==== TABLE OF CONTENTS ====
The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/figures/induction.tex
The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/main.tex
The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/01-introduction.tex
The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/02-algebra.tex
The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/03-massive-index.tex
The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/04-fast-analysis.tex
The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/05-lifts.tex
The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/06-profiles.tex
The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/07-counting.tex
The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/references.bib
===========================
==== BEGIN The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/figures/induction.tex ====
\begin{figure}[!htbp]
\centering
\begin{tikzpicture}[
box/.style={draw=blue!55!black,rounded corners=2pt,align=center,
text width=7.5cm,inner sep=7pt,font=\small},
side/.style={box,text width=4.8cm,draw=black!60,
inner sep=5pt,font=\footnotesize},
flow/.style={-{Stealth[length=2mm]},thick,draw=black!70},
node distance=6mm and 9mm]
\node[box] (input) {Induction input for $2\le n0$ multiplies the commutator term and $m\ge0$
multiplies the linear mass term. Thus
$D_v(1,0)=\sum_\alpha v_\alpha Q_\alpha$.
Section~\ref{sec:algebra} chooses a spinor $u$ and a connected fixing
torus $G$ so that, on gauge and $G$ invariants,
\[
D_u(h,m)^2\ge\frac23\mathcal H(h,m),
\]
where $\mathcal H$ is the averaged squared charge. At fixed positive mass,
the restricted charge is Fredholm. Section~\ref{sec:massive-index}
computes its index as $p(N)$ in the limit $h\downarrow0$ and proves
constancy for all $h>0$.
We then study $h\to\infty$ at mass one. The unitary change of variables
$y=h^{1/3}x$ sends $D_v(h,1)$ to
$h^{1/3}D_v(1,h^{-2/3})$, so this limit is a massless limit at the original
coupling. Internal block states shrink on the physical scale $h^{-1/3}$,
while their relative centers remain on scale one.
\paragraph{Local profiles and their equations.}
Near distinct block centers, a spectral slice separates the block
variables $b$ from the off-block variables. After rescaling the latter,
Section~\ref{fast:section} gives the exact decomposition
\[
D_v(h,m)=\sqrt h\,E_v(b)+D_v^s(h,m)+R_v.
\]
Here $E_v$ acts on the transverse variables and their fermions,
$D_v^s$ is the charge of the block variables, and $R_v$ collects the
remaining slice terms. On the fast space fixed
by the block-center gauge torus, $E_v$ has an even one-dimensional
kernel and a local oscillator gap. A first-order estimate then forces
bounded graph-norm sequences onto this line. In the scalar-block limit,
compressing the charge onto this line gives the ordinary slow charge:
the gauge-spin term cancels the derivative of
the slice density.
To pass the charge equation to a profile, we need more than concentration
of the state. Section~\ref{lift:section} constructs test states whose
charge outputs converge in norm. A correction obtained from the reduced
fast inverse cancels the perpendicular charge output of the leading
oscillator product. Duality then identifies the center equation of each
profile. Fixed physical cutoffs and $L^2$ concentration make these
constructions available without moment bounds on the internal kernels.
\paragraph{Capturing every concentration scale.}
Local profiles leave a possible loss of norm between the internal scale
$h^{-1/3}$ and a vanishing physical radius. Section~\ref{sec:profiles}
excludes that loss by proving, simultaneously with Theorem~\ref{thm:main},
the exterior estimate
\begin{equation}
\norm{rD_u(h,0)F}\ge c_N\norm F,
\qquad
\supp F\subset\{h^{1/3}r>R_N\},\qquad r=|x|.
\label{eq:intro-exterior}
\end{equation}
It applies to compactly supported gauge-invariant vectors in the charge
graph domain, without a rotation restriction. The
smaller-block estimates make proper-cluster profiles account for all
norm on a compact annulus. Their center spaces have dimensions
$9(k-1)\ge9$, so the Euclidean Hardy inequality gives the exterior
estimate at the current size. Only after this step do we extract the
whole-block profile at the origin. The induction therefore uses no
prior existence or dimension statement for $\mathcal K_N$.
Section~\ref{sec:counting} proves a separate massive annular estimate
that prevents escape to infinity. The resulting decomposition preserves
all limiting norms and inner products. Each proper null profile has the
unique even massive center-oscillator vacuum as its center factor. Induction
on the internal blocks and the permutation symmetries then give one
line for each of the $p(N)-1$ proper partitions. Every null profile is even:
Proposition~\ref{prop:singlet-parity} proves this for $\mathcal K_N$
before its dimension is known. Consequently odd massive states have a
uniform charge gap at large $h$. Spectral pairing gives the same gap
on the even complement of the massive kernel, and the index becomes
the exact massive dimension $p(N)$. Extracting these $p(N)$ directions
forces a nonzero whole-block profile. Conversely, recovering any finite
orthonormal whole-block list along with the $p(N)-1$ proper lines and projecting
through the gap bounds that list by one. This closes the induction.
Figure~\ref{fig:induction} records this order of dependencies.
\input{figures/induction}
\clearpage
==== END The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/01-introduction.tex ====
==== BEGIN The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/02-algebra.tex ====
\section{The charge algebra, domains, and rotational invariance}
\label{sec:algebra}
We establish the operator framework needed for the index and limiting
arguments: closed charge domains, a charge coercive on a chosen space
of torus invariants, and even rotationally invariant undeformed kernel
vectors.
We use the real Euclidean space \(\mathfrak k\) of traceless Hermitian
matrices, with scalar product \(\Tr(XY)\) and real Lie bracket
\([X,Y]=(XY-YX)/\ii\). Thus \(\ad(X)\) is a real skew matrix on
\(\mathfrak k\). Uppercase Roman indices denote an orthonormal color
basis and Greek indices denote spinor components;
\(i,j=1,\ldots,9\), \(a,b,c=1,2,3\), and \(d=4,\ldots,9\) are spatial
indices. For \(w\in\R^{16}\) and \(z\in\mathfrak k\), write
\(\theta(w)\cdot z=\sum_{\alpha,A}\theta_\alpha^A w_\alpha z^A\).
All Clifford tensor products are graded tensor products.
Set
\[
T=\gamma^1\gamma^2\gamma^3,\qquad
s_a=2,\quad s_d=-1,
\qquad A_i(v)=\gamma^i v,\quad
B_{ij}(v)=-\gamma^{ij}v,\quad L_i(v)=s_i\gamma^iTv.
\]
Here \(T^t=-T\), \(T^2=-\Id\), and \(T\) commutes with the first
three gamma matrices and anticommutes with the other six. For a unit
real spinor \(v\), coupling \(h>0\), and mass \(m\geq0\), define
\begin{equation}
\label{eq:deformed-charge}
\begin{aligned}
D_v(h,m)&=\sum_i\theta(A_i(v))\cdot p_i
+h\sum_{i3}}\abs{[x_i,x_j]}^2
+m^2\sum_{d=4}^9\abs{x_d}^2,\\
D_v(h,m)^2&=\cH(h,m)+mJ(R(v)),
\qquad R(v)_{ij}=s_jv^t\gamma^i\gamma^jTv.
\end{aligned}
\end{equation}
The cyclic sum uses \((a,b,c)=(1,2,3),(2,3,1),(3,1,2)\).
The matrix \(R(v)\) is skew, preserves \(\R^3\oplus\R^6\), and has
zero spin-basis average. The same formulas hold for an orthogonal
direct sum of compact color algebras and abelian color spaces.
\end{proposition}
\begin{proof}
\emph{The massless square.}
We first compute the kinetic and commutator contributions, including
the gauge cancellation. For a strictly increasing multi-index \(I\), let \(\gamma^I\) denote
the corresponding ordered gamma product. The products of ranks zero
through four form an orthogonal basis of the real \(16\)-by-\(16\)
matrices, for the trace scalar product. Indeed there are
\(1+9+36+84+126=256\) of them. Every nonconstant product of at most
eight distinct gammas is traceless: conjugation by a gamma outside the
product changes its sign for odd rank, and conjugation by a gamma
inside changes its sign for positive even rank. This also proves
orthogonality. The symmetric products have ranks zero, one, and four;
the skew products have ranks two and three.
Put \(\operatorname{sym}M=(M+M^t)/2\) and
\(\operatorname{skew}M=(M-M^t)/2\). The identity needed for the Yukawa
term is
\begin{equation}
\label{alg:spin-identity}
\sum_i\operatorname{sym}\bigl((\gamma^iv)(\gamma^{ij}v)^t\bigr)
=\frac12(v^t\gamma^jv)\Id-\frac12\gamma^j.
\end{equation}
To check it, test its left side against a symmetric monomial
\(M=\gamma^I\) of rank \(r\). Since
\(\sum_i\gamma^iM\gamma^i=(-1)^r(9-2r)M\), its trace pairing is
\[
\bigl((-1)^r(9-2r)-1\bigr)v^tM\gamma^jv.
\]
For \(r=0,1,4\), this is respectively
\(8v^t\gamma^jv\), \(-8\delta_{Ij}\), and zero, which are precisely
the pairings with the right side of \eqref{alg:spin-identity}.
The CAR give \(\theta(w)^2=\abs{w}^2/2\). They therefore give the
kinetic term \(p^2/2\). The square of the bracket multiplier is
\(h^2\sum_{i0\), the graph domain of \(D_h\) embeds compactly into
\(\mathscr H_G\), and its even-to-odd part is Fredholm. Its index is
\begin{equation}
\label{eq:massive-index}
\operatorname{ind}_G D_u(h,1)
:=\dim\Ker(D_h|_{\mathscr H_G^+})
-\dim\Ker(D_h|_{\mathscr H_G^-})=p(N),
\end{equation}
where \(p(N)\) is the number of unordered integer partitions of
\(N\). For all sufficiently small \(h>0\), the massive kernel itself
has dimension \(p(N)\), consists of even vectors, and is separated
from the rest of the spectrum of \(D_h\) by a positive gap uniform
in \(h\).
\end{proposition}
\end{samepage}
The proof occupies this section. It uses the massive deformation to
obtain a Fredholm problem, but makes no assertion yet about its
relation to the undeformed kernel. That relation will require the
global estimates of the subsequent sections.
\subsection{Confinement and continuity at positive coupling}
\begin{lemma}[Coercivity of the massive potential]
\label{lem:massive-potential}
There are \(c>0\) and \(R<\infty\), depending only on \(N\), such
that
\begin{equation}
\label{mass:potential-infinity}
U_{1,1}(x)\geq c\abs{x}^2\qquad(\abs{x}\geq R).
\end{equation}
Moreover, for \(h,m>0\),
\begin{equation}
\label{mass:potential-scaling}
U_{h,m}(x)=\frac{m^4}{h^2}
U_{1,1}\left(\frac hm x\right).
\end{equation}
\end{lemma}
\begin{proof}
The scaling identity follows term by term from
\eqref{eq:square}. To prove the lower bound, suppose otherwise and
choose \(x^{(n)}\) with \(r_n=\abs{x^{(n)}}\to\infty\) and
\(U_{1,1}(x^{(n)})/r_n^2\to0\). After passing to a subsequence,
\(x_i^{(n)}/r_n\to a_i\). The last-six mass terms give \(a_d=0\),
while the first-three squares give
\begin{equation}
\label{mass:escaping-residual}
[x_a^{(n)},x_b^{(n)}]-2x_c^{(n)}=o(r_n)
\quad\text{for cyclic }(a,b,c).
\end{equation}
Dividing by \(r_n^2\) shows that the \(a_a\) commute. Their joint
eigenspaces give an orthogonal decomposition
\(\C^N=\bigoplus_\alpha E_\alpha\), with distinct joint eigenvalue
triples \(\lambda_\alpha\in\R^3\). Choose \(t\in\R^3\) so that
the numbers \(t\cdot\lambda_\alpha\) are distinct, and put
\(y_n=\sum_a t_ax_a^{(n)}\).
The spectral subspaces of \(y_n\) in the corresponding clusters
converge to the \(E_\alpha\). By unitaries tending to the identity
we may identify these subspaces with fixed blocks, without changing
the preceding limits. Their spectral gaps are at least \(c_0r_n\)
for some \(c_0>0\). Equation~\eqref{mass:escaping-residual} implies
\(\norm{[y_n,x_a^{(n)}]}=O(r_n)\). In bases diagonalizing the two
relevant blocks of \(y_n\), an off-block matrix entry is multiplied
in this commutator by a spectral difference of magnitude at least
\(c_0r_n\). Hence
\begin{equation}
\label{mass:off-block-bound}
\norm{(x_a^{(n)})_{\alpha\beta}}=O(1),\qquad\alpha\ne\beta.
\end{equation}
This is a finite-dimensional Hilbert--Schmidt estimate, uniform in
the sequence.
The trace in a diagonal block of \([x_a^{(n)},x_b^{(n)}]\) has zero
contribution from its internal commutator. Its remaining terms are
products of the off-block entries in \eqref{mass:off-block-bound},
so its block trace is \(O(1)\). Divide the block trace of
\eqref{mass:escaping-residual} by \(r_n\). The result is
\(2\dim(E_\alpha)(\lambda_\alpha)_c=0\) for every \(\alpha,c\).
Thus every \(a_i\) vanishes. This contradicts
\(\sum_i\abs{a_i}^2=1\), proving \eqref{mass:potential-infinity}.
Grouping equal limiting joint eigenvalues into a single block was
essential here; no gap was required inside such a block.
\end{proof}
\begin{samepage}
\begin{lemma}[A common massive graph domain]
\label{mass:common-domain}
The domain of \(D_h\), for every \(h>0\), is
\begin{equation}
\label{mass:graph-domain}
\mathcal D=
\left\{\psi\in\mathscr H_G\cap H^1:
\abs{x}\psi\in L^2,\quad [x_i,x_j]\psi\in L^2\ (iR\}}\psi}^2
\leq R^{-2}\norm{\abs{x}\psi}^2
\]
makes the tails uniformly small. The graph embedding is compact.
Thus \(D_h\) has compact resolvent, finite-dimensional kernel, and
closed range, and its odd part is Fredholm.
Writing \(D_h-D_{h_0}=(h-h_0)B\), the bracket bound shows that
\(B:\mathcal D\to\mathscr H_G\) is bounded for the graph norm of
\(D_{h_0}\). Thus
\((D_h-D_{h_0})(D_{h_0}-\ii)^{-1}\) tends to zero in operator norm.
The resolvent identity and a Neumann series imply norm-resolvent
continuity. For completeness, choose a symmetric interval
\((-a,a)\) whose endpoints avoid the spectrum at \(h_0\). Its
finite-rank spectral projection is norm-continuous for nearby
\(h\), and preserves parity. The even and odd ranks of the
projection are therefore locally constant. This is the standard stability
of separated spectral projections; see \cite[Chapter~IV, Section~3.4,
Theorem~3.16]{Kato1995}. The graph estimates above and the parity argument
here are specific to the present operators. On its nonzero spectral
subspace, the odd operator \(D_h\) is an isomorphism between the
even and odd parts, so their rank difference is exactly the graded
kernel dimension. The index is locally constant, and hence
constant on the connected interval \((0,\infty)\).
\end{proof}
\subsection{The minima and their normal oscillators}
The zero set \(Z=\{U_{1,1}=0\}\) is a finite union of compact gauge
orbits. In fact its equations are
\[
x_a=2j_a,\qquad [j_a,j_b]=\epsilon_{abc}j_c,\qquad x_d=0.
\]
Thus \(j\) is a unitary representation of \(\mathfrak{su}(2)\) on
\(\C^N\), and its irreducible dimensions form a partition of \(N\).
Conversely every such representation gives a zero. Conjugacy by
\(\U(N)\) is the same as conjugacy by \(\SU(N)\) on this
configuration space, and the representing matrices have trace zero.
There are therefore exactly \(p(N)\) zero orbits.
These partition labels are the classical vacuum labels of the plane-wave
model~\cite[Sec.~5]{BMN2002}.
For each zero orbit, the required contribution is one even normal
vacuum that survives the stabilizer symmetry. We first specify the
normal Hamiltonian, then calculate its ground line and its character.
Fix a representative \(q=(2j_1,2j_2,2j_3,0,\ldots,0)\in Z\).
Let \(\mathcal O=\U(N)q\), let \(H_q\) be its stabilizer, and set
\(S_q=(T_q\mathcal O)^\perp\subset\mathfrak k^9\). Write
\(U_q^{(2)}\) for the quadratic Taylor term of \(U_{1,1}\) at
\(q\), restricted to \(S_q\). The limiting normal Hamiltonian is
\begin{equation}
\label{mass:normal-Hamiltonian}
H_q^{\mathrm{nor}}
=-\frac12\Delta_{S_q}+U_q^{(2)}(z)
+\frac{\ii}{2}\theta^t
\left(\frac32T-2\ad(j_a)\gamma^a\right)\theta.
\end{equation}
It acts first on the full slice space
\(L^2(S_q)\otimes\mathcal F_N\); gauge transport will impose
\(H_q\)-equivariance there.
The fluctuation frequencies below were computed by
Dasgupta--Sheikh-Jabbari--Van Raamsdonk
\cite[Sections~5.2--5.4]{DasguptaSheikhJabbariVanRaamsdonk2002}.
We rederive them through spin coupling to specify the normal Hessian
and then establish the stabilizer character needed for the invariant
index. In a finite
unitary \(\mathfrak{su}(2)\)-representation, the Hermitian third
generator has a highest weight \(l\). The adjoint raising and
lowering operators produce weights \(l,l-1,\ldots,-l\); on a unit
weight-\(r\) vector the squared lowering norm is
\(l(l+1)-r(r-1)\). This follows by moving a raising operator past a
lowering operator using their commutator. Positivity terminates the
string at \(-l\), so \(2l\) is a nonnegative integer. These strings
split off orthogonally and have uniquely determined ladder
coefficients. Tensoring two strings and adding their weights gives
the usual decomposition into consecutive spins
\(\abs{l-s},\ldots,l+s\). For real skew generators, whose squares
have Casimir \(-l(l+1)\), this implies
\begin{equation}
\label{mass:spin-coupling}
\sum_a\ad(j_a)L_a^{(s)}
=\frac{l(l+1)+s(s+1)-k(k+1)}2
\quad\text{on total spin }k.
\end{equation}
\begin{lemma}[Normal frequencies and the vacuum line]
\label{lem:normal-oscillators}
The quadratic form \(U_q^{(2)}\) is positive definite on \(S_q\).
The oscillator \(H_q^{\mathrm{nor}}\) is nonnegative, has a unique
zero-energy line, and has a positive spectral gap on its orthogonal
complement. Its zero-energy line is even and is fixed by the entire
stabilizer \(H_q\), including exchanges of identical irreducible
summands. It is also fixed by the combined first-three spatial and
compensating gauge rotations, and by \(\Spin(6)\).
\end{lemma}
\begin{proof}
\emph{Bosonic normal modes.}
Decompose \(\mathfrak k_\C\) into irreducible spin-\(l\) summands
for \(\ad(j)\). Half-integral spins are allowed: maps between two
different irreducible summands of the defining representation may
carry such spins. The dimensions calculated in the complexification
give the real dimensions of the original real operators after
summing all their summands.
On the first three spatial components, the linearized residual in
the potential is
\[
2\left(\sum_a\ad(j_a)L_a^{(1)}-\Id\right),
\]
where \(L_a^{(1)}\) generates cross product on \(\R^3\).
Equation~\eqref{mass:spin-coupling} gives the eigenvalues
\(-l,1,l+1\) of \(\sum_a\ad(j_a)L_a^{(1)}\), on coupled spins
\(l+1,l,l-1\), respectively, whenever the indicated representation
exists. Taking the absolute values of the linearized residual gives
normal frequencies
\begin{equation}
\label{mass:boson-frequencies}
\begin{array}{c|c}
\text{frequency}&\text{multiplicity}\\ \hline
2(l+1)&2l+3\\
2l&\max(2l-1,0)\\
2l+1&6(2l+1).
\end{array}
\end{equation}
The last row comes from the last six components, whose squared
frequency is \(1+4l(l+1)=(2l+1)^2\). For \(l>0\), the omitted
zero-frequency spin-\(l\) summand is exactly the gauge tangent:
\(w\mapsto([w,j_a])_a\) is an injective intertwiner from that
color summand, its image is tangent to the zero orbit, and the
dimensions agree. For \(l=0\), only the spin-one first-three
summand exists and has frequency two; there is no gauge tangent.
Thus all frequencies on \(S_q\) are positive. Their bosonic ground
energy per color spin-\(l\) summand is
\begin{equation}
\label{mass:boson-energy}
E_B(l)=\frac12\left((2l+2)(2l+3)
+(2l)(2l-1)+6(2l+1)^2\right)
=16l^2+16l+6.
\end{equation}
The middle term is omitted when its representation is absent; it
is already zero for \(l=0\) and \(l=1/2\).
\smallskip\noindent\emph{Fermionic energy and the ground line.}
The real skew spin-one-half matrices
\(L_a^{(1/2)}=-T\gamma^a/2\) satisfy the same
\(\mathfrak{su}(2)\) relations, and the fermion frequency matrix is
\[
M_q=\frac32T-2\ad(j_a)\gamma^a
=T\left(\frac32-4\sum_a\ad(j_a)L_a^{(1/2)}\right).
\]
The coupled spins \(l+1/2\) and \(l-1/2\) have scalar factors
\(2l+3/2\) and \(-2l-1/2\), respectively, multiplying \(T\).
Since the complexified real sixteen-dimensional spinor has eight
copies of spin one-half, the absolute frequencies and their
complexified multiplicities are
\begin{equation}
\label{mass:fermion-frequencies}
\begin{array}{c|c}
\text{absolute frequency}&\text{complexified multiplicity}\\ \hline
2l+3/2&8(2l+2)\\
2l+1/2&8(2l).
\end{array}
\end{equation}
There are no zero fermion frequencies. A real two-plane with skew
frequency \(\omega\) contributes eigenvalues \(\pm\omega/2\) to
\(\ii\theta^tM_q\theta/2\), by the CAR normalization. The fermionic
ground energy is consequently \(-\Tr\abs{M_q}/4\). From
\eqref{mass:fermion-frequencies} this is \(-E_B(l)\), with
\(E_B(l)\) as in \eqref{mass:boson-energy}. Every bosonic oscillator
and every fermionic two-plane has a unique ground vector, so their
product is the unique zero-energy vector and has a positive gap
above it.
\smallskip\noindent\emph{Stabilizer action and parity.}
The unique ground line contributes to the invariant index only if it
has trivial stabilizer character. We verify that condition in the
fixed exterior realization, which also determines its parity.
Decompose the defining representation as
\[
\C^N=\bigoplus_r(V_{j_r}\otimes M_r),\qquad
H_q=\prod_r\U(M_r),\qquad m_r=\dim M_r,
\]
where the \(j_r\) are distinct and \(V_j\) denotes spin \(j\).
The empty vector in the \(T\)-polarized Fock space is gauge
invariant. The one-particle spin representation under
\(\Spin(3)\times\Spin(6)\) is \(\C^2\otimes\C^4\). Relative to
this polarization, the fermionic ground vector fills precisely the
negative-factor space of \(M_q/T\). On a color spin-\(l\) summand
this is \(V_{l-1/2}\otimes\C^4\), of complex dimension
\begin{equation}
\label{mass:occupied-dimension}
q_l=4(2l)=8l\qquad(l>0).
\end{equation}
Notice the distinction from the doubled multiplicities of the
complexified real frequency matrix in
\eqref{mass:fermion-frequencies}.
The color summand from maps between defining isotypic components
is
\[
\operatorname{Hom}(V_{j_r},V_{j_s})
\otimes\operatorname{Hom}(M_r,M_s),
\]
and its spin list is \(l=\abs{j_r-j_s},\ldots,j_r+j_s\), once
each. Thus the determinant character of its filled space is the
product over that list, omitting \(l=0\), of
\begin{equation}
\label{mass:centralizer-character}
\left((\det g_s)^{m_r}(\det g_r)^{-m_s}\right)^{q_l}.
\end{equation}
The reverse pair \((s,r)\) has the identical spin list and the
inverse character, so the factors cancel. For \(r=s\) the displayed
character is already trivial. Removing the scalar color from
\(\operatorname{End}(\C^N)\) removes only an \(l=0\) summand and
does not affect this argument. Hence the determinant of the full
filled space is the trivial character of \(H_q\). The bosonic
ground Gaussian is also \(H_q\)-invariant. The entire ground line
is therefore fixed by the stabilizer, including its permutation
matrices exchanging identical summands. No ordering sign remains.
Moreover every \(q_l\) is a multiple of four, including when
\(l\) is half-integral. The occupied degree is even, proving the
parity assertion.
\smallskip\noindent\emph{Spatial rotations.}
First-three rotations of \(q\) can be compensated by the
\(\SU(2)\) representation defined by \(j\). Their combined action,
and the action of \(\Spin(6)\), preserve \(S_q\) and commute with
\(H_q^{\mathrm{nor}}\). Each acts on its unique ground line by a
continuous character of a connected semisimple group. Such a
character is trivial: its derivative annihilates the commutator
Lie algebra, which is the entire Lie algebra, and connectedness
then annihilates the character. This proves all the claimed
symmetries.
\end{proof}
\subsection{Small-coupling localization with norm exhaustion}
We next justify the passage from the finitely many normal
oscillators to the spectrum. This is harmonic comparison by
localization, as in semiclassical analysis~\cite{Simon1983}; here the
wells are gauge orbits, so their slice measures and stabilizers
must be retained explicitly. In particular, the argument includes
tails away from their tubes; local oscillator convergence alone
would not determine an index.
\begin{lemma}[Uniform localization near the massive zero orbits]
\label{mass:distance-confinement}
For \(00\),
independent of \(h\), such that
\begin{equation}
\label{mass:distance-bound}
U_{h,1}(x)\geq c\,d_h(x)^2,
\qquad
\cH(h,1)[\psi]\geq
\frac12\norm{\nabla\psi}^2
+\frac c2\norm{d_h\psi}^2-C\norm{\psi}^2.
\end{equation}
In particular, bounded form-energy families with bounded norm have
uniform kinetic bounds and
\(\norm{\mathbf1_{\{d_h>R\}}\psi}^2\leq C'/R^2\).
\end{lemma}
\begin{proof}
The positive normal Hessians in
Lemma~\ref{lem:normal-oscillators}, and the finitely many compact
orbits, imply
\(U_{1,1}\geq c\dist(\cdot,Z)^2\) in a fixed neighborhood of
\(Z\). Indeed normal tubular coordinates have positive quadratic
Taylor term and uniformly cubic Taylor remainder on these compact
orbits. On a compact set outside that neighborhood the same
inequality follows from the positive minimum of
\(U_{1,1}/\dist(\cdot,Z)^2\). At infinity it follows from
Lemma~\ref{lem:massive-potential}, since \(Z\) is bounded. Scaling
then gives its first assertion for all \(x\).
Let \(M=\max_{q\in Z}\abs{q}\). A nearest point of \(h^{-1}Z\)
gives \(h\abs{x}\leq hd_h(x)+M\). Equation~\eqref{mass:fermion-bound}
therefore implies
\[
U_{h,1}(x)+V_F(h,1;x)
\geq c\,d_h(x)^2-C_N(1+M+hd_h(x))
\geq \frac c2d_h(x)^2-C.
\]
Adding the kinetic term proves \eqref{mass:distance-bound}.
In particular the matrix-valued potential plus fermion term is
nonnegative whenever \(d_h\geq R_0\), for a fixed \(R_0\), and
the claimed tail bound follows by integration.
\end{proof}
\begin{lemma}[Normal-slice compactness]
\label{mass:slice-compactness}
Suppose \(h_n\downarrow0\) and \(\psi_n\) are gauge-invariant
vectors of bounded norm and bounded \(\cH(h_n,1)\) form energy.
After a subsequence, there is one profile
\(f_q\in L^2(S_q)\otimes\mathcal F_N\) for each of the \(p(N)\) zero
orbits, with the following properties. Each profile is
\(H_q\)-equivariant and belongs to the oscillator form domain.
With the normalizations described in the proof,
\begin{equation}
\label{mass:slice-exhaustion}
\lim_{n\to\infty}\norm{\psi_n}^2
=\sum_q\norm{f_q}^2,
\qquad
\sum_q H_q^{\mathrm{nor}}[f_q]
\leq\liminf_{n\to\infty}\cH(h_n,1)[\psi_n],
\end{equation}
whenever the subsequence realizes the norm limit and energy
liminf. For any fixed finite family of such sequences, extraction
can be simultaneous and the norm identity polarizes to their inner
products. Parity is preserved.
\end{lemma}
\begin{proof}
Choose disjoint normal tubes of radius \(\rho\) about the compact
orbits in \(Z\). The tubular neighborhood theorem here applies to
the smooth compact embedded orbit \(\U(N)/H_q\) in the Euclidean
configuration space, with its ordinary normal bundle. Scaling its
normal exponential map gives the exact coordinates
\begin{equation}
\label{mass:tube-chart}
\U(N)\times_{H_q}B_{S_q}(\rho/h)
\longrightarrow\mathfrak k^9,
\qquad [k,z]\longmapsto\operatorname{Ad}(k)(q/h+z).
\end{equation}
Within this tube, \(d_h=\abs{z}\). Stabilizer equivariance is kept
on the linear slice: if \(\rho_F\) denotes the gauge action on
\(\mathcal F_N\), then restriction to the slice satisfies
\[
\psi(q/h+\operatorname{Ad}(a)z)=\rho_F(a)\psi(q/h+z),
\qquad a\in H_q.
\]
We do not replace this equivariant linear slice by a singular
quotient.
Let \(d_q=\dim\mathcal O\), and give \(\U(N)/H_q\) a fixed
normalized invariant measure. Integration in
\eqref{mass:tube-chart} gives measure
\(h^{-d_q}J_q(hz)\,\dd z\) after orbit integration, with
\(J_q\) smooth, positive, and \(H_q\)-invariant on
\(B_{S_q}(\rho)\). This follows by differentiating the tube map:
the \(d_q\) orbit columns are
\(h^{-1}[w,q+hz]\), while the normal columns are fixed orthonormal
vectors. Put
\(j_{q,h}(z)=J_q(hz)/J_q(0)\). On every bounded slice set,
\(j_{q,h}\to1\) smoothly and
\(\nabla\log j_{q,h}=O(h)\). Define the flat-measure slice vector
\begin{equation}
\label{mass:slice-normalization}
f_{q,n}(z)=
h_n^{-d_q/2}J_q(0)^{1/2}j_{q,h_n}(z)^{1/2}
\psi_n(q/h_n+z),\qquad \abs{z}<\rho/h_n.
\end{equation}
This identification is isometric on the tube. For general form-domain
vectors we define it by \(L^2\) closure from smooth equivariant
vectors; the derivative estimates below pass by form-core
approximation. Distinct orbit dimensions cause no ambiguity: each
tube has its own displayed constant normalization.
For any smooth equivariant vector, differentiation in an orthonormal
basis of the fixed space \(S_q\) is ordinary ambient
differentiation. Pointwise its squared slice derivative is therefore
at most the full squared ambient gradient. After
\eqref{mass:slice-normalization}, the density derivative changes this
bound only by \(O(h_n)\) on bounded sets. The uniform kinetic bound
in Lemma~\ref{mass:distance-confinement} yields local \(H^1\) bounds
for every \(f_{q,n}\). Rellich compactness, followed by a diagonal
subsequence on expanding balls in the finitely many slices, gives
strong local \(L^2\) convergence and weak local \(H^1\) convergence
to \(f_q\). The distance bound gives both
\(\abs{z}f_q\in L^2\) and uniformly small squared norm tails
\(O(R^{-2})\). For fixed \(R\), all the tubes \(\abs{z}0\) and \(h_0>0\) such that
\begin{equation}
\label{mass:odd-gap}
\norm{D_h\psi}\geq\delta\norm{\psi},
\qquad \psi\in\mathcal D\cap\mathscr H_G^-,\quad 00\), the map
\(\lambda^{-1/2}D_h\) is an isometric parity-changing bijection.
Compact resolvent makes this a statement about a complete discrete
spectrum. Thus \eqref{mass:odd-gap} excludes positive even
eigenvalues below \(\delta^2\) as well, and the kernel is entirely
even. If \(P_h\) denotes its orthogonal projection, we obtain
\begin{equation}
\label{mass:kernel-gap}
\norm{(1-P_h)\psi}\leq\delta^{-1}\norm{D_h\psi},
\qquad 00\).
\end{proof}
The massive signed count is now fixed for every positive coupling. To
compare it with the undeformed kernel, we next analyze large-coupling
states near separated block centers.
==== END The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/03-massive-index.tex ====
==== BEGIN The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/04-fast-analysis.tex ====
\section{The charge near separated centers}
\label{fast:section}
Near a commuting configuration with distinct joint centers, the off-block
coordinates form a fast oscillator. We construct its ground-line
projection and prove concentration on this line, together with bounds
for the charge in the block variables. We then
compute the charge compressed to the line at scalar blocks. These local
estimates do not require invariance under the rotation torus $G$.
Gauge equivariance removes the longitudinal fast momentum and the
central gauge angular momentum in the oscillator square.
The fixed-mass regime supplies cluster profiles, while the growing-mass
regime will exclude escape to infinity after rescaling large annuli.
We use two parameter regimes. In the first, $m\geq0$ is fixed, $h\to\infty$,
and a bounded family has locally bounded graph norm for one fixed charge
$D_v(h,m)$. In the second,
\begin{equation}
m=m_h\longrightarrow\infty,\qquad m_h\leq C_0\sqrt h,
\qquad \|D_{v_\alpha}(h,m_h)\psi_h\|=o(m_h)
\quad(\alpha=1,\ldots,16),
\label{fast:growing-mass}
\end{equation}
where $(v_\alpha)$ is an orthonormal spin basis and
$\sup_h\|\psi_h\|<\infty$. Every assertion with local hypotheses means the
following: multiply by a fixed smooth gauge-invariant cutoff supported in
the specified compact neighborhood, and assume the stated graph bounds
there. Such multiplication preserves the bounds, because the commutator
with a charge is Clifford multiplication by the cutoff gradient and is
independent of $h$ and $m$. All constants below may depend on that
neighborhood and cutoff, $N$, and, in the second regime, $C_0$.
\subsection{Local estimates and spectral slices}
\begin{lemma}[Raw local estimates]\label{lem:raw-estimates}
In either regime, for states supported in a fixed compact set,
\begin{equation}
\|\nabla\psi_h\|\leq C\sqrt h,
\qquad
\sum_{i0\) and \(R_n<\infty\) such that
\begin{equation}\label{eq:exterior-estimate}
\norm{rD_u(h,0)F}\ge c_n\norm{F},
\qquad
\supp F\subset\{h^{1/3}r>R_n\},
\qquad r=|x|,
\end{equation}
for every \(h>0\) and every compactly supported gauge-invariant graph-domain
vector for the \(\SU(n)\) operator. No rotation invariance is required.
\end{proposition}
The proof of Proposition~\ref{prop:exterior-estimate} appears below. At size
\(N\), until that proof is complete, we assume only its conclusions and
\(\dim\cK_n=1\) for \(2\le ne^LR_n\),
\begin{equation}\label{prof:intermediate-bound}
\left(\int_{Mh^{-1/3}0\), first choose \(L\) so that
\(CB/(c_nL)<\delta/2\), then choose \(M_0>e^LR_n\), and finally choose
\(\varepsilon_0\) so that \(e^L\varepsilon_0K/c_n<\delta/2\). This proves
the stated order and uniformity of the limits.
\end{proof}
\subsection{Norm measures and local compactness}
Let \(\pi\) denote the quotient by the compact gauge group. The squared
norm measure of a vector \(\psi\) is the pushforward under \(\pi\) of
\(|\psi(x)|^2\,\dd x\). Bounded sequences of these finite measures have
subsequences converging against continuous compactly supported functions.
The quotient is a locally compact separable metric space: for instance,
the distance between two orbits is the minimum Euclidean distance between
their representatives. Closed bounded quotient sets are compact.
For an unordered partition \(C=(n_1,\ldots,n_k)\) with \(k\ge2\), let
\(\mathcal P_C\) be the profile space of Definition~\ref{def:profile-space}.
Its labeled center space is
\[
\mathcal Z_C=\left\{(c_1,\ldots,c_k)\in(\R^9)^k:
\sum_\alpha n_\alpha c_\alpha=0\right\},
\qquad |c|^2=\sum_\alpha n_\alpha|c_\alpha|^2,
\]
of dimension \(d_C=9(k-1)\). We use the flat measure induced by this metric,
with the fixed normalization for permutations of equal-sized blocks in
Definition~\ref{def:profile-space}. In particular, the quotient norm below
has no extra orbit-volume factor.
\begin{proposition}[Profiles on proper strata]
\label{prop:profile-compactness}
Assume the induction hypotheses at all sizes less than \(N\). Fix \(m\ge0\),
let \(h_j\to\infty\), and suppose that \(\psi_j\) is a bounded sequence of
gauge-invariant vectors with locally bounded \(D_u(h_j,m)\) graph norms.
Here local boundedness means that
\(\norm{D_u(h_j,m)(a\psi_j)}\) is bounded for every smooth compactly
supported gauge-invariant multiplier \(a\).
After a subsequence there is a profile \(F_C\in\mathcal P_C\) on each proper
stratum such that the limiting norm measure restricted to that stratum is
exactly its squared norm density. Moreover, for every test
\(f\in\mathcal T_C\) of Proposition~\ref{prop:test-lifts},
\begin{equation}\label{prof:test-overlap}
\ip{\psi_j}{I_{h_j,u,m}f}\longrightarrow\ip{F_C}{f}.
\end{equation}
The extraction can be made simultaneously for any finite list of
sequences, and then the analogous restrictions of all their polarized
norm measures and test overlaps hold. Parity and rotation symmetries of
the original sequences pass to the profiles under the identifications of
Lemma~\ref{lem:chart-compatibility}.
\end{proposition}
\begin{proof}
Local compactness will construct a kernel-valued profile. The
intermediate-scale estimate is then needed to show that it accounts for
all norm assigned to its stratum. We carry out these two steps before
gluing the local profiles and treating inner products.
\smallskip\noindent\emph{Construction of a local profile.}
On a compact set the fixed mass term is a bounded multiplier, uniformly
in \(h_j\).
The raw estimates of Lemma~\ref{lem:raw-estimates} therefore imply that any
limiting norm measure is supported on commuting configurations. Indeed,
on a compact subset where
\(\sum_{i0\), its mass is \(O(h_j^{-1}/a)\).
Fix a smaller compact set in a separated-center chart, and first multiply
the states by a smooth cutoff supported in the chart and equal to one on
a neighborhood of that set. Write the resulting states again as
\(\psi_j\). All the estimates that follow are uniform on this fixed set.
Lemma~\ref{lem:fast-projection} gives
\[
\psi_j-P_u\psi_j\longrightarrow0,
\qquad P_u\psi_j=\chi_u(b)\eta_j(b),
\qquad \norm{D_u^s(h_j,m)P_u\psi_j}\le C.
\]
The normalized fast section is even and has bounded derivatives as a
vector in the fast Hilbert space. The slow Clifford multipliers act on
the other factor. Consequently the Leibniz identity reads
\begin{equation}\label{prof:section-removal}
D_u^s\bigl(\chi_u\eta_j\bigr)
=\chi_uD_u^s\eta_j+
\sum_\ell a_\ell(\partial_{b_\ell}\chi_u)\eta_j,
\end{equation}
where the \(a_\ell\) are constant slow Clifford matrices. The last term
has norm at most \(C\norm{\eta_j}\). In particular, no factor of \(h_j\)
is introduced when removing the fast section: the large internal bracket
multiplier acts only on slow fermions. Subtracting the bounded mass
multiplier gives
\(\norm{D_u^s(h_j,0)\eta_j}\le C\).
Covariance of \(\chi_u\) means that \(\eta_j\) has the separate gauge
symmetry in each internal \(\SU(n_\alpha)\) factor.
Put \(w_\alpha=b_\alpha-c_\alpha\Id_{n_\alpha}\). On the slow tensor
product, with all graded signs included,
\[
D_u^s(h_j,0)=D_u^C(0)+\sum_{\alpha:n_\alpha>1}
D_{u,\alpha}(h_j,0).
\]
These summands involve disjoint variables and odd Clifford factors, so
their pairwise anticommutators vanish. For compactly supported smooth
sections integration by parts gives
\begin{equation}\label{prof:block-graph-identity}
\norm{D_u^s(h_j,0)\eta_j}^2
=\norm{D_u^C(0)\eta_j}^2+
\sum_{\alpha:n_\alpha>1}
\norm{D_{u,\alpha}(h_j,0)\eta_j}^2.
\end{equation}
The identity extends by closure. One can see this directly here by
approximating after the compact slow cutoff; every potential is bounded
on its support for fixed \(h_j\), and ordinary local smoothing applies.
In particular, each internal graph norm is bounded and
\(\norm{\nabla_c\eta_j}\le C\), since
\(\norm{D_u^C(0)g}^2=\tfrac12\norm{\nabla_cg}^2\).
Rescale the internal variables unitarily by \(y_\alpha=h_j^{1/3}w_\alpha\),
and denote the resulting function by \(\widetilde\eta_j(c,y)\). Then
\begin{equation}\label{prof:rescaled-internal-bounds}
\norm{D_{u,\alpha}(1,0)\widetilde\eta_j}\le Ch_j^{-1/3},
\qquad \norm{\nabla_c\widetilde\eta_j}\le C.
\end{equation}
For an internal cutoff \(\zeta\) of fixed compact support, write the
constant-coefficient kinetic Dirac operator as
\(D_{u,\alpha}(1,0)\) minus its bracket multiplier. That multiplier is
bounded on \(\supp\zeta\). The kinetic norm identity and the bounded
commutator with \(\zeta\) give
\[
\norm{\nabla_{y_\alpha}(\zeta\widetilde\eta_j)}
\le C_\zeta\bigl(\norm{D_{u,\alpha}(1,0)\widetilde\eta_j}
+\norm{\widetilde\eta_j}\bigr).
\]
All spectators are integrated in these norms. Combining these inequalities
for the finitely many blocks with the center derivative estimate gives a
bounded ordinary \(H^1\) norm on every compact subset of the joint
\((c,y)\) space. Its Clifford fiber is finite dimensional. The usual
Rellich theorem therefore gives, by a diagonal subsequence, strong local
\(L^2\) convergence to a function \(\eta_\infty(c,y)\). Compactness has
been applied jointly in all these variables, rather than to functions
with an uncontrolled infinite-dimensional spectator fiber.
Fatou's lemma gives the global \(L^2\) bound for this local chart limit.
Testing \eqref{prof:rescaled-internal-bounds} against compactly supported
smooth functions shows distributionally that
\(D_{u,\alpha}(1,0)\eta_\infty=0\) for each \(\alpha\). Fubini's theorem
and a countable dense family of internal tests show that almost every
slice in the remaining variables is in the corresponding distributional
\(L^2\) kernel. Lemma~\ref{lem:domains} identifies this with
\(\cK_{n_\alpha}\). Equivalently, the orthogonal projection onto each
internal kernel fixes \(\eta_\infty\); the product of these commuting
projections identifies its coefficient space as
\(\bigotimes_\alpha\cK_{n_\alpha}\). This constructs the local profile.
\smallskip\noindent\emph{Exact norm on the stratum.}
We now identify how much limiting norm is assigned to the stratum
\(w=0\). The replacement of \(\psi_j\) by \(P_u\psi_j\) changes integrals against
bounded multipliers by \(o(1)\). The physical fast coordinates tend to zero
in norm measure, since \(z=t/\sqrt{h_j}\) and
\(\norm{t\psi_j}\le C\). On bounded \(y\) sets, smoothness also gives
\(\chi_u(c+h_j^{-1/3}y)-\chi_u(c)\to0\) uniformly in fast Hilbert norm.
It remains to decide whether any internal norm escapes all bounded
\(y\) sets while still reaching \(w=0\).
Here is a precise measure comparison. Take a nonnegative continuous center
test \(a(c)\) of compact support in a smaller chart. Let
\(0\le\kappa_\varepsilon(w)\le1\) be a product of radial internal cutoffs,
equal to one when every \(|w_\alpha|\le\varepsilon\) and zero when some
\(|w_\alpha|\ge2\varepsilon\). Choose them decreasing as
\(\varepsilon\downarrow0\). In testing the quotient measure, include a fixed
physical fast cutoff equal to one near \(z=0\), so that the product extends
continuously by zero inside the outer chart. Its removal changes the limit
by zero, because \(\norm{t\psi_j}\le C\) and \(z=t/\sqrt{h_j}\).
The slice density has already been absorbed unitarily, and
\(\int|\chi_u(b,t)|^2\,\dd t=1\); thus, after removal of the fast cutoff,
these tests of the projected states are exactly the corresponding slow
integrals. For each fixed \(M\), strong local convergence
identifies the integrals on \(\{|y_\alpha|\le M\text{ for all }\alpha\}\).
The difference between such an integral and the integral with
\(\kappa_\varepsilon\) is bounded above, apart from this local convergence
error, by
\begin{equation}\label{prof:measure-tail-bound}
\norm{a}_\infty\sum_{\alpha:n_\alpha>1}
\int_{Mh_j^{-1/3}<|w_\alpha|<2\varepsilon}|\eta_j|^2.
\end{equation}
Each summand satisfies Lemma~\ref{lem:intermediate-scales}, with all other
variables as spectators, because of \eqref{prof:block-graph-identity}.
For the converse comparison, the bounded \(y\) set lies where
\(\kappa_\varepsilon=1\) for all sufficiently large \(j\). Thus its limiting
integral is a lower bound for the one with \(\kappa_\varepsilon\).
To record the order of limits explicitly, put
\[
J_\varepsilon=\int a(c)\kappa_\varepsilon(w)\,\dd\mu,
\qquad
A_M=\int_{\max_\alpha|y_\alpha|\le M}
a(c)|\eta_\infty(c,y)|^2\,\dd c\,\dd y.
\]
For every fixed \(L\ge1\) and
\(M>e^L\max_\alpha R_{n_\alpha}\), the preceding comparisons and
\eqref{prof:intermediate-bound}, after \(j\to\infty\), give
\[
A_M\le J_\varepsilon\le A_M+
\norm{a}_\infty\sum_{\alpha:n_\alpha>1}
c_{n_\alpha}^{-2}\left(2e^L\varepsilon K_\alpha+CB/L\right)^2,
\]
where \(K_\alpha\) are the block graph bounds in
\eqref{prof:block-graph-identity}. Decreasing \(\varepsilon\) restricts
\(J_\varepsilon\) to \(w=0\), by continuity from above and concentration
of the fast coordinates at zero. Next let \(M\to\infty\), keeping its
lower threshold, and finally let \(L\to\infty\). The error vanishes and
\(A_M\) exhausts the profile norm. Thus the restricted measure is exactly
\(|F_C(c)|^2\,\dd c\). If every block is scalar, the sums and internal
cutoffs are absent and strong local center convergence gives the same
conclusion directly.
This argument also covers simultaneous fragmentation in several blocks,
since their error sets are bounded by the finite sum in
\eqref{prof:measure-tail-bound}.
\smallskip\noindent\emph{Compatibility and inner products.}
Local strong convergence and approximation by bounded \(y,t\) sets,
followed by Proposition~\ref{prop:test-lifts}, give the local version of
\eqref{prof:test-overlap}. It also identifies the profile independently
of the outer chart cutoff. If two cutoffs \(\lambda,\widetilde\lambda\)
equal one near the same smaller center support, then
\(\| (\lambda-\widetilde\lambda)I_{h_j,u,m}f\|\to0\) for every test
supported there: its leading lift concentrates in that neighborhood,
and its correction tends to zero in norm. Pairing with the original
bounded sequence proves that the two localizations have identical test
overlaps. These identify the same profile coefficient, including its phase.
There are finitely many partition types and a countable exhaustion of each
stratum by relatively compact charts. Repeating the extraction diagonally
gives profiles on all of them. On overlapping charts,
Lemma~\ref{lem:chart-compatibility} identifies the leading tests and hence,
by their overlaps, the local profiles. These consequently descend with
the required permutation action. Exhaustion of the stratum gives the
global norm bound, and the local overlaps give
\eqref{prof:test-overlap} for every compactly supported profile test.
For a finite list \(\psi_j^1,\ldots,\psi_j^q\), carry out the same
extractions for the list and for the finitely many sums and imaginary
sums needed for complex polarization. Equivalently, take weak limits of
the matrix of measures with entries
\(\pi_*(\ip{\psi_j^a(x)}{\psi_j^b(x)}\,\dd x)\). Positivity of this matrix
and Cauchy--Schwarz bound every entry by its diagonal measures. Linearity
of the local limits then gives, on each stratum, the entry
\(\ip{F_C^a(c)}{F_C^b(c)}\,\dd c\). This proves the polarized assertion.
The same leading identifications commute with the indicated symmetries,
which proves the final assertion.
\end{proof}
\subsection{Passing the charge to the center space}
\begin{lemma}[Removal of collisions]\label{lem:collision-removal}
Let \(S_C\subset \mathcal Z_C\) be the union of the collision subspaces
\(c_\alpha=c_\beta\). If \(F,G\in L^2(\mathcal Z_C)\), with the profile coefficient
fiber, satisfy \(D_u^C(m)F=G\) distributionally on \(\mathcal Z_C\setminus S_C\),
then the same identity holds on \(\mathcal Z_C\). In particular, \(F\) is in the
full Euclidean graph domain. This remains true on the subspace with the
permutation symmetry of the profile space.
\end{lemma}
\begin{proof}
Each collision subspace has codimension nine, also after imposing the
trace-zero constraint. Let \(\rho_\varepsilon\) be the product of smooth
cutoffs that vanish at distance at most \(\varepsilon\) from these finitely
many subspaces and equal one at distance at least \(2\varepsilon\). On
each compact set \(K\subset \mathcal Z_C\), tubular coordinates give
\begin{equation}\label{prof:collision-capacity}
\norm{\nabla\rho_\varepsilon}_{L^2(K)}^2
\le C_K\varepsilon^{-2}\varepsilon^9=C_K\varepsilon^7.
\end{equation}
Intersections of collision subspaces cause no difficulty, since the
derivative of the finite product is bounded by the sum of the individual
derivatives.
For a smooth compactly supported test \(\varphi\), apply the given
distributional identity to \(\rho_\varepsilon\varphi\). The commutator
error has absolute value at most
\[
2^{-1/2}\norm{F}_{L^2(K)}\norm{\varphi}_\infty
\norm{\nabla\rho_\varepsilon}_{L^2(K)},
\]
which tends to zero. All other terms converge by dominated convergence;
the mass multiplier is bounded on \(K\). This proves the global
distributional equation. The maximal and minimal graph domains coincide
by the cutoff and local ellipticity argument of Lemma~\ref{lem:domains},
which applies verbatim to this abelian operator. Tests with values in a
finite-dimensional coefficient subspace suffice, followed by density if
necessary. For permutation-equivariant vectors, averaging the tests over
the finite permutation group gives the same conclusion. There is thus no
additional collision boundary condition.
\end{proof}
\begin{proposition}[Liminf for the center charges]
\label{prop:profile-liminf}
Under the hypotheses of Proposition~\ref{prop:profile-compactness}, suppose
in addition that \(\norm{D_u(h_j,m)\psi_j}\) is globally bounded. On a
subsequence realizing its lower limit, the profiles satisfy
\begin{equation}\label{eq:profile-liminf}
\sum_{C\text{ proper}}\norm{D_u^C(m)F_C}^2
\le\liminf_{j\to\infty}\norm{D_u(h_j,m)\psi_j}^2.
\end{equation}
The derivatives on the left are the full-space graph derivatives.
\end{proposition}
\begin{proof}
Choose any finite family of profile tests \(g_C\in\mathcal T_C\) and put
\(T_jg=\sum_C I_{h_j,u,m}g_C\). Proposition~\ref{prop:test-lifts} and
Lemma~\ref{lem:chart-compatibility} give
\[
\norm{T_jg}^2\longrightarrow\sum_C\norm{g_C}^2,
\qquad
\ip{D_u(h_j,m)\psi_j}{T_jg}
\longrightarrow\sum_C\ip{F_C}{D_u^C(m)g_C}.
\]
For the second identity, move the self-adjoint charge to the tests; its
graph-output error pairs with the bounded norm of \(\psi_j\) and tends
to zero. Cauchy--Schwarz therefore bounds the last functional by
\[
\left(\liminf_j\norm{D_u(h_j,m)\psi_j}\right)
\left(\sum_C\norm{g_C}^2\right)^{1/2}.
\]
The tests are dense in the Hilbert direct sum of the profile spaces.
Riesz representation identifies this bounded functional with the direct
sum of the distributional derivatives, and proves
\eqref{eq:profile-liminf} off the collisions. Lemma~\ref{lem:collision-removal}
extends each derivative across them. This argument uses convergence for
each fixed finite test family; it requires no uniform estimate on the
whole unit ball of test functions.
\end{proof}
\pagebreak
\subsection{Hardy inequality and the current exterior estimate}
We can now prove Proposition~\ref{prop:exterior-estimate} at size \(N\),
without having used any information about \(\cK_N\).
\begin{proof}[Proof of Proposition~\ref{prop:exterior-estimate} at size \(N\)]
Fix a compact annulus \(a\le r\le b\), where \(02\)
gives, for smooth compactly supported \(g\) away from zero,
\[
0\le\left\|\nabla g+\frac{d-2}{2}\frac{c}{|c|^2}g\right\|^2
=\norm{\nabla g}^2-\frac{(d-2)^2}{4}\norm{g/|c|}^2.
\]
Cutoffs and closure extend this Hardy inequality to \(H^1\). Applied on
\(\mathcal Z_C\), with its orthonormal trace coordinates and \(d_C=9(k-1)\ge9\),
it yields
\[
\norm{D_u^C(0)(|c|F_C)}
=2^{-1/2}\norm{\nabla_c(|c|F_C)}
\ge\frac{d_C-2}{2\sqrt2}\norm{F_C}
\ge\frac7{2\sqrt2}\norm{F_C}.
\]
The full-space graph statement needed here is supplied by
Lemma~\ref{lem:collision-removal}. Hence
\eqref{eq:profile-liminf} and \eqref{prof:annulus-norm-exhaustion} imply
\(\liminf_j\norm{D_u(h_j,0)(rF_j)}\ge7/(2\sqrt2)\).
Finally, \(\norm{[D_u,r]F_j}=1/\sqrt2\), proving
\eqref{prof:fixed-annulus-bound}.
Fix a logarithmic width \(A>0\). Since \(5/(2\sqrt2)>1\), a contradiction
sequence in the fixed annulus \(1e^AH_A^{1/3}\). Every member of the partition meeting
\(\supp F\subset\{h^{1/3}r>R_N\}\) is supported in an annulus satisfying
\eqref{prof:scaled-annulus-bound}: its lower radius is at least \(e^{-A}\)
times a radius where it meets \(F\). Summing that bound and using
\eqref{prof:weighted-ims} proves
\(\norm{rD_uF}^2\ge\tfrac12\norm{F}^2\). Taking, for example,
\(c_N=1/\sqrt2\) proves the assertion. The threshold can depend on \(N\).
All identities initially use compact smooth vectors and extend to the
stated domain by Lemma~\ref{lem:domains}.
\end{proof}
\subsection{The full-collapse profile}
\begin{corollary}[Exhaustion for tight sequences]
\label{prof:full-exhaustion}
Under the hypotheses of Proposition~\ref{prop:profile-compactness}, after
Proposition~\ref{prop:exterior-estimate} has been proved at size \(N\),
there is also a full-collapse profile \(F_0\in\cK_N\). Its squared norm is
the mass of the limiting norm measure at zero. If the squared norm
measures are tight, then
\begin{equation}\label{prof:full-norm-exhaustion}
\lim_j\norm{\psi_j}^2
=\norm{F_0}^2+\sum_{C\text{ proper}}\norm{F_C}^2.
\end{equation}
For any finite simultaneously extracted list, one likewise has
\begin{equation}\label{prof:polarized-exhaustion}
\lim_j\ip{\psi_j^a}{\psi_j^b}
=\ip{F_0^a}{F_0^b}+
\sum_{C\text{ proper}}\ip{F_C^a}{F_C^b}.
\end{equation}
The space \(\cK_N\) is allowed to have unknown dimension in this statement.
\end{corollary}
\begin{proof}
Multiply by a fixed compact radial cutoff equal to one near zero.
The resulting undeformed graph norms are bounded, since the mass term is
bounded there. Unitary whole-configuration rescaling by \(h_j^{1/3}\)
gives bounded \(L^2\) functions whose \(D_u(1,0)\) outputs tend to zero.
Local ellipticity and Rellich compactness give a strong local limit.
Fatou's lemma makes that limit globally square integrable, and
Lemma~\ref{lem:domains} puts it in \(\cK_N\). Lemma~\ref{lem:intermediate-scales},
now available at size \(N\), identifies its norm with exactly the atom at
zero by the same inner-ball and shrinking-physical-ball comparison used
in \eqref{prof:measure-tail-bound}. This construction is independent of
the chosen cutoff near zero. It extracts one vector in \(\cK_N\); it
does not require compactness of its unit ball or prior finite
dimensionality.
The commuting quotient is the disjoint union of zero and the proper
strata. Tightness preserves total mass in the weak limit: approximate
the constant function one by compact cutoffs and use the uniform tail
bound. Adding the restricted measures gives
\eqref{prof:full-norm-exhaustion}. Applying this to the finite linear
combinations used for polarization gives
\eqref{prof:polarized-exhaustion}. Rotation and parity pass to \(F_0\)
directly under radial cutoffs and dilation.
\end{proof}
==== END The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/06-profiles.tex ====
==== BEGIN The-unique-threshold-bound-state-of-the-SU-N-BFSS-model//manuscript/sections/07-counting.tex ====
\section{Massive tightness and the kernel count}
\label{sec:counting}
The preceding section accounts for all norm at bounded physical radii.
We first show that bounded massive graph norms cannot escape to infinity
as \(h\to\infty\). We then compare the resulting profiles with the exact
massive index. Throughout this section the induction hypotheses at sizes
less than \(N\) remain in force, and the exterior estimate at size \(N\)
has already been proved. The dimension of \(\cK_N\) is still unknown.
Write
\[
\mathfrak q_{h,m}(\psi)
=\frac1{16}\sum_v\norm{D_v(h,m)\psi}^2
\]
for the form of \(\cH(h,m)\), where the sum is over an orthonormal real
spin basis.
\subsection{Uniform confinement on large annuli}
\begin{samepage}
\begin{proposition}[Massive tightness]\label{prop:massive-tightness}
There are \(a>0\), \(h_0>0\), and \(R_0<\infty\), depending on \(N\),
such that
\begin{equation}\label{count:annular-confinement}
\mathfrak q_{h,1}(\psi)\ge aR^2\norm{\psi}^2
\quad\text{if}\quad
h\ge h_0,\quad R\ge R_0,\quad
\supp\psi\subset\{R<|x|<4R\}.
\end{equation}
This inequality is on gauge invariants, with no rotation restriction.
Consequently, for \(R\ge R_0\), after enlarging the constants if needed,
\begin{equation}\label{count:tail-bound}
\int_{|x|>4R}|\psi|^2
\le\frac{C}{R^2}\mathfrak q_{h,1}(\psi)
+\frac{C}{R^4}\norm{\psi}^2,
\qquad h\ge h_0.
\end{equation}
In particular, every bounded sequence of gauge- and \(G\)-invariant
vectors with \(h\to\infty\) and bounded \(D_u(h,1)\) graph norms has tight
squared norm measures.
\end{proposition}
\end{samepage}
Averaging the abelian charge squares cancels their angular terms. In the
application, each spinor charge acts on a different fast projection of the
same state. The following lemma shows that norm convergence of these
projections suffices for the cancellation.
\begin{lemma}[Center confinement for nearby states]
\label{count:center-confinement}
Fix a center space \(\mathcal Z_C\), an arbitrary spectator Hilbert space
\(\mathcal E\), and a compact set
\(K\subset\{c\in\mathcal Z_C:|c|\ge b_0\}\), where \(b_0>0\).
Let \((v_\nu)_{\nu=1}^{16}\) be an orthonormal real spin basis and
\(m_j>0\) with \(m_j\to\infty\). For each \(\nu\), let
\[
a_{\nu,j}\in L^2(\mathcal Z_C;\mathcal F_C\otimes\mathcal E)
\]
be supported in \(K\), bounded in norm, and in the graph domain of the
center charge \(D_{v_\nu}^C(m_j)\), acting trivially on \(\mathcal E\).
Suppose that
\[
\norm{a_{\nu,j}-a_{1,j}}\longrightarrow0,
\qquad
\norm{D_{v_\nu}^C(m_j)a_{\nu,j}}=o(m_j)
\quad(\nu=1,\ldots,16).
\]
Then \(\norm{a_{\nu,j}}\to0\) for every \(\nu\).
No gauge or rotation invariance is imposed on these center states.
\end{lemma}
\begin{proof}
For brevity write \(a_{v,j}\) for the state belonging to a spinor in the
chosen basis, and put \(a_{0,j}=a_{v_1,j}\). The center mass multiplier
is bounded by \(Cm_j\) on \(K\). Subtracting it from the charge and using
the free center Dirac norm identity gives
\begin{equation}\label{count:center-gradient-bound}
\norm{p_Ca_{v,j}}=\sqrt2\norm{D_v^C(0)a_{v,j}}=O(m_j).
\end{equation}
This identity holds with spectators by integration, or first on finite
spectator sums and then by closure.
Apply the abelian square formula of Proposition~\ref{prop:square} and
divide by \(m_j^2\). Dropping its nonnegative kinetic term yields
\begin{equation}\label{count:abelian-mass-lower-bound}
0\ge \frac12\sum_{i=1}^9s_i^2\norm{c_i a_{v,j}}^2
+\ip{a_{v,j}}{A_{v,j}a_{v,j}}+o(1),
\qquad A_{v,j}=m_j^{-1}p_C\cdot R(v)c.
\end{equation}
The fermion multiplier and the spin part of the rotation generator
contribute only \(O(m_j^{-1})\). The vector field \(R(v)c\) has zero
divergence, so \(A_{v,j}\) is symmetric on these compactly supported
\(H^1\) states.
Bounded center coordinates and \eqref{count:center-gradient-bound} give
\[
\norm{A_{v,j}a_{v,j}}+\norm{A_{v,j}a_{0,j}}\le C.
\]
In the second term, the free Dirac identity for \(v_1\) bounds the full
center gradient of \(a_{0,j}\), and hence bounds the angular operator
for every \(v\). Symmetry therefore gives
\begin{equation}\label{count:angular-replacement}
\begin{split}
&\left|\ip{a_{v,j}}{A_{v,j}a_{v,j}}
-\ip{a_{0,j}}{A_{v,j}a_{0,j}}\right|\\
&\hspace{1cm}\le\norm{a_{v,j}-a_{0,j}}
\bigl(\norm{A_{v,j}a_{v,j}}+\norm{A_{v,j}a_{0,j}}\bigr)=o(1).
\end{split}
\end{equation}
Only norm convergence of the difference is needed here. Bounded
coordinate multiplication also permits replacement by \(a_{0,j}\) in
the mass term with error \(o(1)\).
Average \eqref{count:abelian-mass-lower-bound} over the spin basis.
The identity \(\sum_\nu R(v_\nu)=0\) cancels the common angular
expectation exactly. Since \(s_i^2\ge1\),
\[
0\ge\tfrac12\norm{|c|a_{0,j}}^2+o(1)
\ge\tfrac12b_0^2\norm{a_{0,j}}^2+o(1).
\]
Thus \(a_{0,j}\to0\) in norm. The assumed norm differences give the
same conclusion for every \(a_{v,j}\).
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:massive-tightness}]
Suppose that the eventual uniform assertion
\eqref{count:annular-confinement} is false. Its quantified negation gives
\(h_j\ge j\), \(R_j\ge j\), and unit vectors supported in
\(R_j<|x|<4R_j\) such that
\(\mathfrak q_{h_j,1}(\psi_j)/R_j^2\to0\). Dilation \(x=R_j\xi\),
performed unitarily, and multiplication of the charge by \(R_j\) give
parameters
\begin{equation}\label{count:annular-parameters}
h'_j=h_jR_j^3,\qquad m'_j=R_j^2,\qquad
\frac{h'_j}{m'_j}=h_jR_j\longrightarrow\infty.
\end{equation}
Writing the dilated unit states as \(f_j\), we have
\[
\supp f_j\subset\{1<|\xi|<4\},\qquad
\mathfrak q_{h'_j,m'_j}(f_j)=o((m'_j)^2).
\]
The sixteen nonnegative summands in this form imply
\(\norm{D_v(h'_j,m'_j)f_j}=o(m'_j)\) for each member of the spin basis.
First suppose, along a subsequence, that
\(m'_j/\sqrt{h'_j}\to\infty\). The exact scaling of the bosonic potential
is
\begin{equation}\label{count:potential-scaling}
U_{h',m'}(\xi)=\frac{(m')^4}{(h')^2}
U_{1,1}\left(\frac{h'}{m'}\xi\right).
\end{equation}
Lemma~\ref{lem:massive-potential} and \eqref{count:annular-parameters}
therefore imply
\(U_{h'_j,m'_j}(\xi)\ge c(m'_j)^2|\xi|^2\) on the fixed annulus for all
large \(j\). The fermionic multiplier in Proposition~\ref{prop:square}
has operator norm at most \(C(h'_j+m'_j)\) there, and
\[
\frac{h'_j+m'_j}{(m'_j)^2}\longrightarrow0.
\]
Dropping the nonnegative kinetic energy contradicts the assumed
\(o((m'_j)^2)\) form bound.
Otherwise pass to a subsequence with \(m'_j\le C\sqrt{h'_j}\). This is
the growing-mass regime of Lemmas~\ref{lem:raw-estimates} and
\ref{lem:fast-projection}. The norm measures concentrate on the commuting
part of the compact annulus. Every such configuration has at least two
joint centers. Cover this compact commuting set by finitely many
separated-center charts, chosen small enough for the fast estimates for
all sixteen spinors simultaneously. At each reference configuration its
block-center vector is nonzero. Shrinking the corresponding chart gives
\begin{equation}\label{count:center-away-zero}
|c|\ge b_0>0
\end{equation}
on it, with a possibly different positive constant in each of the finite
charts. A fixed smooth gauge-invariant partition localizes to these
charts and to a complementary region disjoint from the commuting set.
Its charge commutators are \(O(1)=o(m'_j)\). The complementary norm tends
to zero by the raw commutator estimate.
Consider one localized sequence, denoted by \(\psi_j\). In its full
\((b,t)\) Hilbert space set
\(a_{v,j}=P_v(b)\psi_j\). Lemma~\ref{lem:fast-projection} gives
\begin{equation}\label{count:variable-projections}
\norm{a_{v,j}-\psi_j}=o(1),\qquad
\norm{D_v^s(h'_j,m'_j)a_{v,j}}=o(m'_j).
\end{equation}
We retain the fast Hilbert space as a spectator here. The slow center
operator and the massive internal block operators anticommute, since
they act on disjoint variables and odd Clifford factors. Their squared
norm identity, first on a smooth compact slow core and then by closure,
implies
\begin{equation}\label{count:center-small-charge}
\norm{D_v^C(m'_j)a_{v,j}}=o(m'_j).
\end{equation}
This does not assert that \(P_v\) commutes with each block operator.
Rather, it applies the identity for the slow operator to the vector
\(a_{v,j}\), with its full dependence on \(b,t\). Compact slow support
and local ellipticity ensure the individual slow domains, just as in
\eqref{prof:block-graph-identity}.
The projections act only in the fast variables, so all the
\(a_{v,j}\) are supported in the same compact center set as the localized
states. By \eqref{count:center-away-zero}, this set is bounded away
from zero. Regard the internal variables and the fast Hilbert space as
spectators. Equations~\eqref{count:variable-projections} and
\eqref{count:center-small-charge} give every hypothesis of
Lemma~\ref{count:center-confinement}, with mass \(m'_j\).
The lemma therefore makes every \(a_{v,j}\), and hence \(\psi_j\),
tend to zero in norm. Summing the finitely many charts and using the
vanishing complementary norm contradicts \(\norm{f_j}=1\).
This proves \eqref{count:annular-confinement} in both parameter regimes.
For the tail estimate, use a smooth squared radial partition consisting
of an interior cutoff and annular functions \(\chi_j\), supported in
\(R_j4R\), and
\(\sum_j|\nabla\chi_j|^2+|\nabla\chi_{\mathrm{in}}|^2\le C/R^2\).
Averaging the unweighted Dirac localization identity over the spin basis
gives
\[
\mathfrak q_{h,1}(\chi_{\mathrm{in}}\psi)+
\sum_j\mathfrak q_{h,1}(\chi_j\psi)
\le\mathfrak q_{h,1}(\psi)+CR^{-2}\norm{\psi}^2.
\]
The interior form is nonnegative, and the exterior forms are bounded
below by \(aR^2\sum_j\norm{\chi_j\psi}^2\). This proves
\eqref{count:tail-bound}. Graph approximation and monotone exhaustion of
the partition justify the same argument for arbitrary form-domain
vectors. Finally, Lemma~\ref{lem:coercive-charge} gives
\(\mathfrak q_{h,1}(\psi)\le\tfrac32\norm{D_u(h,1)\psi}^2\) on gauge
and \(G\) invariants. The displayed tail bound proves their tightness.
\end{proof}
\subsection{Parity and the exact massive kernel dimension}
\begin{lemma}[Gap at large coupling]\label{lem:large-coupling-gap}
On the gauge- and \(G\)-invariant Hilbert space, there are \(\delta>0\)
and \(h_1<\infty\) such that, for all \(h\ge h_1\),
\begin{equation}\label{count:odd-gap}
\norm{D_u(h,1)\psi}\ge\delta\norm{\psi}
\qquad(\psi\text{ odd}).
\end{equation}
If \(P_h\) is the orthogonal projection onto \(\Ker D_u(h,1)\) on that
space, then
\begin{equation}\label{count:full-gap}
\norm{(1-P_h)\psi}\le\delta^{-1}\norm{D_u(h,1)\psi}.
\end{equation}
Moreover, the massive kernel is entirely even and has dimension \(p(N)\).
\end{lemma}
\begin{proof}
Let \(h_j\to\infty\), \(\norm{\psi_j}=1\), and
\(\norm{D_u(h_j,1)\psi_j}\to0\), with the prescribed invariances.
Proposition~\ref{prop:massive-tightness} and
Corollary~\ref{prof:full-exhaustion} give norm-exhausting profiles. By
Proposition~\ref{prop:profile-liminf}, every proper profile satisfies
\(D_u^C(1)F_C=0\) in the full center graph domain.
Each internal kernel factor is a rotation singlet by
Proposition~\ref{prop:singlet-parity}. The phase and symmetry of the fast
line in Lemma~\ref{lem:fast-line} and the compatibility in
Lemma~\ref{lem:chart-compatibility} therefore make \(G\) act on these
profiles through its ordinary action on the center variables and center
fermions. The abelian instance of Lemma~\ref{lem:coercive-charge}, valid
on the full graph domain by closure, gives
\[
0=\norm{D_u^C(1)F_C}^2
\ge\tfrac23\ip{F_C}{\cH_C(1)F_C}.
\]
This is understood as a form inequality. The mass-only oscillator has
a unique zero-energy line for each center color, as also follows from
the zero-triple case of Lemma~\ref{lem:normal-oscillators}. In orthonormal
center coordinates its bosonic factor is proportional to
\[
\exp\left(-\frac12\sum_{i=1}^9|s_i|\,|c_i|^2\right),
\]
and its fermionic factor is the empty \(T\)-polarized vacuum. Denote the
normalized product by \(\Omega_C\). Its parity is even. Hence each proper
profile belongs to
\begin{equation}\label{count:proper-profile-kernel}
\left(\C\Omega_C\otimes\bigotimes_{\alpha=1}^k\cK_{n_\alpha}\right)
^{\text{equal-size permutations}}.
\end{equation}
The full-collapse profile belongs to \(\cK_N\) and is also even, by
Proposition~\ref{prop:singlet-parity}, regardless of the dimension of that
space.
If every \(\psi_j\) were odd, symmetry transport would make every profile
odd. The preceding identifications would force them all to vanish,
contradicting \eqref{prof:full-norm-exhaustion}. The negation of
\eqref{count:odd-gap} would give exactly such a sequence, by choosing
\(h_j\ge j\) and an odd unit vector with charge norm below \(1/j\).
Thus the eventual uniform odd gap holds.
For each fixed positive \(h\), Proposition~\ref{prop:massive-index} gives
self-adjointness and compact graph embedding on this invariant space.
The square has a discrete spectrum with finite-dimensional eigenspaces.
On the eigenspace of any positive eigenvalue \(\lambda\) of the square,
\(D_u/\sqrt\lambda\) is an isometric bijection between its even and odd
parts. The odd lower bound therefore excludes \(0<\lambda<\delta^2\)
in both parities. Spectral expansion proves
\eqref{count:full-gap}. There are no odd zero vectors; the index formula
\eqref{eq:massive-index} consequently gives
\(\dim\Ker D_u(h,1)=p(N)\), all even, for \(h\ge h_1\).
\end{proof}
\subsection{Partition multiplicities and recovery sequences}
The proper strata are indexed by the unordered partitions of \(N\) with
at least two parts. To see why there is only one stratum for each such
partition, order its block sizes in a fixed way and attach a distinct
center to each listed size. This labeled presentation is identified
only under permutations of entries of equal size. Relabeling unequal
sizes changes the presentation of the same commuting configuration; it
does not define a new partition type. Thus there are exactly
\(p(N)-1\) proper types.
By the induction hypothesis, each internal kernel in
\eqref{count:proper-profile-kernel} has dimension one. Choose the same
normalized kernel vector whenever the block sizes agree. Those vectors
are even. Exchanging two such factors consequently produces no graded
sign. The center Gaussian is unchanged by the corresponding orthogonal
permutation, and the empty center vacuum is fixed by its natural
exterior action. Lemma~\ref{lem:fast-line} supplies the same untwisted
action for the fast section, with positive volume normalizations.
Therefore \eqref{count:proper-profile-kernel} is exactly one dimensional
for each proper partition. Let its normalized vector be \(\Phi_C\).
\begin{lemma}[Recovery of the proper profile lines]
\label{count:proper-recovery}
There are even gauge- and \(G\)-invariant vectors \(v_{C,h}\), one for
each proper partition, such that, as \(h\to\infty\),
\begin{equation}\label{count:proper-recovery-estimates}
\norm{D_u(h,1)v_{C,h}}\longrightarrow0,
\qquad
\ip{v_{C,h}}{v_{C',h}}\longrightarrow\mathbf1_{C=C'}.
\end{equation}
These vectors can be chosen simultaneously with recovery vectors for
any prescribed finite list in \(\cK_N\), with all cross inner products
tending to zero.
\end{lemma}
\begin{proof}
We first approximate \(\Phi_C\) in the center graph norm by admissible
tests. Multiply its center factor by a smooth outer radial cutoff
\(\zeta_R\), equal to one for \(|c|\le R\), zero for \(|c|\ge2R\), and
with derivative bounded by \(C/R\). Multiply also by the collision cutoff
\(\rho_\varepsilon\) of Lemma~\ref{lem:collision-removal}, chosen as a
product over all unordered pairs. These cutoffs are rotation invariant
and have the required permutation symmetry. The resulting vector
\(f_{C,R,\varepsilon}\) belongs to \(\mathcal T_C\).
Since \(D_u^C(1)\Phi_C=0\), only cutoff commutators contribute to its
charge. The outer error tends to zero by the Gaussian tail. For fixed
\(R\), the smooth center factor is bounded on its support, so
\eqref{prof:collision-capacity} bounds the squared collision error by
\(C_R\varepsilon^7\). Its omitted \(L^2\) norm also tends to zero, since
the collisions have measure zero. Consequently there is a sequence of
such tests \(f_{C,\ell}\) with
\begin{equation}\label{count:center-test-approximation}
\norm{f_{C,\ell}-\Phi_C}\longrightarrow0,
\qquad \norm{D_u^C(1)f_{C,\ell}}\longrightarrow0.
\end{equation}
The order is explicit: for tolerance \(1/\ell\), choose the outer radius
first and then choose the collision radius small enough for that fixed
outer support.
For each fixed \(\ell\), Proposition~\ref{prop:test-lifts} produces
\(w_{C,h,\ell}=I_{h,u,1}f_{C,\ell}\), with the prescribed leading norm
and charge output. We impose the remaining symmetries carefully, since
the chartwise corrections need not have been chosen rotation invariant.
Let \(A_G\) be Haar averaging over \(G\), and let
\(E_+=(1+(-1)^F)/2\) be the even projection. For every fixed \(g\in G\),
Lemma~\ref{lem:chart-compatibility} and invariance of the profile give
\[
\ip{w_{C,h,\ell}}{U_gw_{C,h,\ell}}
\longrightarrow\norm{f_{C,\ell}}^2.
\]
These inner products are bounded uniformly in \(g,h\) for fixed
\(\ell\). Haar dominated convergence and the projection identity
\(\norm{A_Gw}^2=\ip{w}{A_Gw}\) show
\(\norm{w_{C,h,\ell}-A_Gw_{C,h,\ell}}\to0\). The identical argument
for the two-element parity action gives
\(\norm{w_{C,h,\ell}-E_+A_Gw_{C,h,\ell}}\to0\).
Moreover, \(D_u\) commutes with \(A_G\) and interchanges the parity
projections, so
\[
\norm{D_u(h,1)E_+A_Gw_{C,h,\ell}}
\le\norm{D_u(h,1)w_{C,h,\ell}}.
\]
Thus imposing these symmetries preserves the leading norms and does not
increase the charge errors.
For fixed \(\ell\), the compact center supports on distinct partition
strata are disjoint compact subsets of the gauge quotient. The leading
identifications concentrate on those supports, so
Lemma~\ref{lem:chart-compatibility} makes the different lifts
asymptotically orthogonal. Their projections have the same property by
the norm approximation just proved.
There are only finitely many proper partitions. Choose an increasing
sequence \(H_\ell\to\infty\) so that for every \(h\ge H_\ell\) the finitely
many norm, overlap, symmetry, and graph-output errors at the fixed test
level \(\ell\) are below \(1/\ell\). Taking that level for
\(H_\ell\le h0\),
\[
\int_{|x|>\eta}|S_h\phi_a|^2
=\int_{|y|>h^{1/3}\eta}|\phi_a(y)|^2\longrightarrow0.
\]
It follows that \(\norm{z_{a,h}-S_h\phi_a}\to0\), proving the Gram
matrix assertion. Lemma~\ref{lem:domains} allows the cutoff product in
the graph domain, and its undeformed charge is
\([D_u,\chi]S_h\phi_a\). The derivative of \(\chi\) is bounded and
supported away from zero, so this norm tends to zero.
The massive multiplier has norm at most \(C|x|\). On the fixed support
of \(\chi\), its norm on \(z_{a,h}\) tends to zero as well: if the support
is contained in \(|x|\le B\), then
\[
\norm{|x|z_{a,h}}^2
\le\eta^2\norm{\phi_a}^2+
B^2\int_{|x|>\eta}|S_h\phi_a|^2.
\]
Take \(h\to\infty\) and then \(\eta\downarrow0\). This proves the
charge estimate without assuming \(|y|\phi_a\in L^2\).
Proposition~\ref{prop:singlet-parity} makes every \(\phi_a\) even and a
rotation singlet; radial cutoff and dilation preserve these symmetries.
At any fixed level of the proper-partition construction, the compact
center supports avoid collisions and hence avoid zero. Its lifts are
therefore asymptotically orthogonal to all the \(z_{a,h}\). There are
only finitely many such pairs. Include these overlaps in the threshold
selection in Lemma~\ref{count:proper-recovery} to obtain the simultaneous
claim there.
\end{proof}
\subsection{Closing the simultaneous induction}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Assume that \(\dim\cK_n=1\) and Proposition~\ref{prop:exterior-estimate}
hold for \(2\le n