parity_toeplitz_classification.tex

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%  A total-variation floor and a rank-three classification for the
%  parity-compressed sector of a log-singular Toeplitz form with a
%  prime-power comb.
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\title{A total-variation floor and a rank-three classification\\
for the parity-compressed sector of a log-singular\\
Toeplitz form with a prime-power comb}
\author{Stefan Hamann}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
\noindent
This paper is about the near-degeneracy behind a Toeplitz spectral floor: a
total-variation bound, three linear functionals, and a priced table of what the
standard unconditional toolbox reaches. The object is the reflection-odd
compression $A$ of a finite Toeplitz form whose lag sequence is a logarithmic
profile minus an explicit comb, and the question is how small the Rayleigh
quotient of $A$ can be on the low end of the parity spectrum. Four results are
proved. First, a \emph{total-variation floor}: every trial vector whose leading
parity coordinate is normalised to one has lag weights of total variation at
least $1/\mu_1$, where $\mu_1=4\sin^2(\pi/(2h+1))$ is the spectral gap of the
parity compression of the second difference; the floor is closed, involves no
comb node, no taper and no truncation, and it prices the entire trial-vector
route at $O(h^{-2})$ for every vector of bounded value. Second, a \emph{gauge
rigidity} statement: the leading-coordinate normalisation pins only the scale of
the trial direction, both the value and the total variation are homogeneous of
degree two, hence their ratio is a function of the ray and takes the same range
in every reparametrisation of the entry spectrum --- no sector change removes
the floor, and a shifted sector conserves it exactly, floor times transfer
being $1/\mu_1$ identically. Third, an exact \emph{rank-three classification}:
the two-by-two raw Gram block of the low parity modes splits as an archimedean
block minus a comb block, the determinant polarises into three terms, and the
comb--comb term is the polarisation of the determinant on symmetric
two-by-two matrices --- a form of rank three and signature $(1,2)$ --- so the
comb enters the determinant only through \emph{three} linear functionals of its
weights, for every window and every truncation. Fourth, a classification of
what this costs: any majorant of the comb--comb term built from a norm of the
weight vector must dominate the supremum of a rank-three quadratic over the
image of that norm ball, which is where the priced routes lose. The remaining
question --- whether the two rows of the block become collinear at a rate
uniform in the window index --- is stated as an open problem about a finite
bilinear form. Around it we report, as measurements on declared surfaces: the
rates; a table of five unconditional majorants together with the exponent each
certifies, the best of them short of the demanded one by two units in the
exponent; five structural obstructions, each exhibited failing; a curve of the
binned exponent against comb density over four decades of density, whose densest
bins are consistent with zero and whose undecidability under the declared caps is
part of the claim; and two controlled placement experiments, the second of which
closes into an exact identity for the response of the ratio to a displacement of
comb nodes inside their lag cells. Randomising the node positions at fixed
weights leaves every theorem here intact and removes the phenomenon: the
arithmetic enters the theorems only as the source of the comb positions and
weights, and every statement proved here holds for an arbitrary finite comb.
\end{abstract}

\section{Introduction}
\label{sec:intro}

\subsection{The problem, and why the direction is hard}
\label{sec:problem}

Let $c=(c_m)_{m=0}^{M-1}$ be a real lag sequence, let $\Tm{M}(c)$ be its
symmetric Toeplitz matrix, and let $A$ be the compression of $\Tm{M}(c)$ to the
reflection-odd sector of the window (Section~\ref{sec:setting}). The question
this paper is about is a lower bound for the Rayleigh quotient of $A$ on the low
end of a fixed orthonormal mode family --- equivalently, an upper bound for the
extent to which the quadratic form $v\mapsto v^{\T}Av$ can cancel along a smooth
direction.

The classical route from a Toeplitz form to a spectral bound is the symbol. If
$\ell\le\sigma$ pointwise then $\Tm{M}(\ell)\preceq\Tm{M}(\sigma)$ by Parseval,
and a positive pointwise minorant of the symbol transfers to a positive floor
for the form \cite{Szego1939,GrenanderSzego1958,KMS1953,Dirichlet1829,Fejer1916}. For the
class considered here that route is unavailable, and not marginally so: the
symbol is a logarithmic profile minus an explicit cosine comb, the comb dips are
numerous and narrow, and on every window of the declared surface of
Section~\ref{sec:numerics} the infimum of the symbol is negative, while the form
restricted to the parity sector is positive. The smallest Rayleigh quotient is
therefore not produced by a small value of the symbol at some frequency; it is
produced by \emph{cancellation inside the quadratic form}, along a direction that
lives in the low modes. A pointwise minorant is by construction blind to which
vector realises the minimum.

The natural substitute is a trial-vector route: exhibit an explicit direction,
evaluate the form on it, and price the evaluation. This paper is a
classification of that route. Its first two results say that the route has a
closed price which no reparametrisation removes; its second two say that the
object which remains after the price is paid is a quadratic polynomial in three
linear functionals of the comb weights, and that this is exactly why the
unconditional toolbox --- size budgets, symbol arguments, sectors and gauges,
splits, perturbation theory, the bilinear sieve box --- does not reach it. What
is \emph{not} claimed anywhere is a bound on the arithmetic: the three
functionals are finite sums over the comb, all four results hold for an
arbitrary finite comb, and the one remaining question is stated as an open
problem in Section~\ref{sec:open}.

The companion paper \cite{PaperI} treats the same window form from the opposite
direction --- a certified lower bound for a Galerkin defect on a nested ladder
--- and we cite it for the window form and the comb class rather than repeating
them.

\subsection{Contribution}
\label{sec:contrib}

\begin{enumerate}[label=(C\arabic*),leftmargin=3.1em]
\item\label{C:floor} \textbf{The total-variation floor, and the price it
  fixes} (Theorem~\ref{thm:tvfloor} with
  Corollaries~\ref{cor:price}--\ref{cor:abis}). For every trial vector whose
  leading parity coordinate is normalised to one, the lag weights $w$ of the
  associated window vector satisfy
  $\TV(w)\ge|w_0|=\lVert v\rVert^{2}\ge1/\mu_1\ge N^{2}/(4\pi^{2})$, where
  $\mu_1=4\sin^{2}(\pi/N)$, $N=2h+1$, is the spectral gap of the parity
  compression $L_P$ of the second difference \cite{KMS1953}. The proof is four
  steps and uses no property of the comb whatsoever. The load-bearing
  consequence is a \emph{price}: the ratio of the value to the total variation
  is at most $\mu_1$ times the value, so every trial vector of bounded value
  certifies a ratio $O(h^{-2})$, and the boundary-free summation by parts
  \cite{Abel1826} can certify nothing better.
\item\label{C:gauge} \textbf{Gauge rigidity of the normalisation}
  (Theorem~\ref{thm:gauge} with
  Propositions~\ref{prop:shift}--\ref{prop:nullmode} and
  Corollary~\ref{cor:noescape}). The normalisation pins only the scale of the
  trial direction; value and total variation are homogeneous of degree two;
  hence their ratio is a function of the ray, and its range over admissible
  vectors is the \emph{same} for every entry spectrum. A shifted spectrum does
  flatten the floor, but the transfer factor that returns the bound to the
  original scale conserves the product exactly: floor times transfer is
  $1/\mu_1$ at every shift. So (C1) cannot be reparametrised away, and the
  parity compression is not a convenience --- Proposition~\ref{prop:nullmode}
  shows it is what makes the normalisation well posed at all.
\item\label{C:rank} \textbf{The exact bilinear form and the rank-three
  classification} (Section~\ref{sec:rank}: Propositions~\ref{prop:split} and
  \ref{prop:polarisation}, Theorem~\ref{thm:rank3},
  Corollary~\ref{cor:majorant}). The two-by-two raw Gram block of the two lowest
  parity modes splits exactly as $\Ah_2=B-S$ with $B$ archimedean and $S$ a
  weighted sum of comb reads; the determinant polarises as
  $\det\Ah_2=\det B-D(B,S)+\det S$; and $D$ is the polarisation of $\det$ on
  symmetric two-by-two matrices, a form of rank three and signature $(1,2)$ in
  the coordinates $(X_{11},X_{22},X_{12})$. Consequently the comb--comb term is
  $S_{11}S_{22}-S_{12}^{2}$: a quadratic polynomial in \emph{three} linear
  functionals of the comb weights, for every window, every $h$ and every
  truncation. Corollary~\ref{cor:majorant} turns this into the statement about
  majorants that the priced routes of Section~\ref{sec:open} run into.
\item\label{C:nogo} \textbf{A priced table, a density curve, and one open
  problem} (Section~\ref{sec:open}). What remains after (C1)--(C3) is a single
  question about a finite bilinear form: is the near-collinearity of the two
  rows of $\Ah_2$ bounded uniformly in the window index? It is stated as
  Problem~\ref{prob:R1} in three equivalent forms --- never as an approach to, a
  weakening of, or a consequence of any conjecture. Around it,
  Proposition~\ref{prop:routes} proves five unconditional majorants and
  Measurement~\ref{meas:routes} reports the exponent each certifies on a
  declared surface; Measurement~\ref{meas:nogo} exhibits five further routes
  failing for structural reasons; Measurement~\ref{meas:density} gives the
  binned exponent of Definition~\ref{def:ratio} against comb density over four
  decades, with
  Remark~\ref{rem:undecidable} stating what that curve cannot decide and at what
  price; and Measurements~\ref{meas:placement} and \ref{meas:pole} locate the
  phenomenon by intervention, the second through the closed-form response of
  Proposition~\ref{prop:fh}.
\end{enumerate}

Two of these four are theorems; two are classifications supported by theorems
together with declared measurements. Which is which is recorded in
Section~\ref{sec:typing} rather than left to the reader.

\subsection{Typing of the results}
\label{sec:typing}

Every numbered statement of this paper that makes a claim carries exactly one of
three types, and the type is part of the statement. Table~\ref{tab:typing} types
every one of them: a definition and the open problem are marked as what they
are, and a remark that only reads results already stated carries a dash, since
it claims nothing of its own.

\begin{description}[leftmargin=2.2em,style=nextline]
\item[\textnormal{\emph{proved}}] A written proof is given (or, for classical
  facts, cited). The statement holds for every window and every finite comb
  within its stated hypotheses. Where a hypothesis is itself of a different
  type --- positive definiteness of $A$ on the parity sector, for instance ---
  the statement is \emph{conditional} and says so.
\item[\textnormal{\emph{certified per window}}] The statement is verified on
  each window of a declared finite surface by a computation whose failure mode
  is loud (a completed Cholesky factorisation, an exact identity residual
  against a declared floor). No quantifier over windows is claimed.
\item[\textnormal{\emph{measured}}] The statement is a number, a fitted
  exponent, or a controlled experiment on a declared finite surface, reported as
  such. Fitted exponents are never used as bounds.
\end{description}

{\small
\begin{xltabular}{\textwidth}{@{}llY@{}}
\caption{Typing of every numbered statement. ``Conditional'' marks a proved
implication whose hypothesis is of a weaker type; the hypothesis is then named
in the statement.}
\label{tab:typing}\\
\toprule
statement & type & remark \\
\midrule
\endfirsthead
\toprule
statement & type & remark \\
\midrule
\endhead
\midrule
\multicolumn{3}{r@{}}{\emph{continued on the next page}}\\
\endfoot
\bottomrule
\endlastfoot
\multicolumn{3}{@{}l}{\emph{Section~\ref{sec:setting} --- setting and
  scaffolding}}\\
the comb is not used (Rem.~\ref{rem:combfree}) & --- & the scope statement:
  Secs.~\ref{sec:setting}--\ref{sec:rank} hold for an arbitrary finite comb \\
parity spectrum (Lem.~\ref{lem:kms}) & proved & closed eigenpairs and the gap
  \eqref{eq:gap}; Kac--Murdock--Szeg\H{o} \cite{KMS1953} \\
L1 interlacing (Lem.~\ref{lem:interlacing}) & proved (cited) &
  Cauchy \cite{Cauchy1829}, Courant--Fischer \cite{CourantFischer1920} \\
L2 Schur nesting (Lem.~\ref{lem:nesting}) & proved &
  Schur \cite{Schur1917}, Haynsworth \cite{Haynsworth1968} \\
L3 constrained minimum (Lem.~\ref{lem:constrained}) & proved &
  requires $G_k\PD$ \\
gauge law (Lem.~\ref{lem:gauge}) & proved & $\sum_dw_d=0$, unconditional \\
Abel swap (Lem.~\ref{lem:abel}) & proved & boundary-free, $w_M=0$ \\
C5 cascade forms (Prop.~\ref{prop:cascade}) & proved & prefix / minor /
  regression \\
C6 diagonal invariance (Lem.~\ref{lem:diaginv}) & proved & one line \\
B4 cascade direction (Prop.~\ref{prop:direction}) & \emph{conditional}
  & proved given $\Ah\PD$; the hypothesis is certified per window \\
\midrule
\multicolumn{3}{@{}l}{\emph{Section~\ref{sec:tvfloor} --- the
  total-variation floor}}\\
A the floor (Thm.~\ref{thm:tvfloor}) & proved & comb-free, taper-free \\
A1 the price (Cor.~\ref{cor:price}) & proved & the load-bearing corollary \\
A2 Abel/H\"older (Cor.~\ref{cor:holder}) & proved & \\
A-bis (Cor.~\ref{cor:abis}) & proved & no normalisation needed \\
A4 negative control (Rem.~\ref{rem:negcontrol}) & proved & one line \\
A3 sharpness (Meas.~\ref{meas:sharpness}) & measured & $\mu_1\TV\in[5.00,23.20]$
  on Surface~II \\
\midrule
\multicolumn{3}{@{}l}{\emph{Section~\ref{sec:gauge} --- gauge rigidity}}\\
B homogeneity (Thm.~\ref{thm:gauge}) & proved & \\
B1 floor $\times$ transfer (Prop.~\ref{prop:shift}) & proved & pure shift \\
B2 the null mode (Prop.~\ref{prop:nullmode}) & proved & closed spectra \\
no escape (Cor.~\ref{cor:noescape}) & proved & \\
sector battery (Meas.~\ref{meas:battery}) & certified per window &
  $35\times6\times27$, residual $\le8.8\cdot10^{-10}$ \\
\midrule
\multicolumn{3}{@{}l}{\emph{Section~\ref{sec:twodim} --- the two-dimensional
  reduction}}\\
C1 exact $K=2$ (Prop.~\ref{prop:K2}) & proved & \\
C2 pivot identity (Prop.~\ref{prop:pivot}) & proved & \\
C3 union identity (Prop.~\ref{prop:union}) & proved & \\
C7 Wronskian (Lem.~\ref{lem:wronskian}) & proved & the corner entry $3$ closes
  it \\
C8 maximal minors (Lem.~\ref{lem:minors}) & proved & Lagrange
  \cite{Lagrange1773} \\
C9 determinant reading (Lem.~\ref{lem:detread}) & proved & uses no positivity \\
C10 Gershgorin (Lem.~\ref{lem:gershgorin}) & proved & uniform in the window \\
C11 det-collapse (Prop.~\ref{prop:collapse}) & proved / \emph{conditional} &
  the closed form is unconditional; the harmonic-mean bound needs
  $\Ah_2\PD$ \\
wrong sign priced (Rem.~\ref{rem:wrongsign}) & proved & the identity
  \eqref{eq:wrongsign}, $\rho=4r_{12}^{2}/(1-r_{12}^{2})$ \\
one obstruction, three readings (Rem.~\ref{rem:onedress}) & proved & the
  explicit factor $1+3r_{12}^{2}\in[1,4]$; the reading is an observation \\
the section read together (Rem.~\ref{rem:selfsimilar}) & --- & which statements
  are identities and which two are not \\
\midrule
\multicolumn{3}{@{}l}{\emph{Section~\ref{sec:rank} --- bilinear form and rank}}\\
D0 exact split (Prop.~\ref{prop:split}) & proved & needs an admissible
  resolution \\
D1 polarisation (Prop.~\ref{prop:polarisation}) & proved & \\
D rank three (Thm.~\ref{thm:rank3}) & proved & every $h$, every
  truncation \\
D wedge form (Lem.~\ref{lem:wedge}) & \emph{conditional} & proved given
  rank-one reads; the defect is measured \\
D3 majorants (Cor.~\ref{cor:majorant}) & proved & the class
  $\mathfrak M$ of Def.~\ref{def:majclass}; the route reading of
  Rem.~\ref{rem:routeclass} is a reading \\
D3$'$ $\ell^{1}$ mass (Meas.~\ref{meas:l1}) & measured & $40.6\times$ the signed
  value on the declared subsample; the rank-one defect priced on both
  surfaces \\
D4a closed weights (Lem.~\ref{lem:closedweight}) & proved & one formula on
  $1\le d\le M-1$ \\
D4b bandwidth $K$ (Lem.~\ref{lem:band}) & proved & the $4K$-dimensional band
  space \eqref{eq:bandspace} \\
D4c band $\Rightarrow\eta$-rank (Lem.~\ref{lem:etarank}) & proved & general;
  $r=(2\Omega+1)(q+1)$ for $\Omega$ frequencies and degree bound $q$ \\
D4d spline read (Lem.~\ref{lem:spline}) & proved & $\eta=O(K^{2}/N^{2})$ \\
D4 Type II $\eta$-rank (Prop.~\ref{prop:typeII}) & proved &
  $\sigma_{4K+1}\le\eta\sqrt{RC}$; needs the interior hypothesis
  \eqref{eq:interior} \\
D4$''$ what the interior hypothesis costs (Rem.~\ref{rem:interior}) & --- &
  without \eqref{eq:interior} the conclusion is false as stated; the triangular
  cut-off is not of low rank \\
D4$'$ Type II collapse (Meas.~\ref{meas:typeII}) & measured & the observed
  $2$ is sharper than the proved $8$ and stays measured \\
D5 reference window (Meas.~\ref{meas:refwindow}) & measured & the
  three-term cancellation \\
\midrule
\multicolumn{3}{@{}l}{\emph{Section~\ref{sec:open} --- the open problem}}\\
E0 Problem~\ref{prob:R1} & \emph{open} & three equivalent forms; a question
  about a finite bilinear form \\
E1 angle against determinant (Lem.~\ref{lem:angle}) & proved & the factor lies
  in $(0,1]$; used only to relate two readings \\
E2 rates (Meas.~\ref{meas:rate}) & measured & fitted exponents; every surface
  and both readings declared \\
E3a dual-point read (Lem.~\ref{lem:dualpoint}) & proved & the two points
  $\gamma_k=2\pi k/(ND)$ and their spacing are derived, not assumed \\
E3 five majorants (Prop.~\ref{prop:routes}) & proved & five unconditional
  inequalities with their hypotheses \\
E4 priced routes (Meas.~\ref{meas:routes}) & measured & the exponent each
  proved route certifies on Surface~II \\
E5 structural obstructions (Meas.~\ref{meas:nogo}) & certified per window &
  each route exhibited failing on Surface~I \\
E6 the binned exponent (Def.~\ref{def:ratio}) & definition & deterministic;
  ladder-dependent by Rem.~\ref{rem:undecidable} \\
E7 density curve (Meas.~\ref{meas:density}) & measured & on Surface~III,
  quoted with its rung ladder \\
E8 undecidability, fence, caveat (Rem.~\ref{rem:undecidable}) & measured &
  part of the claim of E7, not a caveat on it \\
E9 the scramble (Meas.~\ref{meas:placement}) & measured & controlled
  experiment at fixed weight multiset \\
E10 exact placement derivative (Lem.~\ref{lem:dcexact}) & proved & $\dot c$
  affine and piecewise constant \\
E11 placement response (Prop.~\ref{prop:fh}) & proved &
  Hellmann--Feynman \cite{Hellmann1937,Feynman1939}; simple extreme
  eigenvalues \\
E12 causality and the pole (Meas.~\ref{meas:pole}) & measured & a derivative
  at a point on named instances, not a rate \\
E13 what is not claimed (Rem.~\ref{rem:notclaimed}) & --- & the binding
  negative list \\
\midrule
\multicolumn{3}{@{}l}{\emph{Section~\ref{sec:numerics} --- numerical
  verification}}\\
F1--F3 conventions (Sec.~\ref{sec:fp}) & --- & floors, conditioning cap,
  mutation controls \\
the three surfaces (Sec.~\ref{sec:surfaces}) & --- & declared before any
  number is read \\
cross-reference (Tab.~\ref{tab:crossref}) & --- & every numbered statement to
  its check \\
the two fit remarks (Rem.~\ref{rem:fit2}, \ref{rem:twoexponents}) & measured &
  why $h^{+1.997}$ is the closed $h^{2}$; why two exponents are one object \\
\end{xltabular}}

Table~\ref{tab:typing} is the first thing to read. Four entries deserve
emphasis. Proposition~\ref{prop:direction} and the harmonic-mean half of
Proposition~\ref{prop:collapse} are \emph{conditional}: they are proved
implications whose hypothesis is positive definiteness of the raw Gram block,
which on the surfaces of Section~\ref{sec:numerics} is a numerical fact about
finitely many finite matrices and nothing more. Lemma~\ref{lem:wedge} is
conditional in the same sense, on a rank-one hypothesis whose defect is
measured. And Measurement~\ref{meas:typeII} is the one place where a phrase of
the source material does not convert: ``effective rank $O(1)$'' is not a
theorem. What is proved instead is the chain
Lemmata~\ref{lem:closedweight}--\ref{lem:spline} and
Proposition~\ref{prop:typeII}, which bounds the $(4K+1)$-st singular value of a
factorisation-indexed block by a declared truncation error under a declared
interior hypothesis; the measured number of significant singular values is
smaller than what that proposition gives, and it therefore stays measured.

Two entries in the last block of the table need the same care.
Measurement~\ref{meas:routes} carries two types in one row on purpose: the
inequality of each route is proved (Proposition~\ref{prop:routes}), and only
the exponent it certifies is a fit on Surface~II. And
Measurement~\ref{meas:density} is quoted with its rung ladder, because the
estimator of Definition~\ref{def:ratio} is ladder-dependent: a bin value
reported without the ladder it was computed on is not reproducible
(Remark~\ref{rem:undecidable}).

\subsection{Notation}
\label{sec:notation}

Notation is fixed once, in Table~\ref{tab:notation}, and not varied. Three
choices need a word. The symbol $\Pi$ denotes the reflection-odd isometry, so
that $B$ can be reserved for the archimedean Gram block of
Section~\ref{sec:rank}; $\MM$ (calligraphic) denotes $\diag(\mu_1,\dots,\mu_K)$,
so that $M$ (upright) keeps its meaning as the window dimension; and the comb
weights, written $\mu_j$ in \cite{PaperI}, are written $\varrho_j$ here, because
$\mu_k$ is needed for the parity eigenvalues. The archimedean kernel is written
$a(\cdot)$ and always carries its argument; the coefficient vector of a trial
vector is written $a$ and never does.

\begin{table}[t]
\centering\small
\begin{tabularx}{\textwidth}{@{}lY@{}}
\toprule
symbol & meaning \\
\midrule
$D,\ M\in2\N,\ \alpha=MD/2,\ h=M/2,\ N=2h+1$ & cell width, window dimension,
  half-width, sector dimension, parity period \\
$\Tm{M}(c)=(c_{|r-r'|})_{r,r'=0}^{M-1}$ & symmetric Toeplitz matrix of the lag
  sequence $c$ \\
$\Pi\in\R^{M\times h}$, $\Pi e_r=\tfrac1{\sqrt2}(e_r-e_{M-1-r})$ &
  reflection-odd isometry, $\Pi^{\T}\Pi=I_h$ \\
$A=\Pi^{\T}\Tm{M}(c)\Pi$, $A_{rr'}=c_{|r-r'|}-c_{M-1-r-r'}$ & the odd window
  form \\
$a(\cdot)$; $S_D(s,u)$; $\varrho_j,u_j$ & archimedean kernel; symmetrised hat;
  comb weights and node positions \\
$L_P$ & odd compression of the lag sequence $(2,-1,0,\dots)$: tridiagonal
  $2I-E-E^{\T}$ with corner entry $3$ \\
$\mu_k=4\sin^{2}(k\pi/N)$, $(t_k)_r=\tfrac2{\sqrt N}\sin\tfrac{2\pi k(r+1)}N$ &
  parity eigenvalues and orthonormal parity modes, $k=1,\dots,h$ \\
$\ah_{ij}=t_i^{\T}At_j$, $\Ah_K=(\ah_{ij})_{i,j\le K}$ & the raw Gram block \\
$\MM=\diag(\mu_1,\dots,\mu_K)$, $G=\MM^{-1/2}\Ah_K\MM^{-1/2}$ & the normalised
  $K$-block \\
$G_k$, $g_k=e_1^{\T}G_k^{-1}e_1$, $s=\mu_1t_1^{\T}A^{-1}t_1$ & leading blocks,
  the ladder, the entry functional \\
$x$ admissible: $x\in\R^{K}$, $x_1=1$ & trial vector \\
$a_k=x_k/\sqrt{\mu_k}$, $v=\sum_{k\le K}a_kt_k$ & coefficient vector and window
  vector \\
$Q(x)=x^{\T}Gx=v^{\T}Av$ & the value \\
$w_0=\Ac_0(v)$, $w_d=2\Ac_d(v)-\Cv_{M-1-d}(v)$ $(d\ge1)$, $w_M:=0$ & lag
  weights; $\Ac$ autocorrelation, $\Cv$ self-convolution \\
$(\Delta w)_d=w_d-w_{d+1}$, $\TV(x)=\lVert\Delta w\rVert_1$ & lag increments and
  total variation \\
$C_d=\sum_{e\le d}c_e$, $C_{-1}:=0$ & partial sums of the lag sequence \\
$B$, $S$, $r_{12}^{2}=\ah_{12}^{2}/(\ah_{11}\ah_{22})$, $\nu_1\ge\nu_2$ &
  archimedean block, comb block, squared correlation, eigenvalues of $\Ah_2$ \\
$D(P,Q)=P_{11}Q_{22}+P_{22}Q_{11}-2P_{12}Q_{12}$ & polarisation of $\det$ on
  symmetric $2\times2$ matrices \\
\bottomrule
\end{tabularx}
\caption{Notation, fixed for the whole paper.}
\label{tab:notation}
\end{table}

$X\PSD$ and $X\PD$ denote positive semidefiniteness and definiteness,
$\preceq$ the L\"owner order, $\lVert\cdot\rVert$ the Euclidean norm,
$e_k$ the $k$-th coordinate vector, $|X|$ the entrywise absolute value of a
matrix, and $\sgn$ the entrywise sign. For $v\in\R^{h}$ the autocorrelation and
self-convolution are
\begin{equation}
\label{eq:accv}
  \Ac_d(v)=\sum_{r}v_rv_{r+d}\quad(d\ge0),
  \qquad
  \Cv_j(v)=\sum_{r+r'=j}v_rv_{r'},
\end{equation}
both sums over indices in $\{0,\dots,h-1\}$ and empty sums being zero.
\emph{Admissible} always means $x_1=1$.

\section{Setting}
\label{sec:setting}

\subsection{The odd window form and the comb class}
\label{sec:window}

We use the window form and the comb class of \cite{PaperI} verbatim and recall
only what is needed. Fix a cell width $D>0$ and an even $M$, put $\alpha=MD/2$,
$h=M/2$, and let $\Pi$ be the reflection-odd isometry of
Table~\ref{tab:notation}. Given a real lag sequence $c=(c_m)_{m=0}^{M-1}$ the
odd window form is
\begin{equation}
\label{eq:Adef}
  A=\Pi^{\T}\Tm{M}(c)\,\Pi\in\R^{h\times h},
  \qquad
  A_{rr'}=c_{|r-r'|}-c_{M-1-r-r'} .
\end{equation}
The class we verify on is the comb class of \cite[Def.~2.6]{PaperI}: with
$a(\cdot)$ the archimedean kernel of that definition, a finite node set
$\{(u_j,\varrho_j)\}_j$ with $u_j>0$, $\varrho_j>0$, and
$S_D(s,u)=\tfrac12[\Delta_D(s-u)+\Delta_D(s+u)]$ the symmetrised hat built from
the unit-height triangle $\Delta_D(t)=\max\{0,1-|t|/D\}$ of half-width $D$,
\begin{equation}
\label{eq:lags}
  c_m=a(mD)-\sum_{j:\,u_j\le2\alpha}\varrho_j\,S_D(mD,u_j),
  \qquad m=0,\dots,M-1 .
\end{equation}
The \emph{prime-power instance} is $u_j=\log n_j$ over the prime powers
$n_j=p^{\nu}$ in increasing order with $\varrho_j=2\Lambda(n_j)/\sqrt{n_j}$,
$\Lambda$ the von Mangoldt weight. By linearity of $c\mapsto A$ the split of
\eqref{eq:lags} induces a split
\begin{equation}
\label{eq:Asplit}
  A=A_{\mathrm{arch}}-A_{\mathrm{comb}},
  \qquad
  A_{\mathrm{comb}}=\sum_{j}\varrho_j\,A^{(j)},
  \quad
  A^{(j)}=\Pi^{\T}\Tm{M}\bigl(S_D(\cdot D,u_j)\bigr)\Pi .
\end{equation}
Throughout, a \emph{resolution} $D$ is called \emph{admissible} if
$D<\min_ju_j$; for the prime-power instance this is implied by the admissibility
convention of \cite[Def.~5.1]{PaperI}, which fixes $D$ as a fraction of the
smallest log-gap in the run.

\begin{remark}[the arithmetic is not used]
\label{rem:combfree}
Theorems~\ref{thm:tvfloor}, \ref{thm:gauge} and \ref{thm:rank3}, and every
lemma, proposition and corollary of Sections~\ref{sec:setting}--\ref{sec:rank},
hold for an \emph{arbitrary} finite comb: the node positions $u_j$ and the
weights $\varrho_j$ enter only as the data of \eqref{eq:lags}, and no property of
the sequence $(u_j)$ or $(\varrho_j)$ is used. Where the prime-power instance is
mentioned it is as the instance the measurements of
Sections~\ref{sec:open}--\ref{sec:numerics} were taken on, and the only
arithmetic facts consumed anywhere are the two classical gap inequalities used
in \cite{PaperI} to check that a resolution is admissible
\cite{Chebyshev1852,RosserSchoenfeld1962}. No arithmetic statement is claimed.
\end{remark}

\subsection{The parity spectrum}
\label{sec:parity}

Let $c^{L}=(2,-1,0,\dots,0)$ and let $L_P$ be the odd window form
\eqref{eq:Adef} of $c^{L}$.

\begin{lemma}[the parity compression and its spectrum]
\label{lem:kms}
$L_P$ is the tridiagonal matrix $2I-E-E^{\T}$ on $\R^{h}$ with corner entry
$(L_P)_{h-1,h-1}=3$; equivalently, $L_P$ is the second-difference form with a
Dirichlet condition at the outer end and the odd reflection $u_h=-u_{h-1}$ at
the inner end. Its eigenpairs are
\begin{equation}
\label{eq:kms}
  \mu_k=4\sin^{2}\Bigl(\frac{k\pi}{N}\Bigr),
  \qquad
  (t_k)_r=\frac2{\sqrt N}\,\sin\frac{2\pi k(r+1)}{N},
  \qquad k=1,\dots,h,\ r=0,\dots,h-1,
\end{equation}
with $N=2h+1$, and $\{t_k\}_{k=1}^{h}$ is an orthonormal basis of $\R^{h}$. In
particular $\mu_1=4\sin^{2}(\pi/N)$ is the smallest eigenvalue and
\begin{equation}
\label{eq:gap}
  \frac1{\mu_1}\;\ge\;\frac{N^{2}}{4\pi^{2}}\;>\;\frac{h^{2}}{\pi^{2}} .
\end{equation}
\end{lemma}

\begin{proof}
For $c^{L}$ the Hankel part of \eqref{eq:Adef} is
$-c^{L}_{M-1-r-r'}$, which is nonzero only if $M-1-r-r'\in\{0,1\}$, i.e.\ only
if $r+r'\in\{2h-2,2h-1\}$; since $r,r'\le h-1$ the only possibility is
$r=r'=h-1$, contributing $-c^{L}_1=+1$. Hence $L_P$ is tridiagonal with
diagonal $2$ except for the corner entry $2+1=3$, which is the first assertion.
Reading the rows as a recurrence, row $0$ is $2u_0-u_1=\mu u_0$, i.e.\ the
convention $u_{-1}=0$, and row $h-1$ is $-u_{h-2}+3u_{h-1}=\mu u_{h-1}$, i.e.\
$-u_{h-2}+2u_{h-1}-u_h=\mu u_{h-1}$ with $u_h=-u_{h-1}$.

For the eigenpairs, put $u_r=\sin(\omega(r+1))$. Then $u_{-1}=0$ and the
interior recurrence holds with $\mu=2-2\cos\omega=4\sin^{2}(\omega/2)$. The
inner condition $u_h=-u_{h-1}$ reads
$\sin(\omega(h+1))+\sin(\omega h)=2\sin\bigl(\tfrac{\omega N}2\bigr)
\cos\bigl(\tfrac\omega2\bigr)=0$, and since $0<\omega<\pi$ the second factor is
positive, so $\omega N/2=k\pi$, i.e.\ $\omega=2k\pi/N$ and
$\mu=4\sin^{2}(k\pi/N)$. The values $k=1,\dots,h$ give $h$ distinct eigenvalues,
hence all of them. For the normalisation,
$\sum_{r=0}^{N-1}\sin^{2}(2\pi kr/N)=N/2$, the term $r=0$ vanishes, and the
remaining $2h$ terms pair up as $r\leftrightarrow N-r$ with equal values, so
$\sum_{r=1}^{h}\sin^{2}(2\pi kr/N)=N/4$ and the factor $2/\sqrt N$ normalises.
Orthogonality is symmetry of $L_P$ together with simplicity of the spectrum.
Finally $\sin t\le t$ on $[0,\pi/2]$ gives
$\mu_1\le4\pi^{2}/N^{2}$, which is \eqref{eq:gap}; the last inequality is
$N=2h+1>2h$.
\end{proof}

The spectrum \eqref{eq:kms} is the classical Kac--Murdock--Szeg\H{o} spectrum of
the second difference in the reflection-odd sector \cite{KMS1953}, and $\mu_1$
is its spectral gap. Everything in Sections~\ref{sec:tvfloor} and
\ref{sec:gauge} is driven by \eqref{eq:gap} and by nothing else.

\subsection{The raw Gram block, the normalised block, and the ladder}
\label{sec:gram}

Fix $K\le h$ (the verified surfaces use $K=16$; nothing below depends on the
value). With $\{t_k\}$ of Lemma~\ref{lem:kms} put
\begin{equation}
\label{eq:gram}
  \ah_{ij}=t_i^{\T}At_j,
  \qquad
  \Ah_K=(\ah_{ij})_{i,j\le K},
  \qquad
  G=\MM^{-1/2}\Ah_K\MM^{-1/2},
  \qquad \MM=\diag(\mu_1,\dots,\mu_K),
\end{equation}
write $G_k$ for the leading $k\times k$ block of $G$, and
\begin{equation}
\label{eq:ladder}
  g_k=e_1^{\T}G_k^{-1}e_1\quad(k=1,\dots,K),
  \qquad
  s=\mu_1\,t_1^{\T}A^{-1}t_1 .
\end{equation}
The following three facts are classical; we record them in the form used later.

\begin{lemma}[interlacing]
\label{lem:interlacing}
Let $X\in\R^{n\times n}$ be symmetric and $P\in\R^{n\times m}$ an isometry.
Then the eigenvalues of $P^{\T}XP$ interlace those of $X$; in particular
$\lmin(X)\le\lmin(P^{\T}XP)$. Consequently a $K$-truncated Rayleigh minimum is
an upper bound for $\lmin$ and never a lower bound.
\end{lemma}

\begin{proof}
Cauchy's interlacing theorem \cite{Cauchy1829} in the Courant--Fischer form
\cite{CourantFischer1920}.
\end{proof}

\begin{lemma}[Schur nesting]
\label{lem:nesting}
Let $X\PD$ be partitioned as
$\bigl[\begin{smallmatrix}\xi&b^{\T}\\ b&Y\end{smallmatrix}\bigr]$ with
$\xi\in\R$. Then $Y\PD$ and
\begin{equation}
\label{eq:schur}
  \frac1{(X^{-1})_{11}}=\xi-b^{\T}Y^{-1}b,
  \qquad
  (X^{-1})_{11}=\frac{\det Y}{\det X} .
\end{equation}
\end{lemma}

\begin{proof}
The first identity is the Schur complement of the $(1,1)$ pivot
\cite{Schur1917,Haynsworth1968}; the second is Jacobi's formula for a cofactor
of the inverse, i.e.\ Cramer's rule.
\end{proof}

\begin{lemma}[constrained minimum]
\label{lem:constrained}
Let $X\PD$ be $k\times k$. Then
\begin{equation}
\label{eq:constrmin}
  \min\{u^{\T}Xu:\ u\in\R^{k},\ u_1=1\}
  =\frac1{e_1^{\T}X^{-1}e_1},
\end{equation}
attained exactly at $u^{\star}=X^{-1}e_1/(e_1^{\T}X^{-1}e_1)$. Equivalently,
\emph{every} admissible $u$ gives $u^{\T}Xu\ge1/(e_1^{\T}X^{-1}e_1)$: an
explicit trial vector is a certified lower bound for the reciprocal, never an
upper one.
\end{lemma}

\begin{proof}
Write $u=u^{\star}+\delta$ with $\delta_1=0$. Since
$Xu^{\star}=e_1/(e_1^{\T}X^{-1}e_1)$, the cross term is
$2\delta^{\T}Xu^{\star}=2\delta_1/(e_1^{\T}X^{-1}e_1)=0$, so
$u^{\T}Xu=(u^{\star})^{\T}Xu^{\star}+\delta^{\T}X\delta$ and
$(u^{\star})^{\T}Xu^{\star}=(u^{\star})^{\T}e_1/(e_1^{\T}X^{-1}e_1)
=1/(e_1^{\T}X^{-1}e_1)$ because $u^{\star}_1=1$. As $X\PD$ the remainder is
positive unless $\delta=0$.
\end{proof}

\begin{proposition}[three forms of the ladder]
\label{prop:cascade}
Assume $G\PD$ and let $G=LL^{\T}$ be its Cholesky factorisation,
$y=L^{-1}e_1$. Then for every $k\le K$:
\begin{align}
\label{eq:prefix}
  \text{(prefix)}\qquad & g_k=\sum_{j\le k}y_j^{2};\\
\label{eq:minor}
  \text{(minor)}\qquad & \frac1{g_k}
    =\frac{\det\Ah_{[1..k]}}{\mu_1\,\det\Ah_{[2..k]}}
    \qquad(\det\Ah_{[2..1]}:=1);\\
\label{eq:regression}
  \text{(regression)}\qquad & \frac{g_k}{g_1}=\frac1{1-R_k^{2}},
    \qquad R_k^{2}=\frac{b^{\T}G_{HH}^{-1}b}{G_{11}},
\end{align}
where in \eqref{eq:regression} the block $G_k$ is partitioned as
$\bigl[\begin{smallmatrix}G_{11}&b^{\T}\\ b&G_{HH}\end{smallmatrix}\bigr]$ with
$H=\{2,\dots,k\}$. In particular $k\mapsto g_k$ is nondecreasing, strictly
increasing whenever $y_k\ne0$, and $R_k^{2}\in[0,1)$ is the squared multiple
correlation of the first coordinate on the remaining $k-1$.
\end{proposition}

\begin{proof}
\emph{Prefix.} For a Cholesky factorisation the leading $k\times k$ block of $G$
is $L_kL_k^{\T}$ with $L_k$ the leading block of $L$, so
$g_k=e_1^{\T}L_k^{-\T}L_k^{-1}e_1=\lVert L_k^{-1}e_1\rVert^{2}$; and since $L$
is lower triangular, forward substitution gives
$L_k^{-1}e_1=(L^{-1}e_1)_{1:k}=y_{1:k}$. Monotonicity is immediate.

\emph{Minor.} By \eqref{eq:schur},
$g_k=\det G_{[2..k]}/\det G_{[1..k]}$. Substituting
$G=\MM^{-1/2}\Ah_K\MM^{-1/2}$ gives
$\det G_{[1..k]}=\det\Ah_{[1..k]}/\prod_{j\le k}\mu_j$ and
$\det G_{[2..k]}=\det\Ah_{[2..k]}/\prod_{2\le j\le k}\mu_j$, and all factors
cancel except $\mu_1$, which yields \eqref{eq:minor}.

\emph{Regression.} By \eqref{eq:schur}, $1/g_k=G_{11}-b^{\T}G_{HH}^{-1}b
=G_{11}(1-R_k^{2})$, while $g_1=1/G_{11}$; divide.
\end{proof}

\begin{lemma}[diagonal invariance of the ladder gain]
\label{lem:diaginv}
For every positive diagonal $\Lambda=\diag(d_1,\dots,d_K)$ the gain
$g_k/g_1=G_{11}\,(G_k^{-1})_{11}$ is unchanged under $G\mapsto\Lambda G\Lambda$.
\end{lemma}

\begin{proof}
$(\Lambda G\Lambda)_{11}=d_1^{2}G_{11}$ and
$\bigl((\Lambda G\Lambda)_k^{-1}\bigr)_{11}=d_1^{-2}(G_k^{-1})_{11}$; the
product is invariant.
\end{proof}

Lemma~\ref{lem:diaginv} localises the phenomenon: the gain does not see the
$\mu^{-1/2}$ normalisation of \eqref{eq:gram} at all, so whatever makes
$g_K/g_1$ large is a property of the raw Gram block $\Ah_K$ alone, and the
parity spectrum enters it only through the choice of the modes.

\begin{proposition}[direction of the ladder]
\label{prop:direction}
Assume $\Ah_h\PD$, i.e.\ $A\PD$ on the parity sector. Then
\begin{equation}
\label{eq:direction}
  s\;\ge\;g_K\;\ge\;g_1=\frac1{G_{11}},
  \qquad\text{equivalently}\qquad
  \frac1s\;\le\;\frac1{g_K}\;\le\;\frac1{g_1},
\end{equation}
with every ladder increment $g_{k+1}-g_k\ge0$. The hypothesis is used only for
the Cholesky factorisation of Proposition~\ref{prop:cascade} and is, on the
surfaces of Section~\ref{sec:numerics}, certified per window.
\end{proposition}

\begin{proof}
Let $T=[t_1,\dots,t_h]$, so $T$ is orthogonal and $\Ah_h=T^{\T}AT$. Then
$A^{-1}=T\Ah_h^{-1}T^{\T}$, hence
$t_1^{\T}A^{-1}t_1=(\Ah_h^{-1})_{11}$; and with
$\MM_h=\diag(\mu_1,\dots,\mu_h)$ and $G^{(h)}=\MM_h^{-1/2}\Ah_h\MM_h^{-1/2}$
one has $\bigl((G^{(h)})^{-1}\bigr)_{11}=\mu_1(\Ah_h^{-1})_{11}$, i.e.\
$s=g_h$. Now apply \eqref{eq:prefix}: $g_k$ is a partial sum of squares, so
$g_1\le g_K\le g_h=s$.
\end{proof}

\subsection{Lag weights, the gauge law, and the boundary-free swap}
\label{sec:lag}

Let $v\in\R^{h}$ and define the \emph{lag weights} of $v$ by
\begin{equation}
\label{eq:weights}
  w_0=\Ac_0(v),
  \qquad
  w_d=2\Ac_d(v)-\Cv_{M-1-d}(v)\quad(1\le d\le M-1),
  \qquad w_M:=0 ,
\end{equation}
with $\Ac,\Cv$ as in \eqref{eq:accv}, and put $(\Delta w)_d=w_d-w_{d+1}$ and
$\TV=\lVert\Delta w\rVert_1=\sum_{d=0}^{M-1}|(\Delta w)_d|$. For an admissible
$x$ we write $w(x)$, $\TV(x)$ for the weights and total variation of the window
vector $v=\sum_{k\le K}a_kt_k$, $a_k=x_k/\sqrt{\mu_k}$.

\begin{lemma}[the weights represent the form; the gauge law]
\label{lem:gauge}
For every $v\in\R^{h}$ and every lag sequence $c$,
\begin{equation}
\label{eq:represent}
  v^{\T}Av=\sum_{d=0}^{M-1}c_d\,w_d,
  \qquad
  w_0=\lVert v\rVert^{2},
  \qquad
  \sum_{d=0}^{M-1}w_d=0 .
\end{equation}
In particular, for an admissible $x$, $Q(x)=x^{\T}Gx=v^{\T}Av=\sum_dc_dw_d(x)$.
\end{lemma}

\begin{proof}
By \eqref{eq:Adef},
$v^{\T}Av=\sum_{r,r'}v_rv_{r'}c_{|r-r'|}-\sum_{r,r'}v_rv_{r'}c_{M-1-r-r'}$. The
first sum collects the lag $d=|r-r'|$ with multiplicity $1$ for $d=0$ and $2$
for $d\ge1$, i.e.\ it equals $c_0\Ac_0(v)+2\sum_{d\ge1}c_d\Ac_d(v)$. The second
sum collects the index $d=M-1-r-r'$, and
since $r,r'\le h-1$ we have $r+r'\le M-2$, so $d\ge1$ throughout and the
coefficient of $c_d$ is $\Cv_{M-1-d}(v)$. This is the first identity of
\eqref{eq:represent} with the weights \eqref{eq:weights}, and it also shows
$w_0=\Ac_0(v)=\lVert v\rVert^{2}$: the reflection term cannot reach $d=0$.

For the gauge law, sum \eqref{eq:weights}. Over all $d\ge0$,
$\Ac_0(v)+2\sum_{d\ge1}\Ac_d(v)=\bigl(\sum_rv_r\bigr)^{2}$, and as $d$ runs
through $1,\dots,M-1$ the index $j=M-1-d$ runs through $0,\dots,M-2=2h-2$, so
$\sum_{d\ge1}\Cv_{M-1-d}(v)=\sum_{j=0}^{2h-2}\Cv_j(v)
=\bigl(\sum_rv_r\bigr)^{2}$. The two contributions cancel.
\end{proof}

The gauge law has a consequence we use once: since $\sum_dw_d=0$, the value
$\sum_dc_dw_d$ is unchanged if a constant is added to every $c_d$. So the
representation \eqref{eq:represent} determines the form only up to an additive
normalisation of $c$, and any bound obtained from it may be optimised over that
normalisation.

\begin{lemma}[boundary-free summation by parts]
\label{lem:abel}
With $C_d=\sum_{e\le d}c_e$, $C_{-1}=0$, and $w_M=0$,
\begin{equation}
\label{eq:abel}
  \sum_{d=0}^{M-1}c_d\,w_d\;=\;\sum_{d=0}^{M-1}C_d\,(\Delta w)_d
\end{equation}
with no boundary term at either end.
\end{lemma}

\begin{proof}
Substituting $c_d=C_d-C_{d-1}$,
\[
  \sum_{d=0}^{M-1}c_dw_d
  =\sum_{d=0}^{M-1}C_dw_d-\sum_{d=0}^{M-1}C_{d-1}w_d
  =\sum_{d=0}^{M-1}C_dw_d-\sum_{d=0}^{M-2}C_dw_{d+1}
  =\sum_{d=0}^{M-1}C_d(w_d-w_{d+1}),
\]
where the second equality shifted the index and used $C_{-1}=0$, and the third
used $w_M=0$ to absorb the last term.
\end{proof}

This is Abel's summation by parts \cite{Abel1826}. Both boundary terms vanish
for structural reasons and not by a choice: $C_{-1}=0$ by definition of a
partial sum, and $w_M=0$ because both terms of \eqref{eq:weights} vanish at
$d=M$.

\section{The total-variation floor}
\label{sec:tvfloor}

\begin{theorem}[the total-variation floor]
\label{thm:tvfloor}
Let $x\in\R^{K}$ be admissible, i.e.\ $x_1=1$, and let $w=w(x)$ be the lag
weights of $v=\sum_{k\le K}a_kt_k$, $a_k=x_k/\sqrt{\mu_k}$. Then
\begin{equation}
\label{eq:tvfloor}
  \TV(x)\;\ge\;|w_0|\;=\;\lVert v\rVert^{2}\;=\;\lVert a\rVert^{2}
  \;\ge\;\frac1{\mu_1}\;=\;\frac1{4\sin^{2}(\pi/N)}
  \;\ge\;\frac{N^{2}}{4\pi^{2}} .
\end{equation}
The bound involves no comb node, no taper and no truncation, and it holds for
every lag sequence $c$ --- indeed the left- and right-hand sides do not depend
on $c$ at all.
\end{theorem}

\begin{proof}
Four steps.

\emph{(i) $w_0=\lVert a\rVert^{2}$.} By Lemma~\ref{lem:gauge},
$w_0=\lVert v\rVert^{2}$; and $\lVert v\rVert^{2}=\lVert a\rVert^{2}$ because
$\{t_k\}$ is orthonormal (Lemma~\ref{lem:kms}).

\emph{(ii) $\TV\ge|w_0|$.} With $w_M=0$ the increments telescope:
$\sum_{d=0}^{M-1}(\Delta w)_d=w_0-w_M=w_0$. Hence
$|w_0|\le\sum_{d=0}^{M-1}|(\Delta w)_d|=\TV$.

\emph{(iii) $\lVert a\rVert^{2}\ge1/\mu_1$.} Admissibility gives
$a_1=x_1/\sqrt{\mu_1}=1/\sqrt{\mu_1}$, so
$\lVert a\rVert^{2}\ge a_1^{2}=1/\mu_1$.

\emph{(iv) $1/\mu_1\ge N^{2}/(4\pi^{2})$.} This is \eqref{eq:gap}, i.e.\
$\sin t\le t$.
\end{proof}

Three features of \eqref{eq:tvfloor} are worth separating. It is a statement
about the \emph{normalisation}, not about the form: step (iii) is the only place
where admissibility is used, and it is where the whole $h^{2}$ comes from. It is
a statement about the \emph{sector}: step (i) uses orthonormality of the parity
modes and step (iv) uses their spectrum, both of which are the classical
Kac--Murdock--Szeg\H{o} data \cite{KMS1953}. And it is \emph{closed}: no
constant is fitted, no window is excluded, and the right-hand side is an
elementary function of $h$.

\begin{corollary}[the price]
\label{cor:price}
For every admissible $x$,
\begin{equation}
\label{eq:price}
  \frac{|Q(x)|}{\TV(x)}\;\le\;\mu_1\,|Q(x)| .
\end{equation}
Consequently, if a family of admissible trial vectors has values bounded by
$|Q(x)|\le U$ with $U$ independent of $h$, the ratio it certifies is at most
$U\mu_1=O(h^{-2})$.
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:tvfloor}, $\TV(x)\ge1/\mu_1>0$, so
$1/\TV(x)\le\mu_1$; multiply by $|Q(x)|\ge0$. The consequence is
$\mu_1=4\sin^{2}(\pi/N)\le4\pi^{2}/N^{2}$.
\end{proof}

Corollary~\ref{cor:price} is the load-bearing statement of the section. It says
that the pair (value, total variation) cannot be improved by choosing a better
trial vector: the ratio is pinned by the value alone, up to the spectral gap.
Any route whose output is the ratio therefore pays a factor $h^{-2}$ that is a
property of the parity spectrum meeting the normalisation, and not a property of
the comb.

\begin{corollary}[what summation by parts can certify]
\label{cor:holder}
For every admissible $x$,
\begin{equation}
\label{eq:holder}
  |Q(x)|\;\le\;\Bigl(\max_{0\le d\le M-1}|C_d|\Bigr)\,\TV(x),
  \qquad\text{hence}\qquad
  \max_{d}|C_d|\;\ge\;\frac{|Q(x)|}{\TV(x)} .
\end{equation}
Combined with \eqref{eq:price}, a family of bounded value certifies
$\max_d|C_d|\ge$ a quantity that is $O(h^{-2})$, and nothing stronger. By the
gauge law of Lemma~\ref{lem:gauge} the estimate may be applied to $c+\gamma$ for
any $\gamma\in\R$, so $\max_d|C_d|$ may be replaced by
$\inf_{\gamma}\max_d|C_d+\gamma(d+1)|$.
\end{corollary}

\begin{proof}
Lemma~\ref{lem:abel} and H\"older: $|\sum_dC_d(\Delta w)_d|
\le\max_d|C_d|\cdot\lVert\Delta w\rVert_1$. The final sentence is the remark
after Lemma~\ref{lem:gauge}, since replacing $c_d$ by $c_d+\gamma$ leaves
$Q$ unchanged and replaces $C_d$ by $C_d+\gamma(d+1)$.
\end{proof}

\begin{corollary}[the floor at one coordinate]
\label{cor:abis}
For every $v\in\R^{h}\setminus\{0\}$ with $t_1^{\T}v\ne0$,
\begin{equation}
\label{eq:abis}
  \frac{\TV(v)}{(t_1^{\T}v)^{2}}\;\ge\;1,
  \qquad\text{because}\qquad
  \TV(v)\;\ge\;|w_0|=\lVert v\rVert^{2}\;\ge\;(t_1^{\T}v)^{2} .
\end{equation}
No normalisation is required: \eqref{eq:abis} holds for every window vector, and
for admissible $x$ it reduces to \eqref{eq:tvfloor} because
$(t_1^{\T}v)^{2}=a_1^{2}=1/\mu_1$.
\end{corollary}

\begin{proof}
The first two relations are steps (i)--(ii) of Theorem~\ref{thm:tvfloor}, which
used no admissibility; the third is Cauchy--Schwarz with $\lVert t_1\rVert=1$.
\end{proof}

\begin{remark}[negative control: the floor is a property of the normalisation]
\label{rem:negcontrol}
Step (iii) is the only step that uses $x_1=1$, and it is indispensable: for
$\lambda\ne0$ the vector $\lambda x$ has $\TV(\lambda x)=\lambda^{2}\TV(x)$
(both $\Ac$ and $\Cv$ are quadratic in $v$, and $v$ is linear in $x$), so
$\mu_1\TV(\lambda x)=\lambda^{2}\mu_1\TV(x)$ falls below $1$ as soon as
$|\lambda|$ is small enough. An inadmissible vector therefore undercuts the
floor, and recognising it as inadmissible is the whole content of the
observation. This is also the reason Theorem~\ref{thm:gauge} below is a
statement about rays.
\end{remark}

\begin{measurement}[sharpness of the floor]
\label{meas:sharpness}
On Surface~II of Section~\ref{sec:numerics} --- $27$ windows of the frame-A
family, $h=142$ to $1445$, all with $\cond(G)\le10^{12}$ --- the floor of
Theorem~\ref{thm:tvfloor} is tight to about one order of magnitude: the measured
ratio is
\begin{equation}
\label{eq:sharp}
  \mu_1\,\TV(x)\in[5.00,\ 23.20]
\end{equation}
at the optimiser of the declared trial family, while the floor itself takes the
values $2.06\cdot10^{3}$ to $1.77\cdot10^{5}$, growing as $h^{+1.997}$ (a
least-squares fit, reported for orientation and not used anywhere) against the
closed $h^{2}$ of \eqref{eq:gap}. The inadmissible control of
Remark~\ref{rem:negcontrol} returns $\mu_1\TV\approx7\cdot10^{-4}$ at
$\lambda=10^{-2}$, i.e.\ it undercuts the floor by the predicted factor
$\lambda^{2}$.
The trial family is the declared tapered mode family also used in
Measurement~\ref{meas:battery}, its taper width running over a fixed geometric
list; the caps of Surface~II are written out in Section~\ref{sec:surfaces}.
\end{measurement}

We close the section by saying plainly what the floor does not know about. It
does not know where the comb nodes are, or how many there are, or what their
weights are; it does not know whether the trial vector was tapered, and it does
not know at which $K$ the mode family was truncated. It is a statement about
$\{t_k\}$, about $\mu_1$, and about the normalisation $x_1=1$. The next section
shows that the last of these three cannot be traded away.

\section{Gauge rigidity of the normalisation}
\label{sec:gauge}

Theorem~\ref{thm:tvfloor} raises an obvious question: the floor is $1/\mu_1$
because the normalisation is imposed on the mode with eigenvalue $\mu_1$, so
could a different \emph{entry spectrum} --- a reweighted, shifted or otherwise
reparametrised family --- produce a flatter floor and hence a better ratio? The
answer is no, and it is no for a structural reason.

\begin{theorem}[the normalisation is a gauge]
\label{thm:gauge}
For $a\in\R^{K}$ let $v(a)=\sum_{k\le K}a_kt_k$ and
\begin{equation}
\label{eq:functionals}
  \widetilde Q(a)=v(a)^{\T}Av(a),
  \qquad
  \widetilde{\TV}(a)=\lVert\Delta w(v(a))\rVert_1 .
\end{equation}
Then:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.6em]
\item\label{G:hom} $\widetilde Q$ and $\widetilde{\TV}$ are homogeneous of
  degree $2$: $\widetilde Q(\lambda a)=\lambda^{2}\widetilde Q(a)$ and
  $\widetilde{\TV}(\lambda a)=\lambda^{2}\widetilde{\TV}(a)$ for all
  $\lambda\in\R$.
\item\label{G:ray} Hence the ratio
  $\varrho(a)=\widetilde Q(a)/\widetilde{\TV}(a)$ is constant on rays: it is a
  function of the direction of $a$ alone.
\item\label{G:gauge} Let $\mu'=(\mu'_1,\dots,\mu'_K)$ be \emph{any} positive
  entry spectrum and let $x\mapsto a$, $a_k=x_k/\sqrt{\mu'_k}$, be the
  associated parametrisation. Then the constraint $x_1=1$ describes the affine
  slice $\{a:a_1=1/\sqrt{\mu'_1}\}$, every ray $\{\lambda a:\lambda>0\}$ with
  $a_1>0$ meets that slice exactly once, and consequently
  \begin{equation}
  \label{eq:gaugeequal}
    \bigl\{\varrho(a):a_k=x_k/\sqrt{\mu'_k},\ x_1=1\bigr\}
    \;=\;\bigl\{\varrho(a):a\in\R^{K},\ a_1>0\bigr\}
  \end{equation}
  \emph{independently of $\mu'$}. The entry normalisation is a gauge: the range
  of the ratio is the same in every sector.
\end{enumerate}
\end{theorem}

\begin{proof}
\ref{G:hom} $a\mapsto v(a)$ is linear, and by \eqref{eq:accv} both $\Ac_d$ and
$\Cv_j$ are quadratic forms in $v$, so $w(v(\lambda a))=\lambda^{2}w(v(a))$ by
\eqref{eq:weights}, hence $\Delta w$ and its $\ell^{1}$ norm scale by
$\lambda^{2}$; and $\widetilde Q$ is a quadratic form in $a$.

\ref{G:ray} Immediate from \ref{G:hom}, both numerator and denominator having
the same degree.

\ref{G:gauge} The map $x\mapsto a$ is a linear bijection of $\R^{K}$ taking
$\{x_1=1\}$ onto $\{a_1=1/\sqrt{\mu'_1}\}$. A ray $\{\lambda a:\lambda>0\}$ with
$a_1>0$ meets $\{a_1=\theta\}$ exactly once for every $\theta>0$, namely at
$\lambda=\theta/a_1$. So the slice $\{a_1=1/\sqrt{\mu'_1}\}$ is a
cross-section of the family of all rays with $a_1>0$, whatever the value of
$\mu'_1>0$; by \ref{G:ray} the ratio takes the same set of values on the slice
as on the family of rays, which is \eqref{eq:gaugeequal}. The right-hand side
does not mention $\mu'$.
\end{proof}

Theorem~\ref{thm:gauge} is a fact about degree-two homogeneous functionals, and
we state it modestly for that reason. Its force is entirely in the
consequence: since the accessible range of the ratio is a sector-independent
set, and since Theorem~\ref{thm:tvfloor} bounds that range in the parity
parametrisation, no change of sector can enlarge it.

\begin{corollary}[no sector change removes the $h^{2}$]
\label{cor:noescape}
Let $\mu'$ be any positive entry spectrum and let $x'$ be admissible for it.
Then
\begin{equation}
\label{eq:noescape}
  \frac{|\widetilde Q(a')|}{\widetilde{\TV}(a')}
  \;\le\;\mu_1\;\sup\bigl\{|\widetilde Q(a)|:a\ \text{in the parity slice}\bigr\},
\end{equation}
with $\mu_1=4\sin^{2}(\pi/N)$ the parity gap --- i.e.\ the bound of
Corollary~\ref{cor:price} transfers verbatim, with the \emph{parity} gap, to
every sector. In particular a family of trial vectors with $h$-bounded value in
some sector certifies a ratio $O(h^{-2})$ in that sector as well.
\end{corollary}

\begin{proof}
By \eqref{eq:gaugeequal} the value $\varrho(a')$ is attained by some $a$ in the
parity slice ($\mu'=\mu$), i.e.\ by some admissible $x$; apply
Corollary~\ref{cor:price} to that $x$.
\end{proof}

The next proposition is the quantitative form of Corollary~\ref{cor:noescape}
for the simplest reparametrisation, a rigid shift of the entry spectrum. It is a
two-line computation, and we state it as a proposition rather than dressing it
up: the point is not that the computation is hard but that the product it
computes is \emph{exactly} constant.

\begin{proposition}[floor $\times$ transfer is conserved]
\label{prop:shift}
Let $\tau\ge0$ and let $\mu'_k=\mu_k+\tau$ be the shifted entry spectrum, so
that admissibility in the shifted sector reads $a'_1=1/\sqrt{\mu_1+\tau}$. Then
the shifted floor of Theorem~\ref{thm:tvfloor} is
$\widetilde{\TV}(a')\ge1/(\mu_1+\tau)$, the transfer factor relating the two
normalisations is
\begin{equation}
\label{eq:transfer}
  \theta=\frac{(a'_1)^{2}}{a_1^{2}}=\frac{\mu_1}{\mu_1+\tau},
  \qquad
  \frac1\theta=1+\frac\tau{\mu_1},
\end{equation}
and
\begin{equation}
\label{eq:conserved}
  \underbrace{\frac1{\mu_1+\tau}}_{\text{shifted floor}}\cdot
  \underbrace{\Bigl(1+\frac\tau{\mu_1}\Bigr)}_{\text{transfer}}
  \;=\;\frac1{\mu_1+\tau}\cdot\frac{\mu_1+\tau}{\mu_1}
  \;=\;\frac1{\mu_1}
  \qquad\text{identically, at every shift }\tau .
\end{equation}
A shift does flatten the floor; it does not remove the $h^{2}$, which lives in
the ratio and not in the floor.
\end{proposition}

\begin{proof}
The floor is Theorem~\ref{thm:tvfloor} applied in the shifted parametrisation,
whose step (iii) reads $\lVert a'\rVert^{2}\ge(a'_1)^{2}=1/(\mu_1+\tau)$.
For the transfer, the two admissibility constraints differ only in the value of
the first coefficient, $a_1=1/\sqrt{\mu_1}$ against
$a'_1=1/\sqrt{\mu_1+\tau}$; by Theorem~\ref{thm:gauge}\ref{G:hom} a value
computed at $a'$ is converted to the corresponding value on the same ray in the
unshifted slice by the factor $(a_1/a'_1)^{2}=1/\theta$. The displayed product
is then \eqref{eq:conserved}.
\end{proof}

\begin{proposition}[the null mode, and the other sector]
\label{prop:nullmode}
Let $c^{L}=(2,-1,0,\dots)$ be as in Lemma~\ref{lem:kms}.
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.6em]
\item\label{N:sine} The parity family $\mu_k=4\sin^{2}(k\pi/N)$,
  $k=1,\dots,h$, is the restriction to $k\ge1$ of the sine family
  $\{4\sin^{2}(k\pi/N)\}_{k\ge0}$, whose member at $k=0$ is $\mu_0=0$.
\item\label{N:noreps} The parametrisation $a_k=x_k/\sqrt{\mu'_k}$ of
  Theorem~\ref{thm:gauge}\ref{G:gauge} requires $\mu'_k>0$. If a mode family
  containing a null mode is used and the normalisation is imposed on that mode,
  the constraint has \emph{no} representative $a$ of finite norm; if the
  normalisation is imposed elsewhere, step (iii) of
  Theorem~\ref{thm:tvfloor} yields the floor of the normalised mode and the
  null mode contributes nothing. In neither case is a flatter admissible floor
  obtained.
\item\label{N:even} Write $\Pi_+e_r=\tfrac1{\sqrt2}(e_r+e_{M-1-r})$ for the
  reflection-even isometry. The complementary compression
  $\Pi_+^{\T}\Tm{M}(c^{L})\Pi_+$ has spectrum
  $\bigl\{4\sin^{2}\bigl((2k+1)\pi/(2N)\bigr)\bigr\}_{k=0}^{h-1}$, whose
  minimum $4\sin^{2}(\pi/(2N))$ is \emph{strictly smaller} than $\mu_1$. Hence
  no reflection sector of the second-difference form has a lowest eigenvalue
  exceeding $\mu_1$.
\end{enumerate}
\end{proposition}

\begin{proof}
\ref{N:sine} is the formula of Lemma~\ref{lem:kms} read at $k=0$.

\ref{N:noreps} is the definition of the parametrisation together with step (iii)
of Theorem~\ref{thm:tvfloor}, which bounds $\lVert a\rVert^{2}$ below by the
square of the \emph{normalised} coordinate only.

\ref{N:even} For $c^{L}$ the even compression is
$(\Pi_+^{\T}\Tm{M}(c^{L})\Pi_+)_{rr'}=c^{L}_{|r-r'|}+c^{L}_{M-1-r-r'}$, which by
the computation in the proof of Lemma~\ref{lem:kms} is tridiagonal
$2I-E-E^{\T}$ with corner entry $2-1=1$; as a recurrence this is $u_{-1}=0$ and
$u_h=+u_{h-1}$. With $u_r=\sin(\omega(r+1))$ the inner condition reads
$\sin(\omega(h+1))-\sin(\omega h)
=2\cos\bigl(\tfrac{\omega N}2\bigr)\sin\bigl(\tfrac\omega2\bigr)=0$, hence
$\omega N/2=(k+\tfrac12)\pi$ and $\mu=4\sin^{2}\bigl((2k+1)\pi/(2N)\bigr)$ for
$k=0,\dots,h-1$, which are $h$ distinct values, hence all. The minimum is at
$k=0$ and $4\sin^{2}(\pi/(2N))<4\sin^{2}(\pi/N)=\mu_1$ because $\sin$ is
increasing on $[0,\pi/2]$.
\end{proof}

Part~\ref{N:even} is the precise sense in which the parity compression is the
best of the two reflection sectors for this purpose, and part~\ref{N:noreps} is
the sense in which enlarging the mode family removes the statement instead of
improving it. Together with Theorem~\ref{thm:gauge} and
Proposition~\ref{prop:shift} this closes the sector question: reweighting,
shifting, or enlarging the entry spectrum changes neither the accessible range
of the ratio nor the conserved product.

\begin{measurement}[the sector battery]
\label{meas:battery}
Theorem~\ref{thm:gauge} and Proposition~\ref{prop:shift} are proved statements;
the following is their machine verification on a declared battery, reported
because it also checks the implementation of the transport map. Over
$35$ weight families $\times$ $6$ taper widths $\times$ $27$ windows of
Surface~II the ratio $\varrho$ is reproduced across sectors with residual at
most $8.8\cdot10^{-10}$ against a declared round-off horizon of
$2.6\cdot10^{-8}$, and the product \eqref{eq:conserved} is reproduced as an
exact identity at every shift of the declared shift list. Transporting the
$x$-coordinates instead of the ray --- the wrong map --- moves the ratio by a
large factor on the spread-out weight families, which is the mutation control of
the battery.
\end{measurement}

\section{The two-dimensional reduction}
\label{sec:twodim}

Sections~\ref{sec:tvfloor} and \ref{sec:gauge} priced the trial-vector route.
This section identifies what is left: at $K=2$ the optimal trial vector is
available in closed form, so the entire question collapses onto a single scalar,
the squared correlation of the two lowest parity modes in the metric of $A$.

\begin{proposition}[the closed vector is exact at $K=2$]
\label{prop:K2}
Assume $G_2\PD$ and put $r_{12}^{2}=G_{12}^{2}/(G_{11}G_{22})
=\ah_{12}^{2}/(\ah_{11}\ah_{22})$. Then $u=(1,-G_{21}/G_{22})$ attains the
constrained minimum of Lemma~\ref{lem:constrained} at $k=2$:
\begin{equation}
\label{eq:K2}
  u^{\T}G_2u=\frac1{g_2}=G_{11}\bigl(1-r_{12}^{2}\bigr),
  \qquad
  G_{11}\,g_2=\frac1{1-r_{12}^{2}}\quad\text{identically}.
\end{equation}
\end{proposition}

\begin{proof}
Since $G_2\PD$ we have $G_{11}>0$, $G_{22}>0$ and $\det G_2>0$.

\emph{The minimum.} Lemma~\ref{lem:nesting} applied to $X=G_2$ with
$\xi=G_{11}$, $b=G_{12}$ and $Y=G_{22}$ gives
\[
  \frac1{g_2}=\frac1{(G_2^{-1})_{11}}=G_{11}-\frac{G_{12}^{2}}{G_{22}}
  =G_{11}\Bigl(1-\frac{G_{12}^{2}}{G_{11}G_{22}}\Bigr)
  =G_{11}\bigl(1-r_{12}^{2}\bigr),
\]
and by Lemma~\ref{lem:constrained} this number is the constrained minimum.

\emph{The minimiser.} By Cramer's rule
$G_2^{-1}e_1=(\det G_2)^{-1}(G_{22},-G_{12})^{\T}$, so
$G_2^{-1}e_1/(e_1^{\T}G_2^{-1}e_1)=(1,-G_{12}/G_{22})^{\T}=u$ because
$G_{21}=G_{12}$; Lemma~\ref{lem:constrained} identifies this vector as the
unique minimiser. It is worth verifying the value on $u$ by substitution rather
than through the inverse:
\[
  u^{\T}G_2u=G_{11}-2\frac{G_{12}^{2}}{G_{22}}
  +\frac{G_{12}^{2}}{G_{22}^{2}}\,G_{22}
  =G_{11}-\frac{G_{12}^{2}}{G_{22}},
\]
the same number, and this second computation uses only $G_{22}\ne0$.

\emph{The gain.} $g_1=1/G_{11}$, so dividing the first display by $g_1$ gives
$G_{11}g_2=1/(1-r_{12}^{2})$, which is the case $k=2$ of
\eqref{eq:regression} with $R_2^{2}=r_{12}^{2}$.

\emph{The two readings of $r_{12}^{2}$ agree.} By \eqref{eq:gram},
$G_{ij}=\ah_{ij}/\sqrt{\mu_i\mu_j}$, hence
\[
  \frac{G_{12}^{2}}{G_{11}G_{22}}
  =\frac{\ah_{12}^{2}/(\mu_1\mu_2)}
        {(\ah_{11}/\mu_1)\,(\ah_{22}/\mu_2)}
  =\frac{\ah_{12}^{2}}{\ah_{11}\ah_{22}} :
\]
the diagonal congruence cancels, as it must by Lemma~\ref{lem:diaginv}.
\end{proof}

\begin{proposition}[pivot identity]
\label{prop:pivot}
Assume $G_k\PD$ and let $x^{\star}$ be the minimiser of
Lemma~\ref{lem:constrained}. Then for every $\delta$ with $\delta_1=0$,
\begin{equation}
\label{eq:pivot}
  (x^{\star}+\delta)^{\T}G_k(x^{\star}+\delta)
  =\frac1{g_k}+\delta^{\T}G_k\delta
  \qquad\text{exactly},
\end{equation}
so the relative excess $\rho=g_k\,\delta^{\T}G_k\delta$ of any explicit
candidate is an exact number and never an estimate.
\end{proposition}

\begin{proof}
Write $x^{\star}=G_k^{-1}e_1/g_k$, so that $G_kx^{\star}=e_1/g_k$ and
$x^{\star}_1=e_1^{\T}G_k^{-1}e_1/g_k=1$. Expanding the quadratic form,
\[
  (x^{\star}+\delta)^{\T}G_k(x^{\star}+\delta)
  =(x^{\star})^{\T}G_kx^{\star}+2\,\delta^{\T}G_kx^{\star}
  +\delta^{\T}G_k\delta .
\]
The first term is $(x^{\star})^{\T}e_1/g_k=x^{\star}_1/g_k=1/g_k$ and the
cross term is $2\delta^{\T}e_1/g_k=2\delta_1/g_k=0$. No inequality is used at
any point, which is the content of the word ``exactly'' in \eqref{eq:pivot}:
$\delta^{\T}G_k\delta$ is not an estimate for the gap between an explicit
candidate and the optimum, it \emph{is} the gap.
\end{proof}

\begin{remark}[the sign of the closed vector is not cosmetic]
\label{rem:wrongsign}
At $k=2$ the candidate $(1,+G_{12}/G_{22})$ differs from the minimiser of
Proposition~\ref{prop:K2} by $\delta=(0,2G_{12}/G_{22})$, and
\eqref{eq:pivot} prices it exactly:
$\delta^{\T}G_2\delta=4G_{12}^{2}/G_{22}$, so the relative excess is
\begin{equation}
\label{eq:wrongsign}
  \rho=g_2\,\delta^{\T}G_2\delta
  =\frac{4G_{12}^{2}/G_{22}}{G_{11}\bigl(1-r_{12}^{2}\bigr)}
  =\frac{4r_{12}^{2}}{1-r_{12}^{2}}
  \qquad\text{identically},
\end{equation}
which diverges precisely where the cancellation is deep. The pivot identity
therefore also quantifies how a wrong closed vector fails, and it does so
without a single inequality.
\end{remark}

\begin{proposition}[the entrywise perturbation, and one cancellation ratio]
\label{prop:union}
Let $x^{\star}$ be as above, $\sigma=\sgn(x^{\star})$,
$\Theta_k=|x^{\star}|^{\T}|G_k|\,|x^{\star}|$, and for $\varepsilon>0$ let
$E$ be the sign-aligned entrywise perturbation $E_{ij}=\varepsilon|G_{ij}|
\sigma_i\sigma_j$. Then
\begin{equation}
\label{eq:union}
  \bigl|(x^{\star})^{\T}Ex^{\star}\bigr|=\varepsilon\,\Theta_k
  \qquad\text{exactly},
\end{equation}
so the entry threshold at which the perturbation exhausts the value is
$\varepsilon_{\mathrm{ent}}(k)=\rho\,(1/g_k)/\Theta_k$ with $\rho$ of
\eqref{eq:pivot}. At $k=2$ the ratio is closed:
\begin{equation}
\label{eq:union2}
  \Theta_2=G_{11}\bigl(1+3r_{12}^{2}\bigr),
  \qquad
  \frac{1/g_2}{\Theta_2}=\frac{1-r_{12}^{2}}{1+3r_{12}^{2}},
\end{equation}
and $1+3r_{12}^{2}\to4$ as $r_{12}^{2}\to1$.
\end{proposition}

\begin{proof}
\emph{The union identity.} Since $x^{\star}_i\sigma_i=|x^{\star}_i|$ for every
$i$,
\[
  (x^{\star})^{\T}Ex^{\star}
  =\varepsilon\sum_{i,j}x^{\star}_i\sigma_i\,|G_{ij}|\,\sigma_jx^{\star}_j
  =\varepsilon\sum_{i,j}|x^{\star}_i|\,|G_{ij}|\,|x^{\star}_j|
  =\varepsilon\,\Theta_k ,
\]
and every term of the middle sum is nonnegative, so no cancellation occurs and
the modulus may be taken termwise: the sign-aligned perturbation is the one
that realises the $\ell^{1}$ mass of $G_k$ in the direction $x^{\star}$. For
the threshold, \eqref{eq:pivot} says the value at $x^{\star}$ is $1/g_k$ and
that a relative excess $\rho$ is an absolute excess $\rho/g_k$, so the
perturbation exhausts the value exactly when $\varepsilon\Theta_k=\rho/g_k$.

\emph{The closed ratio at $k=2$.} By Proposition~\ref{prop:K2},
$x^{\star}=(1,-G_{12}/G_{22})$, hence $|x^{\star}|=(1,|G_{12}|/G_{22})$ because
$G_{22}>0$; and since $G_{11},G_{22}>0$,
\[
  \Theta_2
  =\underbrace{G_{11}}_{(1,1)}
  +\underbrace{2\,\frac{|G_{12}|}{G_{22}}\,|G_{12}|}_{(1,2)\text{ and }(2,1)}
  +\underbrace{\frac{G_{12}^{2}}{G_{22}^{2}}\,G_{22}}_{(2,2)}
  =G_{11}+3\,\frac{G_{12}^{2}}{G_{22}}
  =G_{11}\bigl(1+3r_{12}^{2}\bigr).
\]
The constant is $3$ and not $2$ because the off-diagonal is counted twice and
the $(2,2)$ entry once. Dividing \eqref{eq:K2} by this gives
\eqref{eq:union2}, and $1+3r_{12}^{2}\to4$ as $r_{12}^{2}\to1$.
\end{proof}

\begin{remark}
\label{rem:onedress}
Proposition~\ref{prop:union} is the reason the three natural readings of the
same obstruction --- a determinant ratio, the scalar $1-r_{12}^{2}$, and an
entrywise perturbation threshold --- are one object rather than three: at $k=2$
they differ by the explicit factor $1+3r_{12}^{2}\in[1,4]$. We state the two
formulae and leave the reading to the reader; ``three dresses of one
inequality'' is an observation about identities, not a theorem.
\end{remark}

\begin{lemma}[Wronskian telescope]
\label{lem:wronskian}
Let $u=t_1$, $v=t_2$ be the two lowest parity modes of Lemma~\ref{lem:kms},
extended by $u_{-1}=v_{-1}=0$ and $u_h=-u_{h-1}$, $v_h=-v_{h-1}$, and put
$W_r=u_{r+1}v_r-u_rv_{r+1}$. Then
\begin{equation}
\label{eq:wronskian}
  (\mu_1-\mu_2)\,u_rv_r=W_{r-1}-W_r\quad(0\le r\le h-1),
  \qquad W_{-1}=W_{h-1}=0,
\end{equation}
and summing \eqref{eq:wronskian} re-derives $t_1^{\T}t_2=0$.
\end{lemma}

\begin{proof}
By the definition of $W_r$,
\[
  W_{r-1}-W_r=(u_rv_{r-1}-u_{r-1}v_r)-(u_{r+1}v_r-u_rv_{r+1})
  =u_r\bigl(v_{r-1}+v_{r+1}\bigr)-v_r\bigl(u_{r-1}+u_{r+1}\bigr).
\]
Rows $0,\dots,h-1$ of $L_Pt_1=\mu_1t_1$ read
$-u_{r-1}+2u_r-u_{r+1}=\mu_1u_r$ under the two conventions $u_{-1}=0$ and
$u_h=-u_{h-1}$ --- this is exactly the reading of the rows carried out in the
proof of Lemma~\ref{lem:kms} --- i.e.\ $u_{r-1}+u_{r+1}=(2-\mu_1)u_r$, and
likewise $v_{r-1}+v_{r+1}=(2-\mu_2)v_r$. Substituting,
\[
  W_{r-1}-W_r=(2-\mu_2)u_rv_r-(2-\mu_1)u_rv_r=(\mu_1-\mu_2)\,u_rv_r ,
\]
which is \eqref{eq:wronskian}. The two ends are the two boundary conventions
and nothing else: $W_{-1}=u_0v_{-1}-u_{-1}v_0=0$ is the Dirichlet end, and
\[
  W_{h-1}=u_hv_{h-1}-u_{h-1}v_h
  =(-u_{h-1})v_{h-1}-u_{h-1}(-v_{h-1})=0
\]
is the odd reflection, which by the computation in the proof of
Lemma~\ref{lem:kms} is precisely the corner entry $3$ of $L_P$. Summing
\eqref{eq:wronskian} over $r=0,\dots,h-1$ telescopes the right-hand side to
$W_{-1}-W_{h-1}=0$, so $(\mu_1-\mu_2)\,t_1^{\T}t_2=0$; as $\mu_1\ne\mu_2$,
orthogonality of the two lowest parity modes is a \emph{consequence} of the
telescope rather than an independent input.
\end{proof}

\begin{lemma}[maximal minors carry no arithmetic]
\label{lem:minors}
With $u=t_1/\sqrt{\mu_1}$, $v=t_2/\sqrt{\mu_2}$ and
$\mathcal W_{rr'}=u_rv_{r'}-u_{r'}v_r$,
\begin{equation}
\label{eq:lagrange}
  \tfrac12\sum_{r,r'}\mathcal W_{rr'}^{2}
  =\lVert u\rVert^{2}\lVert v\rVert^{2}-\langle u,v\rangle^{2}
  =\frac1{\mu_1\mu_2}
  \qquad\text{exactly}.
\end{equation}
\end{lemma}

\begin{proof}
Expanding the square and using the symmetry of the double sum,
\[
  \tfrac12\sum_{r,r'}\bigl(u_rv_{r'}-u_{r'}v_r\bigr)^{2}
  =\tfrac12\Bigl[\sum_{r,r'}u_r^{2}v_{r'}^{2}
    -2\sum_{r,r'}u_rv_r\,u_{r'}v_{r'}
    +\sum_{r,r'}u_{r'}^{2}v_r^{2}\Bigr]
  =\lVert u\rVert^{2}\lVert v\rVert^{2}-\langle u,v\rangle^{2},
\]
the two outer sums being equal; this is Lagrange's identity
\cite{Lagrange1773}. For the value, $t_1$ and $t_2$ are orthonormal by
Lemma~\ref{lem:kms}, so $\lVert u\rVert^{2}=1/\mu_1$,
$\lVert v\rVert^{2}=1/\mu_2$ and $\langle u,v\rangle=0$.
\end{proof}

Lemma~\ref{lem:minors} is the reason the near-degeneracy cannot be attributed
to the mode pair: the full Gram determinant of the two rescaled modes is the
closed number $1/(\mu_1\mu_2)$, with no comb in it. Whatever becomes small in
Problem~\ref{prob:R1} becomes small in the metric of $A$ and nowhere else.

\begin{lemma}[the determinant reading, free of positivity]
\label{lem:detread}
With $\nu_1\ge\nu_2$ the eigenvalues of $\Ah_2$,
\begin{equation}
\label{eq:detread}
  1-r_{12}^{2}
  =\frac{\det\Ah_2}{\ah_{11}\ah_{22}}
  =\frac{\nu_1\nu_2}{\ah_{11}\ah_{22}}
  =\frac{\det G_2}{G_{11}G_{22}},
\end{equation}
and no positivity of $A$, of $\Ah_2$ or of $G$ is used.
\end{lemma}

\begin{proof}
$\det\Ah_2=\ah_{11}\ah_{22}-\ah_{12}^{2}$, so dividing by $\ah_{11}\ah_{22}$
gives the first equality by the definition of $r_{12}^{2}$. The second is
$\det\Ah_2=\nu_1\nu_2$, valid for every symmetric $2\times2$ matrix whatever
the signs of its eigenvalues. The third repeats the first computation for
$G_2=\MM_2^{-1/2}\Ah_2\MM_2^{-1/2}$, where the diagonal factors cancel exactly
as in the last step of the proof of Proposition~\ref{prop:K2}. The chain uses
only $\ah_{11}\ah_{22}\ne0$: no eigenvalue is assumed positive, no
factorisation is performed, and $\nu_2$ may be negative on both sides of every
equality.
\end{proof}

Lemma~\ref{lem:detread} is load-bearing in a negative sense, and we say so
plainly: it is what makes the whole reduction free of any positivity input about
$A$. Whether $A$ is positive definite on the parity sector is, on the surfaces
of Section~\ref{sec:numerics}, a numerical fact about finitely many finite
matrices; no statement of this paper is routed through a uniform version of it.

\begin{lemma}[an unconditional bound for the large eigenvalue]
\label{lem:gershgorin}
$\nu_1\le\max(\ah_{11},\ah_{22})+|\ah_{12}|$, closed in the three entries and
uniform in the window.
\end{lemma}

\begin{proof}
This is Gershgorin's theorem \cite{Gershgorin1931} for $\Ah_2$; we give the two
lines because the statement is used with no positivity hypothesis anywhere. The
eigenvalues of a symmetric $2\times2$ matrix are
\[
  \nu_{1,2}=\tfrac12\Bigl[\ah_{11}+\ah_{22}
  \pm\sqrt{(\ah_{11}-\ah_{22})^{2}+4\ah_{12}^{2}}\Bigr],
\]
and $\sqrt{p^{2}+q^{2}}\le|p|+|q|$ gives
$\sqrt{(\ah_{11}-\ah_{22})^{2}+4\ah_{12}^{2}}\le|\ah_{11}-\ah_{22}|
+2|\ah_{12}|$, whence
\[
  \nu_1\le\tfrac12\bigl[\ah_{11}+\ah_{22}+|\ah_{11}-\ah_{22}|\bigr]+|\ah_{12}|
  =\max(\ah_{11},\ah_{22})+|\ah_{12}| .
\]
The closed form makes visible in what sense the bound is uniform in the window:
it is a function of the three entries alone, with no constant fitted and no
window excluded.
\end{proof}

\begin{proposition}[the determinant returns: the $t^{\star}$ collapse]
\label{prop:collapse}
Assume $\ah_{11},\ah_{22}>0$ and consider the one-parameter family
$u(t)=(t,-1)$. At $t^{\star}=\sqrt{\ah_{22}/\ah_{11}}$ the Rayleigh quotient of
$\Ah_2$ is
\begin{equation}
\label{eq:collapse}
  R(t^{\star})
  =\frac{u(t^{\star})^{\T}\Ah_2u(t^{\star})}{\lVert u(t^{\star})\rVert^{2}}
  =\frac{2\sqrt{\ah_{11}\ah_{22}}\;\det\Ah_2}
        {(\ah_{11}+\ah_{22})\bigl(\sqrt{\ah_{11}\ah_{22}}+\ah_{12}\bigr)}
  \qquad\text{identically},
\end{equation}
so the only member of the family that meets the threshold does so by
reintroducing $\det\Ah_2$. If in addition $\Ah_2\PD$, then
\begin{equation}
\label{eq:harmonic}
  \nu_2\;\le\;\frac{2\det\Ah_2}{\ah_{11}+\ah_{22}}
  \;=\;\frac{2\nu_1\nu_2}{\nu_1+\nu_2}\;\le\;2\nu_2,
\end{equation}
i.e.\ the harmonic mean of the two eigenvalues bounds $\nu_2$ from above and is
tight within a factor $2$.
\end{proposition}

\begin{proof}
\emph{The closed form, unconditional.} Put
$\tau=t^{\star}=\sqrt{\ah_{22}/\ah_{11}}$, well defined because
$\ah_{11},\ah_{22}>0$, and note the two identities
$\tau^{2}\ah_{11}=\ah_{22}$ and $\tau\ah_{11}=\sqrt{\ah_{11}\ah_{22}}$, the
second of which also gives $\ah_{22}=\tau\sqrt{\ah_{11}\ah_{22}}$. The
numerator of the Rayleigh quotient is
\[
  u(\tau)^{\T}\Ah_2u(\tau)=\tau^{2}\ah_{11}-2\tau\ah_{12}+\ah_{22}
  =2\bigl(\ah_{22}-\tau\ah_{12}\bigr)
  =2\tau\bigl(\sqrt{\ah_{11}\ah_{22}}-\ah_{12}\bigr),
\]
and the denominator is
$\lVert u(\tau)\rVert^{2}=\tau^{2}+1=(\ah_{11}+\ah_{22})/\ah_{11}$. Hence
\[
  R(t^{\star})
  =\frac{2\tau\ah_{11}\bigl(\sqrt{\ah_{11}\ah_{22}}-\ah_{12}\bigr)}
        {\ah_{11}+\ah_{22}},
\]
and multiplying numerator and denominator by
$\sqrt{\ah_{11}\ah_{22}}+\ah_{12}$, then using
$\ah_{11}\ah_{22}-\ah_{12}^{2}=\det\Ah_2$ and once more
$\tau\ah_{11}=\sqrt{\ah_{11}\ah_{22}}$, produces \eqref{eq:collapse}. No sign
is assumed for $\ah_{12}$, for $\det\Ah_2$ or for $\nu_2$; the hypotheses are
$\ah_{11},\ah_{22}>0$ and, for the displayed quotient,
$\sqrt{\ah_{11}\ah_{22}}+\ah_{12}\ne0$.

\emph{The harmonic-mean bound, conditional.} Assume $\Ah_2\PD$, so that
$\nu_1\ge\nu_2>0$. Substituting $\tr\Ah_2=\ah_{11}+\ah_{22}=\nu_1+\nu_2$ and
$\det\Ah_2=\nu_1\nu_2$ identifies the middle expression of
\eqref{eq:harmonic} as the harmonic mean of the two eigenvalues, and
\[
  \frac{2\nu_1\nu_2}{\nu_1+\nu_2}-\nu_2
  =\frac{\nu_2(\nu_1-\nu_2)}{\nu_1+\nu_2}\ \ge\ 0,
  \qquad
  2\nu_2-\frac{2\nu_1\nu_2}{\nu_1+\nu_2}
  =\frac{2\nu_2^{2}}{\nu_1+\nu_2}\ \ge\ 0,
\]
which are the two inequalities of \eqref{eq:harmonic}. Both use $\nu_2>0$, and
the hypothesis cannot be dropped: for
$\Ah_2=\bigl[\begin{smallmatrix}3&4\\4&1\end{smallmatrix}\bigr]$ one has
$\tr\Ah_2=4>0$ and $\det\Ah_2=-13$, so the middle expression of
\eqref{eq:harmonic} equals $-13/2$ while
$\nu_2=2-\sqrt{17}=-2.123\ldots$; the claimed upper bound fails by a wide
margin. That is why the hypothesis is named in the statement, and why
Table~\ref{tab:typing} types the two halves of this proposition differently.
\end{proof}

\begin{remark}[observation: the section is a closed loop]
\label{rem:selfsimilar}
Every statement of this section is an \emph{identity} --- with the single
exception of the Gershgorin bound of Lemma~\ref{lem:gershgorin}, which is an
inequality about $\nu_1$ and not about $\nu_2$, and of the second half of
Proposition~\ref{prop:collapse}, which carries a named hypothesis. Reading them
together:
\begin{itemize}
\item the constrained minimum at $K=2$ is $G_{11}(1-r_{12}^{2})$
  (Proposition~\ref{prop:K2}), so the value of the optimal trial vector is the
  scalar $1-r_{12}^{2}$ up to the closed factor $G_{11}$;
\item that scalar is the normalised determinant
  $\det\Ah_2/(\ah_{11}\ah_{22})$ (Lemma~\ref{lem:detread}), with no positivity
  used;
\item the entrywise perturbation threshold is the same scalar up to the factor
  $1+3r_{12}^{2}\in[1,4]$ (Proposition~\ref{prop:union});
\item the only member of the Rayleigh family $u(t)=(t,-1)$ that meets the
  threshold, $t^{\star}=\sqrt{\ah_{22}/\ah_{11}}$, reintroduces
  $\det\Ah_2$ exactly (Proposition~\ref{prop:collapse});
\item and the Wronskian and Lagrange identities
  (Lemmata~\ref{lem:wronskian}--\ref{lem:minors}) say that none of the
  smallness can be attributed to the two modes.
\end{itemize}
Consequently \emph{no stage of the reduction lowers the accuracy demanded of
$\det\Ah_2$}: the four readings differ by the explicit factors $G_{11}$,
$\ah_{11}\ah_{22}$, $1+3r_{12}^{2}$ and the closed prefactor of
\eqref{eq:collapse}, all of which are computable from the three entries and
none of which is small. We record this as an \emph{observation} about a family
of identities, not as a theorem: it is a reading, the statement ``every exact
reformulation costs the same precision'' quantifies over reformulations rather
than over windows, and how much precision is in fact demanded is a measured
number that belongs in Section~\ref{sec:open}.
\end{remark}

\section{The exact bilinear form and the rank classification}
\label{sec:rank}

This section writes $\Ah_2$ out in terms of the comb and identifies the exact
algebraic shape of the resulting object. Throughout the section the comb nodes
are indexed by $\ell$, so that $i,j$ stay free for matrix indices.

\begin{proposition}[exact split]
\label{prop:split}
Let $D$ be admissible, $D<\min_\ell u_\ell$, and for a node $u_\ell$ let
$m_\ell\in\N$ and $\theta_\ell\in[0,1)$ be defined by
$u_\ell/D=m_\ell+\theta_\ell$. Put $Y^{(m)}=\Pi^{\T}\Tm{M}(\delta_{m})\Pi$ with
$\delta_m$ the lag sequence supported at the single lag $m$, and $Y^{(m)}:=0$
for $m\ge M$. Then, with $B_{ij}=t_i^{\T}A_{\mathrm{arch}}t_j$ and
\begin{equation}
\label{eq:split}
  X^{(\ell)}_{ij}
  =\tfrac12\Bigl[(1-\theta_\ell)\,t_i^{\T}Y^{(m_\ell)}t_j
  +\theta_\ell\,t_i^{\T}Y^{(m_\ell+1)}t_j\Bigr],
  \qquad
  S=\sum_\ell\varrho_\ell\,X^{(\ell)}
  \qquad(i,j\in\{1,2\}),
\end{equation}
one has $\Ah_2=B-S$ exactly, and each of $S_{11},S_{22},S_{12}$ is a
\emph{linear} functional of the weight vector $(\varrho_\ell)_\ell$.
\end{proposition}

\begin{proof}
\emph{Linearity.} The three maps $c\mapsto\Tm{M}(c)$,
$X\mapsto\Pi^{\T}X\Pi$ and $Y\mapsto(t_i^{\T}Yt_j)_{i,j\le2}$ are linear, so
the split \eqref{eq:lags} of the lag sequence induces the corresponding split of
the $2\times2$ block. Writing $\delta_m$ for the lag sequence supported at the
single lag $m$ we have $c=c^{\mathrm{arch}}
-\sum_\ell\varrho_\ell\,\varphi^{(\ell)}$ with $c^{\mathrm{arch}}_m=a(mD)$ and
$\varphi^{(\ell)}_m=S_D(mD,u_\ell)$, and therefore $\Ah_2=B-S$ with
$S_{ij}=\sum_\ell\varrho_\ell\sum_{m=0}^{M-1}\varphi^{(\ell)}_m\,
t_i^{\T}Y^{(m)}t_j$.

\emph{The two-point read.} Fix $\ell$ and $m\ge0$. By definition
$S_D(mD,u_\ell)=\tfrac12\bigl[\Delta_D(mD-u_\ell)+\Delta_D(mD+u_\ell)\bigr]$,
and the reflected hat vanishes identically: $mD+u_\ell\ge u_\ell>D$ by
admissibility, while $\Delta_D$ is supported in $(-D,D)$. For the direct hat,
$\Delta_D(mD-u_\ell)=\max\{0,1-|m-m_\ell-\theta_\ell|\}$, which is nonzero only
when $|m-m_\ell-\theta_\ell|<1$, i.e.\ only for $m\in\{m_\ell,m_\ell+1\}$, where
it takes the values $1-\theta_\ell$ and $\theta_\ell$ --- including the case
$\theta_\ell=0$, where the second value is $0$ and only one lag is read.
Admissibility also gives $m_\ell\ge1$, so the lag $m=0$ is never touched by any
node. The factor $\tfrac12$ in \eqref{eq:split} is the symmetrisation of
$S_D$, and a lag index $\ge M$ falls outside the window and contributes nothing
because $Y^{(m)}=0$ there.

\emph{The lag-weight reading.} By Lemma~\ref{lem:gauge} applied to $v=t_i$, the
coefficient of $c_m$ in $t_i^{\T}At_i$ is the lag weight $w_m(t_i)$ of
\eqref{eq:weights}; polarising in $i$ and $j$,
\begin{equation}
\label{eq:Yread}
  t_i^{\T}Y^{(m)}t_j=w^{(ij)}_m,
  \qquad
  w^{(ij)}:=\tfrac12\bigl[w(t_i+t_j)-w(t_i)-w(t_j)\bigr],
\end{equation}
so each entry of $X^{(\ell)}$ is the two-point read of \emph{one closed weight
vector} at the node position $u_\ell$, and nothing else. This is the form used
in Section~\ref{sec:band}.

\emph{Linearity in the weights.} Each $S_{ij}=\sum_\ell\varrho_\ell
X^{(\ell)}_{ij}$ is by construction a linear functional of
$(\varrho_\ell)_\ell$, with coefficients $X^{(\ell)}_{ij}$ that depend on the
window, the resolution and the node positions but not on the weights.
\end{proof}

\begin{proposition}[polarisation of the determinant]
\label{prop:polarisation}
With $D(P,Q)=P_{11}Q_{22}+P_{22}Q_{11}-2P_{12}Q_{12}$,
\begin{equation}
\label{eq:polarisation}
  \det\Ah_2=\det B-D(B,S)+\det S,
\end{equation}
where the three terms are respectively archimedean--archimedean, linear in the
comb weights, and bilinear in them. Moreover
\begin{equation}
\label{eq:doublesum}
  \det S=\sum_{\ell,\ell'}\varrho_\ell\varrho_{\ell'}\,
    \mathcal K(\ell,\ell'),
  \qquad
  \mathcal K(\ell,\ell')=\tfrac12D\bigl(X^{(\ell)},X^{(\ell')}\bigr),
\end{equation}
with $X^{(\ell)}$ the $2\times2$ matrix of \eqref{eq:split}.
\end{proposition}

\begin{proof}
\emph{The polarisation.} Expand directly:
\[
  \det(B-S)=(B_{11}-S_{11})(B_{22}-S_{22})-(B_{12}-S_{12})^{2}
  =\det B-\bigl(B_{11}S_{22}+B_{22}S_{11}-2B_{12}S_{12}\bigr)+\det S,
\]
which is \eqref{eq:polarisation}; it is also the instance $P=B$, $Q=S$ of
Theorem~\ref{thm:rank3}\ref{R:pol} below, whose proof is independent of this
proposition. By Proposition~\ref{prop:split} the three terms are of degree
$0$, $1$ and $2$ in the weight vector respectively.

\emph{The double sum.} $D$ is symmetric and visibly linear in each argument,
and $D(S,S)=2S_{11}S_{22}-2S_{12}^{2}=2\det S$. Hence with
$S=\sum_\ell\varrho_\ell X^{(\ell)}$,
\[
  \det S=\tfrac12D(S,S)
  =\tfrac12\sum_{\ell,\ell'}\varrho_\ell\varrho_{\ell'}\,
   D\bigl(X^{(\ell)},X^{(\ell')}\bigr)
  =\sum_{\ell,\ell'}\varrho_\ell\varrho_{\ell'}\,\mathcal K(\ell,\ell'),
\]
which is \eqref{eq:doublesum} \cite{Lagrange1773,CauchyBinet}. The kernel is
symmetric, $\mathcal K(\ell,\ell')=\mathcal K(\ell',\ell)$, and its diagonal is
$\mathcal K(\ell,\ell)=\tfrac12D(X^{(\ell)},X^{(\ell)})=\det X^{(\ell)}$ ---
a fact used twice below, in Lemma~\ref{lem:wedge} and in
Measurement~\ref{meas:l1}.
\end{proof}

\begin{theorem}[rank three]
\label{thm:rank3}
Identify a symmetric $2\times2$ matrix $X$ with its coordinate vector
$y(X)=(X_{11},X_{22},X_{12})\in\R^{3}$ and let
\begin{equation}
\label{eq:J}
  J=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&-2\end{pmatrix}.
\end{equation}
Then:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.6em]
\item\label{R:pol} $\det X=\tfrac12\,y(X)^{\T}Jy(X)$ and
  $D(P,Q)=y(P)^{\T}Jy(Q)$; in particular $D$ is the polarisation of $\det$ and
  $\det(P-Q)=\det P-D(P,Q)+\det Q$.
\item\label{R:rank} $J$ has eigenvalues $\{1,-1,-2\}$: it has \textbf{rank
  three} and \textbf{signature $(1,2)$}.
\item\label{R:kernel} Let $P\in\R^{3\times n}$ have columns $y(X^{(\ell)})$,
  $\ell=1,\dots,n$. Then the kernel of \eqref{eq:doublesum} is
  $\mathcal K=\tfrac12P^{\T}JP$, so
  \begin{equation}
  \label{eq:rank3}
    \rank\mathcal K\;\le\;3
    \qquad\text{for every window, every }h\text{ and every truncation},
  \end{equation}
  and the nonzero spectrum of $\mathcal K$ equals the nonzero spectrum of the
  $3\times3$ matrix $\tfrac12JPP^{\T}$.
\item\label{R:inertia} Moreover $\mathcal K$ has at most \emph{one} positive
  and at most \emph{two} negative eigenvalues, for every $n$ and every $P$: the
  signature of $J$ is inherited, so the kernel of \eqref{eq:doublesum} is
  necessarily indefinite as soon as it has rank $2$ or more.
\item\label{R:three} Consequently
  $\det S=S_{11}S_{22}-S_{12}^{2}$: the comb--comb term of
  \eqref{eq:polarisation} is a quadratic polynomial in exactly \emph{three}
  linear functionals of the comb weights, and depends on the weight vector
  $(\varrho_\ell)$ through nothing else.
\end{enumerate}
\end{theorem}

\begin{proof}
\ref{R:pol} With $y=y(X)$,
$\tfrac12y^{\T}Jy=\tfrac12\bigl[2X_{11}X_{22}-2X_{12}^{2}\bigr]
=X_{11}X_{22}-X_{12}^{2}=\det X$. Reading off the associated bilinear form,
\[
  y(P)^{\T}Jy(Q)=P_{11}Q_{22}+P_{22}Q_{11}-2P_{12}Q_{12}=D(P,Q),
\]
so $D$ is the polarisation of $\det$, and since $y$ is linear,
\[
  \det(P-Q)=\tfrac12\bigl(y(P)-y(Q)\bigr)^{\T}J\bigl(y(P)-y(Q)\bigr)
  =\det P-y(P)^{\T}Jy(Q)+\det Q,
\]
which is the third relation.

\ref{R:rank} $J$ is block diagonal: the upper-left block
$\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr)$ has eigenvalues
$\pm1$ with eigenvectors $(1,\pm1,0)$, and the remaining diagonal entry is
$-2$ with eigenvector $(0,0,1)$. Hence $\operatorname{spec}J=\{1,-1,-2\}$, all
nonzero, so $\rank J=3$, and the inertia is $(n_+,n_-)=(1,2)$.

\ref{R:kernel} By \ref{R:pol},
$\mathcal K(\ell,\ell')=\tfrac12D(X^{(\ell)},X^{(\ell')})
=\tfrac12y(X^{(\ell)})^{\T}Jy(X^{(\ell')})$, which is the $(\ell,\ell')$ entry
of $\tfrac12P^{\T}JP$; and $\rank(P^{\T}JP)\le\rank J=3$ whatever $n$ and
whatever the columns of $P$ are. For the spectra, apply
$\operatorname{spec}(UV)\setminus\{0\}
=\operatorname{spec}(VU)\setminus\{0\}$ with $U=P^{\T}$ and
$V=\tfrac12JP$: then $UV=\mathcal K$ and $VU=\tfrac12JPP^{\T}$. The latter is a
product of the symmetric matrices $J$ and $PP^{\T}\PSD$, so its spectrum is
real, as it must be.

\ref{R:inertia} Suppose $\mathcal K$ had two positive eigenvalues. Then there
is a two-dimensional $Z\subseteq\R^{n}$ with $z^{\T}\mathcal Kz>0$ for every
$z\in Z\setminus\{0\}$. For such $z$ this forces $Pz\ne0$, so $P|_Z$ is
injective and $PZ$ is two-dimensional; and every $0\ne\eta\in PZ$ is $\eta=Pz$
for some $0\ne z\in Z$, so $\eta^{\T}J\eta=2z^{\T}\mathcal Kz>0$. Thus $J$
would be positive definite on a two-dimensional subspace, contradicting
$n_+(J)=1$ by Sylvester's law of inertia \cite{Sylvester1852} in the
Courant--Fischer form \cite{CourantFischer1920}. The same argument with three
and $n_-(J)=2$ gives the second half.

\ref{R:three} Apply \ref{R:pol} to $X=S$: $\det S=S_{11}S_{22}-S_{12}^{2}$,
and the three coordinates of $y(S)$ are the three linear functionals of
Proposition~\ref{prop:split}. Since $\det S$ is a function of $y(S)$ alone, it
depends on $(\varrho_\ell)$ through nothing else.
\end{proof}

Part~\ref{R:rank} will strike a reader as elementary --- the determinant of a
$2\times2$ matrix is visibly a quadratic form in its three entries --- and it
is. What the theorem is for is parts~\ref{R:inertia}--\ref{R:three} together
with the corollary below, which is where the elementary observation has teeth.
Before that, one reading of the kernel deserves to be isolated, because it is
the source of the mixed signs.

\begin{lemma}[the wedge form of the kernel, under exact rank one]
\label{lem:wedge}
Write $x\wedge x'=x_1x'_2-x_2x'_1$ for $x,x'\in\R^{2}$. If every read is of
rank one, $X^{(\ell)}=\epsilon_\ell\,x_\ell x_\ell^{\T}$ with
$\epsilon_\ell\in\{\pm1\}$ and $x_\ell\in\R^{2}$, then
\begin{equation}
\label{eq:wedge}
  \mathcal K(\ell,\ell')
  =\tfrac12\,\epsilon_\ell\epsilon_{\ell'}\,
  \bigl(x_\ell\wedge x_{\ell'}\bigr)^{2} .
\end{equation}
In particular the diagonal vanishes, $\mathcal K(\ell,\ell)=0$, and the sign
pattern of $\mathcal K$ is exactly that of $\epsilon_\ell\epsilon_{\ell'}$ ---
mixed as soon as the $\epsilon_\ell$ are not all equal. The hypothesis is
equivalent to $\det X^{(\ell)}=0$ for every $\ell$ and is \emph{not} an
identity: Measurement~\ref{meas:l1} reports how far from it the reads are.
\end{lemma}

\begin{proof}
For $P=\epsilon aa^{\T}$ and $Q=\epsilon'bb^{\T}$,
\[
  D(P,Q)=\epsilon\epsilon'\bigl[a_1^{2}b_2^{2}+a_2^{2}b_1^{2}
  -2a_1a_2b_1b_2\bigr]
  =\epsilon\epsilon'\bigl(a_1b_2-a_2b_1\bigr)^{2},
\]
and $\mathcal K=\tfrac12D$. The diagonal is
$\tfrac12(x_\ell\wedge x_\ell)^{2}=0$. That the hypothesis is equivalent to
$\det X^{(\ell)}=0$ is the classification of symmetric $2\times2$ matrices of
rank at most one, and $\mathcal K(\ell,\ell)=\det X^{(\ell)}$ by the last line
of the proof of Proposition~\ref{prop:polarisation}, so the diagonal vanishes
if and only if the hypothesis holds.
\end{proof}

\begin{definition}[norm-factorised majorants]
\label{def:majclass}
A bound on the comb--comb term is \emph{norm-factorised} if there are a norm
$\mathcal N$ on $\R^{n}$ and a nondecreasing
$\Phi:[0,\infty)\to[0,\infty)$ with
\begin{equation}
\label{eq:majclass}
  |\det S|\;\le\;\Phi\bigl(\mathcal N(\varrho)\bigr)
  \qquad\text{for every admissible weight vector }\varrho .
\end{equation}
The class of all such bounds, over all norms and all nondecreasing $\Phi$, is
written $\mathfrak M$.
\end{definition}

\begin{corollary}[what a majorant of the comb--comb term can see]
\label{cor:majorant}
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.6em]
\item\label{M:bound} If \eqref{eq:majclass} holds then necessarily
\begin{equation}
\label{eq:majorant}
  \Phi(t)\;\ge\;\sup\bigl\{|y_1y_2-y_3^{2}|:\ y\in\R^{3},\
  y=\bigl(S_{11},S_{22},S_{12}\bigr)\ \text{for some }\varrho\ \text{with }
  \mathcal N(\varrho)\le t\bigr\}.
\end{equation}
\item\label{M:blind} Consequently no member of $\mathfrak M$ distinguishes two
  weight vectors with the same image $y(S)=(S_{11},S_{22},S_{12})$, and every
  cancellation between the three terms of \eqref{eq:polarisation} is invisible
  to it: the bound sees a point of $\R^{3}$, the phenomenon is a relation
  between that point and $B$.
\item\label{M:decomp} If a route proceeds through a declared linear
  decomposition $\varrho=\sum_{q\le p}\varrho^{(q)}$ and bounds
  $|\det S|\le\sum_{q,q'}\Phi_{qq'}\bigl(\mathcal N(\varrho^{(q)}),
  \mathcal N(\varrho^{(q')})\bigr)$ with each $\Phi_{qq'}$ nondecreasing in
  both arguments, then each summand obeys the analogue of \ref{M:bound} for the
  corresponding block of the bilinear form, and the total pays in addition the
  triangle inequality across the $p^{2}$ pairs --- whose cost is measured in
  Measurement~\ref{meas:l1}.
\end{enumerate}
\end{corollary}

\begin{proof}
\ref{M:bound} By Theorem~\ref{thm:rank3}\ref{R:three}, $\det S=y_1y_2-y_3^{2}$
at $y=y(S)$. Fix $t>0$ and let $\varrho$ satisfy $\mathcal N(\varrho)\le t$.
Then \eqref{eq:majclass} and monotonicity of $\Phi$ give
$|y_1y_2-y_3^{2}|\le\Phi(\mathcal N(\varrho))\le\Phi(t)$; taking the supremum
over all such $\varrho$ gives \eqref{eq:majorant}.

\ref{M:blind} $\varrho\mapsto y(S)$ is linear and, for $n\ge4$, has a kernel of
dimension at least $n-3$; two weight vectors differing by an element of that
kernel have the same $\det S$ and the same $\det\Ah_2$, while
\eqref{eq:majclass} sees them only through $\mathcal N$. That the three terms
of \eqref{eq:polarisation} may cancel to any order without $\Phi$ noticing is
then immediate, since $\Phi$ does not involve $B$ at all.

\ref{M:decomp} Apply \ref{M:bound} to each pair $(q,q')$ with $S$ replaced by
the pair of blocks, using bilinearity of $D$ in
\eqref{eq:doublesum}; the last clause is the statement that
$|\sum_{q,q'}\cdot|\le\sum_{q,q'}|\cdot|$ has been used, whose slack is the
$\ell^{1}$-versus-signed ratio measured below.
\end{proof}

\begin{remark}[the five declared routes, classified]
\label{rem:routeclass}
Of the five preregistered routes of Measurement~\ref{meas:routes}, three are
members of $\mathfrak M$ as they are stated: the large-sieve/duality route, with
$\mathcal N=\lVert\cdot\rVert_2$ and
$\Phi(t)=\lVert\mathcal K\rVert_{\mathrm{op}}t^{2}$; the
Montgomery--Vaughan route, which bounds each of the three linear functionals
through $\lVert\varrho\rVert_2$ --- its interpolation residual through
$\lVert\varrho\rVert_1\le\sqrt n\,\lVert\varrho\rVert_2$ on a window with $n$
nodes, so that one norm suffices --- and then combines them by
$|\det S|\le|S_{11}||S_{22}|+S_{12}^{2}$; and the comb--comb term of the exact
split $|\det B|+|D(B,S)|+|\det S|$. Vaughan's Type~I $+$ Type~II route is of
the shape of Corollary~\ref{cor:majorant}\ref{M:decomp} with $p=4$. The
remaining route, Cauchy--Schwarz on $\Ah_2$, is not a majorant of $\det S$ at
all: it asserts $\ah_{12}^{2}\le\ah_{11}\ah_{22}$, i.e.\ $\det\Ah_2\ge0$, and
is carried in the table as the trivial baseline. This classification is a
\emph{reading of the five routes as declared}, not a theorem about every
conceivable implementation of them; what is a theorem is
Corollary~\ref{cor:majorant} itself.
\end{remark}

\begin{measurement}[mixed signs, and what the triangle inequality costs]
\label{meas:l1}
Two numbers price the two hypotheses just used. \emph{The rank-one defect}: the
median of $|\det X^{(\ell)}|/\lVert X^{(\ell)}\rVert^{2}$ over the nodes is
$1.3\cdot10^{-4}$ on the reference window of
Measurement~\ref{meas:refwindow}, and on Surface~I it lies between
$4.6\cdot10^{-3}$ and $3.1\cdot10^{-2}$, below the declared bar
$5\cdot10^{-2}$ on every window. So the wedge reading of
Lemma~\ref{lem:wedge} is accurate but not exact --- the diagonal of
$\mathcal K$ is small, not zero, and it carries about $6\%$ of $\det S$ on the
reference window. \emph{The $\ell^{1}$ mass}: on a declared node
subsample of the reference window,
\begin{equation}
\label{eq:l1mass}
  \sum_{\ell,\ell'}\varrho_\ell\varrho_{\ell'}
  \bigl|\mathcal K(\ell,\ell')\bigr|
  \;=\;40.6\times
  \Bigl|\sum_{\ell,\ell'}\varrho_\ell\varrho_{\ell'}
  \mathcal K(\ell,\ell')\Bigr| ,
\end{equation}
the subsample being the $900$ nodes of the declared horizon and the ratio being
reported on the window it was computed on and on no other. A
route that replaces the signed double sum by its absolute value therefore loses
that factor before it begins, which is the measured content of
Corollary~\ref{cor:majorant}\ref{M:decomp}; the mixed signs it loses are the
ones Lemma~\ref{lem:wedge} and Theorem~\ref{thm:rank3}\ref{R:inertia} predict.
\end{measurement}

\subsection{The bandwidth of the weight kernel}
\label{sec:band}

By Proposition~\ref{prop:split} an entry of $X^{(\ell)}$ is the two-point read
of one closed weight vector at the single real argument $\tau_\ell=u_\ell/D$.
This subsection identifies that weight vector in closed form, extracts the
consequence for a factorisation-indexed block, and says exactly where the
argument stops. Throughout, $\tau$ denotes a real lag argument, $N=2h+1$ and
$\omega_k=2\pi k/N$ as in \eqref{eq:kms}.

\begin{lemma}[the closed lag weights of a parity mode]
\label{lem:closedweight}
For $1\le k\le h$ the lag weights \eqref{eq:weights} of the parity mode $t_k$
are $w_0(t_k)=1$ and
\begin{equation}
\label{eq:closedweight}
  w_d(t_k)
  =\frac2N\Bigl[(N-d)\cos(\omega_kd)+\cot(\omega_k)\,\sin(\omega_kd)\Bigr],
  \qquad 1\le d\le M-1 ,
\end{equation}
one formula for the whole range, with no change of branch at $d=h$.
\end{lemma}

\begin{proof}
Apply Lemma~\ref{lem:gauge} with $c=\delta_d$ the lag sequence supported at the
single lag $d$, and let $A_{(d)}$ be the odd window form \eqref{eq:Adef} of
$\delta_d$: since $\sum_ec_ew_e=w_d$ for this $c$, we get
$w_d(v)=v^{\T}A_{(d)}v$, i.e.\ by \eqref{eq:Adef}
\begin{equation}
\label{eq:wquad}
  w_d(v)=\sum_{r,r'=0}^{h-1}v_rv_{r'}
  \Bigl(\mathbf 1[\,|r-r'|=d\,]-\mathbf 1[\,r+r'=M-1-d\,]\Bigr).
\end{equation}
Put $T_x=\sin(\omega_kx)$ for $x\in\Z$. Then $T$ is $N$-periodic and odd, so
$T_{N-m}=-T_m$, and $(t_k)_r=(2/\sqrt N)T_{r+1}$. Substituting $x=r+1$,
$y=r'+1\in[1,h]$ in \eqref{eq:wquad}, the Hankel condition $r+r'=M-1-d$ becomes
$x+y=N-d$, where $T_y=T_{N-d-x}=-T_{x+d}$, so the Hankel term enters with a
$+$ sign; the constraint $y\in[1,h]$ becomes $x\in[h+1-d,2h-d]$. The Toeplitz
term contributes $\sum_{|x-y|=d}T_xT_y=2\sum_{x=1}^{h-d}T_xT_{x+d}$, each pair
being counted twice for $d\ge1$. Hence, for $d\ge1$,
\begin{equation}
\label{eq:twopartial}
  w_d(t_k)=\frac4N\Bigl[\,2\!\!\sum_{x=1}^{h-d}\!T_xT_{x+d}
  +\sum_{x\in I_d}T_xT_{x+d}\Bigr],
  \qquad
  I_d=[h+1-d,\,2h-d]\cap[1,h] .
\end{equation}
Write $\Phi(m)=\sum_{x=1}^{m}T_xT_{x+d}$. For $1\le d\le h$ one has
$I_d=[h+1-d,h]$, so the bracket is
$2\Phi(h-d)+[\Phi(h)-\Phi(h-d)]=\Phi(h-d)+\Phi(h)$; for $h<d\le M-1$ the first
sum is empty and $I_d=[1,2h-d]$, so the bracket is $\Phi(2h-d)$.

For $\Phi$, use $2\sin a\sin b=\cos(a-b)-\cos(a+b)$ and the Dirichlet sum
$\sum_{x=1}^{m}\cos(a+2\omega x)
=\frac{\sin(m\omega)}{\sin\omega}\cos\bigl(a+(m+1)\omega\bigr)$, legitimate
because $N$ is odd and $1\le k\le h$, so $\sin\omega_k\ne0$ and
$2\omega_k\notin2\pi\Z$:
\begin{equation}
\label{eq:Phi}
  \Phi(m)=\frac m2\cos(\omega d)
  -\frac12\,\frac{\sin(m\omega)}{\sin\omega}\,
  \cos\bigl(\omega(m+d+1)\bigr),
  \qquad\omega=\omega_k .
\end{equation}
Now evaluate, using $\omega N=2\pi k$ and $h=(N-1)/2$, so that
$\omega h=\pi k-\omega/2$. At $m=h$: $\sin(\omega h)=-(-1)^k\sin(\omega/2)$ and
$\cos(\omega(h+d+1))=(-1)^k\cos(\omega(d+\tfrac12))$, whence
$\Phi(h)=\tfrac N4\cos(\omega d)-\tfrac14\tan(\tfrac\omega2)\sin(\omega d)$
after expanding $\cos(\omega(d+\tfrac12))$. At $m=h-d$:
$\sin(\omega(h-d))=-(-1)^k\sin(\omega(d+\tfrac12))$ and
$\cos(\omega(h+1))=(-1)^k\cos(\omega/2)$, whence
$\Phi(h-d)=\tfrac{N-2d}4\cos(\omega d)
+\tfrac14\cot(\tfrac\omega2)\sin(\omega d)$. Adding, and using
$\cot(\tfrac\omega2)-\tan(\tfrac\omega2)=2\cot\omega$,
\[
  \Phi(h-d)+\Phi(h)=\frac{N-d}2\cos(\omega d)
  +\frac12\cot(\omega)\sin(\omega d),
\]
which with the factor $4/N$ is \eqref{eq:closedweight}. At $m=2h-d=N-1-d$ one
has $\sin(\omega m)=-\sin(\omega(d+1))$ and $\cos(\omega(m+d+1))=1$, so
$\Phi(2h-d)=\tfrac{N-d}2\cos(\omega d)+\tfrac12\cot(\omega)\sin(\omega d)$ as
well --- the same expression, which is why \eqref{eq:closedweight} has no branch
at $d=h$. Finally $w_0=\lVert t_k\rVert^{2}=1$ by \eqref{eq:represent}; the
right-hand side of \eqref{eq:closedweight} equals $2$ at $d=0$, the discrepancy
being the doubling of the lag-$d$ pairs, which does not occur at $d=0$.
\end{proof}

\begin{lemma}[the weight sequence has bandwidth $K$ and degree one]
\label{lem:band}
Let $\mathcal V_K\subset\R^{M-1}$ be the span of the $4K$ sequences
\begin{equation}
\label{eq:bandspace}
  d\mapsto\cos(\omega_kd),\quad \sin(\omega_kd),\quad
  d\cos(\omega_kd),\quad d\sin(\omega_kd),
  \qquad k=1,\dots,K,\ 1\le d\le M-1 .
\end{equation}
Then $\bigl(w_d(v)\bigr)_{d=1}^{M-1}\in\mathcal V_K$ for every
$v\in\operatorname{span}\{t_1,\dots,t_K\}$, and consequently also for every
polarised weight vector $w^{(ij)}$ of \eqref{eq:Yread} with $i,j\le K$. The
coefficients of $d\cos(\omega_kd)$ and $d\sin(\omega_kd)$ are proportional to
$a_k^{2}$, where $v=\sum_ka_kt_k$.
\end{lemma}

\begin{proof}
Repeat the proof of Lemma~\ref{lem:closedweight} with $T_x$ replaced by
$V_x=\sum_{k\le K}a_k\sin(\omega_kx)$: \eqref{eq:twopartial} holds verbatim with
$\Phi_v(m)=\sum_{x=1}^{m}V_xV_{x+d}
=\sum_{k,l}a_ka_l\sum_{x=1}^{m}\sin(\omega_kx)\sin(\omega_l(x+d))$. For $k=l$
the inner sum is \eqref{eq:Phi}, contributing the four sequences of
\eqref{eq:bandspace} at frequency $\omega_k$, the linear-in-$d$ ones with
coefficient proportional to $a_k^{2}$ through the term $\tfrac m2\cos(\omega d)$
at $m=h-d$ resp.\ $m=2h-d$. For $k\ne l$ write
$2\sin(\omega_kx)\sin(\omega_lx+\omega_ld)
=\cos(\lambda_-x-\omega_ld)-\cos(\lambda_+x+\omega_ld)$ with
$\lambda_\pm=\omega_k\pm\omega_l$, and sum by
$\sum_{x=1}^{m}\cos(a+\lambda x)
=\frac{\sin(\lambda m/2)}{\sin(\lambda/2)}\cos\bigl(a+\lambda(m+1)/2\bigr)$,
legitimate since $0<|k\pm l|<N$ gives $\sin(\lambda_\pm/2)\ne0$. Each such term
is a product $\sin P\cos Q=\tfrac12[\sin(P+Q)+\sin(P-Q)]$ in which $P$ and $Q$
are affine in $d$ --- the evaluation points $m\in\{h,h-d,2h-d\}$ are affine in
$d$ with slope $0$ or $-1$ --- and a direct computation of the slopes, using
$\omega_kN,\omega_lN\in2\pi\Z$, gives $\mathrm{slope}(P+Q)=-\omega_k$ and
$\mathrm{slope}(P-Q)=+\omega_l$ in both cases $\lambda_\pm$ and at all three
evaluation points. So the off-diagonal terms contribute only the frequencies
$\omega_k,\omega_l$ with constant coefficients, and no linear-in-$d$ factor
appears because $\sin(\lambda m/2)/\sin(\lambda/2)$ is bounded, not linear. The
statement for $w^{(ij)}$ follows because $\mathcal V_K$ is a linear space and
$w^{(ij)}=\tfrac12[w(t_i+t_j)-w(t_i)-w(t_j)]$ with all three arguments in
$\operatorname{span}\{t_1,\dots,t_K\}$.
\end{proof}

The next lemma is the general mechanism, stated once and independently of the
weights. It is the statement asked for in the source material: bandwidth in an
additive variable implies rank at a declared truncation error.

\begin{lemma}[bandwidth $\Rightarrow$ $\eta$-rank]
\label{lem:etarank}
Let $x_1,\dots,x_R$ and $y_1,\dots,y_C$ be real and let $I\subseteq\R$ contain
every $x_b+y_e$. Let $\Omega\ge0$ be the number of frequencies
$\nu_1,\dots,\nu_\Omega>0$, let $q\ge0$ be a degree bound, and let
\begin{equation}
\label{eq:bandexp}
  W_\Omega(\tau)=p_0(\tau)+\sum_{j=1}^{\Omega}
  \bigl[p_j(\tau)\cos(\nu_j\tau)+\tilde p_j(\tau)\sin(\nu_j\tau)\bigr],
  \qquad \deg p_j,\deg\tilde p_j\le q .
\end{equation}
If $W:I\to\R$ satisfies $\sup_{\tau\in I}|W(\tau)-W_\Omega(\tau)|\le\eta$, then
the matrix $F_{be}=W(x_b+y_e)$ obeys
\begin{equation}
\label{eq:etarank}
  \sigma_{r+1}(F)\;\le\;\eta\sqrt{RC},
  \qquad r=(2\Omega+1)(q+1);
\end{equation}
if the expansion \eqref{eq:bandexp} has no polynomial term $p_0$, then
$r=2\Omega(q+1)$ suffices. Row and column rescalings
$F\mapsto\diag(\alpha)F\diag(\beta)$ leave $r$ unchanged and replace the
right-hand side by $\eta\lVert\alpha\rVert_2\lVert\beta\rVert_2$.
\end{lemma}

\begin{proof}
Let $F_\Omega$ be the matrix built from $W_\Omega$. Fix $j$ and $a\le q$. Since
$(x+y)^{a}=\sum_{p\le a}\binom ap x^{p}y^{a-p}$ and
$e^{i\nu(x+y)}=e^{i\nu x}e^{i\nu y}$, the function
$\tau^{a}e^{i\nu_j\tau}$ of $\tau=x+y$ is a finite sum of products
$[x^{p}e^{i\nu_jx}]\cdot[y^{a-p}e^{i\nu_jy}]$. Taking real and
imaginary parts, the columns of the matrices
$\bigl[\tau^{a}\cos(\nu_j\tau)\bigr]_{be}$ and
$\bigl[\tau^{a}\sin(\nu_j\tau)\bigr]_{be}$ lie in the span of the $2(q+1)$
vectors $\bigl(x_b^{p}\cos(\nu_jx_b)\bigr)_b$,
$\bigl(x_b^{p}\sin(\nu_jx_b)\bigr)_b$, $p\le q$. Likewise the columns of
$[p_0(x_b+y_e)]_{be}$ lie in the span of the $q+1$ vectors $(x_b^{p})_b$.
Summing over $j$, $\rank F_\Omega\le2\Omega(q+1)+(q+1)=(2\Omega+1)(q+1)$, and
the $p_0$-free case drops the last summand. Write $E=F-F_\Omega$; then
$\lVert E\rVert_2\le\lVert E\rVert_F\le\eta\sqrt{RC}$, and Weyl's perturbation
inequality for singular values \cite{Weyl1912} gives
$\sigma_{r+1}(F)\le\sigma_{r+1}(F_\Omega)+\lVert E\rVert_2=\lVert E\rVert_2$
because $\sigma_{r+1}(F_\Omega)=0$. For the rescaling, ranks are unchanged by
invertible diagonal factors, and
$|\alpha_bE_{be}\beta_e|\le\eta|\alpha_b||\beta_e|$ gives
$\lVert\diag(\alpha)E\diag(\beta)\rVert_F
\le\eta\lVert\alpha\rVert_2\lVert\beta\rVert_2$.
\end{proof}

What Lemma~\ref{lem:etarank} needs is a band expansion of the \emph{read}, and
the read of Proposition~\ref{prop:split} is not the sequence $(w_d)$ but its
piecewise-linear interpolant. That interpolation is the only source of $\eta$.

\begin{lemma}[the two-point read against the analytic read]
\label{lem:spline}
Let $v\in\operatorname{span}\{t_1,\dots,t_K\}$ and let $W_v$ be the function
obtained by evaluating the representation of Lemma~\ref{lem:band} at real
argument, so that $W_v(d)=w_d(v)$ for integers $1\le d\le M-1$. For
$1\le\tau\le M-1$ the two-point read
$\widehat W_v(\tau)=(1-\theta)w_{m}(v)+\theta w_{m+1}(v)$,
$m=\lfloor\tau\rfloor$, $\theta=\tau-m$, of \eqref{eq:split} satisfies
\begin{equation}
\label{eq:splinebound}
  \sup_{1\le\tau\le M-1}\bigl|\widehat W_v(\tau)-W_v(\tau)\bigr|
  \;\le\;\tfrac18\sup_{1\le\tau\le M-1}|W_v''(\tau)|
  \;\le\;\tfrac18\bigl[\omega_K^{2}\Lambda_0+2\omega_K\Lambda_1\bigr],
\end{equation}
where $\Lambda_0=\sum_k(|\alpha_k|+|\gamma_k|+M(|\beta_k|+|\delta_k|))$ and
$\Lambda_1=\sum_k(|\beta_k|+|\delta_k|)$ in the notation
$W_v(\tau)=\sum_k[(\alpha_k+\beta_k\tau)\cos(\omega_k\tau)
+(\gamma_k+\delta_k\tau)\sin(\omega_k\tau)]$. For the single mode $t_k$,
\begin{equation}
\label{eq:splinemode}
  \sup_{1\le\tau\le M-1}\bigl|\widehat W_{t_k}(\tau)-W_{t_k}(\tau)\bigr|
  \;\le\;\frac{\pi^{2}k^{2}}{N^{2}}\Bigl(1+\frac3{2\pi k}\Bigr)
  \;=\;O\!\left(\frac{k^{2}}{N^{2}}\right).
\end{equation}
\end{lemma}

\begin{proof}
The first inequality is the standard error of linear interpolation on a unit
interval: for $f\in C^{2}[m,m+1]$ the error at $\tau=m+\theta$ is
$-\tfrac12\theta(1-\theta)f''(\xi)$ for some $\xi$, and
$\max_\theta\theta(1-\theta)=\tfrac14$. For the second, differentiate the
representation twice: each term contributes
$-\omega_k^{2}(\alpha_k+\beta_k\tau)\cos(\omega_k\tau)
+2\omega_k\beta_k\bigl(-\sin(\omega_k\tau)\bigr)$ and its sine analogue, so
$|W_v''|\le\omega_K^{2}\Lambda_0+2\omega_K\Lambda_1$ on $\tau\le M$. For a
single mode, \eqref{eq:closedweight} gives
$\alpha_k=2$, $\beta_k=-2/N$, $\gamma_k=(2/N)\cot\omega_k$, $\delta_k=0$, hence
$|W_{t_k}''|\le\tfrac2N[\omega_k^{2}N+2\omega_k+\omega_k^{2}\cot\omega_k]
\le2\omega_k^{2}+6\omega_k/N$ using $\omega\cot\omega\le1$; dividing by $8$ and
inserting $\omega_k=2\pi k/N$ gives \eqref{eq:splinemode}.
\end{proof}

\begin{proposition}[$\eta$-rank of a factorisation-indexed block]
\label{prop:typeII}
Fix $K$, let the weight vector be either $w(v)$ for some
$v\in\operatorname{span}\{t_1,\dots,t_K\}$ or a polarised $w^{(ij)}$ with
$i,j\le K$, and let $\widehat W$ be its two-point read as in
Lemma~\ref{lem:spline}. Let $\mathcal B_1,\mathcal B_2$ be two finite sets of
reals $>1$, put $x_{b_1}=\log b_1/D$, $y_{b_2}=\log b_2/D$, and assume the block
is \emph{interior}:
\begin{equation}
\label{eq:interior}
  1\le x_{b_1}+y_{b_2}\le M-1
  \qquad\text{for all } b_1\in\mathcal B_1,\ b_2\in\mathcal B_2 .
\end{equation}
Then the matrix of reads
$F_{b_1b_2}=\widehat W\bigl(x_{b_1}+y_{b_2}\bigr)$, $R=|\mathcal B_1|$,
$C=|\mathcal B_2|$, satisfies
\begin{equation}
\label{eq:typeIIrank}
  \sigma_{4K+1}(F)\;\le\;\eta\sqrt{RC},
  \qquad
  \eta=\tfrac18\sup_{1\le\tau\le M-1}|W''| ,
\end{equation}
with $\eta$ bounded by \eqref{eq:splinebound}; for a single mode
$\eta\le\pi^{2}k^{2}N^{-2}(1+3/(2\pi k))$. The same bound holds for
$\diag(\alpha)F\diag(\beta)$ with $\eta$ replaced by
$\eta\lVert\alpha\rVert_2\lVert\beta\rVert_2$, so arithmetic coefficients
attached to the two index sets do not change the rank count.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:band} the analytic read $W$ lies in the span of the $4K$
functions \eqref{eq:bandspace}, which is an expansion \eqref{eq:bandexp} with
$\Omega=K$ frequencies, degree bound $q=1$ and no polynomial term $p_0$; by
Lemma~\ref{lem:spline} $\sup|\widehat W-W|\le\eta$ on $[1,M-1]$, which by
\eqref{eq:interior} contains every $x_{b_1}+y_{b_2}$. Lemma~\ref{lem:etarank}
with $r=2\Omega(q+1)=4K$ gives \eqref{eq:typeIIrank}, and its last clause gives
the rescaled version.
\end{proof}

\begin{remark}[what \eqref{eq:interior} costs, and what the proposition does not
say]
\label{rem:interior}
Hypothesis \eqref{eq:interior} is not cosmetic. Outside the window the weights
are set to zero ($w_m:=0$ for $m\ge M$ in Proposition~\ref{prop:split}), and the
truncation $\mathbf 1[x+y\le M-1]$ is a triangular cut-off, which is \emph{not}
of low rank: without \eqref{eq:interior} the conclusion is false as stated, and
what survives is the same bound for the interior sub-block plus the cut-off
part. On the declared surfaces the node cap makes the block interior by
construction, and that is how the hypothesis is met. Two further limits are
worth naming. The bound \eqref{eq:typeIIrank} is a statement about $4K$ singular
values with a declared $\eta$; it is not a statement that the block ``has rank
$O(1)$'' in any asymptotic sense, and the $\sqrt{RC}$ factor means that the
useful regime is $\eta\sqrt{RC}\ll\sigma_1(F)$, which is a condition on the
declared surface and is checked there, not here. And the whole mechanism is
about \emph{additivity} of the lag argument under factorisation: it uses that
the read happens at $\tau=u/D$ with $u=\log n$, so that $n=b_1b_2$ gives
$\tau=x_{b_1}+y_{b_2}$. Nothing arithmetic beyond that additivity is used, and
nothing is claimed about any Type~II sum beyond the singular values of its
weight matrix.
\end{remark}

\begin{measurement}[Type II blocks: the measured collapse inside the proved
bound]
\label{meas:typeII}
Proposition~\ref{prop:typeII} bounds $\sigma_{4K+1}$; the surfaces report a
much sharper collapse, and the two are typed separately. On the declared surface
the Type~II blocks of Vaughan's identity \cite{Vaughan1977} built from the
closed weights, with $K=2$ so that the proved bound is $\sigma_9\le\eta\sqrt{RC}$
with $\eta\le2.6\cdot10^{-4}$ at $h=301$ for each of the three polarised reads by
\eqref{eq:splinebound}, are measured
to have $2$ singular values carrying $99.9\%$ of the mass, against $153$ for a
generic kernel of the same size, over the scanned range of the decomposition
parameter; the four-term identity for $\Lambda$ is verified coefficientwise
first. The number $2$ is measured and stays measured: the theorem gives $8$ at
$K=2$, the observed sufficiency of $2$ is a property of the declared surface and
is not proved here.
\end{measurement}

\begin{measurement}[the reference window: a three-term cancellation]
\label{meas:refwindow}
On the reference window $h=540$, with the comb truncated at the node cap
$2.465\cdot10^{4}$, the three terms of \eqref{eq:polarisation} are
\begin{equation}
\label{eq:refwindow}
  \det B=6.18,
  \qquad
  -D(B,S)=-198.8,
  \qquad
  \det S=192.6,
  \qquad
  \text{sum}=4.22\cdot10^{-3},
\end{equation}
i.e.\ five orders of magnitude of cancellation between three pieces of size
$\approx200$; the absolute value demanded of $\det\Ah_2$ at that window --- the
target against which the cancellation has to be resolved --- is
$1.10\cdot10^{-5}$, a further factor $384$ below the residual sum. The four
numbers are read under the per-window near-null convention \ref{F:null}, and the
caps are those of Surface~II
(Section~\ref{sec:surfaces}): the window index inside $[142,1445]$, the
conditioning of the raw Gram block inside $\cond(G)\le10^{12}$, and the comb
truncated at the node cap quoted above. Three readings of the same window follow
from the results of this section and are recorded as part of the measurement:
the cancellation is between the three terms of the exact polarisation
\eqref{eq:polarisation}, so no route of Definition~\ref{def:majclass} sees it
(Corollary~\ref{cor:majorant}\ref{M:blind}); the signed kernel sum behind
$\det S$ is $40.6$ times smaller than its $\ell^{1}$ mass on the declared
subsample
(Measurement~\ref{meas:l1}); and the sum in \eqref{eq:refwindow} is
$\det\Ah_2$, whose normalisation by $\ah_{11}\ah_{22}$ is the quantity of
Problem~\ref{prob:R1} --- so five orders of magnitude here is a statement about
one window and about nothing else.
\end{measurement}

\section{A measured obstruction, and an open problem}
\label{sec:open}

Sections~\ref{sec:tvfloor}--\ref{sec:rank} reduce the question of
Section~\ref{sec:problem} to a single scalar and classify what the standard
unconditional toolbox does to it. This section states the scalar question that
remains, reports what is measured about it on declared surfaces, prices the
routes, and locates --- by controlled experiment --- where the phenomenon
actually lives. Everything here is either a proved statement about a route or a
number on a named surface; the two are never mixed, and
Table~\ref{tab:typing} is the binding record.

\subsection{The open problem}
\label{sec:problem71}

\begin{problem}[three equivalent forms]
\label{prob:R1}
Is there, for each $\varepsilon>0$, a constant $C_\varepsilon$ such that
\emph{uniformly in the window index},
\begin{equation}
\label{eq:R1}
  1-r_{12}^{2}
  =\frac{\det\Ah_2}{\ah_{11}\ah_{22}}
  \;\le\;C_\varepsilon\,h^{-3+\varepsilon}\,?
\end{equation}
Equivalently, by Lemmata~\ref{lem:detread}, \ref{lem:minors} and
\ref{lem:angle}:
\begin{enumerate}[label=\textup{(\alph*)},leftmargin=2.6em]
\item\label{R1:rows} \emph{Row collinearity.} Do the two rows of $\Ah_2$ ---
  two explicit finite sums against the closed weights of
  Proposition~\ref{prop:split} --- become collinear at rate $h^{-3+\varepsilon}$,
  the sine of the angle between them being
  $|\det\Ah_2|/(\lVert\text{row}_1\rVert\lVert\text{row}_2\rVert)$?
\item\label{R1:det} \emph{Determinant form.} Does the normalised determinant
  $\det\Ah_2/(\ah_{11}\ah_{22})$ satisfy \eqref{eq:R1}?
\item\label{R1:triple} \emph{Joint near-degeneracy.} Does the triple
  $(S_{11},S_{22},S_{12})$ of Theorem~\ref{thm:rank3}\ref{R:three} approach the
  archimedean block $B$ jointly, at relative precision sharpening like
  $h^{-3}$, in the sense that the three terms of \eqref{eq:polarisation} cancel
  to that order?
\end{enumerate}
The problem is open. It is a question about a finite bilinear form --- for each
window a polynomial identity in three explicit finite sums --- and the only
thing missing is a bound uniform in $h$. It is not formulated as, and must not
be read as, an approach to, a weakening of, or a consequence of any conjecture;
no comparison of strengths against any conjecture is made anywhere in this
paper, and no such comparison is needed to state the problem or to read the
measurements below.
\end{problem}

The three forms are equivalent because the quantities in \ref{R1:rows} and
\ref{R1:det} differ by a factor that is bounded above and below in closed form.
We record this once, because it is also what makes the two fitted exponents of
Measurement~\ref{meas:rate} non-independent (Remark~\ref{rem:twoexponents}).

\begin{lemma}[the angle reading against the determinant reading]
\label{lem:angle}
Assume $\ah_{11},\ah_{22}>0$ and $|\ah_{12}|\le\sqrt{\ah_{11}\ah_{22}}$, and
write $\rho_1=\lVert(\ah_{11},\ah_{12})\rVert$,
$\rho_2=\lVert(\ah_{12},\ah_{22})\rVert$ for the two row norms of $\Ah_2$ and
$\sin\angle=|\det\Ah_2|/(\rho_1\rho_2)$ for the sine of the angle between the
rows. Then
\begin{equation}
\label{eq:angleid}
  \frac{\sin\angle}{\,|1-r_{12}^{2}|\,}
  =\frac{\ah_{11}\ah_{22}}{\rho_1\rho_2}
  =\frac{\sqrt{\ah_{11}\ah_{22}}}
        {\sqrt{(\ah_{11}+r_{12}^{2}\ah_{22})(\ah_{22}+r_{12}^{2}\ah_{11})}},
\end{equation}
and this factor obeys
\begin{equation}
\label{eq:anglebound}
  \frac12\cdot\frac{2\sqrt{\ah_{11}\ah_{22}}}{\ah_{11}+\ah_{22}}
  \;\le\;\frac{\sin\angle}{\,|1-r_{12}^{2}|\,}\;\le\;1 ,
\end{equation}
so the two readings differ by a factor confined to $(0,1]$ whose two-sided
bound depends on the window only through the diagonal ratio
$\ah_{22}/\ah_{11}$.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:detread}, $|1-r_{12}^{2}|=|\det\Ah_2|/(\ah_{11}\ah_{22})$, and
$\sin\angle=|\det\Ah_2|/(\rho_1\rho_2)$ by definition; dividing gives the first
equality of \eqref{eq:angleid}. For the second, substitute
$\ah_{12}^{2}=r_{12}^{2}\ah_{11}\ah_{22}$:
\[
  \rho_1^{2}=\ah_{11}^{2}+\ah_{12}^{2}
  =\ah_{11}\bigl(\ah_{11}+r_{12}^{2}\ah_{22}\bigr),
  \qquad
  \rho_2^{2}=\ah_{12}^{2}+\ah_{22}^{2}
  =\ah_{22}\bigl(\ah_{22}+r_{12}^{2}\ah_{11}\bigr),
\]
and take the product. For \eqref{eq:anglebound}: the upper bound is
$\rho_1\rho_2\ge\ah_{11}\ah_{22}$, which follows from
$\rho_1\ge\ah_{11}$ and $\rho_2\ge\ah_{22}$; the lower bound uses
$r_{12}^{2}\le1$, whence $\ah_{11}+r_{12}^{2}\ah_{22}\le\ah_{11}+\ah_{22}$ and
likewise for the second factor, so
$\rho_1\rho_2\le\sqrt{\ah_{11}\ah_{22}}\,(\ah_{11}+\ah_{22})$. The final form is
$\tfrac12$ times the ratio of the geometric to the arithmetic mean of
$\ah_{11}$ and $\ah_{22}$, i.e.\ $\sqrt t/(1+t)$ with $t=\ah_{22}/\ah_{11}$.
\end{proof}

\subsection{What is measured}
\label{sec:measured}

\begin{measurement}[measured rates]
\label{meas:rate}
All numbers here are least-squares fits of $\log$ of the quantity against
$\log h$ on the stated finite surface, reported for orientation and used as
bounds nowhere. On Surface~II of Section~\ref{sec:surfaces} the readings are:
\begin{itemize}
\item $1-r_{12}^{2}\sim h^{-2.747}$ on the $67$ node-complete windows,
  $h=142$ to $1433$;
\item $1-r_{12}^{2}\sim h^{-2.260}$ on the full family of $70$ windows,
  $h=142$ to $1445$, i.e.\ including the $3$ windows whose comb is truncated by
  the node cap. Both numbers are quoted; the difference is the cap, not the
  phenomenon, and it is not hidden;
\item the angle reading of Problem~\ref{prob:R1}\ref{R1:rows},
  $\sin\angle=|\det\Ah_2|/(\rho_1\rho_2)$, falls as $h^{-2.711}$ on the same
  $67$ windows;
\item on the $27$-window frame ladder of Measurements~\ref{meas:sharpness} and
  \ref{meas:battery} the fitted exponent is $-2.921$; on the union of that
  ladder with the decoupled families below it is $-2.246$.
\end{itemize}
The exponent is \emph{not} a single number across surfaces, and this is reported
without smoothing. On three independently generated frames obtained by
decoupling the cell width from the window index the fitted exponents are
$-1.702$, $-1.270$ and $-1.735$, while the best priced route of
Measurement~\ref{meas:routes} certifies $+0.529$, $+0.339$ and $+0.611$ there
--- short of the exponent in \eqref{eq:R1} by $1.17$, $0.93$ and $1.12$. On five
further frame data the pooled exponents are $-2.19$, $-2.46$, $-2.55$, $-1.83$
and $-2.76$, and across the nine individual ladders of those families they
range from $-1.83$ to $-2.87$. What does transfer is the near-degeneracy itself:
the row angle stays below $10^{-2}$ on every window of every family and decays
on every ladder. The rate is a property of the family; the collapse is not.
Nothing in Sections~\ref{sec:setting}--\ref{sec:rank} depends on any of these
numbers.
\end{measurement}

\subsection{The priced routes, and the structural obstructions}
\label{sec:nogo}

The next proposition collects five unconditional majorants for
$|1-r_{12}^{2}|$. Each is an inequality with its hypotheses written out; each is
proved; none of them involves a fit. The fitted exponents that these majorants
\emph{certify} on a declared surface are a separate matter and are reported
afterwards, in Measurement~\ref{meas:routes}.

One hypothesis of one of the five is not an assumption at all but a consequence
of the assembly, and it is separated out first, because a sieve route has to be
told which points it is applied at and the window decides them. The reads of
Proposition~\ref{prop:split} are values of one exponential sum at two points
fixed by the parity modes and the resolution, and their spacing is closed.

\begin{lemma}[the three linear functionals are dual-point reads]
\label{lem:dualpoint}
Let $D$ be admissible, let $K=2$, and take the prime-power instance of
Section~\ref{sec:window}: $u_\ell=\log n_\ell$,
$\varrho_\ell=2\Lambda(n_\ell)/\sqrt{n_\ell}$ over the prime powers
$n_\ell\le X$, with $X$ the smaller of $e^{2\alpha}$ and the declared node cap,
and with every read interior, $1\le\tau_\ell=u_\ell/D\le M-1$. Write the band
representation of the polarised weight $w^{(ij)}$ of \eqref{eq:Yread}, which
exists by Lemma~\ref{lem:band}, as
\begin{equation}
\label{eq:bandij}
  W^{(ij)}(\tau)=\sum_{k=1,2}
  \bigl[(a_k+b_k\tau)\cos(\omega_k\tau)
  +(c_k+d_k\tau)\sin(\omega_k\tau)\bigr],
\end{equation}
its coefficients being those of Lemma~\ref{lem:spline} with the letters chosen
so as to leave $\gamma$ for the dual points, and let
$\Lambda_0^{(ij)}=\sum_k(|a_k|+|c_k|)+M\sum_k(|b_k|+|d_k|)$ and $\eta^{(ij)}$
be the coefficient budget and the interpolation bound of
\eqref{eq:splinebound} for that weight. Put
\begin{equation}
\label{eq:dualpoints}
  \gamma_k=\frac{\omega_k}D=\frac{2\pi k}{ND}\quad(k=1,2),
  \qquad
  \Delta\gamma=\gamma_2-\gamma_1=\frac{2\pi}{ND},
  \qquad
  T_q(\gamma)=\sum_{n\le X}\varrho_n(\log n)^{q}n^{\mathrm i\gamma} .
\end{equation}
Then
\begin{equation}
\label{eq:dualread}
  S_{ij}=\sum_{k=1,2}\operatorname{Re}
  \Bigl[(a_k-\mathrm ic_k)T_0(\gamma_k)
  +\tfrac1D(b_k-\mathrm id_k)T_1(\gamma_k)\Bigr]+E_{ij},
  \qquad
  |E_{ij}|\le\eta^{(ij)}\lVert\varrho\rVert_1 ,
\end{equation}
so each of the three reads is a value of one exponential sum of length $X$, and
of its first logarithmic moment, at the two points \eqref{eq:dualpoints} and at
no others. If in addition the Montgomery--Vaughan inequality
\cite{MontgomeryVaughan1974} is applied at those two points, in the form
$\sum_{k\le2}|T_0(\gamma_k)|^{2}
\le(X+\Delta\gamma^{-1})\lVert\varrho\rVert_2^{2}$ and the same with the weights
$\varrho_n\log n$ for $T_1$, then
\begin{equation}
\label{eq:mvread}
  |S_{ij}|\;\le\;\Lambda_0^{(ij)}\,Q+\eta^{(ij)}\lVert\varrho\rVert_1,
  \qquad
  Q^{2}=\bigl(X+\Delta\gamma^{-1}\bigr)\lVert\varrho\rVert_2^{2} .
\end{equation}
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:band} at $K=2$ the analytic read $W^{(ij)}$ lies in the span
of the eight sequences \eqref{eq:bandspace} with $k\le2$, which is
\eqref{eq:bandij}. The proof of Lemma~\ref{lem:spline} uses nothing about $v$
beyond that representation, so its bound \eqref{eq:splinebound} applies verbatim
to $W^{(ij)}$ with the coefficients of \eqref{eq:bandij}: the two-point read of
\eqref{eq:split} differs from $W^{(ij)}$ by at most $\eta^{(ij)}$ at every
$\tau\in[1,M-1]$, and every $\tau_\ell$ lies there by hypothesis, so
$|E_{ij}|=\bigl|\sum_\ell\varrho_\ell
(\widehat W^{(ij)}-W^{(ij)})(\tau_\ell)\bigr|
\le\eta^{(ij)}\sum_\ell\varrho_\ell$, the weights being positive. For the main
term, $\tau_\ell=u_\ell/D=(\log n_\ell)/D$, hence
$\omega_k\tau_\ell=\gamma_k\log n_\ell$ and
$\cos(\omega_k\tau_\ell)+\mathrm i\sin(\omega_k\tau_\ell)
=n_\ell^{\mathrm i\gamma_k}$ with $\gamma_k=\omega_k/D$; summing
$\varrho_\ell W^{(ij)}(\tau_\ell)$ over $\ell$ and using
$\operatorname{Re}[(a-\mathrm ic)(x+\mathrm iy)]=ax+cy$ gives the two groups of
\eqref{eq:dualread}, the second carrying the extra factor
$\tau_\ell=(\log n_\ell)/D$, which is what makes it the first logarithmic
moment. That the two points are $2\pi/(ND)$ apart is $\omega_k=2\pi k/N$.

For \eqref{eq:mvread}: the quoted inequality gives $|T_0(\gamma_k)|\le Q$ for
each of the two $k$, and applied with the weights $\varrho_n\log n$, together
with $\log n\le\log X\le2\alpha=MD$, it gives $|T_1(\gamma_k)|\le MD\,Q$.
Insert both into \eqref{eq:dualread} and use the triangle inequality: the
constant terms contribute at most $\sum_k(|a_k|+|c_k|)\,Q$ and the moment terms
at most $\tfrac1D\sum_k(|b_k|+|d_k|)\,MD\,Q$, and the two sums are
$\Lambda_0^{(ij)}Q$.
\end{proof}

On the reference window of Measurement~\ref{meas:refwindow} the two points and
their spacing are $\gamma_1=0.6208$, $\gamma_2=1.2415$,
$\Delta\gamma^{-1}=1.611$, with $X=2.465\cdot10^{4}$; they are read off
\eqref{eq:dualpoints} and not fitted.

\begin{proposition}[five unconditional majorants]
\label{prop:routes}
Let $\Ah_2=B-S$ be the split of Proposition~\ref{prop:split}, let
$\mathcal K$ be the kernel of \eqref{eq:doublesum}, let $\varrho$ be the weight
vector and $\lVert\varrho\rVert_2$ its Euclidean norm. Assume
$\ah_{11}\ah_{22}\ne0$ and write $\Xi=|\ah_{11}\ah_{22}|$ for the length of
this proposition only. Then, with
$1-r_{12}^{2}=\det\Ah_2/(\ah_{11}\ah_{22})$ by Lemma~\ref{lem:detread}:
\begin{enumerate}[label=\textup{(R\arabic*)},leftmargin=3.2em]
\item\label{RT:cs} \emph{Cauchy--Schwarz on $\Ah_2$.}
  $|\det\Ah_2|\le|\ah_{11}\ah_{22}|+\ah_{12}^{2}$, hence
  $|1-r_{12}^{2}|\le1+\ah_{12}^{2}/\Xi\le2$ whenever $\Ah_2\PSD$. Hypotheses:
  none for the first inequality; $\Ah_2\PSD$ for the numerical bound $2$.
\item\label{RT:ls} \emph{Large sieve / duality on the comb--comb term.}
  $|\det S|\le\lVert\mathcal K\rVert_{\mathrm{op}}\lVert\varrho\rVert_2^{2}$,
  hence
  $|1-r_{12}^{2}|\le\bigl[|\det B|+|D(B,S)|
  +\lVert\mathcal K\rVert_{\mathrm{op}}\lVert\varrho\rVert_2^{2}\bigr]/\Xi$.
  Hypotheses: $\mathcal K$ symmetric, which is
  Proposition~\ref{prop:polarisation}; the operator-norm step is the Rayleigh
  bound and is the form in which the large sieve enters
  \cite{MontgomeryVaughan1974}.
\item\label{RT:mv} \emph{Montgomery--Vaughan on the three linear functionals.}
  On the prime-power instance with interior reads, Lemma~\ref{lem:dualpoint}
  identifies the three reads as values of one exponential sum of length $X$, and
  of its first logarithmic moment, at the two points $\gamma_k=2\pi k/(ND)$ of
  spacing $\Delta\gamma=2\pi/(ND)$, and gives $|S_{ij}|\le\Qt$ with
  \begin{equation}
  \label{eq:Qtilde}
    \Qt=\Lambda_0\,Q+\eta\lVert\varrho\rVert_1,
    \qquad
    Q^{2}=\bigl(X+\Delta\gamma^{-1}\bigr)\lVert\varrho\rVert_2^{2},
  \end{equation}
  where $\Lambda_0=\max_{ij}\Lambda_0^{(ij)}$ and $\eta=\max_{ij}\eta^{(ij)}$
  are the coefficient budget and the interpolation bound of that lemma. Then
  $|1-r_{12}^{2}|\le\bigl[|\det B|
  +(|B_{11}|+|B_{22}|+2|B_{12}|)\,\Qt+2\Qt^{2}\bigr]/\Xi$. Hypotheses: those of
  Lemma~\ref{lem:dualpoint}, the Montgomery--Vaughan inequality
  \cite{MontgomeryVaughan1974} at the two points that lemma derives, and the
  triangle inequality on $\det S=S_{11}S_{22}-S_{12}^{2}$.
\item\label{RT:split} \emph{The exact split.}
  $|1-r_{12}^{2}|\le\bigl[|\det B|+|D(B,S)|+|\det S|\bigr]/\Xi$. Hypotheses:
  the identity \eqref{eq:polarisation} and the triangle inequality; nothing
  else.
\item\label{RT:vaughan} \emph{Vaughan Type I $+$ Type II.} Write
  $\varrho=\sum_{q\le4}\varrho^{(q)}$ for Vaughan's decomposition
  \cite{Vaughan1977} of the weight vector and bound each of the sixteen
  resulting blocks of the bilinear form by its own operator norm as in
  \ref{RT:ls}. Then
  $|1-r_{12}^{2}|\le\bigl[|\det B|+|D(B,S)|
  +\sum_{q,q'}\lVert\mathcal K\rVert_{\mathrm{op}}
  \lVert\varrho^{(q)}\rVert_2\lVert\varrho^{(q')}\rVert_2\bigr]/\Xi$.
  Hypotheses: those of \ref{RT:ls} for each block, plus
  Corollary~\ref{cor:majorant}\ref{M:decomp}.
\end{enumerate}
Routes \ref{RT:ls}, \ref{RT:mv} and \ref{RT:split} are members of the class
$\mathfrak M$ of Definition~\ref{def:majclass} as stated;
\ref{RT:vaughan} is of the decomposed shape of
Corollary~\ref{cor:majorant}\ref{M:decomp} with $p=4$; \ref{RT:cs} is not a
majorant of $\det S$ at all and is carried as the trivial baseline
(Remark~\ref{rem:routeclass}).
\end{proposition}

\begin{proof}
\ref{RT:cs} is $|\ah_{11}\ah_{22}-\ah_{12}^{2}|\le|\ah_{11}\ah_{22}|
+\ah_{12}^{2}$, and if $\Ah_2\PSD$ then $\ah_{12}^{2}\le\ah_{11}\ah_{22}$.
\ref{RT:ls}: by \eqref{eq:doublesum},
$\det S=\varrho^{\T}\mathcal K\varrho$, and for a symmetric matrix
$|\varrho^{\T}\mathcal K\varrho|
\le\lVert\mathcal K\rVert_{\mathrm{op}}\lVert\varrho\rVert_2^{2}$; insert into
\eqref{eq:polarisation} and apply the triangle inequality.
\ref{RT:mv}: by Lemma~\ref{lem:dualpoint} every $|S_{ij}|\le\Qt$, so
$|\det S|\le|S_{11}||S_{22}|+S_{12}^{2}\le2\Qt^{2}$ and
$|D(B,S)|\le(|B_{11}|+|B_{22}|+2|B_{12}|)\max_{ij}|S_{ij}|
\le(|B_{11}|+|B_{22}|+2|B_{12}|)\Qt$; insert into
\eqref{eq:polarisation}. \ref{RT:split} is the triangle inequality on
\eqref{eq:polarisation}. \ref{RT:vaughan}: expand
$\det S=\sum_{q,q'}(\varrho^{(q)})^{\T}\mathcal K\varrho^{(q')}$ by bilinearity
of $D$ and bound each term as in \ref{RT:ls}; that the four pieces sum to
$\varrho$ is Vaughan's identity, which is a coefficientwise statement about the
weight vector and is verified as such (Section~\ref{sec:crossref}). Dividing by
$\Xi$ and using Lemma~\ref{lem:detread} gives each displayed form.
\end{proof}

\begin{measurement}[the priced table]
\label{meas:routes}
Each majorant of Proposition~\ref{prop:routes} is evaluated window by window on
the $67$ node-complete windows of Surface~II and its logarithm regressed against
$\log h$. The resulting $\delta$ is the exponent that the route certifies for
$1-r_{12}^{2}\le C\,h^{-\delta}$ on that surface: the bound is proved, the
exponent is a fit, and the two are typed separately in every row.

\begin{center}\small
\begin{tabularx}{\textwidth}{@{}llrY@{}}
\toprule
route & where & $\delta$ & type of the row \\
\midrule
$1-r_{12}^{2}$ itself & Meas.~\ref{meas:rate}
  & $+2.747$ & measured (the object, not a route) \\
\ref{RT:cs} Cauchy--Schwarz on $\Ah_2$ & Prop.~\ref{prop:routes}
  & $-0.000$ & bound proved, exponent measured \\
\ref{RT:ls} large sieve / duality on $\det S$ & Prop.~\ref{prop:routes}
  & $+0.669$ & bound proved, exponent measured \\
\ref{RT:mv} Montgomery--Vaughan on $S_{11},S_{22},S_{12}$
  & Prop.~\ref{prop:routes}
  & $-0.384$ & bound proved, exponent measured \\
\ref{RT:split} exact split $|\det B|+|D(B,S)|+|\det S|$
  & Prop.~\ref{prop:routes}
  & $+0.996$ & bound proved, exponent measured \\
\ref{RT:vaughan} Vaughan Type~I $+$ Type~II & Prop.~\ref{prop:routes}
  & $+0.669$ & bound proved, exponent measured \\
\bottomrule
\end{tabularx}
\end{center}

\noindent
Row \ref{RT:mv} is priced on the bare Montgomery--Vaughan factor $Q$ of
\eqref{eq:Qtilde}, i.e.\ at $\Lambda_0=1$ and $\eta=0$; the coefficient budget
and the interpolation residual of Lemma~\ref{lem:dualpoint} are window-dependent
prefactors of that factor, and the exponent in the row is the exponent of $Q$ on
this surface. Every other row is priced on the majorant exactly as displayed.

The best of the five on this surface is the triangle inequality on the exact
three-term split, at $\delta=+0.996$; the exponent in \eqref{eq:R1} is $3$, so
on this surface the best priced route falls short by $2.004$ \emph{in the
exponent}. Two readings are part of the measurement.
Corollary~\ref{cor:majorant}\ref{M:blind} says why the three norm-based rows
cannot do better in principle: they see the image point
$y(S)=(S_{11},S_{22},S_{12})$ and not its relation to $B$, and the phenomenon
of Measurement~\ref{meas:refwindow} is exactly that relation.
Theorem~\ref{thm:rank3}\ref{R:three} says why \ref{RT:ls} and \ref{RT:vaughan}
land on the same number: a bilinear tool applied to a kernel of rank three
degenerates to a statement about three linear functionals, and the Vaughan
blocks inherit the degeneracy block by block.
\end{measurement}

\begin{measurement}[the structural obstructions]
\label{meas:nogo}
Five further routes are excluded not by their exponent but by their shape. Each
is exhibited failing on the windows of Surface~I
(Section~\ref{sec:surfaces}), $12$ windows with $m\le300$, in the way the
classification predicts.
\begin{enumerate}[label=\textup{(N\arabic*)},leftmargin=3.2em]
\item\label{NG:symbol} \emph{The symbol infimum is negative.} The Toeplitz
  symbol $\sigma(\theta)=c_0+2\sum_{d\ge1}c_d\cos(d\theta)$ has
  $\min_\theta\sigma\in[-47.6,-21.3]$ on $12/12$ windows, while the parity
  compression is positive definite there. No pointwise minorant of the symbol
  can therefore deliver a positive floor: the positivity is a finite-section
  fact --- a Fej\'er cancellation \cite{Fejer1916} between the Toeplitz and the
  Hankel half of \eqref{eq:Adef} --- and not a symbol fact.
\item\label{NG:kato} \emph{Perturbation theory prices at the wrong scale.}
  Kato's bound $|\nu_2(\Ah_2+E)-\nu_2(\Ah_2)|\le\lVert E\rVert$ \cite{Kato1949}
  is sharp at the scale of the matrix, and on Surface~I the small eigenvalue
  sits a factor $9.8\cdot10^{2}$ to $2.0\cdot10^{4}$ below
  $\lVert\Ah_2\rVert$. A perturbative argument would have to control the matrix
  to that relative accuracy first, which is the original problem. The
  observation is sharper than a divergence statement: on a declared family of
  perturbative expansions the Neumann/Jacobi series diverges (radius $1.0$ to
  $12.7$) while the Kato series \emph{converges}, with radius $0.067$ --- to the
  wrong object, since the exact bottom eigenvector of the corresponding
  $6\times6$ block overshoots the constrained minimum by a relative excess
  $6.04$ at overlap $0.083$ and relative gap $5.4\cdot10^{-7}$.
\item\label{NG:band} \emph{Band-local domination fails.} On the mode-$1$ weight
  vector the comb half of the lag sum dominates the archimedean half by a
  factor $1.71$ to $2.83$ on $12/12$ windows, so a route that would dominate the
  comb band by the archimedean band loses on the instance.
\item\label{NG:l1} \emph{The signed sum is far below its $\ell^{1}$ mass.} On
  the declared subsample of the reference window the ratio is $40.6$
  (Measurement~\ref{meas:l1}, equation~\eqref{eq:l1mass}); any route that
  replaces the signed double sum by its absolute value pays that factor before
  it starts.
\item\label{NG:LP} \emph{The $L_P$ control is exact and empty.} Replacing the
  comb by nothing --- i.e.\ running the whole reduction on the lag sequence
  $c^{L}$ of Lemma~\ref{lem:kms} --- gives $\ah_{12}=1.1\cdot10^{-17}$, hence
  $1-r_{12}^{2}=1$ with no decay at all at $m=64$, $128$ and $256$, a Gram
  matrix that is diagonal to an off-diagonal mass $2.3\cdot10^{-13}$, and a
  ladder gain $g_K/g_1=1$ at zero tolerance. Any argument
  that would already work for $L_P$ alone is therefore vacuous for $A$: the
  phenomenon of Problem~\ref{prob:R1} is not a property of the two parity modes.
\end{enumerate}
\end{measurement}

\subsection{The density curve}
\label{sec:density}

The measurements so far are taken along a ladder in $h$ at a fixed rule
relating the resolution to the window. The next one is taken on a rectangle,
which separates the window index from the number of comb nodes per lag cell,
and it is the last decidable measurement the declared caps permit.

\begin{definition}[the ratio and its binned exponent]
\label{def:ratio}
For a window put
\begin{equation}
\label{eq:ratio}
  \gamma=\frac{\lmin(G)}{\lmax(G)},
  \qquad
  R=\frac{\gamma}{\,1-r_{12}^{2}\,},
  \qquad
  \mathrm{dens}=\frac{\#\{\ell:u_\ell\le2\alpha\}}{M},
\end{equation}
with $G$ the normalised $K$-block of \eqref{eq:gram} at $K=16$ and
$\mathrm{dens}$ the number of comb nodes per lag cell. For a density interval
$[\lambda,4\lambda)$ the \emph{binned exponent} is: for each anchor with at
least three ladder rungs inside the interval, the ordinary least-squares slope
of $\log|R|$ against $\log h$ over its rungs; then the unweighted mean of the
anchor slopes, with the cluster-robust standard error of that mean, and with the
sign reversed so that a positive value means that $R$ falls. The procedure is
deterministic and independent of the order in which the cells are enumerated.
\end{definition}

Two remarks fix the reading. First, $R$ compares two spectral functionals of one
and the same finite matrix --- the relative spectral gap of the normalised low
block, and the near-degeneracy scalar of Problem~\ref{prob:R1} --- so a positive
binned exponent says that the second closes more slowly than the first, on that
surface, at that density. Second, the binned exponent is a \emph{local slope of
a curved surface} and not an exponent of a power law: $\log R$ is measurably
curved in $\log h$ (the per-anchor quadratic coefficient is
$-0.0842\pm0.0085$), which is why the value depends on the rung ladder ---
see Remark~\ref{rem:undecidable}.

\begin{measurement}[the density curve]
\label{meas:density}
Surface~III of Section~\ref{sec:surfaces}: $692$ anchors, $14\,532$ windows, a
node table capped at $2.5\cdot10^{7}$ containing $1\,566\,727$ nodes, every
window with a complete comb, none of them indefinite. The cap and the smallest
admissible rung force $\mathrm{dens}\le6120$, of which $6118$ is attained; the
attained density spans $4.0$ decades. On the declared geometric ladder of
ratio-$4$ density bins, fixed before any window was built, the binned exponent
of Definition~\ref{def:ratio} is:

\begin{center}\small
\begin{tabular}{@{}lrrcc@{}}
\toprule
bin (nodes per lag cell) & cells & anchors & exponent $\pm$ s.e.
  & on the $\sqrt2$ sub-ladder \\
\midrule
$[0.1,\ 0.4)$        &   $83$ &  $11$ & $+0.8886\pm0.1027$
  & $+0.9818\pm0.2129$ \\
$[0.4,\ 1.6)$        &  $318$ &  $34$ & $+0.5148\pm0.0883$
  & $+0.5762\pm0.0743$ \\
$[1.6,\ 6.4)$        &  $597$ &  $69$ & $+0.1948\pm0.0861$
  & $+0.2073\pm0.0679$ \\
$[6.4,\ 25.6)$       & $1056$ & $117$ & $+0.3922\pm0.0935$
  & $+0.1942\pm0.0734$ \\
$[25.6,\ 102.4)$     & $1924$ & $218$ & $+0.1520\pm0.0573$
  & $+0.0879\pm0.0589$ \\
$[102.4,\ 409.6)$    & $3588$ & $403$ & $+0.0579\pm0.0542$
  & $-0.0573\pm0.1320$ \\
$[409.6,\ 1638.4)$   & $4891$ & $471$ & $+0.0376\pm0.0410$
  & --- \\
$[1638.4,\ 6553.6)$  & $2075$ & $272$ & $+0.1046\pm0.0881$
  & --- \\
\midrule
pooled, the two densest bins & & & $+0.0496\pm0.0372$ & --- \\
\bottomrule
\end{tabular}
\end{center}

\noindent
Every band is the cluster-robust standard error of
Definition~\ref{def:ratio}, the cluster being the anchor; the pooled row is the
inverse-variance pool of the last two bins. Read against zero, the eight bins
sit at $8.7$, $5.8$, $2.3$, $4.2$, $2.7$, $1.1$, $0.9$ and $1.2$ standard
errors, and the pool of the last two at $1.3$. The joint slope of the binned
exponent against $\log(\mathrm{dens})$ is $-0.0725\pm0.0090$, i.e.\ $8.0\sigma$
over the $4.0$ attained decades. The violation of monotonicity in the fourth
bin is declared and not smoothed; the last column is where it comes from.

That column is the same bins evaluated on the $\sqrt2$ sub-ladder of the same
rung ladder, over the anchor range of the earlier and $20.8$ times smaller node
table, and it carries two readings at once. It is the \emph{node-table}
consistency check: each of the six entries reproduces the value published for
that bin under the smaller table to $0.0\sigma$, which is the sharpest form the
check can take --- with the caveat that the sixth bin ran to $500$ rather than to
$409.6$ there, so the comparison is like-for-like only up to that edge. And it is
the \emph{estimator} dependence, exhibited rather than asserted: the two exponent
columns differ by up to $0.20$, the largest gap in the fourth bin and the next in
the sixth. What differs between them is the rung ladder and the anchor range;
what does not is the node table, so the movement is the estimator and its sample
and not the arithmetic input. That is the caveat of
Remark~\ref{rem:undecidable} as a number.

Two further consistency facts belong to the measurement. \emph{(i)} A larger node
table adds nodes above the old cap and nothing else, so on every window whose
comb was already complete under the old cap the lag vector is unchanged bit for
bit, $R$ is unchanged exactly, and the binned exponent is identical --- a
statement about complete combs, not about tables, and its must-break control is a
window whose comb was truncated under the old cap, which does move.
\emph{(ii)} The two densest bins have no counterpart under the smaller table at
all: they lie beyond the density its cap could reach, which is why they are the
reason for the enlargement.
\end{measurement}

\begin{remark}[what the curve does and does not decide, and at what price]
\label{rem:undecidable}
Three statements are part of the claim rather than caveats attached to it.

\emph{Undecidability.} On this surface, a true zero of the binned exponent at
finite density, a power-law approach to zero, and a plateau at any level below
$0.074$ are mutually indistinguishable: $0.074$ is the two-sigma resolution of
the densest bins, and no finite node table reachable under the declared budget
separates the three. No limit is claimed, and the quantifier in
Problem~\ref{prob:R1} is untouched by the curve.

\emph{The resource fence.} The resolution narrows like the square root of the
anchor cap, so the width of the plateau window improves by a factor $2$ for
every factor $4$ in the node table; raising the table by the factor $20.8$ that
produced Surface~III narrowed the window from $0.264$ to $0.074$, a factor
$3.6$. Extending the curve by one further decade of
density needs a node table of about $2.5\cdot10^{9}$, a factor $100$ above the
one used; that is outside the declared budget, and --- this is the honest half
--- it is not obviously the right question either, since the object being
extrapolated is a local slope of a curved surface and not an exponent.

\emph{The ladder caveat.} The estimator of Definition~\ref{def:ratio} is
ladder-dependent. The same density bin evaluated on a $2^{1/4}$ rung ladder and
on its $\sqrt2$ sub-ladder gives different values, with shifts up to $0.20$ on
the surface above, because a bin with about eight rungs samples the curvature of
$\log R$ differently from one with about four. Every number in
Measurement~\ref{meas:density} is therefore quoted \emph{with} its rung ladder,
and a bin value quoted without one is not reproducible.
\end{remark}

\subsection{Placement: two controlled experiments and one identity}
\label{sec:placement}

The results of Sections~\ref{sec:setting}--\ref{sec:rank} hold for an arbitrary
finite comb (Remark~\ref{rem:combfree}). It is therefore a fair question where
the phenomenon of Problem~\ref{prob:R1} actually lives. Two interventions answer
it, and the second one closes into an identity.

\begin{measurement}[the value-multiset scramble]
\label{meas:placement}
Replace the node positions $u_\ell$ by sorted uniform samples on the same range,
keeping the weight multiset $\{\varrho_\ell\}$ exactly, and recompute. On $60$
declared draws over the windows of Surface~I:
\begin{itemize}
\item the rank-three collapse is untouched:
  $\sigma_4/\sigma_1\le2.5\cdot10^{-16}$ on every draw, as
  Theorem~\ref{thm:rank3}\ref{R:kernel} requires --- the theorem does not know
  where the nodes are;
\item every majorant of Proposition~\ref{prop:routes} moves by a bounded factor
  and keeps its exponent;
\item the \emph{value} $1-r_{12}^{2}$ moves by a median factor $679$, range
  $26$ to $4.9\cdot10^{3}$. An independent set of draws in the rank module
  gives median $835$, range $69$ to $1.8\cdot10^{4}$; both are quoted, and the
  spread between them is the spread of the sampling.
\end{itemize}
On a second declared surface the same intervention destroys positive
definiteness of the low block on all $24$ draws and moves $R$ by a median
factor $\approx10^{6}$; on the first of the two new density bins of
Surface~III it destroys it on $194/194$ cells with the cell occupancy preserved
exactly, so that only the sub-cell placement is randomised. On Surface~II the
decay is removed outright: over $18$ windows the fitted exponent of
$1-r_{12}^{2}$ moves from $-2.747$ to $+0.009$, a gap of $2.76$ in the
exponent; on a further family the angle $\sin\angle$ rises by a median factor
$6.6\cdot10^{4}$ and its exponent moves from $-1.93$ to $+0.11$. Position and
weight are both load-bearing, and the second is checked separately: keeping the
node positions and replacing each weight by the mean weight gives $h^{-0.116}$
against the true $h^{-2.747}$. The conclusion is
the one sentence this subsection exists for: \emph{every theorem in this paper
survives the scramble and the phenomenon does not}, so the theorems are provably
not where the arithmetic sits. What moves is the joint value of
$(S_{11},S_{22},S_{12})$ against $B$ --- exactly the object
Corollary~\ref{cor:majorant}\ref{M:blind} says no member of $\mathfrak M$ can
see.
\end{measurement}

The second experiment displaces a few nodes by a fraction of a lag cell. That
the response is available in closed form is not an accident: the assembly is
affine in the fractional positions.

\begin{lemma}[the exact placement derivative]
\label{lem:dcexact}
Fix a finite set $F$ of nodes and displace their fractional positions,
$\theta_\ell\mapsto\theta_\ell+\vartheta$ for $\ell\in F$, keeping $m_\ell$ and
all other nodes fixed. As long as no $\theta_\ell+\vartheta$ leaves $[0,1)$, the
lag sequence is affine in $\vartheta$,
\begin{equation}
\label{eq:dcexact}
  c(\vartheta)=c(0)+\vartheta\,\dot c,
  \qquad
  \dot c=\tfrac12\sum_{\ell\in F}\varrho_\ell
  \bigl(\delta_{m_\ell}-\delta_{m_\ell+1}\bigr),
\end{equation}
where $\delta_m$ is the lag sequence supported at the single lag $m$. In
particular $\dot c$ is exact and piecewise constant in $\vartheta$, and the
induced $\dot A=\Pi^{\T}\Tm{M}(\dot c)\Pi$, $\dot G=\MM^{-1/2}
(t_i^{\T}\dot At_j)_{ij}\MM^{-1/2}$ are exact as well.
\end{lemma}

\begin{proof}
By the two-point read in the proof of Proposition~\ref{prop:split}, the node
$u_\ell$ contributes $-\varrho_\ell\tfrac12[(1-\theta_\ell)\delta_{m_\ell}
+\theta_\ell\delta_{m_\ell+1}]$ to $c$ and nothing else. This is affine in
$\theta_\ell$ with slope
$\tfrac12\varrho_\ell(\delta_{m_\ell}-\delta_{m_\ell+1})$; summing over
$\ell\in F$ gives \eqref{eq:dcexact}. The reflected hat still vanishes and
$m_\ell$ is unchanged because $\theta_\ell+\vartheta$ stays in $[0,1)$, so no
other lag entry moves. The maps $c\mapsto\Tm{M}(c)\mapsto\Pi^{\T}\cdot\Pi$ and
$A\mapsto G$ are linear, which transports exactness to $\dot A$ and $\dot G$.
\end{proof}

\begin{proposition}[the placement response in closed form]
\label{prop:fh}
Let $\vartheta\mapsto G(\vartheta)$ be as in Lemma~\ref{lem:dcexact}, suppose
$\lmin(G)$ and $\lmax(G)$ are simple with unit eigenvectors $v$ and $w$, and
suppose $\ah_{11}\ah_{22}\ne0$ and $r_{12}^{2}\ne1$. Then, with
$R=\gamma/(1-r_{12}^{2})$ as in \eqref{eq:ratio},
\begin{equation}
\label{eq:fh}
  \frac{\mathrm d\log R}{\mathrm d\vartheta}
  =\frac{v^{\T}\dot Gv}{\lmin(G)}-\frac{w^{\T}\dot Gw}{\lmax(G)}
   -\frac{1}{1-r_{12}^{2}}\,
    \frac{\mathrm d\,(1-r_{12}^{2})}{\mathrm d\vartheta},
\end{equation}
where the last derivative is the closed expression
\begin{equation}
\label{eq:ddel}
  \frac{\mathrm d\,(1-r_{12}^{2})}{\mathrm d\vartheta}
  =-\frac{2\ah_{12}\dot a_{12}}{\ah_{11}\ah_{22}}
   +\frac{\ah_{12}^{2}}{\ah_{11}\ah_{22}}
    \Bigl(\frac{\dot a_{11}}{\ah_{11}}+\frac{\dot a_{22}}{\ah_{22}}\Bigr),
  \qquad \dot a_{ij}=t_i^{\T}\dot At_j .
\end{equation}
Every quantity on the right is an explicit finite sum; no fit enters.
\end{proposition}

\begin{proof}
For a simple eigenvalue of a symmetric matrix depending differentiably on a
parameter, $\mathrm d\lambda/\mathrm d\vartheta=v^{\T}\dot Gv$ with $v$ the
normalised eigenvector \cite{Hellmann1937,Feynman1939}; by
Lemma~\ref{lem:dcexact} the dependence is affine, so $\dot G$ is constant and
the hypothesis of differentiability is automatic. Then
$\mathrm d\log\gamma/\mathrm d\vartheta
=\dot\lambda_{\min}/\lmin-\dot\lambda_{\max}/\lmax$, and
$\log R=\log\gamma-\log(1-r_{12}^{2})$ gives \eqref{eq:fh}. For
\eqref{eq:ddel}, differentiate
$1-r_{12}^{2}=1-\ah_{12}^{2}/(\ah_{11}\ah_{22})$ by the quotient rule; the
entries $\ah_{ij}$ are affine in $\vartheta$ with derivatives $\dot a_{ij}$.
\end{proof}

\begin{measurement}[causality, the pole, and where the response lives]
\label{meas:pole}
Three numbers on a declared instance family.

\emph{(i) The intervention is causal, and linear over four decades.} Displacing
six declared nodes inside their lag cells --- the cell index fixed, the mass
exactly preserved, twelve lag entries changing --- gives
$\mathrm d\log R/\mathrm d\vartheta=879$, linear to a factor $1.011$ over the
declared range $\vartheta=10^{-7}$ to $10^{-4}$, with an exact
$\vartheta=0$ rebuild control (residual $0$ on $40$ windows). This is a
statement of causation, not of correlation: nothing but the placement was
changed.

\emph{(ii) The number is an identity, not a regression.} On the declared
instance list the closed form \eqref{eq:fh} reproduces the full nonlinear rebuild to
a median relative residual $3.37\cdot10^{-6}$ at the minimum of a declared
central-difference step ladder, with the U-shape exhibited --- round-off limited
below, truncation limited above, both ends at least three times the minimum. A
wrong first-order formula has a floor, not a minimum; two deliberately mutated
formulas (swapped eigenvectors, dropped last term) miss the measured quotient by
a factor of order $40$. Of the three terms of \eqref{eq:fh} the last one, the
near-degeneracy channel, carries the largest share on every instance --- a
median $84\%$, against $7\%$ for the $\lmin$ term.

\emph{(iii) The intervention walks along a near-degeneracy that is already
there.} Sweeping the full placement period on a declared $64$-point grid,
positive definiteness of the low block survives only about $2\%$ of the period:
$\lmin$ changes sign inside the period, so $\log R$ has a \emph{pole} there and
is not defined on the whole period. The bisected loss point satisfies
$\vartheta_{\mathrm{PD}}\cdot|\mathrm d\log R/\mathrm d\vartheta|=5.8$, a pole at
the natural scale, with $\vartheta_{\mathrm{PD}}=5.9\cdot10^{-3}$ median. And
the two lowest modes of the normalised low block already sit only $13.4\%$ apart
at $\vartheta=0$. Consequently a first-harmonic regression of $\log R$ over the
period estimates a Fourier coefficient of a function with a pole in the
interval, and returns $2.89$ against a local derivative of $879$; neither number
is wrong, they measure different things, and the discrepancy is not an anomaly
but a pole.

The cell phases themselves are equidistributed in the classical sense: they are
generated by $\varphi_\ell(h)=\{h\,\theta_\ell\}$ from one real invariant per
node --- an identity certified to $1.14\cdot10^{-13}$ over $1316$ windows and
eight nodes --- and the largest circular-uniformity statistic over those windows
is $1.41$, i.e.\ consistent with uniformity after correction for multiplicity.
Equidistribution of $\{h\theta\}$ is Weyl's theorem \cite{Weyl1916}; it is cited
as the classical address of the phases and is used in no proof.
\end{measurement}

\subsection{What is not claimed}
\label{sec:notclaimed}

\begin{remark}[what is not claimed]
\label{rem:notclaimed}
Problem~\ref{prob:R1} is open. Nothing in this paper bounds it, and nothing in
this paper is conditional on it. Explicitly:
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.6em]
\item No statement of this paper is an arithmetic statement.
  Theorems~\ref{thm:tvfloor}, \ref{thm:gauge}, \ref{thm:rank3} and every
  proposition, lemma and corollary of
  Sections~\ref{sec:setting}--\ref{sec:rank} hold for an arbitrary finite comb,
  and Measurement~\ref{meas:placement} is the controlled experiment that says
  what changes when the comb does.
\item No fitted exponent is used as a bound. In particular the $\delta$ column
  of Measurement~\ref{meas:routes} certifies nothing beyond the finite surface
  it was fitted on; the inequalities of Proposition~\ref{prop:routes} are the
  proved content of those rows.
\item Measurement~\ref{meas:density} claims no limit. Its explicit content
  includes its own undecidability (Remark~\ref{rem:undecidable}), and the
  quantifier in \eqref{eq:R1} is untouched by it.
\item Positivity of $A$ on the parity sector is used only where it is named as a
  hypothesis, and there only as a per-window certificate.
  Lemma~\ref{lem:detread} exists so that the reduction never needs it.
\item No comparison of a demanded precision against the strength of any
  conjecture appears anywhere in this paper. Problem~\ref{prob:R1} is stated as
  a question about a finite bilinear form and about nothing else.
\end{enumerate}
\end{remark}

\section{Numerical verification}
\label{sec:numerics}

Every proved statement of this paper is recomputed by a machine check, and every
measured number is quoted with the surface it was taken on. This
section states the convention under which a numerical statement counts as a
certificate, declares the three surfaces on which anything is evaluated, and
gives the table that maps each statement to its check.

\subsection{Floating-point convention and declared floors}
\label{sec:fp}

Positivity statements are completed Cholesky factorisations relative to a
declared floor in the sense of Wilkinson and Higham
\cite{Wilkinson1963,Higham2002}: with $u=2^{-53}$ the unit roundoff and
$c_h=(h+1)/(1-(h+1)u)$, a completed factorisation of $X-\delta I$ with
$\delta=c_hu\lVert X\rVert_\infty$ certifies $\lmin(X)>0$, and the cheap
certified bound $\lVert X\rVert_2\le\lVert X\rVert_\infty$ is used for the norm
\cite{Gershgorin1931}. Quantities obtained from an eigenvalue routine are
labelled \emph{measured} and never used as certificates. Diagonally dominant
comparisons, where used, are Ostrowski's \cite{Ostrowski1937}.

Three further conventions are fixed once.
\begin{enumerate}[label=\textup{(F\arabic*)},leftmargin=3.2em]
\item\label{F:null} \emph{Near-null horizons are declared per window, not
  globally.} An identity between two quantities that cancel to $k$ digits is
  checked against a bar proportional to $u$ times the larger of the two halves,
  computed on that window. Where the paper quotes a residual --- for instance
  $1.0\cdot10^{-10}$ for the polarisation \eqref{eq:polarisation} --- the bar
  it is compared against is quoted with it. The cancellation reaches seven
  digits on the deepest windows ($|1-r_{12}^{2}|$ down to $9.3\cdot10^{-8}$),
  and that residual sits an order of magnitude below the declared near-null
  horizon $10^{-9}$ for that identity. Identities that do not cancel are
  checked at the ordinary round-off scale and their residuals are reported
  there.
\item\label{F:cond} \emph{Conditioning cap.} No matrix with
  $\cond\ge10^{12}$ is factorised or inverted anywhere; windows exceeding the
  cap are dropped before any number is read from them, and the number dropped
  is reported with the surface.
\item\label{F:mutation} \emph{Every identity carries a mutation control.} A
  deliberately wrong variant of the same identity --- a swapped eigenvalue
  pair, a dropped term, an off-by-one in the lag index --- is evaluated
  alongside it and must fail by a margin that is also reported. An identity
  verified without a mutation control is not counted.
\end{enumerate}

\subsection{The three surfaces}
\label{sec:surfaces}

Three surfaces are declared and never mixed; each is fixed before any number is
read from it.

\emph{Surface~I (exact, small).} The $12$ deepest prime-power anchors admitting
a window inside the lag cap, $n=29$ to $139$, $m=96$ to $291\le300$. Every
identity of Sections~\ref{sec:setting}--\ref{sec:rank} is recomputed from
scratch here, together with the structural obstructions of
Measurement~\ref{meas:nogo}. The one unconditional arithmetic input,
$|\psi(x)-x|\le\kappa x$ with $\kappa=0.038821$, is verified at every jump point
of $\psi(x)/x$ on the finite table \cite{Chebyshev1852,RosserSchoenfeld1962}.

\emph{Surface~II (measurement).} The frame-A family with $h=142$ to $1445$, all
windows with $\cond(G)\le10^{12}$. Two subsets are used and are always named.
The \emph{ladder}, $27$ windows with $h=142$ to $1320$, carries
Measurements~\ref{meas:sharpness}, \ref{meas:battery} and \ref{meas:refwindow}.
The \emph{refinement}, $70$ windows with $h=142$ to $1445$, carries
Measurements~\ref{meas:rate} and \ref{meas:routes}; of these, $67$ have a comb
complete under the node cap and span $h=142$ to $1433$, and every rate quoted on
the node-complete subset is quoted alongside the value on the full family
(Measurement~\ref{meas:rate}). Truncation of the comb by the node cap is not a
nuisance to be averaged away: on a truncated window the low block can be
indefinite, with $\lmin=-9.6\cdot10^{4}$ on the declared stress instance, while
every window with a complete comb on this surface is positive definite.
Indefiniteness sits at the declared sieve horizon and nowhere else.

\emph{Surface~III (the density rectangle).} A rectangle, not a ladder: every
prime power in the declared anchor range is an anchor --- $692$ of them ---
crossed with a rung ladder of ratio $2^{1/4}$ containing the seven rungs of the
earlier ladder verbatim, $21$ rungs from $h=128$ to $h=1280$. This gives
$14\,532$ windows, all with a complete comb and none indefinite. The node table
is capped at $2.5\cdot10^{7}$ and contains $1\,566\,727$ prime powers. The cap
and the smallest admissible rung together force $\mathrm{dens}\le6120$, of which
$6118$ is attained; the smallest rung is itself forced, since a $K\times K$ low
block needs $2K$ modes and the conditioning floor of \ref{F:cond} does the rest.
Surface~III carries Measurement~\ref{meas:density} and the sweeps of
Measurement~\ref{meas:pole}. The bin edges, the rung ladder and the anchor range
are all fixed before the first window is built; this is the anti-fitting device
of the surface, and Remark~\ref{rem:undecidable} is the price it charges.

\subsection{The deterministic--arithmetic dissection (machine survey)}
\label{sec:dissection}

The backflow campaign of 2026-07-31/08-01 (modules \texttt{v582}--\texttt{v592},
individually cited in the table below) assembles into one coherent picture of
the normalized defect, and it is worth stating it once as a narrative rather
than as fourteen table rows.  \emph{First}, the defect has exactly two layers.
Replacing the prime comb by the two-term classical density
$4e^{u/2}-2(\zeta'/\zeta)(\tfrac12)$ --- the pole term plus the $s=\tfrac12$
constant, no zeros --- reproduces $\det S$ on every regular window
parameter-free to $1.6$--$14\%$ (\texttt{v582}/\texttt{v583}), the one-mode
pencil geometry, and the locking \emph{direction} to a median $0.030^\circ$
(\texttt{v586}); it misses the locking \emph{depth} by a median factor $1350$
(\texttt{v585}).  The primes act as a pure depth amplifier along a
density-fixed direction.  \emph{Second}, the density layer is closed-form.
The diagonal lag weights obey the single exact expression
$W_{kk}(d)=\tfrac{2}{N}[(N{-}1{-}d)\cos(\omega_kd)+\sin(\omega_k(d{+}1))/\sin\omega_k]$
(\texttt{v587}), both sides of the defect are one exact geometric-trig
function $G_{ij}(\beta)$ --- the comb side at the pole exponent, the
archimedean side summed over the trivial-zero ladder $\beta_n=(2n{+}\tfrac12)D$
(\texttt{v588}) --- and the pole part is \emph{exactly} rank-one, giving the
closed locking-direction law $v_2/v_1=-(a^2{+}16\pi^2)/(2(a^2{+}4\pi^2))$ with
limit $-\tfrac12$ and the closed determinant law
$\det S\sim640\pi^2e^a/a^3$ (\texttt{v591}/\texttt{v592}); the measured
dominance growth of Problem~\ref{prob:R1}'s floor reduction tracks the closed
formula at correlation $0.9985$.  \emph{Third}, the arithmetic layer is
identified: the residual mass oscillation is the explicit-formula zero comb
(correlation $0.82$ with $200$ damped zeros), the comb recovers a factor $5.7$
of the lock depth at the smallest window, and the recovery plateau is
structural --- finite zero sums cannot replace the bound (\texttt{v589}).
What remains of Problem~\ref{prob:R1} after this dissection is therefore
precisely: (a) the finite-$N$ (Euler--Maclaurin/edge) expansion of the closed
density layer, elementary in character; and (b) a bound on
$\sum_\gamma\sin(\gamma u)/\gamma$ at window resolution with amplitude
accuracy set by the lock depth --- explicit-formula analytic number theory.
None of this is claimed as progress toward any conjecture beyond this paper's
scope; the typing discipline of Table~\ref{tab:typing} applies verbatim.

The rank-3 chain of 2026-08-03 (modules \texttt{v683}--\texttt{v685} of the
accompanying suite) carries this dissection to a theorem-level statement about
the sign of $\det S$ on the declared family.  \emph{First}, the prime influence
on the load-bearing $2\times2$ block is exactly three linear functionals
$S_1,S_2,S_3$ of the comb weights (assembly against an independent sieve to
$4.9\times10^{-16}$); every entrywise absorption route for the razor-thin
margin of Problem~\ref{prob:R1} is typed circular (the margin/budget ratio
$4.7\times10^{-7}$ demands ${\sim}10^{6}\times$ more than per-entry inputs
deliver, the measured collective-cancellation gain being $378$--$712$),
while the surviving det-level compression carries $\kappa_{\max}=0.0895\ll1$
(\texttt{v683}).  \emph{Second}, the one determinant functional is treated by
the explicit formula --- residuum $\le6.9\times10^{-6}$ with $22\,491$
budget-certified zeros to $T=2\times10^{4}$ and the exact per-zero envelope
$2C_G/\gamma^2$ --- giving the \emph{unconditional}
$\kappa_{\mathrm{unc}}=0.039$--$0.190<1$ on all five declared windows; the
pretentious escape is blocked quantitatively by the classical zero-free strip
$\gamma\ge14.134$ (damping $\times634$; \texttt{v684}).  \emph{Third}, the
uniformity quantifier closes on the surface: a symbolic envelope gives
$K_{\mathrm{env}}(a,h)\le0.9798<1$ for all $62$ complete family windows with
$a\ge3.434$ (exact $U_0$ rebooking $M'=M+\Delta$, term-algebra-exact weights,
det-minorant $\ge0.949$), the five windows below carry finite certificates
$\kappa'\le0.108$ --- so $\det S>0$ holds \emph{unconditionally on the entire
declared window surface} (\texttt{v685}).  This is a surface statement about
the declared family, not a bound uniform in the window scale $a$; the
absorption margin of Problem~\ref{prob:R1} itself (the $h^{-3+\varepsilon}$
rate) remains open, and nothing here is claimed as progress toward any
conjecture.

The T-B chain of the same date (modules \texttt{v692}--\texttt{v693})
closes most of what the previous paragraph left open, with the input
typing stated exactly.  \emph{The typing} (\texttt{v692}): the deployed
$2\times2$ lock block decomposes exactly as $\hat
A_2=G_Z+P+\mathrm{tail}$ --- a positive zero-Gram plus a psd rank-1
pole layer --- and the margin identity $\det(G_Z+P)=\det
G_Z+c_P(s_\perp^{\mathsf T}G_Zs_\perp)$ exhibits the razor-thin
absorption margin of Problem~\ref{prob:R1} as the \emph{transverse
zero mass}, a sum of squares; on-line zeros can only increase the
determinant ($2\times2$ psd superadditivity), so the question
compresses to off-line quadruples above the cited verification
height: a quantified partial-RH tail input, neither an envelope
question nor a renaming of the window-positivity conjecture.
\emph{The closure} (\texttt{v693}): with the strongest published
explicit bounds --- Platt--Trudgian $3\times10^{12}$, the explicit
Hasanalizade--Shen--Wong count, functional-equation halving, and the
explicit Ingham-form zero density (arXiv:2507.15184) --- built into a
sharpened safe-side envelope (sinh difference pairing, $V(0)=0$,
on-line subtraction; penalty down $\times6.5$--$14.4$), the
determinant of the lock block is positive
\emph{unconditionally-modulo-citations on $60$ of the $70$ complete
family windows}; the open remainder is exact (nine windows with
required heights $T^*\in[1.0\times10^{13},3.0\times10^{13}]$ and
$h=5690$ at $8.5\times10^{14}$), and the extension beyond the
declared family grows like $e^{3.74a}$.  This closes the
\emph{block-determinant} reading of Problem~\ref{prob:R1} on most of
the family; the $h^{-3+\varepsilon}$ margin rate itself and full
window positivity remain open, and nothing here is claimed as
progress toward any conjecture.

The transverse strike of the same date (module \texttt{v712}) closes
nine of the ten remaining windows.  The margin identity of
\texttt{v692} is \emph{scalarized along the transverse direction}:
the safe-side penalty is paid only by the transverse zero mass
$s_\perp^{\mathsf T}G_Zs_\perp$ rather than by the full matrix norm,
which tightens the certificates by $\times4.23$ (median) exactly on
the open windows.  The census now stands at \emph{$69$ of the $70$
complete family windows closed unconditionally-modulo-citations at
the cited height $3\times10^{12}$}; the single remainder is
$h=5690$, whose required verification height drops from
$8.5\times10^{14}$ to $T^*=1.6\times10^{14}$.  A deck-triplication
lever tested alongside is dead for this purpose (triangle
inequality: channel suprema can only lose).  The zeros
$\le2\times10^{4}$ enter the margin computation only (their count
checked against the Riemann--von Mangoldt formula); the penalty uses
no zero data.  As before, the $h^{-3+\varepsilon}$ margin rate and
full window positivity remain open, and nothing here is claimed as
progress toward any conjecture.

\subsection{Cross-reference table}
\label{sec:crossref}

One reading is quoted in this paper and is not in the table below, because no
module recomputes it: the fitted exponents of Measurement~\ref{meas:rate}. It is
a quantity read off an eigenvalue routine on a surface declared above, is typed
\emph{measured} in Table~\ref{tab:typing}, and is not used as a bound anywhere;
what the modules certify are the identities the reading is a reading of. (The
tightness range of Measurement~\ref{meas:sharpness} and the four numbers of
Measurement~\ref{meas:refwindow}, originally in this list, are now recomputed on
their identical declared surfaces by the readout module \texttt{v563}, which
also exhibits the rung-ladder caveat of Remark~\ref{rem:undecidable} as a
structure fact: the shift is anchor selection.) The definitions and
remarks are likewise absent: a definition is a procedure and a remark is a
reading, and both are typed as such in Table~\ref{tab:typing}.

{\footnotesize\relpenalty=0 \binoppenalty=0
\begin{xltabular}{\textwidth}{@{}llY@{}}
\caption{Every proved statement and every measurement carried by a module, with
the check that recomputes it. Module identifiers refer to the verification suite
accompanying this work; item numbers are those of the module documentation.}
\label{tab:crossref}\\
\toprule
statement & check & what the check recomputes \\
\midrule
\endfirsthead
\toprule
statement & check & what the check recomputes \\
\midrule
\endhead
\midrule
\multicolumn{3}{r@{}}{\emph{continued on the next page}}\\
\endfoot
\bottomrule
\endlastfoot
Lem.~\ref{lem:kms} (parity spectrum)
  & \texttt{v555}--\texttt{v559}, parity control
  & $L_Pt_k=\mu_kt_k$ with an orthonormal basis;
    $\Pi^{\T}\Tm{M}(c^{L})\Pi=L_P$ at zero tolerance \\
Lem.~\ref{lem:interlacing}, \ref{lem:nesting} (interlacing, nesting)
  & \texttt{v559} links~1, 4
  & the two-block Schur floor; a $K$-truncated Rayleigh minimum as an upper
    bound for $\lmin$ and never a lower one \\
Lem.~\ref{lem:gauge} (gauge law)
  & \texttt{v555} item~(4); \texttt{v559} link~3
  & $\sum_dw_d=0$; $\sum_dc_dw_d=x^{\T}Gx$ \\
Lem.~\ref{lem:abel} (Abel swap)
  & \texttt{v555} item~(4); \texttt{v559} link~9
  & the boundary-free identity at levels $k=1,2,3$ \\
Prop.~\ref{prop:cascade} (three ladder forms)
  & \texttt{v557} item~(1) I1--I3; \texttt{v559} link~12
  & prefix from one Cholesky; minor form; regression form \\
Lem.~\ref{lem:diaginv} (diagonal invariance)
  & \texttt{v557} item~(2)
  & invariance of $g_K/g_1$ under $G\mapsto\Lambda G\Lambda$ \\
Prop.~\ref{prop:direction} (ladder direction)
  & \texttt{v556} item~(5); \texttt{v559} link~4
  & $s\ge g_K\ge g_1$ with strictly positive increments \\
Thm.~\ref{thm:tvfloor} (TV floor)
  & \texttt{v555} item~(2), steps T1--T3
  & $w_0=\lVert a\rVert^{2}$; $\TV\ge|w_0|$; $\mu_1\TV\ge1$ \\
Cor.~\ref{cor:holder} (Abel/H\"older)
  & \texttt{v555} item~(4); \texttt{v559} link~9
  & $|Q|\le\TV\max_d|C_d|$ \\
Cor.~\ref{cor:abis} (floor at one coordinate)
  & \texttt{v556} item~(4b)
  & $\TV\ge\lVert v\rVert^{2}\ge(t_1^{\T}v)^{2}$ \\
Cor.~\ref{cor:price} (the price)
  & \texttt{v555} item~(2), step T3
  & \eqref{eq:price} is $\mu_1\TV\ge1$ divided by $\TV$, and that inequality is
    what the step certifies \\
Rem.~\ref{rem:negcontrol} (negative control)
  & \texttt{v555} item~(2), control
  & an inadmissible vector undercuts the floor \\
Thm.~\ref{thm:gauge} (gauge)
  & \texttt{v556} item~(1); \texttt{v559} link~10
  & degree-two homogeneity; ratio equal across sectors \\
Prop.~\ref{prop:shift}, Cor.~\ref{cor:noescape} (conservation, no escape)
  & \texttt{v556} item~(2)
  & floor $\times$ transfer $=1/\mu_1$ at every shift of the declared shift
    list, which is also the corollary \\
Meas.~\ref{meas:battery} (the sector battery)
  & \texttt{v556} item~(1)
  & the ratio across the declared battery of weight families, taper widths and
    shifts; the wrong transport map failing loudly \\
Prop.~\ref{prop:nullmode} (null mode, other sector)
  & \texttt{v555}--\texttt{v559}, parity control
  & the sine family at $k\ge1$ with an orthonormal basis; its parts
    \ref{N:noreps} and \ref{N:even} are closed forms of the same recurrence \\
Prop.~\ref{prop:K2} ($K=2$ exact)
  & \texttt{v557} item~(3); \texttt{v559} link~13
  & the closed vector attains $1/g_2$; $G_{11}g_2=1/(1-r_{12}^{2})$ \\
Prop.~\ref{prop:pivot}, \ref{prop:union} (pivot, entrywise)
  & \texttt{v557} item~(4)
  & the pivot identity; $|x^{\star\T}Ex^{\star}|=\varepsilon\Theta_K$ \\
Lem.~\ref{lem:constrained} (trial direction)
  & \texttt{v557} item~(5); \texttt{v559} links~4, 7
  & every admissible trial is a lower bound for $1/g_K$ \\
Lem.~\ref{lem:wronskian}, \ref{lem:minors} (Wronskian, minors)
  & \texttt{v558} item~(1) I3
  & the telescope with $W_{-1}=W_{h-1}=0$;
    $\tfrac12\sum\mathcal W^{2}=1/(\mu_1\mu_2)$ \\
Lem.~\ref{lem:detread} (determinant reading)
  & \texttt{v558} items~(2), (6); \texttt{v559} link~14
  & the three equal expressions, no positivity used \\
Lem.~\ref{lem:gershgorin} (Gershgorin)
  & \texttt{v558} item~(2)
  & $\nu_1\le\max(\ah_{11},\ah_{22})+|\ah_{12}|$ \\
Prop.~\ref{prop:collapse} ($t^{\star}$ collapse)
  & \texttt{v558} item~(3)
  & the closed form \eqref{eq:collapse}; the harmonic-mean bound \\
Prop.~\ref{prop:split}, \ref{prop:polarisation} (split, polarisation)
  & \texttt{v558} item~(4) TH1--TH3; \texttt{v559} link~15
  & $\Ah_2=B-S$; \eqref{eq:polarisation}; the kernel $\mathcal K$ \\
Thm.~\ref{thm:rank3} (rank three)
  & \texttt{v558} item~(5) TH4; \texttt{v559} link~15
  & $J$, its spectrum, $\rank\mathcal K\le3$, the inertia bound \\
Lem.~\ref{lem:wedge}, Meas.~\ref{meas:l1} (wedge, $\ell^{1}$)
  & \texttt{v558} items~(4), (6)
  & the wedge reading, and its rank-one defect against the declared bar
    $5\cdot10^{-2}$; the $\ell^{1}$-to-signed ratio itself is measured on the
    declared subsample and is not recomputed \\
Cor.~\ref{cor:majorant} (majorants)
  & \texttt{v558} item~(6)
  & the image of a norm ball; the scramble discriminator \\
Lem.~\ref{lem:closedweight}--\ref{lem:spline} (weights, band, spline)
  & \texttt{v558} item~(5), band checks
  & \eqref{eq:closedweight} at zero tolerance on $1\le d\le M-1$; the fit of
    $(w_d)$ in \eqref{eq:bandspace}; the interpolation bound
    \eqref{eq:splinemode} \\
Prop.~\ref{prop:typeII}, Meas.~\ref{meas:typeII} (Type II)
  & \texttt{v558} item~(5) TH5
  & $\sigma_{4K+1}\le\eta\sqrt{RC}$ on an interior block; the measured
    singular-value profile against a generic kernel \\
Prob.~\ref{prob:R1} (the open problem)
  & \texttt{v559} link~16
  & the \emph{shape} is certified per window; the rate and the
    quantifier stay open \\
Lem.~\ref{lem:angle} (angle against determinant)
  & \texttt{v558} item~(2)
  & the ratio \eqref{eq:angleid} and its two-sided bound
    \eqref{eq:anglebound} \\
Lem.~\ref{lem:dualpoint} (dual-point read)
  & \texttt{v558} items~(4), (5)
  & the three reads as two-point reads of the polarised closed weights against
    the direct block; those weights seeing a node only through $\log n$; the
    parity frequencies $\omega_k$ at zero tolerance, the two points of
    \eqref{eq:dualpoints} being $\omega_k/D$ \\
Prop.~\ref{prop:routes} (five majorants)
  & \texttt{v558} items~(5), (6); \texttt{v559} NG1--NG8
  & each majorant re-derived from its hypotheses; Vaughan's identity
    coefficient by coefficient \\
Meas.~\ref{meas:routes} (priced routes)
  & \texttt{v559} NG1--NG8
  & each classified route exhibited failing as classified \\
Meas.~\ref{meas:nogo} (structural obstructions)
  & \texttt{v559} NG1--NG8; \texttt{v558} item~(6)
  & the symbol infimum; the Kato scale and the two radii; the band-local
    ratio; the $L_P$ control at zero tolerance \\
Meas.~\ref{meas:density} (the density curve)
  & \texttt{v562} items~(3), (4)
  & the binned exponent as a deterministic procedure; the ladder
    dependence; the node-table invariance on complete combs, with its
    must-break \\
Lem.~\ref{lem:dcexact}, Prop.~\ref{prop:fh} (placement response)
  & \texttt{v562} item~(1)
  & $\dot c$ exact and piecewise constant; \eqref{eq:fh} against a full
    rebuild on a step ladder with its U-shape and two mutants \\
Meas.~\ref{meas:pole} (causality, the pole)
  & \texttt{v562} items~(1), (2)
  & the slope and its linearity; the sign change inside the period; the
    near-degeneracy at $\vartheta=0$ \\
Meas.~\ref{meas:placement} (the scramble)
  & \texttt{v558} item~(6); \texttt{v560} item~(5); \texttt{v562} item~(2)
  & rank and majorants unchanged; the value moved; the low block lost at
    fixed cell occupancy \\
Meas.~\ref{meas:refwindow} (the reference window)
  & \texttt{v563} item~(1)
  & the declared scan bit for bit ($h=540$, $X=157^{2}$); the four quoted
    numbers; split and polarisation at round-off, with two mutants \\
Meas.~\ref{meas:l1} (the $\ell^{1}$ mass)
  & \texttt{v563} item~(2)
  & the $40.6\times$ ratio on the declared subsample; the kernel double sum
    against $\det S$ (Cauchy--Binet) at round-off \\
Meas.~\ref{meas:sharpness} (the tightness range)
  & \texttt{v563} item~(3)
  & the envelope $[5.00,\,23.20]$ with both ends attained; the floor on all
    $1122$ trial vectors; the inadmissible vector undercutting by exactly
    $\lambda^{2}$ \\
Rem.~\ref{rem:undecidable} (the ladder caveat)
  & \texttt{v563} items~(4), (5)
  & the curve endpoints on identical bin edges; the shift reproduced as
    anchor selection ($13$ vs $12$ anchors), curvature transport small,
    a wrong curvature $11.9\times$ louder \\
Prob.~\ref{prob:R1} (structure typing on the surface)
  & \texttt{v569} items~(1)--(6)
  & the relative pencil $Z=B^{-1}S$ on all $70$ declared windows: the three
    pencil identities at $2\times10^{-14}$; the locking mode $\lambda_2$
    hugging $1$ (median $1.000021$) with the spectator uniformly separated
    ($\min|1-\lambda_1|=3.53$, always above $1$) --- the near-degeneracy
    reduces to the single scalar $1-\lambda_2$ on the reachable surface;
    $h=540$ reads $1-\lambda_2=-2.27\times10^{-5}$; the scramble explodes it
    by $\times1.9\times10^{6}$ (typing only: the uniform quantifier of
    Prob.~\ref{prob:R1} is untouched) \\
Prob.~\ref{prob:R1} (the spectator separation, certified)
  & \texttt{v570} items~(1)--(5)
  & the closed per-window floor $\lambda_1\ge\sqrt{\det S/\det B}$ (or
    $\operatorname{tr}Z$) clears the margin on all $70$ windows,
    $\min F=2.130\ge1.05$ --- the one-mode reduction rests on closed
    scalars, not on an eigenvalue routine; the $h$-uniform dominance
    $\det S\ge(1+c)^2\det B$ stays open (the cheap two-atom witness route
    is killed: negative pair mass up to $13.7\times\det S$) \\
Lem.~(new) the cross-weight formula and the edge Loewner structure
  & \texttt{v576} items~(1)--(6)
  & the polarized cross weights closed in trigonometric form (exact,
    symbolic $N\le11$ + $10^{-11}$ surface); the reflection-edge read a
    scaled Loewner matrix of $U_{s-1}$ (rank $\le s-1$; $s=2$ rank one);
    the exact edge-defect coefficient
    $-\tfrac{3\pi^4}{175}(s^2-4)(11s^2+6)N^{-4}$; the pair-kernel sign
    polynomial $P(s,z)$ with $P(2,3)=0$; the macro kernels
    $W^\infty(\sigma)$ \\
Prob.~\ref{prob:R1} (the locking direction, censused)
  & \texttt{v577} items~(1)--(5)
  & the locking mode aligns with the null ray $(2,-1)$ at
    $23.5$--$33.8^\circ$ on all $70$ windows (below random; scramble
    $+21^\circ$), declining slowly ($h^{-0.10}$): a real partial
    carrier --- the macro regime bends the direction \\
Prob.~\ref{prob:R1} (Regime B, geometric half closed)
  & \texttt{v579} items~(1)--(5)
  & $\operatorname{sign}K_\infty(\sigma,\tau)$ matches the real two-lag
    kernel on every resolved grid cell ($332/332$ at $h=300$); the
    diagonal $\det W^\infty(\sigma)$ negative on ${\sim}72\%$
    (same-scale coupling negative, analytically); the wrong polarisation
    breaks the match --- the $\Lambda$-weighted arithmetic occupation
    stays open \\
Prob.~\ref{prob:R1} (the arithmetic occupation, measured)
  & \texttt{v580} items~(1)--(4)
  & the lag-side identity $\det S=\tfrac12\sum c_{\mathrm{at}}
    c_{\mathrm{at}}D(W,W)$ exact; every significant occupation cell
    sign-matches $K_\infty$ on every tested window --- the primes
    populate the geometric sign map; the open content is quantitative \\
Prob.~\ref{prob:R1} (the transport lemma, killed)
  & \texttt{v581} items~(1)--(4)
  & $40$ doubling pairs: locking-direction drift $0.3$--$6.7^\circ$ per
    octave but growing ($h^{+0.36}$ vs the $h^{-1-\delta}$ bar) --- not
    summable; no fixed-direction scalar transport \\
Prob.~\ref{prob:R1} (the 1D lock projection)
  & \texttt{v596} items~(1)--(4)
  & along the closed direction the zero functional cancels the closed
    density value to median $7.7\times10^{-4}$ (decay $h^{-1.01}$);
    coupling spectrum collective (max share $5.9\%$, $N_{50}=56$);
    the arithmetic content is one explicit identity family \\
Prob.~\ref{prob:R1} (the mapping completion)
  & \texttt{v595} items~(1)--(3)
  & the v593 residual was the $\sigma$ mapping; with
    $a_{\mathrm{eff}}=ND/2$ the closed forms match entries and
    determinant at identity level ($1.00000$--$1.00017$); kernel piece
    $9\times10^{-6}$; the density layer is a finite set of exact
    elementary evaluations \\
Prob.~\ref{prob:R1} (cutoff completion; unconditional certificate)
  & \texttt{v593}/\texttt{v594}
  & the v592 finite-$N$ gap was the integration cutoff: the
    $\sigma_0$-corrected closed forms match the determinant to
    $0.2$--$1\%$ (density layer closed end-to-end, residual
    $O(1/N^2)$); and the entry-level arithmetic deviation is bounded
    unconditionally via B\"uthe 2018 (sup $17.4$, certificate margin
    $565\times$) --- the first unconditional statement \\
Prob.~\ref{prob:R1} (the continuum determinant law)
  & \texttt{v592} items~(1)--(4)
  & $\det S(a)$ closed with the $e^{2a}$ coefficient identically zero;
    exact leading law $640\pi^2e^a/a^3$; the measured dominance tracks
    the closed formula at correlation $0.9985$; entry-level arithmetic
    certificate $\sup|\mathrm{Osc}|\times\mathrm{TV}$ with margin
    $47.6\times$, conditional on the measured oscillation sup \\
Prob.~\ref{prob:R1} (the rank-one pole term)
  & \texttt{v591} items~(1)--(3)
  & the pole part of the closed entries is exactly rank-one
    ($\det\equiv0$ symbolically); the locking direction obeys
    $v_2/v_1=-(a^2+16\pi^2)/(2(a^2+4\pi^2))$ with limit $-1/2$,
    matching the measured directions to $0.04$--$1.4\%$ --- the
    near-collinearity of Prob.~7.1 is the correction theory of a
    rank-one matrix \\
Prob.~\ref{prob:R1} (the zero comb)
  & \texttt{v589} items~(1)--(3)
  & the arithmetic depth layer is identified as the zeta-zero comb
    (correlation $0.82$ with $200$ damped zeros; the comb recovers a
    factor $5.7$ of the lock depth at $h=184$); the zero-oscillation
    bound is a quantified explicit-formula target; no bound claimed \\
Prob.~\ref{prob:R1} (the closed deterministic defect)
  & \texttt{v588} items~(1)--(4)
  & one exact function $G_{ij}(\beta)$ carries both sides -- the pole
    exponent ($S$) against the trivial-zero ladder ($B$); closed $B$
    to $2\times10^{-5}$, the closed defect matches the density layer
    to $0.2\%$ on all $70$ windows with census decay $h^{-1.40}$ ---
    the deterministic half of Prob.~7.1 in closed form \\
Prob.~\ref{prob:R1} (the entry closed forms)
  & \texttt{v587} items~(1)--(4)
  & a new exact formula closes the diagonal lag weights:
    $W_{kk}(d)=(2/N)[(N{-}1{-}d)\cos(\omega_k d)+\sin(\omega_k(d{+}1))
    /\sin\omega_k]$, machine-exact for all lags; the exact cell-sum
    reproduces the prime-free entries and det to $0.03\%$ --- the
    deterministic layer's $S$-side is closed-form \\
Prob.~\ref{prob:R1} (the locking direction, density-fixed)
  & \texttt{v586} items~(1)--(4)
  & the real locking eigenvector agrees with the prime-free model's to
    $0.030^\circ$ median (max $0.331^\circ$, decay $h^{-1.16}$) while
    the depth differs by $1350\times$: the primes are a pure depth
    amplifier; the deterministic direction certifies the defect within
    $\le2.9$; the Prob.~7.4-adjacent near-collinearity question gets
    its direction from the density model \\
Prob.~\ref{prob:R1} (the locking, two-layer split)
  & \texttt{v585} items~(1)--(4)
  & the prime-free model misses the normalised defect by $2$--$4$
    orders (median $1350\times$) while reproducing $\det S$: the
    dominance is density, the locking depth is the primes; layers
    $h^{-1.43}$ (density) $\times$ $h^{-1.08}$ (arithmetic) $=$
    $h^{-2.51}$ real, sign overshoots at $h=1219,1445$ \\
Prob.~\ref{prob:R1} (the dominance, prime-free closed form)
  & \texttt{v583} items~(1)--(5)
  & the two-term classical law $4e^{u/2}-2(\zeta'/\zeta)(1/2)$ (no
    zeros) reproduces $\det S$ on all $69$ regular windows
    parameter-free to $1.016$--$1.143$; the $h=1292$ sign flip is
    genuine prime-fluctuation content; the dominance splits into a
    prime-free two-integral inequality plus a zero-oscillation bound \\
Prob.~\ref{prob:R1} (the dominance, density-driven)
  & \texttt{v582} items~(1)--(4)
  & the smoothed comb reproduces $\det S$ to $99.8$--$99.9\%$ ($2\%$
    window) on every regular window; scrambled control $70\%$ --- the
    real comb is anomalously smooth at these scales; the $h$-uniform
    dominance relocates to PNT-level regularity of the smooth density
    functional; boundary $h=1292$ named \\
Prob.~\ref{prob:R1} (the dominance, typed long-range)
  & \texttt{v573} items~(1)--(5)
  & the pair-band anatomy on a declared subset: every near band
    ($<16$ lag cells) and the diagonal drag negative; the far band
    carries the whole excess with stable net ($[1.55,1.91]\times\det S$,
    gross growing to $13\times$); scale-locked ($\ge h/2$ pairs) and
    weight-concentrated --- the top-$10\%$-weight pairs alone net
    $0.85$--$1.00\times\det S$, essentially $\det S$ itself --- with a
    flat phase profile; the scramble explodes $\det S$
    $\times49$ and collapses the far share --- a certificate for the
    dominance must control heavy-pair window-scale coherence \\
\end{xltabular}}

\subsection{Two remarks on the fits}
\label{sec:fits}

\begin{remark}[the floor exponent against the closed exponent]
\label{rem:fit2}
Measurement~\ref{meas:sharpness} reports the floor of
Theorem~\ref{thm:tvfloor} growing as $h^{+1.997}$ on Surface~II, while
\eqref{eq:gap} gives the closed value $1/\mu_1=1/(4\sin^{2}(\pi/N))$ with
$N=2h+1$, whose growth is exactly $h^{2}$ up to the discrete correction. The
two agree, and the small deficit is that correction: expanding the sine,
$1/\mu_1=(2h+1)^{2}/(4\pi^{2})\,(1+O(h^{-2}))$, so a least-squares fit of
$\log(1/\mu_1)$ against $\log h$ over a finite window range returns
$2-c/\log h_{\mathrm{eff}}$ for a positive $c$ coming from the $+1$ in $2h+1$.
The fit is therefore a consistency check on the surface and not an independent
statement; the closed form is what is used, and the fitted number is quoted
because suppressing it would hide the size of the discretisation.
\end{remark}

\begin{remark}[two reported exponents, one measured object]
\label{rem:twoexponents}
Measurement~\ref{meas:rate} reports $h^{-2.747}$ for $1-r_{12}^{2}$ and
$h^{-2.711}$ for the sine of the angle between the rows of $\Ah_2$, on the same
$67$ windows. These are not two independent readings. By
Lemma~\ref{lem:angle} the two quantities differ by the factor
$\ah_{11}\ah_{22}/(\rho_1\rho_2)$, which lies in $(0,1]$ and equals
$\tfrac12$ times the ratio of the geometric to the arithmetic mean of the two
diagonal entries when $r_{12}^{2}$ is close to $1$; the diagonal ratio
$\ah_{22}/\ah_{11}$ stays in a bounded interval on the surface, so the factor
is bounded above and below there and can shift the fitted exponent only by the
slow drift of that ratio. The observed difference, $0.036$ in the exponent, is
of exactly that size. The two numbers are reported because both readings of
Problem~\ref{prob:R1} are in circulation and a reader should be able to match
either; no regression theorem is claimed, and neither number is used as a
bound.
\end{remark}

\subsection{Reproducibility}
\label{sec:repro}

The checks are grouped into thirty modules, \texttt{v555} to \texttt{v563},
\texttt{v569}--\texttt{v570}, \texttt{v573}, \texttt{v576}--\texttt{v583} and
\texttt{v585}--\texttt{v596}, run in a single suite; \texttt{v563} is the readout closure of
this paper --- it recomputes every number quoted here from an exploration probe
without a recomputing module, on the identical declared surface ---
\texttt{v569} reads the same surface through the relative pencil
$Z=B^{-1}S$: the near-degeneracy of Problem~\ref{prob:R1} is carried by a
single eigenvalue locking to $1$ while the other stays uniformly separated
(a typing of the problem's structure on the reachable surface, not a bound)
--- \texttt{v570} certifies that separation from closed scalars (the
floor $\sqrt{\det S/\det B}$, resp.\ $\operatorname{tr}Z$, clears the margin
on every window), so the one-mode structure rests on determinant/trace
dominance rather than on an eigenvalue computation --- \texttt{v573}
types \emph{where} that dominance lives: every near band of the pair kernel
drags negative and the whole excess is long-range ($\ge16$ lag cells), so a
certificate must control far-pair coherence, not near-diagonal mass ---
\texttt{v576} closes the parity read block analytically (the cross-weight
formula; the Chebyshev--Loewner edge structure with the exact $N^{-4}$
defect coefficient; the pair-kernel sign polynomial: the analytic prototype
of the long-range pattern) --- and \texttt{v577} censuses the locking
direction against the rank-one null ray (correlated on every window,
declining slowly, scramble-sensitive: a partial carrier).
Each module rebuilds its surface from the declared caps
rather than reading stored data, verifies the arithmetic input at every jump
point of the finite table it uses, and carries the mutation controls of
\ref{F:mutation}. Total runtime is under two seconds for the identity modules
and about two minutes for the module that builds Surface~III. Every matrix that
is factorised or inverted anywhere in the suite satisfies the cap of
\ref{F:cond}; the two exceptions --- the truncated-comb stress instance of
Surface~II and the deliberately inadmissible vector of
Remark~\ref{rem:negcontrol} --- are must-break controls, are declared as such,
and no number is read from them except the failure they are built to exhibit.

\section{Limitations, and what is not claimed}
\label{sec:limits}

\begin{enumerate}[label=\textup{(L\arabic*)},leftmargin=3.2em]
\item\label{L:uniform} \textbf{No uniformity in the window index.} Every
  statement typed \emph{certified per window} in Table~\ref{tab:typing} is
  verified window by window on a declared finite surface, and no quantifier over
  windows is claimed. In particular the hypothesis of
  Proposition~\ref{prop:direction} and the positivity hypothesis of
  \eqref{eq:harmonic} are certificates, not theorems.
\item\label{L:positivity} \textbf{Positivity facts are facts about finitely
  many finite matrices.} That $A$ is positive definite on the parity sector on
  the tested windows is a numerical statement with a declared floor. It is not
  routed through, compared with, or used to suggest anything about any
  criterion; Lemma~\ref{lem:detread} exists precisely so that the reduction
  never needs it.
\item\label{L:fits} \textbf{Fits are fits.} Every exponent in
  Measurements~\ref{meas:sharpness}, \ref{meas:rate}, \ref{meas:routes} and
  \ref{meas:density} is a least-squares fit on a declared surface, reported for
  orientation. No fit is used as a bound anywhere. In particular the $\delta$
  column of Measurement~\ref{meas:routes} is a property of the surface: what is
  proved about those five routes is the inequalities of
  Proposition~\ref{prop:routes}, and nothing else.
\item\label{L:exponent} \textbf{The exponent is not one number.} On the
  surfaces of Measurement~\ref{meas:rate} the fitted exponent of
  $1-r_{12}^{2}$ ranges from $-1.27$ to $-2.95$, and the two readings of
  Problem~\ref{prob:R1} give different fits on the same windows
  (Remark~\ref{rem:twoexponents}). Any sentence of the form ``the quantity
  decays like $h^{-\delta}$'' is a statement about a declared surface and a
  declared reading, and this paper never makes it in any other form.
\item\label{L:corner} \textbf{The dense corner is exhausted only up to the
  declared caps.} Measurement~\ref{meas:density} states its own
  undecidability, and that statement is part of the claim: a true zero, a
  power-law approach, and a plateau below $0.074$ are indistinguishable on
  Surface~III. Extending the curve by one decade costs a node table a factor
  $100$ larger, which is outside the declared budget.
\item\label{L:ladder} \textbf{The curve estimator is ladder-dependent.} The
  binned exponent of Definition~\ref{def:ratio} shifts by up to $0.20$ between
  the declared rung ladder and its coarse sub-ladder, because $\log R$ is
  curved in $\log h$ (measured per-anchor curvature $-0.0842\pm0.0085$). Every
  value in Measurement~\ref{meas:density} is quoted with its rung ladder, and a
  value quoted without one is not reproducible. A ladder-robust estimator ---
  integrating the curvature rather than sampling it --- is not attempted here.
\item\label{L:errors} \textbf{Not every standard error is reproduced.} For the
  six older bins of Measurement~\ref{meas:density} the source reports the
  values and the joint density slope but not the individual standard errors,
  and the table says so rather than filling the column. The declared
  monotonicity violation in the fourth bin is likewise reported and not
  smoothed.
\item\label{L:D4} \textbf{One phrase did not convert.} The bandwidth
  $\Rightarrow$ $\eta$-rank mechanism is proved
  (Lemmata~\ref{lem:closedweight}--\ref{lem:spline},
  Proposition~\ref{prop:typeII}), but it delivers $4K$ singular values at a
  declared truncation error and only for blocks satisfying the interior
  hypothesis \eqref{eq:interior}; it does not deliver ``effective rank $O(1)$'',
  and the smaller number actually observed stays a measurement
  (Measurement~\ref{meas:typeII}). Remark~\ref{rem:interior} states what the
  interior hypothesis excludes.
\item\label{L:arith} \textbf{The comb's arithmetic is not used by any
  theorem.} Remark~\ref{rem:combfree} is meant literally: replacing the
  prime-power instance by an arbitrary finite comb changes no proof in
  Sections~\ref{sec:setting}--\ref{sec:rank}. What changes is the \emph{value}
  of the three functionals of Theorem~\ref{thm:rank3}\ref{R:three}, and
  Measurement~\ref{meas:placement} is the controlled experiment that says so.
\item\label{L:placement} \textbf{The placement results are local.}
  Proposition~\ref{prop:fh} is an identity, but
  Measurement~\ref{meas:pole} evaluates it on a declared instance list and at a
  declared displacement scale; $\log R$ has a pole inside the placement period,
  so no global statement over the period is available, and none is made. The
  number $879$ is a derivative at a point on named instances, not a rate.
\item\label{L:scope} \textbf{What is not claimed.} Problem~\ref{prob:R1} is
  open; nothing here bounds it and nothing here is conditional on it. No
  statement of this paper is an arithmetic statement, no comparison against the
  strength of any conjecture appears anywhere, and
  Measurement~\ref{meas:density} claims no limit.
  Remark~\ref{rem:notclaimed} states this in full, and
  Table~\ref{tab:typing} is the binding record of what is claimed at what
  strength: where the prose and the table disagree, the table governs.
\end{enumerate}

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\end{document}