The remaining possibility is $a=1,w=1$,
that is, $u=z_I=\pm1$. Its upper bound is two, and
$d_i=4(z_I-u)=0$.
For two sides, type $R$ again has $w\le0$, while type $S$ has $w<1$,
hence $w\le1/2$. If $u$ is even, the $t_b-u/2$ are integers with two
odd values, so their squared sum is at least two. If $u$ is odd, all
four are half-integers, contributing at least one; both $h_i-u$ are odd,
so the final sum in \eqref{eq:indices:square-both} contributes at least
one more. In both cases
\[
Q\le2a-2a^2-2\le-2.
\]
This proves every case, including the stated exceptional alternative.
\end{proof}
The exception requires an actual loss unit. Its side was selected among
whole tests, so property (ii) of
Definition~\ref{def:indices:analytic-hypotheses} applies to $d_i=0$.
This use of a whole side is the reason the metric construction stops using
a side before its sphere representative switches to the $J$ rule.
\subsection{Incidence assignments and fixed-type runs}
Consider an interior filled component between true $J$-cuts. Let $L_i$
be its number of cells, and $m_i$ its number of positive cells. By
Equation~\eqref{eq:stack:interior-topology},
\begin{equation}\label{eq:indices:interior-data}
p_i=L_i-1,\qquad b_i^+=m_i,\qquad
\Theta_i=m_i+\frac{\Delta_i}{2}.
\end{equation}
The bit differences $\Delta_i$ telescope on contiguous runs, so their
sum is the difference of two bits and belongs to $\{-1,0,1\}$.
Each negative cell contributes minus one half the sum of the squares of
its two even half-sphere evaluations.
We specify the assignment convention carefully. Reserve two potential
slots for each positive cell, one at each adjacent test. A slot is
occupied by the whole test or by the half test assigned to that cell's
side of a $J$-break. Let $s_P\le2m_i$ be the number
of the component's assigned tests occupying these slots, and put
$s_N=s_i-s_P$. All other assigned tests use the negative rule: charge one
nonzero even half value in the component, or, if all the relevant half
values vanish, charge a unit of $\ell_i$. A nonzero even half contributes
at most $-2$ to $v^2$.
The negative halves used with the clamp in
Lemma~\ref{lem:indices:positive-square} are reserved along with the
positive-cell slots, and are never used by the negative rule. Each half
belongs to one adjacent test, so no half is charged twice. Positive
cells have disjoint adjacent macros at the prescribed spacing. A test
assigned across a $J$ to a neighboring negative component uses the
negative half in that component; the positive-cell slot on the other side
is then empty. This convention also applies when only one of those
components belongs to a run under consideration. At an outside cap,
the bordering negative trace half is available to the negative rule.
During switching the negative rule only needs a nonzero half or the
incidence unit when both half values vanish, exactly as in
Definition~\ref{def:indices:analytic-hypotheses}.
\begin{lemma}\label{lem:indices:component-score}
For an interior fixed component of type $R$ or $S$ satisfying the
incidence property (ii) of
Definition~\ref{def:indices:analytic-hypotheses},
\begin{equation}\label{eq:indices:charge-lower}
8\kappa_i\ge\max(4s_i-4m_i,\,4m_i),
\end{equation}
and
\begin{equation}\label{eq:indices:q-score}
q_i+4\ge
\max(2s_i+L_i-7m_i,\,m_i+L_i-2s_i).
\end{equation}
More precisely, if $z_0$ negative-rule assignments have no nonzero half
option and $e_0$ positive cells have the exception of
Lemma~\ref{lem:indices:positive-square}, then
\begin{equation}\label{eq:indices:incidence-charge}
\ell_i\ge z_0+e_0,\qquad
8\kappa_i\ge
4(s_N-z_0)+4m_i-8e_0+8\ell_i
\ge4s_N+4m_i.
\end{equation}
\end{lemma}
\begin{proof}
The zero negative-rule tests and the exceptional whole tests are distinct
assigned tests with disjoint sphere supports. Property (ii) therefore
charges distinct units and gives $\ell_i\ge z_0+e_0$.
The negative rule contributes at most $-2(s_N-z_0)$ to $v_i^2$.
The positive-cell estimates contribute at most $-2m_i+4e_0$.
Every remaining negative square can be discarded in an upper bound.
Multiplying by $-2$ in \eqref{eq:indices:ell} gives the middle inequality
of \eqref{eq:indices:incidence-charge}. Substitution of
$\ell_i\ge z_0+e_0$ gives the last one (in fact it leaves an additional
$4z_0$). Now $s_N\ge s_i-2m_i$ and $s_N\ge0$, proving
\eqref{eq:indices:charge-lower}. Finally, since there are no cap
insertions on an interior piece,
\[
q_i+4=8\kappa_i+L_i-3m_i-2s_i.
\]
Using the two charge bounds gives \eqref{eq:indices:q-score}.
\end{proof}
\begin{lemma}\label{lem:indices:fixed-run}
For any nonempty contiguous run of interior components of types $R,S$,
\begin{equation}\label{eq:indices:fixed-run}
\sum_{i\text{ in the run}}(q_i+4)\ge-2.
\end{equation}
\end{lemma}
\begin{proof}
Let $L,m,s$ be the totals on the run. Summing each of the two separate
bounds in \eqref{eq:indices:q-score} gives
\[
\sum(q_i+4)\ge\max(2s+L-7m,\,m+L-2s).
\]
Every internal adjacency contributes one assigned test to the run, while
only its two endpoints can contribute an additional test. Thus
$L-1\le s\le L+1$. Corollary~\ref{cor:stack:interior} gives
$L\ge4m-3$ for $m\ge1$.
If $m\ge2$, the first bound is at least
$3L-7m-2\ge5m-11\ge-1$. For $m=1,L\ge3$ it is at least $3L-9\ge0$.
The remaining short possibilities are
\[
\begin{array}{c|c|c}
(m,L)&\text{possible }s&
\max(2s+L-7m,\,m+L-2s)\\ \hline
(1,1)&0,1,2&2,0,-2\\
(1,2)&1,2,3&1,-1,1.
\end{array}
\]
For $m=0$ the first bound is at least $L>0$. Every case is at least $-2$.
\end{proof}
\subsection{The outside caps and the large exceptional evaluations}
The constants in Proposition~\ref{prop:estimates:cap} depend only on the
old caps $W$, before the extra blow-ups. They are independent of both the
new exceptional evaluations and the number of intervals. We use both
of its square bounds, one for $v$ and one for $K$.
\begin{lemma}\label{lem:indices:end-exclusion}
The delays before the first and after the last positive bridge, followed
by the odd values $\Lambda(E_{\rm exc})$, can be chosen so that no outside
component has type $R$, and every outside component of type $S$ satisfying
properties (i) and (ii) of
Definition~\ref{def:indices:analytic-hypotheses} has
\begin{equation}\label{eq:indices:end-score}
q_i+4\ge8.
\end{equation}
This holds also for a single component containing both outside caps.
The choices precede the number of middle repetitions and all finite
metric isolation sizes.
\end{lemma}
\begin{proof}
The exclusion of $R$ is part of Proposition~\ref{prop:estimates:cap}.
Consider type $S$. Let $L$ count its internal cells and $m$ its positive
cells. Rational orthogonal splitting separates the old cap parts, the
new exceptional summands, the bordering negative trace halves, and the
internal cells. Applying the incidence argument of
Lemma~\ref{lem:indices:component-score}, including the bordering trace
halves, gives
\begin{equation}\label{eq:indices:end-square}
8\kappa_i\ge4s_i-4m
+2\sum_{E_{\rm exc}\text{ in }i}v(E_{\rm exc})^2-C_1.
\end{equation}
To see that $C_1$ is uniform, the old-cap terms in $v^2$ are bounded above
by the sum of their fixed constants $C_W$, the exceptional summands
contribute $-v(E_{\rm exc})^2$, and all unused negative squares can only
improve the estimate. The same $z_0,e_0$ incidence argument cancels the
possible positive-cell exceptions. Only one or two fixed cap pieces
are involved.
On an end component $p_i=L+O(1)$, $s_i\ge L-O(1)$, and
$b_i^+=m+O(1)$; all the constants, including $z_i$, depend only on the
fixed caps. Substituting \eqref{eq:indices:end-square} in
\eqref{eq:indices:q-and-i} therefore gives a single fixed $C$ with
\begin{equation}\label{eq:indices:end-lower}
q_i\ge3L-7m
+2\sum_{E_{\rm exc}\text{ in }i}v(E_{\rm exc})^2-C.
\end{equation}
For example, the intermediate expression is
$2s_i+L-7m+2\sum v(E_{\rm exc})^2-C'$; using $s_i\ge L-O(1)$
gives the displayed form. Spacing gives $m\le(L+3)/4$, so its first two
terms are at least $(5L-21)/4$. Choose a fixed $L_*$ so large that
$L\ge L_*$ already implies $q_i\ge4$.
Delay the first and last positive cells beyond $L_*$. An end component
with $L<L_*$ then has no positive cell. Suppose, seeking the only
remaining obstruction, that $q_i<4$. All its topological dimension terms
are bounded independently of the new values of $\Lambda(E_{\rm exc})$,
so \eqref{eq:indices:q-and-i} bounds $\kappa_i$ above. There are no
positive cells, and the cap estimate for $v$ gives $v_i^2\le C_v$,
independently of those values. Thus
\[
0\le\ell_i=\kappa_i+\frac{v_i^2}{4}\le C_\ell.
\]
The tangential necessary condition
\eqref{eq:indices:tangential-bound} now gives
\[
K_i^2\ge2\chi_i+3\sigma_i-4p_i-16\ell_i\ge-C_K.
\]
There are only boundedly many negative cells and fixed caps in these
short end components, so $C_K$ is uniform. The \emph{second} cap estimate,
for $K$, and negativity of all the other summands give
\[
K_i^2\le C'_K-\sum_{E_{\rm exc}\text{ in }i}K(E_{\rm exc})^2.
\]
It follows that every $|K(E_{\rm exc})|$ is bounded by a fixed number $B$,
independent of $\Lambda(E_{\rm exc})$. But
\[
|v(E_{\rm exc})|
=|K(E_{\rm exc})-\Lambda(E_{\rm exc})|
\ge|\Lambda(E_{\rm exc})|-B.
\]
Choose the odd exceptional evaluations at both ends so large that
\eqref{eq:indices:end-lower} is incompatible with $q_i<4$ even for
$L<L_*$. A component containing either end contains its exceptional
summand, so the argument covers both single-cap and two-cap components.
This proves \eqref{eq:indices:end-score}.
The selection just made uses only the old-cap constants and the bounded
short-end topologies: first $L_*$ and the delays, then the exceptional
values. The middle repetitions can subsequently be chosen to make
$n_D^{\rm tot}>0$ and the ordinary count nonzero. Only after that finite
problem is fixed are the isolation lengths and perturbation sizes chosen
as in Corollary~\ref{cor:estimates:choices}. In particular the cap bounds
are not being asserted uniformly over unboundedly many new field
problems at one fixed isolation length.
\end{proof}
\begin{proposition}\label{prop:indices:fixed-exclusion}
With the preceding cap choices, a fixed-type tuple satisfying properties
(i) and (ii) of Definition~\ref{def:indices:analytic-hypotheses} has no
true $J$-cut. No mixed projection regularity is assumed in this assertion.
It is a single type $A$ component with no lens end, particles, or other
positive loss. Its cut-down instanton dimension is zero.
\end{proposition}
\begin{proof}
Suppose first that $k>0$, so the two outside components are distinct.
Each is type $A$ or $S$. Let $a$ be the number of all type $A$ components,
and $b\in\{0,1,2\}$ the number of type $S$ outside components.
There are $a+b\ge2$ distinguished components. They contribute at least
$4a+8b$ to $\sum(q_i+4)$. The remaining components form at most
$a+b-1$ contiguous interior runs of types $R,S$, each contributing at
least $-2$ by Lemma~\ref{lem:indices:fixed-run}. Thus
\[
\sum_i(q_i+4)\ge4a+8b-2(a+b-1)
=2a+6b+2\ge6,
\]
contradicting \eqref{eq:indices:virtual-sums}. A run adjacent to an
outside $S$ uses only its interior portion in this count; the outside
component has already contributed its separate score of eight.
If $k=0$, there is a single component and $q_1=0$. It cannot be type $R$
by the cap exclusion, or type $S$ by \eqref{eq:indices:end-score}.
It is therefore type $A$. Property (i) rules out every lens end,
particle, and allocated trajectory loss, since any of these would force
$q_1>0$. Equation~\eqref{eq:indices:phase-count} is exactly its ordinary
cut-down dimension-zero condition.
\end{proof}
\subsection{Projection away from reducible zero-spinor components}
For the mixed calculation only type $R$ is discarded. The sharper square
bound for that type has no exceptional positive-cell case.
\begin{lemma}\label{lem:indices:mixed-run}
For a nonempty contiguous run of $r$ interior components of type $R$,
\begin{equation}\label{eq:indices:mixed-run}
\sum_{i\text{ in the run}}(4+i_i-p_i)\ge-2.
\end{equation}
More precisely, let $s,m$ be the total assigned tests and positive cells,
let $s_P$ count occupied positive-cell slots, and let
$\mathrm{miss}=2m-s_P$. With $\Delta$ the telescoping endpoint bit
difference,
\begin{equation}\label{eq:indices:mixed-run-sharp}
\sum(4+i_i-p_i)
\ge s-4m+r+\Delta+3\,\mathrm{miss}.
\end{equation}
\end{lemma}
\begin{proof}
From the definitions and \eqref{eq:indices:interior-data},
\[
i_i-p_i=6\kappa_i-3-m_i+\Delta_i-2s_i.
\]
With $e_0=0$, Equation~\eqref{eq:indices:incidence-charge} gives
$6\kappa_i\ge3s_{N,i}+3m_i$. Consequently
\[
4+i_i-p_i\ge
1+s_i-4m_i+\Delta_i+3(2m_i-s_{P,i}),
\]
which proves \eqref{eq:indices:mixed-run-sharp} after summing.
If $m=0$, its right side is $s+r+\Delta\ge0$ because
$r\ge1$ and $\Delta\ge-1$. Suppose $m>0$. Write $a,b\ge0$ for the
numbers of cells before the first and after the last positive cell in
the run, and $\epsilon_a,\epsilon_b\in\{0,1\}$ for the two endpoint
test assignments into the run. Then
\[
L\ge4m-3+a+b,\qquad s=L-1+\epsilon_a+\epsilon_b.
\]
An endpoint with zero offset and no assignment leaves its boundary
positive-cell slot empty. It therefore contributes one to
$\mathrm{miss}$. At the left endpoint,
\[
a+\epsilon_a+3\,\mathbf1_{\{a=0,\epsilon_a=0\}}\ge1,
\]
and the identical statement holds at the right endpoint. The two
empty boundary slots are distinct, including when there is only one
positive cell. Applying these inequalities to
\eqref{eq:indices:mixed-run-sharp} gives
\[
\sum(4+i_i-p_i)
\ge r+\Delta-4+a+\epsilon_a+b+\epsilon_b+3\,\mathrm{miss}
\ge r+\Delta-2\ge-2.
\]
When a boundary test is assigned to a negative component outside the run,
its cost is charged there by its negative half; inside the run the empty
positive slot gives precisely the correction just used. No charge from
an exterior component is borrowed in this calculation.
\end{proof}
\begin{proposition}\label{prop:indices:mixed-exclusion}
A tuple containing type $P$, satisfying the type $R$ incidence in
property (ii) and the projection property (iii) of
Definition~\ref{def:indices:analytic-hypotheses}, has no true $J$-cut.
It has no particles or other integer losses. Apart from an unbroken
free main field, the only possible case is one separated lens end of
effective trace charge $1/4$, a single main component of type $P$, and no other
lens end or loss.
\end{proposition}
\begin{proof}
Write $N$ for the number of non-$R$ main components, and
$N_{\mathrm{lens},i}$ for the number of lens ends on component $i$.
Since every lens penalty in
\eqref{eq:indices:projection-input} is at least one, the necessary
projected dimension after phase and phase cuts is at most
\begin{align}
&\sum_{i\notin R}(i_i-N_{\mathrm{lens},i}-2w_i)
+\sum_{i\in R}p_i-2\eta-1
\notag\\
&\qquad=5-4N-\sum_{i\in R}(4+i_i-p_i)
-\sum_{i\notin R}(N_{\mathrm{lens},i}+2w_i).
\label{eq:indices:projected-dimension}
\end{align}
Indeed substitute $\sum i_i-2\eta=2-4k$ and
$k+1=N+\#R$ to obtain the equality. This is a dimension of a necessary
projection, so forgetting interpiece matching can only enlarge the
candidate space. Its negative value excludes the original tuple
without any regularity assertion for the discarded $R$ fields.
If $k>0$, both outside components are non-$R$ by
Lemma~\ref{lem:indices:end-exclusion}, and hence $N\ge2$.
There are at most $N-1$ contiguous $R$-runs. Applying
Lemma~\ref{lem:indices:mixed-run} in
\eqref{eq:indices:projected-dimension} bounds it above by
\[
5-4N+2(N-1)=3-2N\le-1.
\]
Thus $k=0$. There is only one main component, and because the tuple
contains $P$ it has type $P$. The projected dimension is at most
$1-N_{\mathrm{lens},1}-2w_1$, so $w_1=0$ and
$N_{\mathrm{lens},1}\le1$. If there is a lens end, use the actual penalty
$\delta_{\rm sp}-1$ in \eqref{eq:indices:projection-input}. A central
cap costs at least five, while a trace cap of charge $\frac14+j$ costs
$1+6j$. Nonnegative dimension therefore forces trace charge exactly
$1/4$. For an actual limiting tuple,
Lemma~\ref{lem:indices:lens-charge} also excludes particles or
breaks inside that cap and its cylindrical end. These are all the
possibilities asserted.
\end{proof}
\begin{theorem}[Numerical exclusion]\label{thm:indices:exclusion}
Choose the old-cap data, outside delays, exceptional evaluations, finite
stack, metric sizes, and sufficiently small equation data in the order
of Lemma~\ref{lem:indices:end-exclusion} and
Corollary~\ref{cor:estimates:choices}, with $n_D^{\rm tot}>0$.
Assume the regularity, incidence, and projection properties of
Definition~\ref{def:indices:analytic-hypotheses}. Then the only fixed-type tuples
meeting the nonphase cuts are the legitimate unbroken isolated type $A$
instantons. The free part after the $\eta=n_D^{\rm tot}-1$ phase cuts
and phase quotient has expected dimension one. Its only possible
true-face or positive-loss limiting tuples are single trace-minimum
lens faces, with one type $P$ main component and no other losses.
Consequently the only boundary types to be constructed for its
compactified one-dimensional count are links of the legitimate instantons
and those trace-minimum ends.
\end{theorem}
\begin{proof}
Proposition~\ref{prop:indices:fixed-exclusion} gives the fixed-type
assertion, and Equation~\eqref{eq:indices:phase-count} gives the dimension.
Proposition~\ref{prop:indices:mixed-exclusion} gives the asserted free
degenerations. Lemma~\ref{lem:indices:lens-charge} describes the minimal
cap. The analytic realization of these exclusions and of the instanton
links is proved in Section~\ref{sec:analysis}; regular trace-end gluing
and orientation comparison are proved in Section~\ref{sec:gluing}.
\end{proof}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/08-indices.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/09-analysis.tex ====
\section{Representatives, compactness, and regularization}
\label{sec:analysis}
We fix one of the finite topological problems constructed in
Section~\ref{sec:stack}, with the metrics, artificial lengths, backgrounds,
and tolerances chosen in Sections~\ref{sec:geometry} and
\ref{sec:estimates}. Only the actual $J=S^3$ and lens cuts may now have
infinite length. In particular, a very long $Y$ seam is still a finite
part of this problem. Constants below may depend on this finite problem.
Connections have the specified integral determinant, transitions are fixed,
and all gauges have determinant one. The letters $A,R,S,P$ in this section
denote the field types of Definition~\ref{def:indices:types}, not the capped
Seifert surface $A$.
For the spinor regularization, the ordinary equation and representative
data are based at the choices
of Proposition~\ref{prop:stack:composition}, with the zero-winding and
segmentation prescriptions already installed. Work inside its open
neighborhood of genuine-face-preserving data on which the ordinary
count is $\Omega$. The weighted perturbation norm below maps
continuously into the finitely many norms defining this neighborhood.
Every subsequent generic choice is made in a sufficiently small relative
open ball satisfying the harmonic-reducer, bit-identification, and
parameter-locality conditions. These conditions leave the local
variation spaces used below available. Thus regularizing the ideal
problems never requires leaving the neighborhood that preserves $\Omega$.
The ordinary library proposition itself is independent of this choice:
it applies to arbitrary ordinary data and assumes neither the metric
clamps nor a nonzero count.
Here is the analytic statement used in the dimension calculation. Its
ordinary assertion is also available before the spinor construction and
does not assume the nonvanishing of any count.
\begin{theorem}[Compatible regularization]\label{thm:analysis:regularization}
For the fixed finite family, equation perturbations and representatives
can be chosen arbitrarily small in the prescribed norms with the following
properties.
\begin{enumerate}
\item Ordinary irreducible equations and the labelled sphere, surface,
and point tests are transverse, including the relevant ideal and broken
strata. Their representatives have the exact support and persistence
properties of Lemma~\ref{lem:analysis:incidence}. They are compatible with
fixed determinant gluing, marked translation on cylinders, and all true
faces. Prescribed harmonic reducers on separated caps are preserved.
\item Single-component, single-valued equivariant perturbations give the
fixed-type regularity in Definition~\ref{def:indices:analytic-hypotheses}.
Every tuple of fixed type is consequently excluded except the legitimate
unbroken cut-down instantons.
\item A finite collection of rational multisection terms, in addition to
single-valued terms, eliminates every forbidden mixed ideal stratum by
the negative dimension of its necessary projection. No equation or
retained cut on this projection depends on discarded $R$ fields or their
particles. The allowed free stratum is regular of dimension one, and the
single trace-end main problem is regular of dimension zero. Choices on
the exterior of a separated lens cap agree under the corresponding bit
flip.
\item Solutions have the compactness, removable-singularity, integral
particle-charge, and exponential true-end properties proved below.
Near each legitimate instanton, the phase-cut link has oriented count
$2^\eta$ times its instanton sign, where $\eta=n_D^{\rm tot}-1$.
\end{enumerate}
The finite branch lists have compatible weighted germs and total weight
one. After the no-particle end gluing of Section~\ref{sec:gluing}, their
oriented one-dimensional zero sets obey the ordinary rational Stokes
rule.
\end{theorem}
We construct the representatives, prove compactness, and regularize the
fixed and forbidden strata in this section. The proof of the theorem is
completed in Section~\ref{sec:gluing}, after the remaining lens ends have
been given regular collars. Closed-manifold monopole compactness
provides useful background \cite{FeehanLenessPU2I}, but does not by itself
give the face, support, and projected-regularity assertions above.
The proof order is summarized in Table~\ref{tab:analysis:stages}.
Sections~\ref{sec:ordinary}--\ref{sec:stack} construct the nonzero count
$\Omega$; the ordinary library used there is independently available.
The remaining stages preserve this count and close the spinor boundary
argument. In particular, the regular end construction in
Section~\ref{sec:gluing} precedes the final interior regularization there.
\begin{table}[htbp]
\centering
\small
\begin{tabular}{@{}p{.27\textwidth}p{.67\textwidth}@{}}
\toprule
Stage & Output used next \\
\midrule
Ordinary representatives & An independent ordinary library with exact
support rules. Later choices remain in the neighborhood preserving
the count constructed in Sections~\ref{sec:ordinary}--\ref{sec:stack}. \\
Energy and fixed types & Ideal compactness, exclusion of unwanted fixed
types, and the legitimate instanton links. \\
Projected finite induction & Exclusion of forbidden mixed candidates
and regular zero-dimensional trace-main problems. \\
Regular end gluing & Section~\ref{sec:gluing} constructs exhaustive
collars for the remaining no-particle lens ends, preserving branch
weights and orientations. \\
Final free stratum & Section~\ref{sec:gluing} completes the interior
one-dimensional regularization and applies the weighted boundary rule
to the resulting compact count. \\
\bottomrule
\end{tabular}
\caption{Stages of the analytic construction. The numerical
inequalities of Section~\ref{sec:indices} are applied as their
regularity and incidence hypotheses become available.}
\label{tab:analysis:stages}
\end{table}
\subsection{Function spaces and allowable supports}
Let $X$ be a completed component of an open face, and choose product
coordinates $(t,y)$ on each end. Use a smooth reference configuration
which is the prescribed flat connection with zero spinor sufficiently
far out; exponentially convergent references give the same spaces. Write
$c^{a,\alpha}$ for the closure of smooth sections in the $C^{a,\alpha}$
norm, with $a\geq3$ and $0<\alpha<1$. For $\delta>0$ put
\begin{equation}\label{eq:analysis:weighted-norm}
\|u\|_{a,\alpha,\delta}
=\|u\|_{c^{a,\alpha}(X_{\rm core})}
+\sup_{j\geq0}e^{\delta j}
\|u\|_{c^{a,\alpha}([j,j+2]\times Y)},
\end{equation}
summing the last expression over the finitely many ends. Take the
closure of compactly supported smooth sections in this norm. We use
$E_a=c^{a,\alpha}_\delta(\Lambda^1\otimes\ad E\oplus W^+\otimes E)$
for field differences and $F_a=c^{a-1,\alpha}_\delta$ for the curvature,
Dirac, and gauge-slice equations. On finite inserted cylinders use the
corresponding uniform strip norms; exponential weights are measured from
the ends when a gluing norm is needed. Choose $\delta$ smaller than the
positive normal spectral gaps of the finitely many limiting flat states.
The specific asymmetric lens norm is described in
Section~\ref{sec:gluing}.
When a Sobolev cokernel test is used, choose a second weight
$0<\delta_S<\delta_H=\delta$, both below the same positive indicial
gaps. Then $c^{a,\alpha}_{\delta_H}$ embeds in
$W^{1,p}_{\delta_S}$ for every finite $p$: on the $j$th strip the
weighted integrand is bounded by a constant times
$e^{-p(\delta_H-\delta_S)j}$. We do not use inclusion at equal
weights, which need not hold. Elliptic end estimates transfer
surjectivity back to weight $\delta_H$ because no indicial root lies
between these two weights, as detailed below.
The local gauge group consists of $c^{a+1,\alpha}$ determinant-one gauges
whose difference from a parallel limiting constant is in
$c^{a+1,\alpha}_\delta$. Constants at different ends are independent.
Thus the gauge Lie algebra is the decaying space together with a finite
sum of cut-off parallel end sections. It is not the gauge group with a
chosen frame fixed at every end. The usual local slice is obtained by
adjoining the adjoint of the infinitesimal gauge action to the field
equations. For a non-$R$ field its gauge stabilizer is finite. At an $R$
field we use the Abelian equations and its parallel stabilizer instead.
If $\mathfrak s$ is its Lie algebra of global parallel stabilizers,
the augmented gauge target in this latter convention is
\[
\operatorname{ann}(\mathfrak s)
=\left\{f\in c^{a-1,\alpha}_\delta:
\int_X\langle f,\sigma\rangle=0\text{ for every }
\sigma\in\mathfrak s\right\},
\]
and gauge parameters are taken modulo $\mathfrak s$. An augmentation
using the full target instead retains its $\mathfrak s$ cokernel,
as on the cap in Section~\ref{sec:gluing}. All universal
surjectivity assertions below concern non-$R$ fields; in particular
the nonzero active spinor of type $S$ kills its infinitesimal
determinant-one gauge stabilizer.
The weighted elliptic operator on each fixed-asymptote slice is Fredholm;
the extra limiting constants implement precisely the index convention
of Section~\ref{sec:ordinary}. The fixed-operator weighted Sobolev
Fredholm theory is classical
\cite[Theorems~6.2 and~8.1, Lemmas~7.1 and~7.3]{LockhartMcOwen1985}.
The transfer to the present little-H\"older spaces and the estimates
uniform in neck length are established below. The local analytic
inputs are the small-curvature gauge theorem
\cite[Theorems~1.3 and~2.1, Corollary~2.2]{Uhlenbeck1982Connections},
the usual gauge slices~\cite{DonaldsonKronheimer1990}, and local
elliptic parametrices and Sobolev/Schauder estimates
\cite[Corollary~2.1.B and Theorem~2.2.C, revised author text]{Taylor1991}.
The local sampling terms used below read holonomies and transported
spinor values on compact graphs of loops and paths, and return outputs
supported in compact balls. We first specify how these supports behave
when necks separate.
The parameter space is treated stratum by stratum. Near a corner its
length coordinates $T_j$ are independent, and the embeddings of each
compact subset of a limiting component commute. A datum on a face is
extended using these embeddings and a product of parameter cutoffs.
A connected sampling graph and every ball in its energy cutoff belong
to the same component. A term requiring transport through a cut is
turned off before that cut separates. A nonstationary term far down an
inserted collar is either zero there or has an exponentially small
coefficient, with the same statement for its derivatives. We may impose
any fixed finite exponential rate required subsequently. This is a
condition on terms extending to increasing depths, not a restriction on
compactly supported tests on any given open face.
Here is an explicit form of the exponential requirement. For an
elementary term $p$, let $p^{[D]}$ mean its contribution whenever an
input or output graph reaches depth at least $D$ in an inserted
collar, or crosses a neck of length at least $2D$; include any
nonstationary error in this contribution. Put
\begin{equation}\label{eq:analysis:exponential-seminorm}
N_{k,\mu}(p)=\sup_{D\geq0}e^{\mu D}
\|p^{[D]}\|_{C^k_{\rm par,field,space}},\qquad \mu>0,
\end{equation}
on each of the bounded configuration and parameter sets used for
the library norms. The norm includes the relevant output operator
derivatives, and its supremum is uniform in the other length
coordinates. Each term has bounded-depth compact graphs on its
limiting part, and its incompatible crossing extension is cut off
at a finite length, so these seminorms are finite. Include integral
$k,\mu$ in the countable norm list below. The summed contributions
affected by flattening a middle of depth $D$ are then
$O(e^{-\mu D})$ at every fixed required order. Fixed polynomial
factors from derivatives can be absorbed by decreasing $\mu$.
Parallel end constants do not supply a nondecaying forcing term:
their temporal part is removed by the parallel-group gauge ODE,
and its endpoint rotations are allowed independently.
There are two further restrictions. First, a single-component term
between two $J$ cuts uses only the metric parameters retained on that
component: those in the intervals from its left bounding interval to its
right bounding interval, inclusive. It vanishes near any intermediate
$J$ separation incompatible with that component. Second, terms reaching
a separated lens cap $N$ never couple it to another component. The
Dirac perturbation on $N$ is zero, the curvature perturbation is zero on
its Abelian locus, and exterior data are equal under the bit flip.
Inside $N$ use the tensor-equivalent ASD data and sphere tests. These
requirements are simultaneous at corners, by the product collars of
Theorem~\ref{thm:geometry:family}. They leave all compact probes on the
relevant main components available.
Ordinary perturbations on a finite $Y$ seam are time-independent gradients
of smooth invariant functions of simultaneous solid-torus holonomies,
averaged with a transverse disk density. The thickenings are submersive,
their values are bounded, and their spatial Hessians are bounded on
$L^2$. Retain their ordinary transversality choices and extend them
independently of the spinor. They are supported away from the true
cylinders. Their gradient nature, rather than a bound integrated over
the seam length, is what is used in the energy estimate of
Lemma~\ref{lem:estimates:cylinder}.
For the spinor family a fixed finite $Y$ seam is part of the compact
body. Additional equation probes may be supported in any of its
compact balls; their norms are included in the small nongradient
remainder after the seam lengths have been fixed. This permits full
local output variation there. The equation on a genuine ordinary
Floer cylinder, where the length is unbounded in Floer theory, retains
the separate time-independent gradient prescription.
\subsection{Exact representatives}
\begin{lemma}[The zero-winding modification]\label{lem:analysis:zero-winding}
Fix $1/2<b<1$. On the Hilbert manifold of $H^b$ paths
$g:[0,1]\longrightarrow\SU(2)$ with endpoints $1$ there is a smooth
conjugation-equivariant operation $g\mapsto\widehat g$, homotopic to the
identity operation relative to the endpoints, such that every commuting
path of winding zero has image a uniform positive distance from $-1$.
The operation fixes commuting paths of nonzero winding. It uses only
the given path.
\end{lemma}
\begin{proof}
Identify $\SU(2)$ with the unit quaternions. For a nonconstant commuting
path define
\[
M(g)=\int_0^1\operatorname{Im}g(t)\otimes
\operatorname{Im}g(t)\,dt.
\]
This is a rank-one positive matrix: if its eigenvalue were zero, continuity
would force $g$ to be the constant path $1$. Near such a path its largest
eigenvalue remains simple and determines a smooth, unoriented real line
$L(g)$. Orthogonal projection onto $\R\oplus L(g)$ followed by
normalization gives a path $h_g$ in the circle subgroup associated to
$L(g)$. Shrinking the neighborhood makes this projection nonzero at
every time. Sobolev embedding $H^b\hookrightarrow C^0$ and the Banach
algebra property for $b>1/2$ show that all these operations are smooth.
In the winding-zero neighborhood, the lift of $h_g$ starting at zero is
a unique $H^b$ path $s_g(t)\in L(g)$ with $s_g(0)=s_g(1)=0$ and
$h_g=\exp s_g$. This definition is independent of an orientation of
$L(g)$. Let $C$ be the set of commuting winding-zero paths meeting
$\{\operatorname{Re}\leq0\}$. It is closed in the full based path space.
Indeed axes have a convergent subsequence, uniform convergence preserves
commutativity, and a limiting path meeting this hemisphere is
nonconstant; the degree of its projected circle path is locally
constant. An invariant open neighborhood $U$ of $C$ can consequently
be chosen on which $L(g)$ and $s_g$ are defined and which contains no
commuting path of nonzero winding.
Smooth separation on a Hilbert manifold gives a function $f$, equal to
one on $C$, with support in $U$ and $0\leq f\leq1$. Averaging over
conjugation preserves these properties. Set
\[
\widehat g(t)=g(t)\exp(-f(g)s_g(t)),
\]
with the second factor equal to one outside $U$. Multiplying the exponent
by a parameter in $[0,1]$ supplies the required homotopy. On $C$ the
modified path is constant. A commuting winding-zero path outside $C$
lies in the positive-real hemisphere; its lifted angles lie in
$(-\pi/2,\pi/2)$ and partial multiplication replaces them by
$(1-f)$ times those angles. It remains in that hemisphere. Thus all
such modified paths lie in the closed positive-real hemisphere, whose
distance from $-1$ is positive. The stated behavior at nonzero winding
follows from the choice of $U$.
\end{proof}
Apply the lemma to the normalized holonomy sweep of each sphere map.
Parallel transport compares the frames, and the continuous square root
of the determinant holonomy, starting at one, makes the sweep
$\SU(2)$-valued. The determinant evaluations on these spheres are even,
so the sweep ends at one. At a parallel splitting its eigen-winding is
$v(S)/2$. The equation $\widehat g(t)=-1$ has real codimension three with
one moving mark $t$, hence degree two. Subsequent equivariant cut
variations are absolutely smaller than half the avoidance gap in
Lemma~\ref{lem:analysis:zero-winding}. Include all comparison paths in
the basic support. The support is the image of a compact space of
dimension at most two, and no thickening is added to it for purposes of
incidence.
\begin{lemma}[Segmentation at a primary cut]\label{lem:analysis:segmentation}
The sphere representatives can be chosen so that, at a true $J$ face,
their event is the disjoint union of the events on the two half spheres,
with the prescribed signs. During the change from the whole-sphere rule
to this rule, a commuting path whose two half degrees vanish retains a
uniform avoidance gap. Every event on the face is intrinsic to its
assigned side; no derivative of transport along the growing neck enters
its definition.
\end{lemma}
\begin{proof}
The punctured traces are simply connected. The Hurewicz identification
of $\pi_2$ with $H_2$ shows that the sphere class and the signed sum of
the two half classes are represented by homotopic sphere maps in the
interval under consideration. Realize the latter map by concatenated
sweeps with identity intervals and fixed whiskers. Use the whole-path
operation until concatenation has been reached.
At concatenation write $s$ for the whole logarithm and $s_1,s_2$ for
the logarithms of the two zero-degree segments, extended by zero on
the other segment. On commuting data with both degrees zero,
$s=s_1+s_2$. Interpolate the multipliers by
\[
\exp(-(1-u)f s)\,
\exp(-u f_1s_1)\,\exp(-u f_2s_2),\qquad 0\leq u\leq1.
\]
At a point of a segment meeting the nonpositive hemisphere its own cutoff
is one, as is the whole cutoff. The exponent therefore cancels its
whole angle. If a segment remains in the positive hemisphere, any
partial cancellation still keeps its angle between its original value
and zero. This proves the same uniform gap throughout the interpolation.
The interpolation is equivariant and is defined on neighborhoods by the
same cutoffs as before. The metric construction places this switch in
the regimes where the old whole-sphere clamp is no longer required on
that side.
After switching, keep marks off the identity intervals. A whisker to
the second sweep and its inverse conjugate that sweep by a transport
independent of its internal mark. The equation that its value is $-1$
and the modification are conjugation-invariant, so they may instead be
defined in a frame intrinsic to that half. Variations of its cut use
the same intrinsic description. Whiskers and the earlier homotopy lie
in the original interval interior. At the actual split there is thus
no graph crossing $J$ and no length-dependent frame comparison.
\end{proof}
To retain support incidences through finite switches, retain the earlier compact
homotopy-support domains on their closed finite parameter ranges and
the intrinsic half-support domains on their subsequent closed ranges.
At a switching endpoint use the union of their limiting supports.
This is a finite collection of closed incidence charts of dimension
at most two in each parameter fiber; transversality is imposed also
on their boundary strata, whose parameter dimension can only decrease.
Consequently existing hits persist across every finite switch.
In a neighborhood of the true face only the later intrinsic half
domains remain. This convention changes neither the smoothly
switching event equation nor the boundary of the actual zero set.
It does not thicken a support or increase the released-degree budget.
For a cap surface or point use two complex sections of the rank-three
universal adjoint bundle at the evaluation point, and require the
resulting $3\times2$ matrix to have rank at most one. Its rank-one
stratum has complex codimension two; the rank-zero stratum has complex
codimension six. This is the usual degeneracy representative of
$c_2(\ad_{\C}E)=-p_1(\ad E)$, with the sign and normalization for the
chosen insertion kept outside the count. A variable point on an
immersed surface reduces the real degree from four to two. Choose
distinct surface representatives pairwise transverse with no triple
incidence; fixed cap points are general. All sphere supports are
disjoint from one another and from the cap tests. This is the exact
evaluation representative: the sampling graphs used to construct its
sections do not enlarge its basic incidence support.
Phase cuts are sections of character two for the original spinor phase
$\mathbb T$. They are determinant-one gauge invariant. Equation
perturbations use none of the sweep marks or surface evaluation variables.
A cut variation uses at most its own such variable, and phase sections
use none. These restrictions ensure the index accounting in
Section~\ref{sec:indices}.
\subsection{Sampling libraries and the finite isotropy construction}
For each connected graph choose one base frame. Its data consist of
finitely many based holonomies and spinor values transported to that
frame. Use bounded spinor samples, for example
$\Phi/(1+|\Phi|^2)^{1/2}$, when a global bound is needed. An output is
a fixed smooth form or spinor supported in a ball, transported from the
base, times a smooth bounded equivariant function of these data. The
same construction gives section values and small group-valued variations
by exponentiation. Include ordinary compact cutoffs in sample and
parameter charts.
For a finite small-ball cover of every input graph and output region
choose functions $\chi_\nu$ which are one on smaller balls still covering
these sets. Multiply the term by
\begin{equation}\label{eq:analysis:energy-cutoff}
\prod_\nu\vartheta_\nu\left(
\int_X\chi_\nu\bigl(|F_a^0|^2+|\Phi|^4\bigr)
+8\pi^2\sum_j m_j\chi_\nu(x_j)\right),
\end{equation}
where $\sum_jm_j[x_j]$ is the current ideal label, and each
$\vartheta_\nu$ is supported below a small regular-gauge threshold.
Its value can be one on the smaller neighborhood being tested. The
threshold is also smaller than a charge-one curvature atom. A finite
ball cover can always be made sufficiently fine at a smooth field and
away from specified particles. Crucially, each $\chi_\nu$ is confined
to the component of its associated graph. Weighted atoms enter this
formula additively, including at collisions.
Enumerate a dense countable library of these single-valued terms by
$p_j$. Choose positive weights $w_j$ as follows. List the desired
fixed-order operator bounds on bounded configuration and parameter sets,
the corresponding active-graph regularity bounds on nested balls, and
the exponential end bounds, as a countable list of seminorms $N_k$.
Arrange
\begin{equation}\label{eq:analysis:weights}
w_j\geq 2^j\max(1,N_1(p_j),\ldots,N_j(p_j)),
\qquad
\|\lambda\|_{\mathfrak P}=\sum_j w_j|\lambda_j|.
\end{equation}
For each fixed $k$ the finitely many ratios $N_k(p_j)/w_j$ with $j<k$
have a finite maximum $C_k$; for $j\geq k$ the ratio is at most $2^{-j}$.
Thus $\sum_j\lambda_jp_j$ converges in every listed norm, with a bound
depending on $k$, uniformly on bounded sets, and tails tend to zero there.
The parameter space $\mathfrak P$ is a separable weighted $\ell^1$ space.
Taking a small ball in it makes any prescribed finite collection of
equation norms as small as required. The field derivative of each term is compact from $E_a$
to $F_a$: graph evaluation and transport are of order zero, inclusion
from field order $a$ to equation order $a-1$ is compact on its compact
support, and derivatives of the energy cutoffs are finite-rank scalar
functionals. The tail estimates preserve compactness for the sum.
\begin{lemma}[Finite samples and available variations]
\label{lem:analysis:samples}
Finite framed samples, avoiding any specified finite particle set,
detect the stabilizer of a smooth field. They give arbitrary curvature
and retained cut variations at type $A$, and arbitrary tangential
curvature and active-line spinor variations at type $S$. With a type
$P$ witness they give arbitrary full equation and retained cut variations
at every non-$R$ target, using finite weighted branch lists if needed.
Phase-cut values can be varied using the witness alone.
\end{lemma}
\begin{proof}
A gauge stabilizing a connection is parallel and is determined by its
base value, which commutes with all based holonomies. It must also fix
all transported spinor values, up to the prescribed common phase when
phase is included. For $\SU(2)$ one noncentral holonomy reduces its
centralizer to a circle and two noncommuting ones reduce it to the
center. Spinor values impose finite-dimensional linear eigenspace
conditions. This proves finite detection; equivalently the stabilizer
is the common stabilizer of finitely many matrices and vectors in these
fixed finite-dimensional representations. Removing finitely many
points does not change the parallel-transport criterion. Paths can be
perturbed to avoid these points, retaining the finitely many strict
conditions that detect the stabilizer.
At $A$ the frame stabilizer is the center, which acts trivially on the
adjoint target. At $S$ the combined frame and phase stabilizer preserves
the ordered active line. Its invariant outputs are exactly diagonal
curvature and spinors in that line. In a slice tube for the compact
sample group an arbitrary invariant output at the center extends to an
equivariant function: first choose it on the slice, average over its
stabilizer, and extend by the group action. Cutoffs supported inside
the tube globalize it. Outputs in arbitrarily small balls span all
smooth compactly supported directions in the stated target. Separate
cut coefficients give their values independently of equation outputs.
The sample and parameter cutoffs may be supported away from the
zero-spinor reducible locus. At a central end, choices of a parallel
line create no extra parameter after unframing: its full connected
parallel group acts transitively on these lines and its elements extend
over the collar. The residual group is the splitting-preserving group
already used for the tangential equations.
For the mixed assertion use independent frames for the target and the
witness, together with the common phase. If a parallel gauge on a
$P$ field compensates a phase $z\ne\pm1$, its distinct eigenspaces are
parallel lines and the spinor lies in one of them. That would be type
$S$. Thus the projection of its stabilizer to phase is exactly
$\{1,-1\}$. With phase so restricted an $A$ target has only its central
stabilizer and an $S$ target only a finite stabilizer. The product sample
group therefore has finite isotropy $H$.
In a tube $G\times_HD$, choose an output $s$ on the slice and include
all translates $h\cdot s(h^{-1}\cdot)$, $h\in H$, each with weight
$1/|H|$. Multiply by an invariant cutoff whose support lies inside
the tube. On a neighborhood of its boundary every branch is zero;
extend outside by $|H|$ repeated zero branches. A change of slice
representative permutes the list with multiplicities, so this is a
well-defined weighted germ. Choose the coefficient space to contain
the whole $H$-span of each chosen output space. Every branch then has
the same full coefficient range at the center, giving surjectivity
branch by branch. A witness alone has stabilizer acting trivially on
the phase-character-two target, so arbitrary phase-section values are
available there even single-valuedly.
\end{proof}
\begin{proposition}[Ordinary exact-support library]
\label{prop:analysis:ordinary-library}
The preceding construction at zero spinor supplies the ordinary
representatives and perturbations used in Sections~\ref{sec:ordinary}
and \ref{sec:caps}. It is independent of the existence or value of the
closed cap count. It permits simultaneous regularity on the countably
many required lower strata and normal derivative regularization at the
specified harmonic reducers while fixing their values.
\end{proposition}
\begin{proof}
At an irreducible instanton all required adjoint outputs are available
by Lemma~\ref{lem:analysis:samples}. Universal surjectivity follows by
pairing a cokernel with arbitrary compactly supported smooth outputs
away from the finitely many labelled points; these outputs are dense
in $L^p$. Independent variations give the full value matrix of each
retained rank test and the three holonomy-event directions. The
transversality argument below applies on each position, rank, parameter,
and topology stratum. It uses no nonvanishing input.
For a specified harmonic reducer take a sample slice with its circle
stabilizer singled out by noncentral data. Infinitesimal based
holonomies detect a connection variation modulo infinitesimal gauge:
if every loop variation is simultaneous infinitesimal conjugation,
transport that infinitesimal base value along paths; the loop identity
makes the result independent of the path and its covariant derivative
is the given variation. A finite number of samples thus injects the
finite-dimensional normal kernel. Localized output forms detect its
finite-dimensional cokernel. Slice-linear equivariant variations
therefore prescribe any complex-linear kernel-to-cokernel map in the
normal circle representation. They vanish on the commuting slice and
leave the Abelian equation unchanged. A generic arbitrarily small map
makes a complex index-zero block invertible, or a complex index-one
block surjective. Independent cut variations prescribe the corresponding
normal cut derivative. Subsequent irreducible variations are supported
off the reducible locus.
On an ordinary cylinder use the time-independent gradient library for
the equation, whose irreducible transversality is established in
Section~\ref{sec:ordinary}. For a cut based at a moving mark translate
its graph with that mark. This is joint translation invariance of the
marked problem; the equation still does not depend on the mark.
Energy cutoffs and labelled atoms translate with it. The face and end
prescriptions above make these choices agree with compact probes on
the pieces and with their exponentially small extensions. The gauge
group and the fixed transition identifications are those already
specified, so this construction introduces neither an $\SO(3)$ lift
sum nor a new matching factor. Persistence and particle accounting
are proved in Lemma~\ref{lem:analysis:incidence}.
\end{proof}
\begin{lemma}[Independence from discarded fields]
\label{lem:analysis:R-independence}
The coupled terms can be supported so that they vanish on every tuple
of fixed type. On an exact $J$ face a term involving a connected graph
in an $R$ component vanishes on a neighborhood of that sample. After
discarding $R$ fields, their particles, and their tests, every remaining
equation and cut is a function only of the retained fields and particles
and of the retained parameter variables. The same assertion holds for
separated lens-cap data.
\end{lemma}
\begin{proof}
Support a mixed term compactly inside the finite-isotropy locus of its
independent-frame sample space. In a tuple of fixed type the common
phase can be compensated independently on every $S$ component, while
it acts trivially on every zero-spinor component. The combined
stabilizer is therefore continuous. An $R$ input or output has a
continuous independent frame stabilizer even when another component
contains a witness. Both types of sample are outside the closed
support of the term.
No frame is compared between the disconnected graphs. On a split
face the only nonzero coupled terms consequently sample retained
components. The energy cutoffs of those terms have balls on precisely
those components, so the atomic summands in
\eqref{eq:analysis:energy-cutoff} contain no discarded positions.
First-stage terms are single-component and already have this property.
Parameter cutoffs use only allowed variables; the parameters of an
$R$ component may remain in the projection, as stipulated in its
dimension count. Lens-cap sampling and coupling have been disabled
separately. This proves independence for the entire formula, including
the scalar cutoff factors, not just for its visible output.
\end{proof}
\subsection{Differentiability and universal regularity}
Moving a sampling path is not a smooth operation at unchanged field
regularity. We use the following precise version of the regularity
argument, rather than differentiating translations indefinitely on one
Banach space.
\begin{lemma}[Compatible implicit charts]\label{lem:analysis:scale}
Let $E_s=c^{a+s,\alpha}_\delta$ and
$F_s=c^{a+s-1,\alpha}_\delta$, with the finite-dimensional equation,
parameter, and cut variables included. For the sampling formulas above,
the equation map at level $s$ is $C^1$. For every prescribed $j$ it is
$C^j$ from sufficiently high field order to $F_s$. At a smooth
universally surjective zero its universal solution set has $C^j$ local
charts, for every prescribed finite $j$, with the same perturbation
space $\mathfrak P$ and the same smooth finite-dimensional kernel
coordinates at all levels.
\end{lemma}
\begin{proof}
The derivative of evaluation along a moving path contains one derivative
of the field; transport satisfies the same assertion by differentiating
its ordinary differential equation. Little-H\"older translation is
continuous, so field order $a+s$ gives a continuous first derivative
with target order $a+s-1$. Additional parameter differentiations use
additional field derivatives. Smooth functions, finite products,
integration in \eqref{eq:analysis:energy-cutoff}, and the summability
in \eqref{eq:analysis:weights} preserve these assertions. Metrics and
bundles are compared in fixed local identifications, so their variation
only changes smooth coefficients of the differential operators.
Write the field linearization as $L_s:E_s\to F_s$. It is elliptic plus
a compact derivative. At a smooth solution its kernel is smooth and
independent of $s$, and higher-order right-hand sides give higher-order
solutions. For the graph terms this is the same elliptic bootstrap as
below: their field derivatives have order zero and the cutoff derivatives
are scalar functionals multiplying smooth outputs. Pick finitely many
smooth perturbation directions $Q\subset\mathfrak P$ spanning the
cokernel at the lowest level. Linear regularity shows that
$\widetilde L_s:E_s\oplus Q\to F_s$ is surjective at every level and
has a common smooth finite-dimensional kernel $K$. Choose a projection
onto $K$ using finitely many compactly supported smooth pairings and
coordinates on $Q$. Its kernels $V_s$ are compatible complements, and
$\widetilde L_s:V_s\to F_s$ is an isomorphism.
Write $\mathfrak P=Q\oplus\mathfrak P'$ and
$Z=K\oplus\mathfrak P'$. The $C^1$ implicit theorem at each level
gives a graph $v_s:U_s\subset Z\to V_s$. On sufficiently small
common domains uniqueness of the lowest-level graph makes the graphs
agree. For any finite list of higher levels we may intersect their
domains; thus the common graph $v$ is $C^1$ into every level needed in
a given finite-order argument.
Here is the higher-order step. In graph coordinates write
$\mathcal F(v(z),z)=0$ and
$B_s(z)=D_v\mathcal F(v(z),z):V_s\to F_s$. It remains invertible
near the base point. If $v$ is $C^r$ into the requisite higher levels,
its differentiated equation has the form
\[
B_s(z)H_r(z)=R_r(z),\qquad H_r=D^rv,
\]
where $R_1=-D_z\mathcal F$, and for $r\geq2$ the right-hand side
uses derivatives of $v$ only up to order $r-1$. Choose a level $t$
high enough that $\mathcal F:V_t\times Z\to F_s$ is $C^{r+1}$.
Then $R_r$ and $B_s|_{V_t}$ are $C^1$, while $H_r$ is continuous
into $V_t$. Subtracting the equations at $z+h$ and $z$ gives
\begin{align*}
B_s(z)\bigl(H_r(z+h)-H_r(z)\bigr)
={}&R_r(z+h)-R_r(z)\\
&-\bigl(B_s(z+h)-B_s(z)\bigr)H_r(z+h).
\end{align*}
Apply the fixed inverse $B_s(z)^{-1}$. The right-hand side has a
first-order expansion with $o(\|h\|)$ remainder, using continuity of
$H_r$ at level $t$. This proves differentiability into $V_s$; the
same expression proves continuity of the derivative. Induction gives
$C^{r+1}$ charts at the lower level. Only finitely many higher charts
are needed for any specified order. In particular this proof does not
differentiate the inverse repeatedly as a same-order operator.
\end{proof}
To test universal surjectivity, first use the augmented linearization
$W^{1,p}_{\delta_S}\to L^p_{\delta_S}$ with $p>4$ at a fixed smooth
zero in the weight-$\delta_H$ configuration space.
The perturbation outputs span the curvature, Dirac, and cut targets,
not the added gauge-adjoint target. The latter cokernel component is
removed first: pairing with infinitesimal gauge variations gives the
gauge Laplacian $G^*G$, where $G$ is the infinitesimal gauge action.
A dual solution $\xi$ has weight $-\delta_S$. On an end the
limiting gauge operator is $-\partial_t^2+\Delta_\gamma$.
Its spectral decomposition and elliptic regularity give
\[
\xi(t)=a_e+t b_e+O(e^{-\epsilon t}),\qquad
a_e,b_e\in\ker\Delta_\gamma,
\]
with exponential decay in the complementary modes. Growing normal
modes are excluded by the choice of weight. Exponentially decreasing
coefficient errors preserve this expansion, after decreasing
$\epsilon$ to absorb polynomial factors. For every independently
allowed cut-off parallel end constant $\chi_e c$, pair the cokernel
with the field variation $G(\chi_e c)$. Gauge equivariance kills
the equation and cut rows. Green's formula for the gauge row gives
\[
0=\int_X\langle\xi,G^*G(\chi_e c)\rangle
=-\langle b_e,c\rangle_{L^2(Y_e)}.
\]
Thus every $b_e$ is zero. The dual solution is bounded with decaying
derivative, so a second integration by parts gives
$\|G\xi\|_2^2=0$. It is a global infinitesimal stabilizer and is
zero on the non-$R$ slices used here. More generally the extended
gauge Laplacian has kernel and cokernel $\mathfrak s$, and induces
an isomorphism from the gauge domain modulo $\mathfrak s$ to
$\operatorname{ann}(\mathfrak s)$. This also explains the target
convention specified above. The argument applies to tangential
non-$R$ slices and to infinitesimal local lifts of finite branches.
Compactly supported smooth outputs away from finitely many particles
are dense in its equation target. The variations in
Lemma~\ref{lem:analysis:samples} consequently annihilate no nonzero
cokernel in the required full or tangential target. Field evaluation
on the compact paths is continuous at this level. Independent section
variations give the retained finite-dimensional cut targets.
Linear elliptic regularity gives the same surjectivity at higher orders and
weight $\delta_H$. To justify the weight step explicitly, an
$L^p_{\delta_S}$ solution with right-hand side in
$c^{a-1,\alpha}_{\delta_H}$ satisfies on each sufficiently remote
end the limiting spectral equation plus a small decreasing-coefficient
error and the summable graph tails. The positive and negative spectral
integrals improve its decay to $\delta_H$, since the interval
$[\delta_S,\delta_H]$ contains no indicial root; parallel gauge
modes have just been treated by their separate end constants.
The error is absorbed in the weight-$\delta_H$ norm, and graph tails
in \eqref{eq:analysis:exponential-seminorm} are chosen faster than
$\delta_H$. Interior Schauder estimates then give the asserted
H\"older order. The same argument identifies the smooth kernels and
the finite-dimensional cokernel quotients at both weights, including
the finite extended gauge coordinates. The cokernel identification
uses the range and regularity of solutions, not smooth representatives
of the full adjoint cokernel: adjoints of thin samples can have path-
or point-supported distribution terms. Density of smooth outputs
detects those cokernels without any smoothness assertion.
It does not presume that equal-weight H\"older
closure embeds in weighted Sobolev space.
Equivariance makes gauge directions redundant, also on local branch
lifts. Apply parametric transversality in the charts of
Lemma~\ref{lem:analysis:scale}; the differentiability order can always
be chosen larger than both zero and the relevant Fredholm index
\cite[Theorem~1.3 and Corollary~1.5]{Smale1965}.
Finite isotropy is handled on the local finite slice
covers. Rank-one and rank-zero matrix strata are treated separately,
with variations of the full matrices available before either restriction.
\subsection{Energy, regularity, and ideal limits}
\begin{lemma}[Compactness on bodies and removable particles]
\label{lem:analysis:compactness}
For the fixed finite family, and for perturbations in a sufficiently
small bounded set of the norms above, the following statements hold
uniformly, including coefficient-convergent sequences of already ideal
solutions.
\begin{enumerate}
\item The trace-free curvature and covariant spinor gradient are bounded
in $L^2$, and the spinor is bounded in $L^4$.
\item The measure $|\Phi|^4\,d\Vol$ has no atoms in a weak limit,
including a limit obtained by translating along a true cylinder.
Neither does $|F_a^{0,+}|^2\,d\Vol$.
\item On compact sets, a subsequence converges modulo gauge away from
finitely many points. The limiting field extends across these points
and solves the same equations with the ideal cutoff formula
\eqref{eq:analysis:energy-cutoff}. The lost charges are positive
integers, and each charge-$m$ atom has mass $8\pi^2m$ in the stated
curvature norm.
\item On any fixed ideal stratum, if there is no further bubbling or
end loss, convergence is in the completed higher-order field charts.
At a new bubble there is still convergence in $C^{0,\beta}$ on surviving
exact cut paths for any fixed $\beta<1$.
\end{enumerate}
\end{lemma}
\begin{proof}
The integrated inequality of Lemma~\ref{lem:estimates:local}, combined
with Proposition~\ref{prop:estimates:energy} and Chern--Weil, gives
\begin{equation}\label{eq:analysis:global-energy}
\|F_a^0\|_2^2+\|\nabla_a\Phi\|_2^2+\|\Phi\|_4^4\leq C.
\end{equation}
Here the negative zeroth-order coefficients have uniformly bounded
squared integral. The true collar middles have positive scalar
curvature, asymptotically flat background, and exponentially small
forcing; the remaining finite body, including all finite $Y$ seams,
has a fixed bound. The perturbation values have bounded squared
integral by construction. On completed ends first use decaying cutoffs
and then pass to the limit. The integrated Dirac identity uses
$\|P_D\|_2^2$, not a derivative of $P_D$. The curvature identity is
\[
\|F_a^0\|_2^2=2\|F_a^{0,+}\|_2^2+8\pi^2\kappa.
\]
The charge bound and the positive quartic term therefore give all
three terms in \eqref{eq:analysis:global-energy}.
For the nonconcentration statement take a cutoff equal to one on $B_r$
and supported in $B_{2r}$. The local inequality gives, with harmless
positive constants and uniformly bounded local geometry,
\begin{equation}\label{eq:analysis:no-atom}
c\int_{B_r}|\Phi|^4
\leq C r^{-2}\int_{B_{2r}\setminus B_r}|\Phi|^2+o_r(1)
\leq C\left(\int_{B_{2r}\setminus B_r}|\Phi|^4\right)^{1/2}
+o_r(1).
\end{equation}
The lower-order and bounded inhomogeneous terms tend to zero on these
shrinking balls; a uniform $L^4$ bound and H\"older's inequality give
this also for a bounded coefficient times $|\Phi|^2$. First pass to
a weak measure limit at radii with zero boundary mass, then let $r$
tend to zero along such radii. The limiting annulus masses tend to
zero even if there were an atom at the center. Equation
\eqref{eq:analysis:no-atom} forces that atom to vanish. The same local
argument applies after translation on a true cylinder. Since
$F_a^{0,+}$ is quadratic in $\Phi$ plus a bounded perturbation,
its squared norm has no atom either.
Fix a ball outside curvature concentration points. Small-energy
Coulomb gauge gives a $W^{1,2}$ connection form $a$ small in $L^4$
after choosing the ball and the energy threshold small enough. The
spinor has $W^{1,2}$ control from
\eqref{eq:analysis:global-energy} and is small in $L^4$ on this ball,
by the nonconcentration just proved. In these gauges the augmented
first-order system for $u=(a,\Phi)$ is
\begin{equation}\label{eq:analysis:critical-system}
\mathcal D u=C(u)u+b,
\qquad |C(u)|\leq C|u|,
\end{equation}
where $b$ is uniformly bounded; bounded lower-order linear terms may
be included in $\mathcal D$ or $b$. For $2<p<4$, the critical Sobolev
estimate gives on a smaller ball
\[
\|u\|_{W^{1,p}}
\leq C_p\bigl(\|u\|_{L^p}+\|b\|_{L^p}
+\|C(u)\|_4\|u\|_{4p/(4-p)}\bigr).
\]
Absorb the last term by making the $L^4$ coefficient small. For a
weak field the same step is justified by a local parametrix: freeze
the principal symbol on a sufficiently small chart, absorb its small
variation and the $L^4$ potential, and solve at exponents $2$ and $p$.
The cutoff commutator contains $u\in L^4$, hence belongs to $L^p$;
uniqueness of the small-potential inverse identifies the two solutions.
Choose $p>8/3$, so $4p/(4-p)>8$. The quadratic terms are then in an
$L^q$ space with $q>4$. A second elliptic estimate gives local
boundedness and a positive H\"older exponent. Equation
\eqref{eq:analysis:critical-system} now gives $W^{1,q}$ for every
finite $q$, hence $C^{0,\beta}$ for every fixed $\beta<1$ on smaller
balls. This part of the argument needs only bounds on perturbation
values, so it precedes any differentiation of their graph formulas.
For an active graph, its cutoff balls have this regularity with uniform
bounds and the graph lies in their smaller interiors. Transport and
sample formulas consequently acquire their first regularity bounds.
When there is no ordinary $Y$ term sampling a developing concentration
point, differentiating the equations and using nested smaller radii gives
the next elliptic bounds. Repeating this proves every fixed order. The
diagonal choice \eqref{eq:analysis:weights} includes the constants at
each step, so the countable sum obeys the same bounds. Transition
gauges have corresponding bounds from the connection transformation
equation. This gives full bounds on a no-new-bubbling stratum and
smoothness of a single weak solution once it extends over the labelled
points. It does not assert uniform high derivatives for an ordinary
$Y$ term whose loop family passes near a new particle.
There is one separate convergence issue for the ordinary solid-torus
$Y$ terms. Thicken every transport leg, including any comparison
whisker, with the transverse disk of bases; no fixed common whisker
or common point is left in the family. On each leg the map from
its three-dimensional path-and-disk domain to the spatial three-manifold
is submersive in its interior. The preimage of a spatial particle
is discrete there, so its projection to the disk is a null set;
the moving endpoint maps give the same conclusion on boundary pieces.
There are only finitely many legs in a term. Alternatively, for
these slice-wise gradient terms, all input paths avoid a finite
four-dimensional particle set at every time except its finitely many
time coordinates, which already suffices for almost-everywhere
spacetime convergence. On the good loops transport converges, and
the bounded integrands give strong local $L^p$
convergence of the averaged perturbation values by dominated
convergence. Thus the equations and local elliptic estimates give
strong first-order convergence off the particles. Interpolation with
the uniform higher-integrability bounds gives $C^{0,\beta}$ convergence
on compact good sets for any prescribed $\beta<1$, in particular with
$\beta>b$ as needed for the modified sweeps. If no new bubbling
occurs, the full graph bootstrap, including these thickened-loop terms,
gives convergence at every order. We require only the stated lower
order convergence at a new bubble.
Choose the concentration threshold before the graph cutoff thresholds.
Every new concentration point carries at least that fixed positive
amount of curvature energy. A graph meeting it has a covering cutoff
which equals one there and is therefore disabled. For any other graph,
weak convergence of the energy measures gives exactly the scalar
cutoff limit in \eqref{eq:analysis:energy-cutoff}. This proves
convergence of the limiting equations wherever they are defined;
no transport or evaluation at an atom is required. The energy bound
leaves only finitely many new points.
For completeness, the limiting punctures are removable in the required
perturbed problem. On a sequence of shrinking spherical annuli the
limiting $L^2$ curvature tends to zero. Rescaled annular Coulomb gauges
are then small in $W^{1,2}$ and $L^4$; the flat annular limit is trivial
since the annulus retracts to $S^3$. Cutting the connection to zero
inside such an annulus costs arbitrarily small $L^2$ curvature: the
cutoff derivative is controlled by the rescaled $L^4$ bound. Apply
small-ball gauges to these fillings and pass weakly to obtain a
$W^{1,2}$ extension in an extended bundle. On the fixed outer collar
the gauges give the same original connection up to a bundle
identification. The spinor extends in $W^{1,2}$, and point cutoffs show
that the equations extend distributionally. Terms whose graphs meet
the labelled atom are already off. The preceding weak bootstrap makes
the extension regular. This is the local small-energy removal
argument of \cite{Uhlenbeck1982Removable} with the bounded inhomogeneous
terms retained.
Compare the Chern integrals of the original sequence and this extension
on a surrounding sphere collar of strong $W^{1,p}$ convergence, with
$p>4$. Fubini's theorem selects a boundary sphere on which a subsequence
converges in tangential $W^{1,p}$; the Chern--Simons boundary integrals
then converge. More precisely the difference of the two ball charge
integrals is the degree of their determinant-one clutching map
$g_j:S^3\to\SU(2)$ plus the difference of the boundary
Chern--Simons integrals in the convergent gauges. The latter tends
to zero. Bounded charge bounds the integer degrees, so a subsequence
has constant degree and the limiting charge difference is that integer.
Since the self-dual part has no
atom, Chern--Weil identifies the lost curvature mass with $8\pi^2$
times that integer. The mass is positive, so the integer is positive.
Previously labelled weights simply add to new weights; collisions give
the same cutoff expression because the summands $m_j\chi(x_j)$ add.
The limiting equations on every fixed position stratum are therefore
smooth, particle-dependent equations of the stipulated form.
All arguments are uniform for coefficients converging in the weighted
seminorms: approximate by a finite library, use the finite convergence
argument, and use the uniform tail bounds. If no further degeneration
occurs, the full bootstrap and the true-end estimate below give
convergence in the completed field charts. This proves the lemma.
\end{proof}
\begin{lemma}[True necks and exponential completion]
\label{lem:analysis:neck}
Along a sequence in which genuine neck lengths tend to infinity, a
subsequence has finitely many particles and nonconstant cylinder levels.
Every translated cylinder level has zero spinor and is ASD. Nonconstant
levels have a discrete positive charge cost. The remaining necks carry
no diffuse loss and converge to matching flat states with exponential
decay of all derivatives. On a no-loss stretch of length $T$ the
deviation from the flat orbit is bounded by exponentially decreasing
contributions from its two ends and from the prescribed forcing.
\end{lemma}
\begin{proof}
Deep translation eliminates the background splice errors and all
nonstationary graph terms. The limit is an unperturbed solution on a
positive-scalar-curvature cylinder with flat fixed-line background.
The integrated Weitzenb\"ock inequality with cutoffs and finite energy
forces its spinor to be zero. It is therefore ASD. We first prove
the small-strip estimate and completion of any one such finite-energy
level, without assuming a finite extraction count.
The flat orbits on $S^3$ and on the lens space form a finite list.
For a lens this follows by diagonalizing the representation of its
finite cyclic fundamental group. Their adjoint $H^1$ vanishes;
one may also obtain this by averaging over the finite covering group.
Positive scalar curvature kills the flat Dirac kernel. Thus their
normal boundary operators have a common positive gap $\beta$.
On any stretch close to a flat $\gamma$, put the spatial connection
$a=\gamma+\xi$ in the slice $d_\gamma^*\xi=0$. Use the parallel
group to remove the parallel component of the temporal connection.
The remaining temporal component $\alpha$ is determined, on the
complement of constants, by an equation of the form
\[
d_\gamma^*d_a\alpha=d_\gamma^*\bigl(Q(\xi,\Phi)+f_+\bigr),
\]
where $Q$ is quadratic. Its inverse from $H^{-1}$ to $H^1$ gives
$\|d_a\alpha\|_2\leq C(\|Q\|_2+\|f_+\|_2)$, uniformly for
small $\xi$. Hence the projected nonlinear term is of order zero
in the low norm. Writing $u=(\xi,\Phi)$, the projected equation is
\begin{equation}\label{eq:analysis:neck-evolution}
(\partial_t+B_\gamma)u=N(u)+f(t),
\qquad \|N(u(t))\|_{L^2(Y)}
\leq C\varepsilon\|u(t)\|_{L^2(Y)}.
\end{equation}
The transformed forcing $f$ has the same strip Sobolev bounds as the
original forcing. The operator $B_\gamma$ is the co-closed
$*_3d_\gamma$ block together with the flat Dirac block; all parallel
gauge constants have been separated. Higher-order tame estimates
give $\|N(u)\|_{H^{k-1}(I)}\leq C_k\varepsilon\|u\|_{H^k(I)}$
on a fixed-size strip, with a fixed enlargement of $I$ if required.
Smallness in the needed local norms follows from small-energy
regularity on these genuine collars.
Here are the norms in the decay estimate. Move the ends by less
than one, absorbing these bounded collars into the end norms, so
the length of the stretch is an integer $M\geq6$. Put
$I_j=[j,j+1]\times Y$ and
$I_j^+=[\max(0,j-2),\min(M,j+3)]\times Y$, for $0\leq j<M$.
Set
\[
A_j=\|u\|_{H^k(I_j)},\qquad
F_j=\|f\|_{H^{k-1}(I_j^+)},\qquad
A_-=\|u\|_{H^k([0,3]\times Y)},\quad
A_+=\|u\|_{H^k([M-3,M]\times Y)}.
\]
For any sufficiently small $0<\delta<\beta$ one has
\begin{align}\label{eq:analysis:neck-decay}
A_j\leq{}&C_k\left(e^{-\delta j}A_-
+e^{-\delta(M-1-j)}A_+\right)\notag\\
&+C_k\sum_{r=0}^{M-1}e^{-\delta|j-r|}F_r.
\end{align}
Its constants are independent of $M$. To prove it, first propagate
the positive spectral subspace of $B_\gamma$ from the left and the
negative subspace from the right in $L^2(Y)$. The two semigroups
have $L^2$ operator norm at most $e^{-\beta|t-s|}$. Integrating over
unit strips and using Cauchy--Schwarz yields, with
$L_j=\|u\|_{L^2(I_j)}$, the discrete bound
\[
L_j\leq C\left(e^{-\beta j}A_-+e^{-\beta(M-1-j)}A_+\right)
+C\sum_r e^{-\beta|j-r|}(\varepsilon L_r+F_r).
\]
End traces may be chosen in the first and last unit strips by
Fubini, or bounded by $A_\pm$. The exponentially weighted sequence
norm with rate $\delta<\beta$ bounds this convolution uniformly in
$M$, so sufficiently small $\varepsilon$ absorbs its $L_r$ term.
Next, the interior elliptic estimate for $\partial_t+B_\gamma$ on
overlapping strips gives
\[
A_j\leq C_k\left(\|u\|_{L^2(I_j^+)}+F_j
+\varepsilon\|u\|_{H^k(I_j^+)}\right)
\]
away from the two end collars. Only a fixed number of neighboring
$A_r$ enter the last norm. In the same exponentially weighted
sequence norm this finite-neighbor operator is bounded uniformly in
$M$; absorb it after further decreasing $\varepsilon$. Substituting
the low-norm bound proves \eqref{eq:analysis:neck-decay}, with a
possibly smaller $\delta$. This two-stage argument uses a nonsingular
$L^2$ semigroup estimate and then elliptic regularity. It does not
assert a bounded $H^{k-1}\to H^k$ semigroup kernel at $t=s$.
Increasing $k$ gives all required spatial and temporal derivatives.
Now consider either tail of one finite-energy ASD cylinder level.
Its unit-strip energies tend to zero. Local small-energy compactness
therefore puts all sufficiently remote strips near the finite set of
flat orbits. Overlapping strips and continuity force an entire tail
to remain near one orbit, since distinct sufficiently small flat
neighborhoods are disjoint. The slice and estimate just proved apply
on arbitrary finite subintervals of that tail. Sending their remote
endpoint to infinity gives exponential convergence to the flat;
boundedness of its end norm follows from small-energy regularity.
Only now apply the Chern--Simons identity to its completed ends.
The finite flat list has discrete Chern--Simons differences modulo
integer charge, and a nonconstant ASD level has positive energy.
Consequently these levels have a uniformly positive charge cost.
Together with the integral particle masses and
\eqref{eq:analysis:global-energy}, this bounds the extraction count.
After those finitely many extractions every remaining unit strip is
close to a flat orbit; a violating strip translated to the origin
would produce a further level or particle. The same slice argument
and \eqref{eq:analysis:neck-decay} apply to each residual stretch.
They keep it in one flat neighborhood and give exponential decay
from its two ends, plus the exponentially weighted forcing sum.
There is no diffuse charge in this residual stretch. In the slice
neighborhood choose one lift of Chern--Simons; its difference between
two slices tending to the same flat tends to zero. This is the
topological charge between them. The positive local Weitzenb\"ock
estimate, or \eqref{eq:analysis:neck-decay}, controls the spinor and
self-dual energy there. Chern--Weil then controls the remaining
curvature energy. Thus no energy is missed by the discrete extractions.
Exponential forcing gives exponential completion and, in a no-loss
middle, exponential smallness. Comparisons by the compact residual
parallel groups at the ends are allowed by our gauge convention.
The forcing bounds are uniform under arbitrary simultaneous approaches
to true faces by \eqref{eq:analysis:exponential-seminorm} and the
product collars. The removed parallel temporal component is a pure
gauge mode, so a slowly varying comparison of end constants cannot
produce a residual polynomial forcing term.
This lemma concerns isolated rational flat states. The harmonic
flat-slice argument at an ordinary $S^1\times S^2$ auxiliary neck is
the separate one in Section~\ref{sec:ordinary}; no assertion about
that positive-dimensional flat variety is being deduced from the
spectral gap used here.
\end{proof}
\begin{lemma}[Exact support and incidence budget]
\label{lem:analysis:incidence}
At an ideal limit a basic sphere, surface, or point test persists unless
a particle meets its exact basic support. In that case one may drop
the test with its auxiliary evaluation variable, while retaining the
particle's incidence with that support. For every particle of weight
$m$, the dimension of its permitted position plus the degrees released
by all such dropped tests is at most $4m$. The zero-evaluation sphere
rules in Definition~\ref{def:indices:analytic-hypotheses} hold, with
different charged tests using distinct loss units.
\end{lemma}
\begin{proof}
If no particle meets the basic sphere support, the sphere path converges in
$C^{0,\beta}$ with $\beta>b$, hence in the $H^b$ topology needed in
Lemma~\ref{lem:analysis:zero-winding}. The modified sweep and every
surviving correction converge. A correction graph hitting an atom
vanishes by \eqref{eq:analysis:energy-cutoff}; this does not invalidate
the limiting section formula. Thus the test persists unless its basic
sweep support itself is hit. At a $J$ face the sweep has already been
segmented; the mark lies on exactly one nonidentity segment and assigns
the event to that side. Identity segments are uniformly disjoint from
the event and cannot be limits of its marks. Cap sections at an
unhit evaluation point have the same convergence and cutoff argument.
Phase sections and all other sampled cut terms converge by their
cutoff formulas; at a fixed tuple phase cuts are not needed for the
fixed-type exclusion.
The position and released-degree counts are explicit. A sphere or
surface hit gives at most two position parameters and releases degree
two. For a possibly degenerate sphere map, retain a preimage in its
compact two-dimensional parameter domain; injectivity is unnecessary
for this upper bound. A fixed point gives no position parameter and
releases four degrees. Two distinct cap surfaces meet transversely,
so a simultaneous hit has zero position dimension and releases four
degrees. There are no triple incidences, competing sphere supports,
or sphere--cap incidences. An unconstrained particle has four position
parameters and releases no degree. These cases give at most four for
weight one; multiplicities only increase the allowed bound $4m$.
The lower-rank matrix stratum has higher codimension and cannot increase
this budget.
For a parallel field, a whole test with zero evaluation is impossible
without a support hit by Lemma~\ref{lem:analysis:zero-winding} and the
absolute smallness of its variations. During segmentation the same
conclusion applies when both half evaluations vanish, by
Lemma~\ref{lem:analysis:segmentation}. At the split it applies to the
assigned half separately. At a lens face the whole test lies in $N$;
the charge-zero extendibly central reference has the same avoidance,
and the index and charge table of
Proposition~\ref{prop:indices:lens-index} charges any remaining permitted
alternative to the stated excess. Disjoint basic sphere supports
prevent one particle unit from paying for two such tests. Assign a
unit to each witnessing particle on a main support, and assign the
necessary excess in a tested lens cap to that cap's single sphere.
The different caps are disjoint. Near a $J$ separation each incident
particle remains on the intrinsic compact support of its assigned
half, so it cannot escape into the connector while carrying a second
test. A particle carried into a newly separated $N$ transfers its
integer weight to that cap's excess; when its evaluation is zero the
reference charge is zero, so this weight still pays for its one test.
Unconstrained particles escaping to a connector carry their charge
there but create no additional zero-degree support incidence.
Colliding labels add their weights and never identify two charge units.
These assignments, with each connector charge allocated once in the
virtual decomposition, prove the distinct-unit assertion.
In particular a main monopole particle removes six field dimensions
and restores at most four through incidence and position variables.
Its net penalty is at least two. The ordinary instanton loss is eight,
so its corresponding penalty is at least four.
\end{proof}
\subsection{Candidate main fields and their degenerations}
\label{subsec:analysis:candidates}
The dimension argument of Section~\ref{sec:indices} will be applied to
a larger class than the actual ideal limits. We keep the equations on
every main component but replace the lens-cap and cylinder fields by their
possible charge assignments and impose no matching across true cuts.
The retained equations will give ideal compactness for these main fields;
the assigned losses will bound the number of degenerations. Neither
assertion will require reconstruction of the omitted fields.
A \emph{candidate} consists of completed main fields, their parameter
values, weighted main particles, retained cut variables, and a discrete
charge assignment. The main fields solve their equations and all
retained nonphase cuts for the current perturbation data. When phase
sections have been installed, we also impose those cuts. For
single-valued data this is a single system of equations; after finite
branch lists have been added, a candidate solves one local lift of that
library. The class includes every actual ideal boundary tuple for the
same data, but its omitted lens caps and matchings need not be reconstructible.
The charge and incidence data are as follows. At a $J$ connector
record its additional nonnegative integer charge; at a lens record an
effective cap charge in the range and congruence determined by the
main flat state and its required sphere test. Retain the total charge
condition. At a labelled point on a basic support allow the
corresponding test and its auxiliary evaluation variable to be dropped,
but retain its exact support incidence
as in Lemma~\ref{lem:analysis:incidence}. There are finitely many
assignments of a segmented test to a side at each $J$ face.
At a finite switching value use the closed support charts and limiting
union specified after Lemma~\ref{lem:analysis:segmentation}; near a true
$J$ face only the intrinsic half-support charts occur. Include the
boundary strata of these closed parameter ranges among the finite
support-stratum choices.
We first bound the candidate fields and charges. The main charges
need not be nonnegative, so the total charge condition alone is not
an energy bound for each piece. Apply the integrated Weitzenb\"ock
inequality on each completed main piece first, using
Young's inequality on its negative zeroth-order terms. Bounded
squared costs of those terms and of the forcing give uniform bounds
on $\|\Phi_i\|_4^4$ and $\|\nabla\Phi_i\|_2^2$, independently of
the charge. The curvature equation then gives
$\|F_i^{0,+}\|_2^2\leq B_i$. Write
$\kappa_i^{\rm main}$ for the actual charge of this smooth completed
main field, excluding particles and omitted end fields; this is
distinct from the virtually filled $\kappa_i$ of
Section~\ref{sec:indices}. Thus
\begin{align}\label{eq:analysis:candidate-energy-order}
\kappa_i^{\rm main}&\geq-\frac{B_i}{4\pi^2},\qquad
\mathcal L\leq\kappa^{\rm tot}+\sum_i\frac{B_i}{4\pi^2},\notag\\
\kappa_i^{\rm main}&\leq\kappa^{\rm tot}+
\sum_{j\ne i}\frac{B_j}{4\pi^2}.
\end{align}
Here $\kappa^{\rm tot}=\sum_i\kappa_i^{\rm main}+\mathcal L$ and
$\mathcal L\geq0$ is the omitted effective end and particle loss.
Finally
$\|F_i^0\|_2^2\leq2B_i+8\pi^2\kappa_i^{\rm main}$ gives the full curvature
bound. The finitely many component and face types have uniform
constants $B_i$. This argument requires only the completed main
equations and the assigned nonnegative losses. It supplies the energy
bound for the degeneration argument that follows.
\begin{lemma}[Finiteness of degeneration data]
\label{lem:analysis:finite-data}
For this finite problem the degeneration data needed for candidate
exclusion have finite ranges.
Order the data by: deeper true faces;
larger total effective end and particle loss; fewer distinct main
particle locations at fixed loss; more dropped tests; and more $R$
components. Treat finite assignment, closed support-stratum, and rank
choices together at each step. In a fixed finite branch library, if the single-valued
coefficients converge in $\mathfrak P$ and the additional finite
coefficients converge, every sequence of candidates has a subsequence
which either converges modulo the completed field slices at the same
step or limits to an earlier step or a fixed-type tuple.
\end{lemma}
\begin{proof}
There are finitely many true faces and flat-state assignments. The
energy bound gives a uniform upper bound on all nonnegative end
losses and particle weights. These charges lie in discrete sets
(quarters suffice for the lens data and integers for particles and
$J$ losses), so their ranges and the possible numbers of positions
are finite. There are finitely many basic tests and components.
Relative bundle sectors are also finite after inessential boundary
degree relabellings. With determinant fixed and limiting flats fixed,
relative $c_2$ is an integer and is bounded by Chern--Weil and the
candidate curvature bound just proved. For split fields, every possible
completed core has rational-homology-sphere boundary, so compactly
supported two-forms detect all of its real second cohomology. A finite
dual collection $\beta_j$ satisfies
\[
|\langle v,[\beta_j]\rangle|
\leq (2\pi)^{-1}\|F_L-F_M\|_2\|\beta_j\|_2.
\]
Thus the full curvature bound bounds every coordinate of the integral
splitting lattice; torsion gives only finitely many further choices.
The finitely many split cores and compactly supported representatives
are fixed in the current metric problem. Holonomy gauges of nonzero
degree only relabel the relative sectors with the same stated charge
data. These observations give the asserted finite ranges.
Apply body compactness and neck extraction. At a newly appearing cut,
Lemmas~\ref{lem:analysis:compactness} and \ref{lem:analysis:neck}
give the stated charge assignments. True cuts persist, and a
new cut is an earlier step. A new bubble or positive cylinder level
increases the total loss. An already separated lens may acquire a
different main asymptote only through an extracted cylinder level.
If the old state is $\gamma$, the new state is $\gamma'$, and the
extracted ASD cylinder runs from $\gamma'$ to $\gamma$ with charge
$e>0$, replace its assigned cap charge by
\begin{equation}\label{eq:analysis:cap-charge-transfer}
\kappa_N'=\kappa_N+e.
\end{equation}
The Chern--Simons identity gives exactly the charge congruence for
$\gamma'$, with the fixed determinant transitions. The previous
tested cap charge is positive. Hence the new charge is at least
$1/4$ if the new state is trace, and at least one if it is central,
by its respective congruence. It satisfies the same allowed range
and lies at an earlier loss step. This argument needs the cylinder
extracted from the main solution, not a reconstructed old cap.
Several levels add their charges; a pure parallel-frame change has
zero charge and does not change the flat state.
Escape of an already labelled particle into
an already separated end transfers its weight to effective end loss;
if the total loss stays constant, the number of main locations
decreases. Collisions also decrease that number. A surviving basic
test can fail only by an exact hit, which adds a dropped test; existing
hits persist by closedness of their incidence parameterizations on
that face. In particular an incident particle cannot escape down a
true $J$ middle: the rule has already switched there, and the assigned
half test has a compact intrinsic support in its retained component.
It remains incident on that half in the limit. At a newly appearing
lens face the whole test lies in the separated $N$ core and is
omitted from the main problem altogether; a particle incident to it
transfers its integer weight to that cap charge. A decrease in the
number of dropped \emph{main} tests here is harmless, since the
new true face is earlier in the ordering. At an already separated
end every retained main test has compact support, so only particles
without such an incidence can escape down that end. Thus no missing
main test is restored without justification. Earlier whiskers used
during a switching homotopy are absent in the neighborhood of the
true face and cannot carry an incidence for its current test.
The marks cannot escape through identity segments of sphere sweeps.
The $R$ condition is closed under smooth completed convergence: a
normalized parallel splitting has a convergent subsequence in the
compact space of base lines and remains parallel, and the spinor
remains zero. A new $R$ component therefore moves earlier.
If none of these worsenings occurs, there is no new body or end loss.
The strong bootstrap and exponential completion give convergence in
the same high-order field charts. Changes among $A,S,P$ cause no
singularity in the full non-$R$ field model as long as at least
one $P$ remains: the finite-isotropy construction and compatible
charts of Lemmas~\ref{lem:analysis:samples} and \ref{lem:analysis:scale}
apply throughout. If the last $P$ disappears, the limit is a fixed-type
tuple. Rank-one tests may limit to rank zero; cover them using full
matrix-value neighborhoods and impose transversality to each rank
stratum there. This avoids an unjustified compactness assertion for
the open rank-one stratum alone.
\end{proof}
\subsection{The fixed stage and legitimate instanton links}
\begin{proposition}[First-stage regularity]
\label{prop:analysis:first-stage}
There is an arbitrarily small single-valued, single-component choice
for which type $A$ equations with all retained nonphase tests are
transverse on every relevant ideal and parameter stratum, and type
$S$ tangential equations are transverse. The first two conditions of
Definition~\ref{def:indices:analytic-hypotheses} follow. The only
fixed-type tuples are a finite set of legitimate unbroken cut-down
instantons, and the coupled complex Dirac operator can also be made
surjective at each of them without altering the instanton equations.
\end{proposition}
\begin{proof}
Use the countable single-valued library with the locality and bit
identifications imposed above. Lemma~\ref{lem:analysis:samples} and
the universal-surjectivity argument give regularity on each $A$
equation and retained-cut problem and on each pure $S$ tangential
problem. There are countably many relative topologies, coordinate
charts, and finite position and rank strata. The intersection of
their residual regularity sets in the arbitrarily small allowed open
ball of $\mathfrak P$ is residual. At a true $J$ face, a component's
equations use only its retained parameters, so this assertion does not
give it spurious parameter directions from another component.
For $A$, the index after the reference fillings is $q_i$. Each assigned
effective lens charge supplies the correction $\delta_I$, while removal
of its one parameter
and degree-two test restores one, giving the penalty $\delta_I-1$.
The permitted cap charge and test exclude the zero-charge central
case, and all remaining lens penalties are positive by
Proposition~\ref{prop:indices:lens-index}. A main instanton particle
has net penalty at least four by Lemma~\ref{lem:analysis:incidence},
and an allocated positive integer $J$-trajectory charge has positive
index penalty. A regular nonempty problem therefore has $q_i\geq0$,
strictly if any of these losses occurs.
For $S$ retain only the pure tangential equations. The active line is
ordered by its nonzero spinor. Use the sharp reference fillings of
Proposition~\ref{prop:indices:lens-index}, whose tangential correction,
including the limiting line stabilizer, is zero. The actual pure-line
problem has index $d_s+p_i$, with the prescribed ideal labels adding
at most $4\ell_i$ position directions when tests are ignored.
Transversality implies $d_s+p_i+4\ell_i\geq0$. This is precisely the
tangential bound required; it makes no assertion of full normal
regularity at an arbitrary $S$ field. The incidence condition is
Lemma~\ref{lem:analysis:incidence}.
Apply Proposition~\ref{prop:indices:fixed-exclusion}, which uses only
these fixed-stage bounds and the metric clamps. Every fixed tuple
except an unbroken, loss-free, cut-down type $A$ field is excluded.
Compactness from Lemmas~\ref{lem:analysis:compactness} and
\ref{lem:analysis:neck} now makes that zero-dimensional regular
instanton set compact, hence finite. This reasoning uses compactness
of sequences and the already proved fixed-type bounds; it does not
assume compactness of the final monopole bordism.
At such an instanton $a$, phase equivariance forces $P_D(a,0)=0$,
but its spinor derivative may be any complex-linear compact operator.
Let $K_a$ and $C_a$ be the kernel and cokernel of its current complex
Dirac operator. Finitely many transported spinor samples inject
$K_a$, and finitely many localized smooth output spinors detect
$C_a$. Hence their compositions give every map $K_a\to C_a$.
The irreducible connection has only its central gauge stabilizer;
complex-linear spinor operators respect both that center and phase.
They extend from its sample slice with compact cutoffs. Since
$\dim_{\C}K_a-\dim_{\C}C_a=n_D^{\rm tot}>0$, a generic arbitrarily
small such variation makes the Dirac operator surjective. The value
at zero spinor remains zero and the ASD equation and its cuts are
unchanged. Treat the finitely many instantons in disjoint configuration
neighborhoods.
These additional finite conditions are open. They can be combined
with the preceding residual choices; alternatively make the small
finite changes and then choose generically within their open
surjectivity neighborhoods. Absence of the other fixed tuples persists
under sufficiently small changes by ideal compactness. Existing true
cuts and positive losses persist in a limit. An unbroken $R$ or $S$
sequence retains a parallel reduction and cannot tend to a legitimate
irreducible instanton. The finitely many legitimate instantons have
regular isolated ASD neighborhoods, and every nearby instanton lies
in their union, since an outside sequence would have an excluded
ideal limit. Their Dirac surjectivity is an open condition there.
Consequently the finite adjustments and the subsequent generic choice
preserve the fixed-type exclusion. This completes the first stage.
\end{proof}
Fix this ultimate first-stage choice before constructing the quadratic
phase sections below. Its finitely many instantons and their
surjective Dirac kernels are therefore the centers of those charts.
Subsequent mixed and interior regularizing variations are identically
zero in smaller fixed neighborhoods of the instanton collars. They
do not move the centers or their quadratic leading sections. Thus
no assertion that genericity on every ideal component stratum is open
is needed: after each preliminary adjustment genericity is reselected
inside the open neighborhood preserving the forbidden-tuple absence,
the finite ordinary zeros, and their Dirac surjectivity; the link data
are installed only after that selection is complete.
\begin{proposition}[The instanton link and its phase degree]
\label{prop:analysis:instanton-link}
Put $n_*=n_D^{\rm tot}>0$ and $\eta=n_*-1$. At each legitimate
cut-down instanton, the normal spinor link before phase quotient is
$S^{2n_*-1}/\{\pm1\}$ and its phase quotient is $\CP^{n_*-1}$.
The phase-character-two line restricts to $\mathcal O(2)$ on this
projective space. The $\eta$ phase sections can be chosen so that
their cut-down link is regular and has signed count
\begin{equation}\label{eq:analysis:instanton-degree}
2^\eta\,\operatorname{sign}(a).
\end{equation}
There is one fixed overall boundary-orientation convention in this
formula, the same for every instanton. Subsequent mixed branch terms
may be kept zero near these links.
\end{proposition}
\begin{proof}
The cut-down ASD and metric problem is a regular isolated point and
the complex Dirac operator is onto. Its kernel $K$ has dimension
$n_*$. The implicit theorem, with the cut marks included among the
finite variables, solves the transverse field, metric, and cut
coordinates in terms of the small spinor kernel. Curvature outputs
are phase-invariant, so their first spinor derivative at zero is zero;
Dirac outputs have phase weight one. The normal model is consequently
a smooth equivariant deformation of $K$, with the gauge center
identifying $\psi$ and $-\psi$. Equivalently write
$\Phi=r\psi$, $\|\psi\|=1$, and divide the Dirac equation by $r$.
At $r=0$ it is the surjective kernel problem, and radial blow-up gives
boundary $S(K)/\{\pm1\}$. Dividing by phase gives $\mathbb P(K)$.
Its orientation is the complex orientation multiplied by the ordinary
instanton sign.
Choose phase sections quadratic to leading order in $\Phi$. Finitely
many spinor samples inject the finite-dimensional space $K$, so any
complex quadratic form on $K$ extends to a quadratic function of these
samples. The connection sample slice makes this extension equivariant;
the center acts trivially on squares. Put the support in an interior
parameter chart near the instanton. Division by $r^2$ extends its
section smoothly over the blow-up and gives the chosen quadratic form
on the projective boundary. The freedom is independent of the other
marks and of the equation perturbations.
More explicitly, a homogeneous function $q(z)$ of phase weight two
satisfies $q(\lambda z)=\lambda^2q(z)$ and defines a section of
$\mathcal O(2)$ on $\mathbb P(K)$. The center quotient does not turn
this into $\mathcal O(1)$: the principal bundle
$S(K)/\{\pm1\}\to\mathbb P(K)$ has doubled Hopf transition
functions when its effective circle is identified with a standard
circle. Thus its associated weight-two line for the original phase
has $c_1=2h$. This agrees with the link calculation in
\cite[Lemma~3.28 and Proposition~3.29]{FeehanLenessPU2II}.
For a direct regular choice, in coordinates
$[z_0:\cdots:z_{n_*-1}]$ take
$z_j^2-a_jz_0^2$, $1\leq j\leq n_*-1$, with all $a_j\ne0$.
Every common zero has $z_0\ne0$, there are exactly $2^{n_*-1}$
such points, and their complex Jacobian is invertible. This also
proves the intersection number
$\langle(2h)^{n_*-1},[\CP^{n_*-1}]\rangle=2^{n_*-1}$.
For $n_*=1$ there are no phase cuts and the same formula is one.
The implicit theorem extends these boundary points to regular collars.
Choose every later coupled term outside smaller neighborhoods of these
finitely many collars. Where such a term is zero its repeated branches
have total weight one, so no extra rational factor changes
\eqref{eq:analysis:instanton-degree}.
\end{proof}
\subsection{Mixed projections and finite exclusion}
\begin{lemma}[The necessary mixed projection]
\label{lem:analysis:projection}
For a candidate containing a $P$ field, delete the $R$ fields, their
particle positions, and their tests, but retain their parameter
variables. Quotient the remaining fields by their independent gauges
and the common phase. This defines a Fredholm problem with finite
isotropy near its candidate projections. Its dimension, before phase
quotient and phase cuts, is at most
\begin{equation}\label{eq:analysis:projected-dimension}
\sum_{i\notin R}\left(
i_i-\sum_{\text{lens ends of }i}(\delta_{\rm sp}-1)-2w_i\right)
+\sum_{i\in R}p_i.
\end{equation}
The phase cuts and quotient subtract $2\eta+1$. At every such
candidate projection, admissible local finite-branch variations make
the universal equations and retained cuts transverse.
\end{lemma}
\begin{proof}
Lemma~\ref{lem:analysis:R-independence} proves that the equations
factor through this projection, including every energy-cutoff atom.
The $P$ witness makes its isotropy finite by
Lemma~\ref{lem:analysis:samples}. Particle positions satisfying a
dropped sphere incidence can be parameterized by preimages in the
sphere domain; a surface incidence uses its surface domain, and two
surface incidences use their transverse fiber product. These are the
finite-dimensional position spaces used in computing the index.
Surviving test marks are retained separately. Singularities of a
possibly nonimmersed sphere map do not increase that parameter-space
dimension.
The virtually filled field index is $i_i$. Removal of a lens filling
costs $\delta_{\rm sp}$, while omitting its one parameter and degree-two
test restores one. Its penalty is therefore the full
$\delta_{\rm sp}-1$, not merely one per lens. Every unit of main
particle charge loses six field dimensions and recovers at most four
through the position and incidence budget. Allocated integer
connector losses lose at least as much as the bound two charged here.
The discarded $R$ equations contribute no fields or cuts to the
projection, only their $p_i$ remaining parameters. This proves
\eqref{eq:analysis:projected-dimension}; higher-codimension rank strata
only decrease it. Finally the common phase has finite stabilizer and
the $\eta$ complex phase sections impose $2\eta$ real conditions.
Full equation outputs at every retained target, full matrices at cap
tests, holonomy-event values, and independent phase-section values
are available by Lemma~\ref{lem:analysis:samples}. The
universal-surjectivity argument and Lemma~\ref{lem:analysis:scale}
apply on each finite branch lift. No transversality statement on an
arbitrary removed $R$ field is used.
\end{proof}
\begin{proposition}[Finite exclusion of mixed candidates]
\label{prop:analysis:finite-induction}
After the first-stage choices, finitely many additional local
multisection terms with arbitrarily small coefficients exclude all
forbidden mixed candidates. The only remaining mixed candidate types are
the unbroken free part and the single trace-minimum lens-end main
problem of Proposition~\ref{prop:indices:mixed-exclusion}. The latter
is regular of dimension zero. The unbroken free problem has expected
dimension one; its regularization follows the lens gluing in
Section~\ref{sec:gluing}. The additional choices agree on paired lens
exteriors and are zero near the legitimate instanton collars.
\end{proposition}
\begin{proof}
The inequalities in the proof of
Proposition~\ref{prop:indices:mixed-exclusion}, applied to
\eqref{eq:analysis:projected-dimension}, give strictly negative
projected index for each forbidden candidate type. Here the numerical
argument is used to compute the index, before asserting that a
negative index eliminates a zero. The preceding fixed stage already
excluded all unwanted fixed-type tuples. We now establish the
regularity that turns these negative indices into absence.
Use the finite ordering of Lemma~\ref{lem:analysis:finite-data}.
Suppose all earlier steps have been excluded at the current
coefficients. The set needed at this step is the image under
projection of the candidates of Section~\ref{subsec:analysis:candidates},
with their $R$ fields still present before projection.
Lemma~\ref{lem:analysis:finite-data} makes this image compact modulo
local slices and its finite assignment choices: a sequence has a limit
at this step, at an already excluded earlier step, or at an already
excluded fixed tuple. It is not necessary, and is not asserted, that
every zero of the loosened equations after deleting the $R$ fields
belongs to a compact set.
A forbidden candidate cannot instead converge to a legitimate clean
instanton. Any existing cut or positive loss persists in the
degeneration order. It cannot become the single permitted trace end
either: already separated end loss cannot decrease, and a particle
escaping into a newly separated lens carries at least one unit,
whereas the permitted total lens charge is $1/4$. The unbroken free
stratum with no degeneration is not one of the exclusion steps.
These facts justify the use of the already excluded fixed locus and
of protected neighborhoods of the legitimate instanton collars.
At each point of the compact candidate image choose finitely many
supported variations from Lemma~\ref{lem:analysis:projection}
spanning the missing universal directions. Take a finite subcover
and the finite sum $V$ of those coefficient spaces. There are only
finitely many previously chosen branch lists. At each local lift
include all their branch choices and all translates in the newly
chosen coefficient spaces. The invariant-span prescription in
Lemma~\ref{lem:analysis:samples} makes the universal map transverse
for every branch separately. At zero new coefficients a new list
merely repeats the old list with total weight one.
Surjectivity persists near this compact image for sufficiently small
$v\in V$. Moreover every new candidate for sufficiently small $v$
has its projection in this neighborhood. Otherwise take $v_j\to0$
and candidates outside it. Compactness gives either a forbidden
earlier limit or a candidate at the current step whose projection is
inside the neighborhood, both contradictions. The same argument
shows that absence at all completed steps is open under sufficiently
small changes.
Apply finite-dimensional parametric transversality to $V$, in the
countably many projected field charts and to each local branch and
rank stratum. The universal zero set is smooth by
Lemma~\ref{lem:analysis:scale}; the regular parameter values are
residual (equivalently apply Sard--Smale to its parameter projection).
At any such value, every projected negative-index zero set in the
chosen neighborhood is empty. Since every candidate projects there,
the current step is excluded. Choose the coefficients small enough
to preserve all previous exclusions, the metric tolerances, and the
protected instanton collars. There are finitely many steps, each
adding finitely many terms, so the total product of branch choices is
finite. This turns every negative projected index used in
Proposition~\ref{prop:indices:mixed-exclusion} into absence, proving
its exclusions. Regularity of the remaining strata is a separate step.
For the single trace-end main problem, use the same construction to
obtain regularity instead of emptiness.
Its compact candidate set is isolated from forbidden corners by what
was just proved. It has a single type $P$ main component; its effective
sample action permits ordinary single-valued variations. Choose these
simultaneously on the paired lens exteriors, using their exact tensor
identification. Its expected dimension is zero.
All additions use the corner and bit-flip extension prescriptions,
so the required exterior equalities are preserved. This completes
the forbidden-candidate exclusion and trace-main regularization.
\end{proof}
\begin{lemma}[The weighted boundary rule]\label{lem:analysis:stokes}
A compact oriented one-dimensional zero set built from the finite
branch lists above in the surviving free stratum has total weighted
oriented boundary zero. Its
weights are rational, and a region on which every additional branch
term is zero has the weight of the original single-valued problem.
\end{lemma}
\begin{proof}
In this application the unbroken stratum has one $P$ component.
After quotienting the ineffective diagonal center, its gauge and
phase action is free. Thus its local base charts are ordinary
manifold charts; no additional effective-orbifold integration factor
is being omitted.
In a sample slice tube every $H$ translate has weight $1/|H|$.
Changing the slice lift permutes these translates and preserves their
multiplicities. To check orientations, choose the finite samples to
detect the exact combined stabilizer at the central candidate. The
$P$ witness restricts phase to $\{\pm1\}$. A nonzero-spinor target
with such a phase has scalar parallel gauge $\pm1$: in $\SU(2)$
an eigenvalue $\pm1$ forces both eigenvalues to be that scalar.
An $A$ target already has only central gauges. Thus $H$ is a
subgroup of a product of central groups $\{\pm1\}$, with the
common phase, modulo its ineffective kernel. Nearby tube stabilizers
are conjugate to subgroups of $H$ and are still central. They act
trivially on the real adjoint, parameter, and cut directions and
complex-linearly by signs on the spinor blocks. Deform the
linearization equivariantly to its real ASD part and complex Dirac
part. These actions preserve the determinant orientation there,
and hence preserve it throughout the deformation. Each cutoff is
zero on a neighborhood of the tube boundary, so extension by repeated
zeros creates no boundary of a branch zero set. Finite products have
product weights, and the sum of their weights is one. Consequently
the weighted oriented germs agree on chart overlaps, even where
branches coincide or separate.
Choose a partition of unity on a finite cover of a compact remainder
and apply the one-dimensional Stokes formula on each oriented local
lift with its weight. Terms involving derivatives of the partition
sum to zero because the weighted germs agree. The remaining terms
are exactly the weighted boundary points. This proves their total
is zero. At a zero perturbation all repeated branches represent the
same germ and their total weight is one, proving the final assertion.
\end{proof}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/09-analysis.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/10-gluing.tex ====
\section{Regular lens ends and the final boundary count}
\label{sec:gluing}
Fix the finite stack, its determinant transitions, its metrics, and the
small perturbations obtained from Propositions~\ref{prop:analysis:first-stage}
and \ref{prop:analysis:finite-induction}. These choices exclude the
forbidden fixed and mixed candidates and make the trace-main problems
regular. We construct their lens collars before completing the
regularization of the unbroken free stratum.
All artificial isolations and all surgery seams have finite length.
Only the primary three-sphere cuts and the lens cuts are boundary
faces of the parameter space. The only end types left by the exclusions
are the legitimate unbroken instantons and a single separated minimal
trace cap with a free main component. The latter limits have no
particles, cylinder losses, additional true cuts, or corners.
The instanton collars are supplied by
Proposition~\ref{prop:analysis:instanton-link}; we now construct and
exhaust the remaining lens collars.
\subsection{The framed minimal cap}
Let $N$ be the cylindrical completion of the disk bundle of Euler number
$-4$, with zero section $S$, and fix a representative $\gamma$ of the
trace-zero limiting flat connection. Its determinant-one parallel group
is
\[
H=\{\diag(e^{i\theta},e^{-i\theta}):\theta\in\R\}
\cong\U(1).
\]
The cap $N$ has exactly one matching end. On $N$, \emph{framed}
means quotienting by gauge transformations tending
to the identity at this fixed representative. Thus $H$ still acts on the
framed configuration space. On the main component we instead allow
parallel limiting gauge transformations. The determinant connections
and all identifications in this distinction are fixed.
\begin{proposition}[Regular minimal cap]
\label{prop:gluing:regular-cap}
At each of finitely many regular isolated main-component limits on a
trace face, the cap data can be chosen arbitrarily small, compatibly with
the preceding exclusions, so that the following statements hold.
The framed cap problem of charge $1/4$, with its sphere-sweep mark and
the normalized holonomy condition $-1$, is a finite set of regular
points. Every point is a crossing on the unique harmonic Abelian
connection with ordered summand difference $v(S)=2$. These points are
fixed by $H$, and their oriented sum has absolute value one.
The spinor is zero, and the coupled Dirac operator is invertible at
each point. The choices for the two determinant lifts are related by
the prescribed cap tensor operation.
\end{proposition}
\begin{proof}
The boundary eigenline of the ordered splitting is prescribed by its
degree modulo four. With that eigenline fixed, the Abelian ASD extension
is unique: the harmonic curvature is determined by the relative class,
the difference between two such connections is a closed one-form modulo
gauge, and $H^1(N;\R)=0$. The vanishing of $b^+(N)$ gives surjectivity
of its tangential ASD deformation problem. A different parallel
boundary identification is absorbed by the parallel stabilizer of this
same connection. In particular, the two eigenlines are not an extra
pair of framed cap solutions.
Corollary~\ref{cor:indices:trace-minimum} gives the following dimensions
before the sweep cut: the tangential framed ASD part has real index
zero, the normal ASD part has complex index one, and the coupled Dirac
part has complex index zero. Positive scalar curvature and the
specified ASD fixed-line curvature kill the Dirac kernel; its index
then kills its cokernel. The same Weitzenb\"ock argument kills every
cap spinor for the allowed small curvature perturbations. There is no
Dirac forcing on the separated cap.
We explain why the two remaining normal regularity requirements can
be imposed by the allowed perturbations. Write $T$ for the complex
normal ASD operator at the reducer. Its kernel and cokernel are
finite-dimensional complex representations of $H$, all with the
off-diagonal character. Derivatives of based holonomies detect
$\ker T$ modulo infinitesimal gauge. Indeed, suppose a deformation has
on every based loop only the derivative produced by the same
infinitesimal conjugation. Subtract that conjugation and compare
parallel transport from the base point along paths. Independence of
the path, which follows from the vanishing loop derivatives, defines
an infinitesimal gauge transformation producing the deformation.
Its off-diagonal part tends to zero on the end, since $\gamma$ has
no off-diagonal parallel section. It is therefore a framed gauge
direction. Consequently the holonomy derivatives separate the normal
kernel. Finitely many suffice: successively reduce their common
kernel, whose dimension is finite.
Include a noncentral holonomy in this finite sample to fix the torus.
In a slice for the sample orbit, a torus-equivariant linear function
of the normal sample coordinates, followed by a localized
off-diagonal two-form output, vanishes on commuting data and has the
prescribed normal derivative. Multiplication by an invariant cutoff
extends it to an allowed perturbation. Smooth compactly supported
outputs detect every nonzero cokernel element, so finitely many
outputs suffice as well. These constructions realize all complex
linear maps from $\ker T$ to $\coker T$.
For completeness, split domain and range into kernel, cokernel, and
complements on which $T$ is invertible. The Schur complement of a
small perturbation is a finite-dimensional map
$\ker T\longrightarrow\coker T$. Surjective such maps are dense
because $\dim_{\C}\ker T-\dim_{\C}\coker T=1$.
The preceding perturbations supply all their first derivatives.
We may thus make $T$ surjective by an arbitrarily small perturbation
whose value on the reducer remains zero. Its kernel is now a complex
line.
In the determinant-root normalization the reducer's sphere sweep is
a loop in $H$. The eigenline has degree $v(S)/2=1$, so the sweep has
winding $1$ or $-1$, according to its fixed orientation. Perturb its
toral component slightly, relative to the collapsed ends, so that its
crossings of $-1$ are transverse. Their signed sum is this winding.
At a crossing, left translation identifies the target tangent space
with
\[
T_{-1}\SU(2)=\R\tau\oplus\C_{\mathrm{off}},
\]
where $\tau$ generates the diagonal Lie algebra. The sweep-mark
derivative is nonzero in $\R\tau$. On the complex normal kernel, the
off-diagonal cut derivative is $H$-equivariant and hence complex linear.
The same finite sample construction, now with output in the cut
target, varies this map arbitrarily. Choose it nonzero at every
crossing. Equation and cut variations are independent, so this does
not undo surjectivity of $T$. The combined cut derivative on the
kernel and mark is therefore an isomorphism
\begin{equation}
\ker T\oplus\R\ \longrightarrow\
\C_{\mathrm{off}}\oplus\R\tau.
\label{eq:gluing:cap-cut}
\end{equation}
Its complex factor preserves orientation, leaving precisely the
toral crossing sign. The framed dimension calculation here is
\begin{equation}
2+1-3=0.
\label{eq:gluing:cap-dimension}
\end{equation}
This is different from the ordinary interval calculation
$1+1+1-3=0$: the latter uses the unframed index of $W'$, an interval
parameter, and a sweep mark. No interval parameter is being added
to the fixed framed cap in \eqref{eq:gluing:cap-dimension}.
It remains to exclude additional cap points. Near every reducer
crossing, \eqref{eq:gluing:cap-cut} and the implicit function theorem
give a unique framed solution for the fixed tangential data. It is
the reducer crossing just constructed. Away from the reducers,
ordinary sample variations give full equation and cut transversality.
The residual $H$-orbit of a framed irreducible has positive dimension,
whereas its regular framed cut-down space would have dimension zero.
Thus no such irreducible point can remain. Equivalently, the
irreducible quotient by $H$ has negative expected dimension and the
same local variations make its universal problem transverse.
There is no compactness boundary in this argument: a particle costs
an integer charge; a nonconstant lens-cylinder loss at total charge
$1/4$ leaves a flat cap, which misses the sphere cut. These facts
follow from Lemma~\ref{lem:indices:lens-charge} and the compactness
of Lemma~\ref{lem:analysis:compactness}. After the regular neighborhoods
of the reducer crossings are fixed, the remaining candidate set is
compact, so finitely many ordinary perturbation neighborhoods suffice.
The oriented sum of all points is therefore the winding sum.
There are only finitely many relevant main limits and finitely many
local branches there. Cap adjustments can be supported in their
tangential parameter neighborhoods, away from the other true faces,
and copied to the flipped cap by tensoring. They can also avoid
the legitimate instanton charts. The established forbidden-stratum
exclusions persist under sufficiently small such adjustments. To
see the required openness, a sequence of newly appearing forbidden
zeros would have an ideal limit by Lemma~\ref{lem:analysis:compactness}.
Further breaks and losses remain forbidden. A fixed-type tuple has
a fixed-type limit; an unbroken reducible or active-line configuration
cannot converge smoothly to a legitimate irreducible zero-spinor
instanton because its parallel reduction persists. These alternatives
are exactly the closed degenerations treated in
Proposition~\ref{prop:analysis:finite-induction}. This proves the
compatibility asserted in the proposition.
\end{proof}
\subsection{The augmented operators at a matching pair}
Fix a main solution $u_M$ and a cap crossing $u_N$ as above, in a
single local branch. All remaining metric parameters, cut marks,
and cap evaluation variables are included in the constrained
operators. The neck length itself is held fixed during this
linearization. Fix the effective phase on the main component by
the argument of a nonvanishing local sample function of original
phase character two. Such a function exists at a free main solution.
Its differential is an isomorphism on the effective phase direction.
Let $G$ denote infinitesimal determinant-one gauge action and augment
the equation-and-cut derivative by $G^*$. Use sufficiently high
little-H\"older completions; the field target has one less derivative
than the field domain. On an end with coordinate $s\geq0$, a positive
weight means that $e^{\delta s}$ times the field is in the unweighted
space. The number $\delta>0$ is less than the absolute value of every
nonzero eigenvalue of the boundary operator. At the trace flat the
only zero mode of the augmented connection operator is the temporal
one-form $\tau\,ds$. The adjoint spatial first cohomology vanishes,
the off-diagonal flat has no parallel sections, and the boundary
Dirac is invertible.
\begin{lemma}[The scalar obstruction and the weight change]
\label{lem:gluing:blocks}
After fixing phase as above, the positive-weight main augmented
constrained operator $L_M^+$ is an isomorphism. On the cap the
positive-weight augmented constrained operator $L_N^+$ has zero
kernel and has precisely the one-dimensional cokernel
\begin{equation}
\ell(f)=\int_N\langle\tau,f_{\mathrm{gauge}}\rangle.
\label{eq:gluing:scalar-cokernel}
\end{equation}
Here $f_{\mathrm{gauge}}$ is its $G^*$-component and $\tau$ is the
parallel diagonal gauge generator on the cap. Changing the cap
field weight from $+\delta$ to $-\delta$ gives an isomorphism $L_N^-$.
\end{lemma}
\begin{proof}
On the main completion gauge transformations may tend to parallel
constants. Analytically their infinitesimal domain includes cutoff
extensions of these constants, in addition to decaying sections.
This extension is essential. The gauge Laplacian satisfies
\[
\langle G^*G\xi,\xi\rangle
=\|d_a\xi\|_2^2+\|\xi\Phi\|_2^2.
\]
The boundary term is zero for an extended constant with decaying
derivative. A vector in its kernel is parallel and annihilates the
nonzero main spinor, and is consequently zero. In the complementary
weight, a homogeneous cokernel representative can have a constant
or linear parallel asymptote. Pairing with the cutoff parallel
constants in the extended gauge domain forces the linear
coefficient, namely its boundary flux, to vanish. The remaining
bounded representative satisfies the same integrated identity
and is zero. One can
also obtain this directly on the cylindrical constant mode by
solving the scalar second-order equation with a free limiting
constant; the nonconstant boundary modes have the spectral gap.
Thus $G^*G$, on the extended gauge domain, is invertible.
It follows that the gauge augmentation gives a genuine slice for
unframed fields. The main cut-down problem, with phase fixed, is
regular and isolated. Its equation-and-cut derivative on that slice
is therefore an isomorphism. Combining it with the gauge inverse
proves the assertion for $L_M^+$.
On the reducer cap, write $A_N$ for the reference connection.
The parallel gauge $\tau$ acts trivially on its field. For a
variation $v=(a,\psi,\nu)$, including its mark $\nu$, integration
by parts gives
\[
\ell(L_N^+v)=\int_N\langle\tau,d_{A_N}^*a\rangle=0
\]
for decaying variations; the spinor is zero. The scalar gauge
equation consequently has the compatibility condition
\eqref{eq:gluing:scalar-cokernel}. There is no other one.
Indeed, the cap gauge Laplacian $Q=G^*G$ on decaying gauge
parameters has zero kernel by integration. In its complementary
weight, an adjoint homogeneous solution has a constant or linear
parallel asymptote; the other modes have the boundary spectral
gap. On this one-ended cap, integration against the global
parallel $\tau$ forces the linear flux to vanish. Integrating
the resulting bounded solution against itself shows that it is
parallel. Thus the adjoint kernel is exactly $\R\tau$, and
$\im Q$ consists exactly of scalar targets of integral zero.
Given equation-and-cut target $y$, framed regularity from
\eqref{eq:gluing:cap-cut} and the invertible Dirac supply a field
and mark variation $u$ realizing $y$. For any scalar target
$f_{\mathrm{gauge}}$ with integral pairing zero against $\tau$,
solve
\[
Q\xi=f_{\mathrm{gauge}}-G^*u
\]
in decaying cap gauges. Adding $G\xi$ corrects the scalar row
and leaves the equation-and-cut target unchanged, by equivariance
at the solution. Conversely a zero of the augmented operator is
a framed gauge direction by constrained regularity, and then
$Q\xi=0$ makes it zero. This gives zero kernel and full range
in $\ker\ell$. Thus
$\ind L_N^+=-1$ with the full scalar target used here.
Only the eigenvalue zero is crossed in changing the weight. Its
multiplicity is one, so $\ind L_N^-=0$. A vector in the kernel of
$L_N^-$ has an expansion
\[
a=c\tau\,ds+a_0,\qquad a_0=O(e^{-\epsilon s}),\quad\epsilon>\delta,
\]
with the same decay for derivatives and with decaying spinor; the
remaining finite-dimensional variables are the cap mark variables.
This expansion follows by the spectral decomposition on the end,
or by integrating each mode of the cylindrical operator. Pair its
gauge equation with $\tau$ on the truncation $N_R$. Since $\tau$ is
parallel, Green's formula gives
\[
0=\int_{N_R}\langle\tau,d_{A_N}^*a\rangle
=-\int_{\partial N_R}\langle\tau,a(\nu)\rangle.
\]
Letting $R$ tend to infinity forces
$c\,\|\tau\|_{L^2(\partial N)}^2=0$. The vector hence lies in the
positive-weight domain and is zero by the preceding kernel assertion.
Index zero now proves surjectivity and the asserted inverse.
\end{proof}
Main parameters can also change the cap equations. This does not
make the separated operator diagonal automatically. The next
observation records the needed change of variables.
\begin{lemma}[Eliminating shared parameter derivatives]
\label{lem:gluing:triangular}
Let $p$ denote the main parameter and auxiliary-variable directions
which also occur in the cap equations or cap cut. Locally the
limiting representative $\gamma$, collar gauges, and link metric
can be kept fixed. In these choices, the limiting coupled derivative
can be reduced to $L_M^+\oplus L_N^-$ by a triangular change of variables.
The field part of its cap correction to $p$ decays exponentially.
\end{lemma}
\begin{proof}
Write $B_Np$ for the derivative of the cap equation and cut in these
directions. The augmentation is defined at the reference field
and does not vary with $p$; thus $(B_Np)_{\mathrm{gauge}}=0$.
Moreover these variations do not change the limiting flat, so their
other components decay. By \eqref{eq:gluing:scalar-cokernel},
$\ell(B_Np)=0$. Lemma~\ref{lem:gluing:blocks} therefore gives a unique
positive-weight correction
\begin{equation}
Kp=-(L_N^+)^{-1}_{\ker\ell}B_Np.
\label{eq:gluing:parameter-correction}
\end{equation}
In this formula the inverse means the inverse from $\ker\ell$ to
the cap domain, including its mark variable.
Substituting $v_N=w_N+Kp$ changes the cap row
$L_Nv_N+B_Np$ into $L_Nw_N$. The main row is unchanged, giving
the triangular elimination. Elliptic regularity and the boundary
gap give exponential decay of the field component of $Kp$, with
derivatives, at some rate $\beta>0$ independent of a subsequent
choice of smaller $\delta$. This uses the fixed boundary spectral
gap and the compact support or prescribed rapid decay of $B_Np$.
The mark component is an ordinary finite-dimensional correction
on the cap core. All these
statements are uniform on the finite set of main limits and local
branches under consideration.
\end{proof}
\subsection{A finite-length inverse and the nonlinear gluing map}
\begin{proposition}[Regular gluing and exhaustion]
\label{prop:gluing:collar}
For every matching pair $(u_M,u_N)$ above and every sufficiently
large finite lens-neck length $D$, there is exactly one nearby
unbroken zero in a fixed local branch, modulo determinant-one gauge
and effective phase. These zeros depend continuously, and smoothly
in finite-order solution charts, on $D$. Every sequence of unbroken
zeros converging to this pair belongs to this collar for all
sufficiently large lengths. The construction preserves the finite
branch lists, their multiplicities, and their weights.
\end{proposition}
\begin{proof}
Use an oriented cylinder $[0,D]\times\partial N$, with coordinate
$t$ increasing from the main body to the cap body. Cut off the
exponentially decaying tails of $u_M$ and $u_N$ near the middle and
splice through $\gamma$. Perform the same extensions for all
equation perturbations and cut representatives. Lemma~\ref{lem:analysis:neck}
and the prescribed perturbation tails give a preglued field $u_D^0$
with residual, including the cuts,
\begin{equation}
\|\mathcal F_D(0)\|\le C e^{-bD}
\label{eq:gluing:residual}
\end{equation}
for some $b>0$, after decreasing $b$ to absorb fixed polynomial
factors. Here $\mathcal F_D$ is the augmented, phase-fixed local
equation in field and finite-dimensional variables centered at
$u_D^0$. The preglued fields have uniform bounds of every fixed
local differentiability order. The perturbation library was
chosen with summable bounds at each such order.
\emph{Linear estimate.}
Choose $\delta>0$ below the boundary gap and as small as needed
below. On the neck use the weight $w_D(t)=e^{\delta t}$, smoothly
continued as $1$ on the main core and $e^{\delta D}$ on the cap
core. The cap's own mark variables and cut targets are scaled by
$e^{\delta D}$ as well. The main variables are not scaled.
Viewed from the cap with outward coordinate $s=D-t$, the relative
weight is $e^{-\delta s}$; viewed from the main side it is
$e^{\delta t}$. The limiting operators in these norms are
therefore exactly $L_M^+$ and $L_N^-$.
Figure~\ref{fig:gluing:weight} shows the two outward coordinates and
their induced weights.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[font=\small,>=Stealth]
\draw[rounded corners=2pt,fill=blue!5] (0,0) rectangle (2,1.05);
\draw[rounded corners=2pt,fill=blue!5] (8,0) rectangle (10,1.05);
\draw (2,.2)--(8,.2) (2,.85)--(8,.85);
\node at (1,.55) {main core};
\node at (9,.55) {cap core};
\node at (5,.55) {$w_D(t)=e^{\delta t}$};
\draw[->] (2.2,1.4)--(7.8,1.4) node[midway,above] {$t$};
\node[above] at (2,1.4) {$0$};
\node[above] at (8,1.4) {$D$};
\draw[->] (7.8,-.35)--(2.2,-.35)
node[midway,below] {$s=D-t$};
\node[align=center,below] at (1,-.15)
{$w_D=1$\\$L_M^+$};
\node[align=center,below] at (9,-.15)
{$w_D=e^{\delta D}$\\$L_N^-$};
\node[below] at (5,-1.05)
{cap-relative weight: $e^{-\delta D}w_D=e^{-\delta s}$};
\end{tikzpicture}
\caption{The asymmetric weight on a finite lens neck. Measured from
the main core it induces the positive completion weight; after dividing
by the cap-core factor $e^{\delta D}$ it induces the negative completion
weight in the cap's outward coordinate. The cap mark and cut target
carry the same cap-core factor.}
\label{fig:gluing:weight}
\end{figure}
First transplant the triangular change of
Lemma~\ref{lem:gluing:triangular}, cutting off $Kp$ on the long
cylinder. Its cutoff error is exponentially small, even with
the displayed weight, when $\delta$ is small compared with its
decay rate. Write $\widetilde L_D$ for the resulting derivative.
There is a constant $C_\delta$, independent of large $D$, such that
\begin{equation}
\|v\|_{X_{D,\delta}}
\le C_\delta\|\widetilde L_Dv\|_{Y_{D,\delta}}.
\label{eq:gluing:weighted-inverse}
\end{equation}
Here one may use the supremum of weighted local H\"older norms on
unit strips, with the usual norms on the compact cores and on the
finite-dimensional variables. These are Banach norms on the
finite cylinder and give the same elliptic Fredholm problem.
We give an estimate proof to account for possible concentration in
the middle. After conjugating by $w_D$, the model there is
$\partial_t+B-\delta$, with $B$ the self-adjoint augmented boundary
operator. For every spectral value $\lambda$ its scalar inverse
on the full line integrates toward $+\infty$ when
$\lambda-\delta<0$ and toward $-\infty$ when
$\lambda-\delta>0$. The respective kernels decay as
$e^{-|\lambda-\delta||t-s|}$.
Since $\dist(\delta,\operatorname{spec}B)>0$, spectral summation
and local elliptic estimates give a bounded full-cylinder inverse,
also on the uniformly local norms just specified. In particular,
this estimate includes the parallel scalar mode $\lambda=0$.
Cover the glued manifold by the main completion model, the cap
completion model, and this cylindrical model. Take cutoff
functions with transitions of a fixed, large width $R$, constant
on the compact cores and cut supports. Apply the two inverse
estimates of Lemma~\ref{lem:gluing:blocks} and the cylinder estimate
to these cutoffs of $v$. The commutators are at most
$C_\delta R^{-1}\|v\|$, in the corresponding one-order-lower norm.
Coefficient errors at the far ends tend to zero as $R$ increases;
errors from the actual splicing tend to zero as $D$ increases.
The compact graph terms have no unaccounted interaction between
the patches: terms supported on a body have its cutoff equal to
one, and tails or terms reaching the deep cylinder have arbitrarily
small exponentially weighted operator bounds. The same tail
bounds control the finite-dimensional parameter forcing outside
its body's cutoff. Shared parameter
terms have already been removed by
\eqref{eq:gluing:parameter-correction}. Choose $R$ to absorb the
commutators and end errors, and then choose $D$ large to absorb the
remaining errors. This proves
\eqref{eq:gluing:weighted-inverse}. It also explains why merely
discarding the zero boundary eigenvalue would give an incorrect
estimate.
The unbroken constrained family has index one after fixing effective
phase. Fixing the lens-neck length $D$ removes one parameter, so the
finite-length constrained problem has index zero. Thus
\eqref{eq:gluing:weighted-inverse} implies that its derivative is
invertible. Undoing the weight and the triangular change gives
an inverse $Q_D$ in the original norms with
\begin{equation}
\|Q_D\|\le C e^{a\delta D}
\label{eq:gluing:inverse-loss}
\end{equation}
for a fixed constant $a$. A fixed polynomial in $D$ can be
absorbed by increasing $a$, for every fixed $\delta>0$.
No bound independent of $D$ in the original symmetric norms is
being asserted.
\emph{Nonlinear estimate.}
Set
$L_D=D\mathcal F_D(0)$ and
$R_D(v)=\mathcal F_D(v)-\mathcal F_D(0)-L_Dv$.
In a small ball of radius $r$ in the original field and variable
norms,
\begin{equation}
\|R_D(v)-R_D(v')\|
\le C e^{a'\delta D}r\,\|v-v'\|.
\label{eq:gluing:nonlinear}
\end{equation}
The local quadratic field terms have this estimate by multiplication
in the chosen high-order spaces. We spell out its moving-path
point because arbitrary same-order smoothness of translation is
not needed. If $T_p$ denotes pullback along a smooth family of
paths or local transports, then with one derivative lost,
\[
\|(T_p-T_0)h\|_{k-1,\alpha}
\le C|p|\,\|h\|_{k,\alpha}.
\]
For the parameter derivative, subtract its value at the smooth
base field $u_D^0$ by first replacing the field by $u_D^0$ and
then replacing $p$ by zero. The first difference is bounded by
$C\|u-u_D^0\|_{k,\alpha}$; the second is bounded by
$C|p|\|u_D^0\|_{k+1,\alpha}$.
Only the base field needs the extra derivative. Differentiating
the transport ordinary differential equation gives these same
bounds for holonomies. Products, smooth functions of finitely
many samples, localized outputs, and the energy-cutoff scalars
preserve them. Consequently the derivative of the entire graph
formula differs from its derivative at the smooth preglued base
by at most $Cr$. Integration on the segment from $v$ to $v'$
gives \eqref{eq:gluing:nonlinear}. Little-H\"older completions
justify this first-derivative calculation; the compatible
higher-order solution charts are those of
Lemma~\ref{lem:analysis:scale}. The prescribed summable
library bounds and the rescaling introduce at most the exponential
factor written in \eqref{eq:gluing:nonlinear}.
Choose $\delta$ so small that
$(2a+a')\delta<b/2$. Put
$r_D=2C^2e^{-(b-a\delta)D}$, enlarging its fixed coefficient if
necessary. Equations~\eqref{eq:gluing:residual}--\eqref{eq:gluing:nonlinear}
show that
\[
v\longmapsto-Q_D\bigl(\mathcal F_D(0)+R_D(v)\bigr)
\]
maps the radius-$r_D$ ball into itself and has Lipschitz constant
less than $1/2$ for all large $D$. It therefore has a unique fixed
point. The same estimates give uniqueness on any ball of radius
$e^{-cD}$ for which
$(a+a')\delta<c<b-a\delta$; the inverse-scaled pregluing residual
$Q_D\mathcal F_D(0)$ then fits in that ball. Dependence on finite $D$ follows by the
parameterized implicit theorem, with elliptic regularity giving
the compatible smooth solution charts.
\emph{Exhaustion.}
Let exact unbroken zeros tend to the specified matching pair while
$D\to\infty$. There is no loss in their necks. By
Lemma~\ref{lem:analysis:neck}, gauges on their two halves can be
chosen in which the deviation from $\gamma$ and all its derivatives
decay exponentially from the two bodies. Flattening each half
near the middle then gives fields on the completed main and cap
pieces with exponentially small residuals. Initially they are
$o(1)$-close to $u_M$ and $u_N$ in small positive decay weights:
on any fixed core this is convergence to the limit, and the
uniform exponential tail estimate controls the complementary
ends in any smaller weight.
Put these flattened fields in the fixed completed gauge slices
before applying the augmented estimates. The completed main
gauge inverse and phase-fixing function give its slice; on the
cap the scalar gauge error has integral zero and the decaying
gauge inverse onto that range gives the framed slice. These
are fixed-piece slice maps with constants independent of $D$.
Their gauge and phase rows now vanish exactly. Equivariance
preserves the exponentially small equation-and-cut residuals.
The inverse estimate for $L_M^+$, followed by absorption of the
local nonlinear remainder, improves the main field and its
parameter differences from $o(1)$ to $O(e^{-b_1D})$ for some
$b_1>0$. On the cap, $L_N^+$ is injective with closed range, so
it has the corresponding estimate on its domain even for a
residual not known beforehand to satisfy $\ell=0$. Apply this
estimate to the cap difference after subtracting $Kp$; the already
controlled parameter difference and the flattening error are
exponentially small. A second absorption therefore gives the
same exponential improvement on the cap. Shrink $b_1$ if needed
to include all cutoff errors.
The half gauges may differ by an element of $H$. The cap reducer
is fixed by $H$, so that mismatch is absorbed on the cap without
changing the matching point. Splicing the half gauges makes the
global difference exponentially small. Passing to the global
gauge slice costs at most a polynomial in $D$. Here is a direct
bound: on a fixed main-core ball the limiting spinor has positive
norm bounded away from zero, and the gauge-action norm there
controls the norm of a Lie-algebra element. Kato's inequality
and the one-dimensional Poincar\'e inequality along the tube give
\[
\|\xi\|_2^2\le C(1+D)^2
\bigl(\|d_a\xi\|_2^2+\|\xi\Phi\|_2^2\bigr).
\]
Local elliptic estimates upgrade this bound for the inverse of
$G^*G$ to the required field orders. The small nonlinear slice
correction has the same polynomial loss, which exponential
closeness absorbs. Hence the exact zeros lie in an
$O(e^{-b_2D})$-neighborhood of the pregluing for some $b_2>0$.
The fixed-piece half estimates give this positive rate before
the final small weight is selected. Choose $\delta$ small
enough and then choose $c$ so that
\[
(a+a')\delta<c<\min\{b-a\delta,b_2\}.
\]
The uniqueness ball of radius $e^{-cD}$ contains both the
constructed solution and all these exact zeros for large $D$.
They must therefore be the same zeros. This proves
exhaustion, not just existence of a gluing map.
\emph{Branches and matching multiplicity.}
The cap crossing is fixed by $H$. If both pieces are framed,
the diagonal $H$ quotient identifies every matching rotation
with the same pair. If the main piece is unframed from the
outset, this is exactly the same quotient. There is one glued
zero per cap crossing; no $H$-volume, eigenline choice, or extra
factor of two is introduced. The effective phase was fixed on
the main piece, and the zero-spinor cap adds no phase parameter.
Each finite multisection is, in a local sample slice, a finite
list with multiplicities and weights. At the separated pair
the active graph data belong to the appropriate parts. A graph
which would compare frames across the true cut is absent there;
its finite-length tail is either zero or exponentially small.
Thus on each local branch the limiting equations are precisely
the matching branch equations used above. The contraction
construction applies to each member of this finite list, with
the same thresholds after taking a maximum. It commutes with
permutations of local lifts, and repeated branches keep their
multiplicities. Accordingly it pulls the matching list back to
the list of unbroken zeros with exactly the original weights.
No new choice of branches, or averaging over matching rotations,
is made during gluing.
\end{proof}
\subsection{Completion of the compatible regularization}
\label{subsec:gluing:completion}
The lens collars constructed above complete the local description of
all permitted ends. We can now regularize the remaining free stratum
relative to those collars and the instanton collars.
\begin{proof}[Proof of Theorem~\ref{thm:analysis:regularization}]
The ordinary construction is Proposition~\ref{prop:analysis:ordinary-library},
and its exact support and lower-level behavior follow from
Lemmas~\ref{lem:analysis:compactness} and \ref{lem:analysis:incidence}.
For ordinary non-PSC Floer ends the compactness and gradient-regularity
arguments are those in Section~\ref{sec:ordinary}; the isolated true-neck
argument is needed only for the present spinor family.
Proposition~\ref{prop:analysis:first-stage} verifies the fixed-type
requirements using single-valued data. Lemma~\ref{lem:analysis:projection}
and Proposition~\ref{prop:analysis:finite-induction} exclude the forbidden
mixed candidates and give regular zero-dimensional trace-main problems.
The estimates and ideal limits are Lemmas~\ref{lem:analysis:compactness}
and \ref{lem:analysis:neck}. Proposition~\ref{prop:analysis:instanton-link}
gives the legitimate instanton collars and their contribution.
Propositions~\ref{prop:gluing:regular-cap} and \ref{prop:gluing:collar}
now give exhaustive regular collars at the remaining no-particle lens
ends. On the unbroken part use a countable small single-valued library
supported off these collars and the protected instanton collars, or
regularize compact remainders successively. Universal surjectivity there
again uses its single $P$ component, and the expected dimension is one.
Such small final choices preserve the forbidden-stratum exclusions by
the compactness argument of
Proposition~\ref{prop:analysis:finite-induction}.
There are no other ideal boundaries by the numerical exclusions, now
with their analytic hypotheses proved. The lens construction uses the
local cap operator and the exponential estimates, with no gluing at an
excluded positive-particle stratum. With the instanton and lens collars
added, sequential compactness gives the claimed compact weighted
one-dimensional object. Lemma~\ref{lem:analysis:stokes} supplies its
boundary rule. All choices were made in arbitrarily small prescribed
norms and obeyed the support, parameter-locality, and tensor-identification
conditions from the start.
\end{proof}
\subsection{The local comparison of determinant orientations}
Table~\ref{tab:gluing:orientations} collects the conventions used in the
comparison. The sign $s$ is the sign of the full real cap determinant
line, including its framing, mark, and cut factors.
\begin{table}[htbp]
\centering
\footnotesize
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{.15\textwidth}>{\raggedright\arraybackslash}p{.24\textwidth}>{\raggedright\arraybackslash}p{.32\textwidth}>{\raggedright\arraybackslash}p{.19\textwidth}@{}}
\toprule
Datum & Ordinary convention & Spinor continuation & Reference \\
\midrule
Determinant and gauges & Determinant connection and transition maps fixed;
determinant-one gauges & Same fixed exterior data under a local cap
flip & Convention~\ref{conv:ordinary:index} \\
Framing & Unframed main; framed one-ended cap & The cap stabilizer
absorbs matching rotations, with no extra multiplicity &
Proposition~\ref{prop:gluing:collar} \\
Local dimensions & Interval: $1+1+1-3=0$ & Fixed framed cap:
$2+1-3=0$ & Theorem~\ref{thm:negative:unit};
Proposition~\ref{prop:gluing:regular-cap} \\
Real cap line & Ordered parameter, mark, cut, and framing factors;
comparison sign $s$ & The same full real comparison;
$\varepsilon_1=-s\varepsilon_0$ &
Proposition~\ref{prop:negative:local-sign} \\
Dirac factor & Absent from the ordinary operator & Canonical complex
orientation contributes the separate real sign $+1$ &
Proposition~\ref{prop:gluing:signs} \\
Phase and cuts & Ordinary instanton sign retained & Original character
two; effective phase quotients by $\{1,-1\}$; the link cut is
$\mathcal O(2)$ & Proposition~\ref{prop:analysis:instanton-link} \\
Branch weights & The integral count uses the fixed lift weights &
Lens matching preserves each branch weight; near instanton links the
total branch weight is one &
Propositions~\ref{prop:gluing:collar} and
\ref{prop:analysis:instanton-link} \\
\bottomrule
\end{tabular}
\caption{The ordinary conventions and their spinor continuation.
The interval count varies the metric and the sweep mark; the cap count
fixes the separating face and retains the framed cap directions.
The full comparisons, rather than a prescribed value of $s$, determine
the cancellation.}
\label{tab:gluing:orientations}
\end{table}
Let $\varepsilon(\mathbf e)$ denote the relative lift weights already
used in the integral ordinary composition, where
$\mathbf e=(e_1,\ldots,e_n)$. They are fixed before constructing
the spinor problem. The next proposition shows that these same
weights cancel the lens ends.
\begin{proposition}[The ordinary weights cancel the spinor lens ends]
\label{prop:gluing:signs}
For every regular trace-face main solution and every local branch,
the two caps obtained by flipping its lens bit have opposite
oriented contributions after multiplication by
$\varepsilon(\mathbf e)$. This comparison is independent of the
main solution, respects all remaining gluings, and preserves the
branch weights. In particular it is the relative sign of
Proposition~\ref{prop:negative:local-sign}, with no new sign choice.
\end{proposition}
\begin{proof}
Fix a regular trace-face main solution and one local branch. The
exterior data agree under the lens-bit flip by
Proposition~\ref{prop:analysis:finite-induction}, and
Proposition~\ref{prop:gluing:regular-cap} chooses tensor-related cap
data. We first recall the relative cap operation from
Proposition~\ref{prop:negative:local-sign} in these matching conventions.
If $x\in H^2(N;\Z)$ has $x(S)=1$, then
\[
x^2=-\tfrac14,\qquad \PD(S)|_N=-4x.
\]
Tensoring on the cap by the line with first Chern class $-2x$
changes the rank-two determinant by $-4x$.
Globally, before this bit is flipped, $c\cdot S=0$ and
$S^2=-4$. Thus
\begin{equation}
\Delta c^2=-4,
\qquad \Delta c_2=-1,
\qquad \Delta\bigl(c_2-c^2/4\bigr)=0.
\label{eq:gluing:flip-charge}
\end{equation}
The same flip adds $\PD(S)$ to $\Lambda$. This is a local cap
replacement with the identical exterior bundle and identical
relative exterior charge; it is not a global choice of a
square root on the exterior.
The tensor line has the order-two flat boundary character. In
the fixed common exterior determinant normalization, it
sends $\diag(i,-i)$ to $\diag(-i,i)$, a conjugate trace
representative. Choose once a determinant-preserving parallel
boundary conjugation returning to the fixed $\gamma$.
Its ambiguity lies in the connected group $H$ and is absorbed
by the cap stabilizer. At charge $1/4$ the relative cap degree
is fixed, and \eqref{eq:gluing:flip-charge} identifies the desired
cap sectors. Use the intrinsic cap root obtained by tensoring
the old root with the twisting line, continued from the collapsed
sweep loop as in Proposition~\ref{prop:negative:local-sign}.
With these roots the normalized sphere path is unchanged before
boundary conjugation, which preserves its central target $-1$.
The adjoint data and sphere cut therefore correspond; copying
the cap curvature and cut perturbations gives the paired crossings
and full framed deformation-and-cut complexes of
Proposition~\ref{prop:gluing:regular-cap}.
Fix the ordering of parameter, mark, cut, and phase factors used
in the ordinary and spinor counts. At a product-flat lens
splitting the instanton orientation line factors by determinant
gluing as
\begin{equation}
\lambda_I(\mathrm{total})
\cong \lambda_I(M;\mathrm{unframed})
\otimes\lambda_I(N;\mathrm{framed}).
\label{eq:gluing:excision}
\end{equation}
The framed cap factor includes the boundary-stabilizer orientation
which would otherwise occur in matching. Use the full cap lines
$K_e$ and gluing isomorphisms
$\Gamma_e:\lambda_V\otimes K_e\longrightarrow\lambda_e$
of Proposition~\ref{prop:negative:local-sign}; $K_e$ includes the
framing, sweep-mark, and holonomy-target factors. The comparison
and weights in \eqref{eq:negative:orientation-comparison} and
\eqref{eq:negative:lift-weights} are
\[
\Phi=\Gamma_1(\id\otimes\tau_{\mathrm{cap}})\Gamma_0^{-1},
\qquad \Phi(o_0)=s\,o_1,
\qquad \varepsilon_1=-s\,\varepsilon_0,
\]
where $o_e$ are the previously chosen ordinary sector orientations.
Thus $\Phi(\varepsilon_0o_0)=-\varepsilon_1o_1$.
The map $\tau_{\mathrm{cap}}$ includes every framing and cut factor,
including the possible complex conjugation of an ordered
off-diagonal coordinate and reversal of the toral generator under
the boundary Weyl conjugation. Only this full real determinant-line
sign $s$ is used. The products of these comparisons are exactly
the weights $\varepsilon(\mathbf e)$ in the ordinary composition.
Proposition~\ref{prop:negative:local-sign} proves that this comparison
extends with constant sign over the relative configuration sectors:
the connection spaces are connected and the integral-lift orientation
systems have trivial monodromy on the determinant-one gauge quotient.
The fixed charge $1/4$ excludes a change of internal boundary degree.
The same proposition proves naturality under other gluings and
multiplicativity on disjoint caps; the constrained cap factors have
even parity. These are comparisons of orientation torsors and require
no path of solutions. They apply here because the exterior bundle,
relative degree, cap framing, and mark and cut ordering are precisely
the ones just identified.
Finally, homotope the spinor deformation operator, keeping its
symbol, to the direct sum of the instanton deformation operator
and the coupled complex Dirac operator. Its orientation is the
instanton orientation times the complex Dirac orientation, followed
by the chosen parameter, cut, and effective-phase factors.
Tensoring the cap and applying the chosen boundary conjugation act
complex linearly on the spinor bundles, so the canonically oriented
Dirac determinant contributes the separate real sign $+1$.
This remains true if the interior Dirac operators are joined by a
homotopy rather than identified by an identical formula. The exterior
phase, parameter, and all other cut factors agree in the paired
problems. Thus the full spinor comparison has exactly the ordinary
relative sign above.
The perturbation prescription ties each exterior finite branch list
under the flip, rather than only its averaged value. The chosen main
branch therefore has the same paired branch and rational weight.
Proposition~\ref{prop:gluing:collar} preserves those lists and weights
under gluing, with one match per cap crossing and no additional
$H$-multiplicity. The opposite orientation comparison consequently
holds for each weighted lens-boundary contribution.
\end{proof}
\subsection{Closing the one-dimensional count}
\begin{theorem}[Contradiction from the boundary count]
\label{thm:gluing:contradiction}
The finite stack of Theorem~\ref{thm:stack:nonzero} cannot exist
under an orientation-preserving identification $Y_{-2}\cong Y_2$.
Consequently the remaining oriented cosmetic pair of slopes
$-2$ and $2$ is impossible.
\end{theorem}
\begin{proof}
Let $\Omega\ne0$ be the oriented ordinary instanton count of
Theorem~\ref{thm:stack:nonzero}, with the fixed lift weights
$\varepsilon(\mathbf e)$ and the prescribed cap insertions.
Put
\[
r=n_D^{\mathrm{tot}}>0,\qquad \eta=r-1.
\]
The ordinary dimension equation is
$D_I^{\mathrm{tot}}+n-2n-z=0$. Adding the spinor directions,
dividing by effective phase, and imposing the $\eta$ complex
phase cuts gives
\begin{equation}
D_I^{\mathrm{tot}}+2r+n-2n-z-1-2\eta=1.
\label{eq:gluing:one-dimensional}
\end{equation}
By Theorem~\ref{thm:analysis:regularization} the free zero set
is therefore an oriented one-dimensional finite weighted
branch space, with the exclusion and compactness properties
already proved there.
The instanton-link contribution is supplied by
Proposition~\ref{prop:analysis:instanton-link}. At a legitimate
regular cut-down instanton the phase quotient of the normal link is
$\CP^{r-1}$, and the $\eta=r-1$ original-phase-character-two cuts
are sections of $\mathcal O(2)$. With $h=c_1(\mathcal O(1))$,
\begin{equation}
\left\langle(2h)^{r-1},[\CP^{r-1}]\right\rangle
=2^{r-1}=2^\eta.
\label{eq:gluing:link-multiplicity}
\end{equation}
For $r=1$ this means the single point of $\CP^0$ and no phase cuts.
The proposition gives the complex orientation multiplied by the ordinary
instanton sign, up to one common boundary-orientation convention.
The mixed and final interior perturbations vanish in these charts,
and their repeated zero branches have total weight one.
Consequently the total instanton-link boundary contribution is
$\sigma_0 2^\eta\Omega$, for one common
$\sigma_0\in\{1,-1\}$.
Choose small disjoint radial neighborhoods of these finitely
many instantons, and cut each of the finitely many regular lens
collars at a sufficiently large finite $D$.
Proposition~\ref{prop:gluing:collar} identifies every end of the
latter type with its matching main point and cap crossing.
The remaining part of the free zero set is compact. Indeed,
an escaping sequence has an ideal limit by
Lemma~\ref{lem:analysis:compactness};
Theorem~\ref{thm:indices:exclusion} and
Proposition~\ref{prop:analysis:finite-induction} exclude every
limit except the neighborhoods just removed. The exhaustion
assertion of Proposition~\ref{prop:gluing:collar} and the instanton
blow-up charts preclude escape even to those limits from the
truncated remainder.
Apply Lemma~\ref{lem:analysis:stokes} to this compact oriented
one-dimensional remainder. Its finite local branch lists have the
compatible weighted germs proved there, and the instanton and lens
collars preserve their weights. Branches extend by the same repeated
zero list across a perturbation-support boundary, so no artificial
boundary point is introduced. The total weighted oriented boundary
is therefore zero.
Every lens contribution cancels with its bit-flipped partner
by Proposition~\ref{prop:gluing:signs}, using the same
$\varepsilon(\mathbf e)$ as in $\Omega$. The exclusions have
left no other boundary. Stokes therefore gives
\[
0=\sigma_0 2^\eta\Omega.
\]
This contradicts $\Omega\ne0$ over $\Q$. All neighborhoods
used here are the regular zero-particle neighborhoods proved
above; no gluing assertion at a positive-particle level has
entered the boundary argument.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
The reduction in Section~\ref{sec:introduction} shows that a
distinct orientation-preserving cosmetic pair on a nontrivial
knot in $S^3$ would have slopes $-2$ and $2$ and genus two.
The meridional slope is already excluded by the absence of a
second sphere filling, and zero cannot pair with a different
slope because of first homology. An oriented homeomorphism
in the remaining case can be taken to be a diffeomorphism,
giving the identification used throughout the construction.
Theorem~\ref{thm:gluing:contradiction} excludes that case.
Thus any orientation-preserving homeomorphism between two
fillings in the statement has equal slopes.
\end{proof}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//sections/10-gluing.tex ====
==== BEGIN Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//references.bib ====
@inproceedings{Gordon1991,
author = {Gordon, C. McA.},
title = {{Dehn} surgery on knots},
booktitle = {Proceedings of the International Congress of Mathematicians, Kyoto 1990},
editor = {Satake, Ichiro},
volume = {I},
year = {1991},
pages = {631--642},
url = {https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.1/ICM1990.1.ocr.pdf}
}
@article{GordonLuecke1989,
author = {Gordon, C. McA. and Luecke, J.},
title = {Knots are determined by their complements},
journal = {Journal of the American Mathematical Society},
volume = {2},
number = {2},
year = {1989},
pages = {371--415},
doi = {10.2307/1990979}
}
@article{NiWu2015,
author = {Ni, Yi and Wu, Zhongtao},
title = {Cosmetic surgeries on knots in {$S^3$}},
journal = {Journal f{\"u}r die reine und angewandte Mathematik},
volume = {706},
year = {2015},
pages = {1--17},
doi = {10.1515/crelle-2013-0067},
eprint = {1009.4720},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/1009.4720v2}
}
@article{Hanselman2023,
author = {Hanselman, Jonathan},
title = {{Heegaard Floer} homology and cosmetic surgeries in {$S^3$}},
journal = {Journal of the European Mathematical Society},
volume = {25},
number = {5},
year = {2023},
pages = {1627--1669},
doi = {10.4171/JEMS/1218},
eprint = {1906.06773},
archivePrefix = {arXiv},
url = {https://ems.press/content/serial-article-files/32846}
}
@article{Sivek2015,
author = {Sivek, Steven},
title = {{Donaldson} invariants of symplectic manifolds},
journal = {International Mathematics Research Notices},
volume = {2015},
number = {6},
year = {2015},
pages = {1688--1716},
doi = {10.1093/imrn/rnt345},
eprint = {1301.0377},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/1301.0377v2}
}
@misc{DaemiLidmanMillerEismeier2025,
author = {Daemi, Aliakbar and Lidman, Tye and Miller Eismeier, Mike},
title = {Filtered instanton homology and cosmetic surgery},
year = {2025},
eprint = {2410.21248},
archivePrefix = {arXiv},
howpublished = {\href{https://arxiv.org/abs/2410.21248v2}{arXiv:2410.21248v2}},
note = {Version 2, 21 March 2025},
url = {https://arxiv.org/abs/2410.21248v2}
}
@book{DonaldsonKronheimer1990,
author = {Donaldson, S. K. and Kronheimer, P. B.},
title = {The Geometry of Four-Manifolds},
publisher = {Clarendon Press},
address = {Oxford},
year = {1990},
url = {https://academic.oup.com/book/52942}
}
@book{Donaldson2002,
author = {Donaldson, S. K.},
title = {{Floer} Homology Groups in {Yang--Mills} Theory},
series = {Cambridge Tracts in Mathematics},
volume = {147},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2002},
note = {With the assistance of M. Furuta and D. Kotschick},
doi = {10.1017/CBO9780511543098}
}
@article{Floer1988,
author = {Floer, Andreas},
title = {An instanton-invariant for 3-manifolds},
journal = {Communications in Mathematical Physics},
volume = {118},
number = {2},
year = {1988},
pages = {215--240},
doi = {10.1007/BF01218578}
}
@misc{GhoshSivekZentner2023,
author = {Ghosh, Sudipta and Sivek, Steven and Zentner, Raphael},
title = {Rational homology 3-spheres and {$\mathrm{SL}(2,\mathbb C)$} representations},
year = {2023},
eprint = {2310.17965},
archivePrefix = {arXiv},
howpublished = {\href{https://arxiv.org/abs/2310.17965v1}{arXiv:2310.17965v1}},
note = {Version 1, 27 October 2023},
url = {https://arxiv.org/abs/2310.17965v1}
}
@article{AtiyahSingerIII,
author = {Atiyah, M. F. and Singer, I. M.},
title = {The index of elliptic operators. {III}},
journal = {Annals of Mathematics},
volume = {87},
number = {3},
year = {1968},
pages = {546--604},
doi = {10.2307/1970717}
}
@article{AtiyahPatodiSingerI,
author = {Atiyah, M. F. and Patodi, V. K. and Singer, I. M.},
title = {Spectral asymmetry and {Riemannian} geometry. {I}},
journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
volume = {77},
number = {1},
year = {1975},
pages = {43--69},
doi = {10.1017/S0305004100049410},
url = {https://webhomes.maths.ed.ac.uk/~v1ranick/papers/aps001.pdf}
}
@article{FintushelStern1997,
author = {Fintushel, Ronald and Stern, Ronald J.},
title = {Rational blowdowns of smooth 4-manifolds},
journal = {Journal of Differential Geometry},
volume = {46},
number = {2},
year = {1997},
pages = {181--235},
doi = {10.4310/jdg/1214459932},
eprint = {alg-geom/9505018},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/alg-geom/9505018v1}
}
@article{Gabai1987,
author = {Gabai, David},
title = {Foliations and the topology of 3-manifolds. {III}},
journal = {Journal of Differential Geometry},
volume = {26},
number = {3},
year = {1987},
pages = {479--536},
doi = {10.4310/jdg/1214441488}
}
@book{EliashbergThurston1998,
author = {Eliashberg, Yakov M. and Thurston, William P.},
title = {Confoliations},
series = {University Lecture Series},
volume = {13},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {1998},
doi = {10.1090/ulect/013}
}
@article{Eliashberg2004,
author = {Eliashberg, Yakov},
title = {A few remarks about symplectic filling},
journal = {Geometry \& Topology},
volume = {8},
year = {2004},
pages = {277--293},
doi = {10.2140/gt.2004.8.277},
eprint = {math/0311459},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/math/0311459v2}
}
@article{KronheimerMrowka2004,
author = {Kronheimer, P. B. and Mrowka, T. S.},
title = {{Witten}'s conjecture and {Property P}},
journal = {Geometry \& Topology},
volume = {8},
year = {2004},
pages = {295--310},
doi = {10.2140/gt.2004.8.295},
eprint = {math/0311489},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/math/0311489v5}
}
@article{KronheimerMrowka1995,
author = {Kronheimer, P. B. and Mrowka, T. S.},
title = {Embedded surfaces and the structure of {Donaldson}'s polynomial invariants},
journal = {Journal of Differential Geometry},
volume = {41},
number = {3},
year = {1995},
pages = {573--734},
doi = {10.4310/jdg/1214456482},
url = {https://people.math.harvard.edu/~kronheim/structhm.pdf}
}
@article{Donaldson1999,
author = {Donaldson, S. K.},
title = {{Lefschetz} pencils on symplectic manifolds},
journal = {Journal of Differential Geometry},
volume = {53},
number = {2},
year = {1999},
pages = {205--236},
doi = {10.4310/jdg/1214425535}
}
@article{Taubes1995,
author = {Taubes, Clifford Henry},
title = {More constraints on symplectic forms from {Seiberg--Witten} invariants},
journal = {Mathematical Research Letters},
volume = {2},
number = {1},
year = {1995},
pages = {9--13},
doi = {10.4310/MRL.1995.v2.n1.a2}
}
@article{AurouxKatzarkov2008,
author = {Auroux, Denis and Katzarkov, Ludmil},
title = {A degree doubling formula for braid monodromies and {Lefschetz} pencils},
journal = {Pure and Applied Mathematics Quarterly},
volume = {4},
year = {2008},
pages = {237--318},
note = {Cited theorem locator refers to arXiv:math/0605001v1.},
eprint = {math/0605001},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/math/0605001v1}
}
@article{Smith2001,
author = {Smith, Ivan},
title = {{Lefschetz} pencils and divisors in moduli space},
journal = {Geometry \& Topology},
volume = {5},
year = {2001},
pages = {579--608},
doi = {10.2140/gt.2001.5.579},
url = {https://msp.org/gt/2001/5-2/gt-v5-n2-p04-p.pdf}
}
@article{KronheimerMrowka2010,
author = {Kronheimer, Peter and Mrowka, Tomasz},
title = {Knots, sutures, and excision},
journal = {Journal of Differential Geometry},
volume = {84},
number = {2},
year = {2010},
pages = {301--364},
doi = {10.4310/jdg/1274707316},
eprint = {0807.4891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/0807.4891v2}
}
@article{Munoz1999Ring,
author = {Mu{\~n}oz, Vicente},
title = {Ring structure of the {Floer} cohomology of {$\Sigma\times S^1$}},
journal = {Topology},
volume = {38},
number = {3},
year = {1999},
pages = {517--528},
doi = {10.1016/S0040-9383(98)00028-7},
eprint = {dg-ga/9710029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/dg-ga/9710029v1}
}
@article{Munoz1999FukayaFloer,
author = {Mu{\~n}oz, Vicente},
title = {{Fukaya--Floer} homology of {$\Sigma\times S^1$} and applications},
journal = {Journal of Differential Geometry},
volume = {53},
number = {2},
year = {1999},
pages = {279--326},
eprint = {math/9804081},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/math/9804081v3}
}
@article{FeehanLenessPU2I,
author = {Feehan, Paul M. N. and Leness, Thomas G.},
title = {{PU}(2) monopoles. {I}: Regularity, {Uhlenbeck} compactness, and transversality},
journal = {Journal of Differential Geometry},
volume = {49},
number = {2},
year = {1998},
pages = {265--410},
doi = {10.4310/jdg/1214461020},
eprint = {dg-ga/9710032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/dg-ga/9710032v1}
}
@article{FeehanLenessLinksI,
author = {Feehan, Paul M. N. and Leness, Thomas G.},
title = {{PU}(2) monopoles and links of top-level {Seiberg--Witten} moduli spaces},
journal = {Journal f{\"u}r die reine und angewandte Mathematik},
volume = {538},
year = {2001},
pages = {57--133},
doi = {10.1515/crll.2001.069},
eprint = {math/0007190},
archivePrefix = {arXiv},
url = {https://sites.math.rutgers.edu/~feehan/preprints/crelle01a.pdf}
}
@article{FeehanLenessPU2II,
author = {Feehan, Paul M. N. and Leness, Thomas G.},
title = {{PU}(2) monopoles. {II}: Top-level {Seiberg--Witten} moduli spaces and {Witten}'s conjecture in low degrees},
journal = {Journal f{\"u}r die reine und angewandte Mathematik},
volume = {538},
year = {2001},
pages = {135--212},
doi = {10.1515/crll.2001.064},
eprint = {dg-ga/9712005},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/dg-ga/9712005v3}
}
@incollection{Dolgachev1982,
author = {Dolgachev, Igor},
title = {Weighted projective varieties},
editor = {Carrell, James B.},
booktitle = {Group Actions and Vector Fields},
series = {Lecture Notes in Mathematics},
volume = {956},
year = {1982},
pages = {34--71},
publisher = {Springer},
address = {Berlin, Heidelberg},
doi = {10.1007/BFb0101508},
url = {https://sites.lsa.umich.edu/idolga/wp-content/uploads/sites/1334/2024/08/weighted82.pdf}
}
@article{Abreu1998,
author = {Abreu, Miguel},
title = {{K{\"a}hler} geometry of toric varieties and extremal metrics},
journal = {International Journal of Mathematics},
volume = {9},
year = {1998},
pages = {641--651},
doi = {10.1142/S0129167X98000282},
eprint = {dg-ga/9711014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/dg-ga/9711014v1}
}
@article{Guillemin1994,
author = {Guillemin, Victor},
title = {{K{\"a}hler} structures on toric varieties},
journal = {Journal of Differential Geometry},
volume = {40},
year = {1994},
pages = {285--309},
doi = {10.4310/jdg/1214455538}
}
@article{Uhlenbeck1982Connections,
author = {Uhlenbeck, Karen K.},
title = {Connections with {$L^p$} bounds on curvature},
journal = {Communications in Mathematical Physics},
volume = {83},
number = {1},
year = {1982},
pages = {31--42},
doi = {10.1007/BF01947069}
}
@article{Uhlenbeck1982Removable,
author = {Uhlenbeck, Karen K.},
title = {Removable singularities in {Yang--Mills} fields},
journal = {Communications in Mathematical Physics},
volume = {83},
number = {1},
year = {1982},
pages = {11--29},
doi = {10.1007/BF01947068}
}
@article{Smale1965,
author = {Smale, Stephen},
title = {An infinite dimensional version of {Sard}'s theorem},
journal = {American Journal of Mathematics},
volume = {87},
number = {4},
year = {1965},
pages = {861--866},
doi = {10.2307/2373250}
}
@incollection{Kirby1997,
author = {Kirby, Rob},
title = {Problems in low-dimensional topology},
booktitle = {Geometric Topology},
editor = {Kazez, William H.},
series = {AMS/IP Studies in Advanced Mathematics},
volume = {2.2},
publisher = {American Mathematical Society and International Press},
year = {1997}, pages = {35--473}
}
@incollection{BleilerHodgsonWeeks1999,
author = {Bleiler, Steven A. and Hodgson, Craig D. and Weeks, Jeffrey R.},
title = {Cosmetic surgery on knots},
booktitle = {Proceedings of the Kirbyfest},
publisher = {Geometry \& Topology Publications},
series = {Geometry \& Topology Monographs},
volume = {2}, year = {1999}, pages = {23--34},
doi = {10.2140/gtm.1999.2.23},
url = {https://msp.org/gtm/1999/02/gtm-1999-02-002p.pdf}
}
@article{BL90,
author = {Boyer, Steven and Lines, Daniel},
title = {Surgery formulae for {Casson}'s invariant and extensions to homology lens spaces},
journal = {Journal f{\"u}r die reine und angewandte Mathematik},
volume = {405},
year = {1990},
pages = {181--220},
doi = {10.1515/crll.1990.405.181}
}
@article{OS11,
author = {Ozsv{\'a}th, Peter S. and Szab{\'o}, Zolt{\'a}n},
title = {Knot {Floer} homology and rational surgeries},
journal = {Algebraic \& Geometric Topology},
volume = {11},
number = {1},
year = {2011},
pages = {1--68},
doi = {10.2140/agt.2011.11.1},
eprint = {math/0504404},
archivePrefix = {arXiv},
primaryClass = {math.GT}
}
@article{Wang06,
author = {Wang, Jiajun},
title = {Cosmetic surgeries on genus one knots},
journal = {Algebraic \& Geometric Topology},
volume = {6},
year = {2006},
pages = {1491--1517},
doi = {10.2140/agt.2006.6.1491}
}
@article{Wu11,
author = {Wu, Zhongtao},
title = {Cosmetic surgery in {$L$}-space homology spheres},
journal = {Geometry \& Topology},
volume = {15},
number = {2},
year = {2011},
pages = {1157--1168},
doi = {10.2140/gt.2011.15.1157}
}
@article{IchiharaWu2019,
author = {Ichihara, Kazuhiro and Wu, Zhongtao},
title = {A note on {Jones} polynomial and cosmetic surgery},
journal = {Communications in Analysis and Geometry},
volume = {27}, number = {5}, year = {2019}, pages = {1087--1104},
doi = {10.4310/CAG.2019.v27.n5.a3},
eprint = {1606.03372}, archivePrefix = {arXiv},
url = {https://arxiv.org/abs/1606.03372v1}
}
@article{Detcherry2026,
author = {Detcherry, Renaud},
title = {A quantum obstruction for purely cosmetic surgeries},
journal = {Annales de l'Institut Fourier},
volume = {76}, number = {1}, year = {2026}, pages = {229--247},
doi = {10.5802/aif.3673},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3673/}
}
@article{FuterPurcellSchleimer2025,
author = {Futer, David and Purcell, Jessica S. and Schleimer, Saul},
title = {Excluding cosmetic surgeries on hyperbolic {$3$}-manifolds},
journal = {Journal of Computational Geometry},
volume = {16}, number = {1}, year = {2025}, pages = {694--736},
doi = {10.20382/jocg.v16i1a19},
eprint = {2403.10448}, archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2403.10448v3}
}
@article{Ren2025,
author = {Ren, Qiuyu},
title = {Cosmetic surgery on satellite knots},
journal = {Bulletin of the London Mathematical Society},
volume = {57}, number = {12}, year = {2025}, pages = {3934--3940},
doi = {10.1112/blms.70188},
eprint = {2411.13828}, archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2411.13828v2}
}
@article{KotelskiyLidmanMooreWatsonZibrowius2024,
author = {Kotelskiy, Artem and Lidman, Tye and Moore, Allison H. and Watson, Liam and Zibrowius, Claudius},
title = {Cosmetic operations and {Khovanov} multicurves},
journal = {Mathematische Annalen},
volume = {389}, number = {3}, year = {2024}, pages = {2903--2930},
doi = {10.1007/s00208-023-02697-5},
url = {https://link.springer.com/article/10.1007/s00208-023-02697-5}
}
@article{StipsiczSzabo2021,
author = {Stipsicz, Andr{\'a}s I. and Szab{\'o}, Zolt{\'a}n},
title = {Purely cosmetic surgeries and pretzel knots},
journal = {Pacific Journal of Mathematics},
volume = {313}, number = {1}, year = {2021}, pages = {195--211},
doi = {10.2140/pjm.2021.313.195},
eprint = {2006.06765}, archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2006.06765v1}
}
@article{TaoCables2019,
author = {Tao, Ran}, title = {Cable knots do not admit cosmetic surgeries},
journal = {Journal of Knot Theory and Its Ramifications},
volume = {28}, number = {4}, year = {2019}, pages = {1950034},
doi = {10.1142/S0218216519500342},
eprint = {1804.02210}, archivePrefix = {arXiv},
url = {https://arxiv.org/abs/1804.02210v1}
}
@article{TaoConnected2022,
author = {Tao, Ran}, title = {Knots admitting purely cosmetic surgeries are prime},
journal = {Topology and its Applications},
volume = {322}, year = {2022}, pages = {108270},
doi = {10.1016/j.topol.2022.108270},
eprint = {1909.05048}, archivePrefix = {arXiv},
url = {https://arxiv.org/abs/1909.05048v1}
}
@misc{PidstrigachTyurin1995,
author = {Pidstrigach, Victor and Tyurin, Andrei},
title = {Localisation of the {Donaldson}'s invariants along {Seiberg--Witten} classes},
year = {1995},
eprint = {dg-ga/9507004},
archivePrefix = {arXiv},
note = {Version 1, 24 July 1995},
url = {https://arxiv.org/abs/dg-ga/9507004v1}
}
@article{OkonekTeleman1996,
author = {Okonek, Christian and Teleman, Andrei},
title = {Quaternionic monopoles},
journal = {Communications in Mathematical Physics},
volume = {180},
number = {2},
year = {1996},
pages = {363--388},
doi = {10.1007/BF02099718},
eprint = {alg-geom/9505029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/alg-geom/9505029v4}
}
@article{CullerDaemiXie2020,
author = {Culler, Lucas and Daemi, Aliakbar and Xie, Yi},
title = {Surgery, polygons and {$SU(N)$}-{Floer} homology},
journal = {Journal of Topology},
volume = {13},
number = {2},
year = {2020},
pages = {576--668},
doi = {10.1112/topo.12137},
eprint = {1712.09910},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/1712.09910v1}
}
@article{Kawasaki1979,
author = {Kawasaki, Tetsuro},
title = {The {Riemann--Roch} theorem for complex {V}-manifolds},
journal = {Osaka Journal of Mathematics},
volume = {16}, number = {1}, year = {1979}, pages = {151--159},
doi = {10.18910/8716},
url = {https://ir.library.osaka-u.ac.jp/repo/ouka/all/8716/ojm16_01_13.pdf}
}
@article{Gramain1973,
author = {Gramain, Andr\'e},
title = {Le type d'homotopie du groupe des diff\'eomorphismes d'une surface compacte},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
series = {4}, volume = {6}, number = {1}, year = {1973}, pages = {53--66},
doi = {10.24033/asens.1242},
url = {https://www.numdam.org/article/ASENS_1973_4_6_1_53_0.pdf}
}
@article{LockhartMcOwen1985,
author = {Lockhart, Robert B. and McOwen, Robert C.},
title = {Elliptic differential operators on noncompact manifolds},
journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze},
series = {4}, volume = {12}, number = {3}, year = {1985}, pages = {409--447},
url = {https://www.numdam.org/article/ASNSP_1985_4_12_3_409_0.pdf}
}
@article{Aronszajn1957,
author = {Aronszajn, Nachman},
title = {A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order},
journal = {Journal de Math\'ematiques Pures et Appliqu\'ees},
series = {9}, volume = {36}, year = {1957}, pages = {235--249},
url = {https://sites.math.washington.edu/~blwilson/Nodal/Aronszajn.pdf}
}
@book{Taylor1991,
author = {Taylor, Michael E.},
title = {Pseudodifferential Operators and Nonlinear {PDE}},
series = {Progress in Mathematics}, volume = {100},
publisher = {Birkh\"auser}, address = {Boston}, year = {1991},
doi = {10.1007/978-1-4612-0431-2},
note = {Revised author text available online; cited locators refer to that text},
url = {https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/NLIN.pdf}
}
@inproceedings{Giroux2002,
author = {Giroux, Emmanuel},
title = {G\'eom\'etrie de contact: de la dimension trois vers les dimensions sup\'erieures},
booktitle = {Proceedings of the International Congress of Mathematicians, Beijing 2002},
volume = {II}, publisher = {Higher Education Press},
year = {2002}, pages = {405--414},
eprint = {math/0305129}, archivePrefix = {arXiv},
url = {https://arxiv.org/abs/math/0305129v1}
}
@article{AkbulutOzbagci2002,
author = {Akbulut, Selman and Ozbagci, Burak},
title = {On the topology of compact {Stein} surfaces},
journal = {International Mathematics Research Notices},
volume = {2002},
year = {2002}, number = {15}, pages = {769--782},
eprint = {math/0103106}, archivePrefix = {arXiv},
url = {https://arxiv.org/abs/math/0103106v2}
}
@article{Massuyeau2011,
author = {Massuyeau, Gw{\'e}na{\"e}l},
title = {An introduction to the abelian {Reidemeister} torsion of three-dimensional manifolds},
journal = {Annales math{\'e}matiques Blaise Pascal},
volume = {18}, number = {1}, year = {2011}, pages = {61--140},
doi = {10.5802/ambp.294},
eprint = {1003.2517}, archivePrefix = {arXiv},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.294/}
}
==== END Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere//references.bib ====