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\title{Completed Weil positivity through support log 7}
\author{Kenan}
\date{2026-07-18}

\begin{document}
\maketitle

\begin{abstract}
The completed centered Weil quadratic form is strictly positive for every nonzero real even test function supported in $[-R,R]$ when $R\le (\log 7)/2$. The proof identifies the form with a finite-source radial continuum operator, certifies positivity at the chamber endpoints, and transports the endpoint bounds to smaller supports by an exact dilation. A related exact matrix family is uniformly eventually positive on a shorter parameter interval.
\end{abstract}

\[
\mathcal Q_\xi(\phi)>0\quad\left(0\ne\phi\ \text{real even},\ \operatorname{supp}\phi\subset[-R,R],\ R\le\tfrac12\log 7\right)
\]

\section{What is proved}

Write $q=4/t$ for the autocorrelation-support parameter. The complete radial continuum form $\mathfrak A_t^{\rm full}$ is strictly positive for every $t\ge 4/\log 7$, which is equivalent to completed centered Weil positivity for real even tests supported in $[-R,R]$ with $R\le (\log 7)/2$.

For the exact Jacobi-whitened matrices $W_n(tn)$, there is also a uniform eventual-positivity theorem for $t\in[4/\log 6,7.2056493]$. Every fixed ray $t\ge4/\log2$ is eventually positive as well.

\[
\mathfrak A_t^{\rm full}\succ0\qquad\left(t\ge\frac4{\log7}\right)
\]

\[
\exists N\ \forall n\ge N\ \forall t\in\left[\frac4{\log6},7.2056493\right],\quad W_n(tn)\succ0
\]
\section{The Weil-operator dictionary}

The Euler-free operator $\mathfrak A_t^+$ is corrected by the finitely many prime-power translations whose logarithms lie below $q=4/t$. With $b_d=\Lambda(d)/(2\sqrt d)$ and $a_d(t)=t\log d/2$, the active terms are compressed even translations $C_{a_d(t)}$.

An exact intertwining identity identifies this radial form with one half of the completed centered Weil form. Thus the operator inequality is not an analogy or a Galerkin surrogate: it is the bounded-support Weil statement itself.

\[
\mathfrak A_t^{\rm full}=\mathfrak A_t^+-\sum_{\log d<4/t}\frac{\Lambda(d)}{2\sqrt d}\,C_{t\log(d)/2}
\]

\[
\mathfrak A_t^{\rm full}(f)=\tfrac12\mathcal Q_\xi(\phi_{t,f})
\]
\section{Endpoint certificates and chamber closure}

At the endpoint $q=\log7$, the active atoms are $2,3,4,5$. Their exact piecewise action on shifted-Legendre modes is used to form the head matrix, the head-to-tail Gram term, and a Feshbach-Schur complement. High-precision interval congruence proves a positive tail floor and a positive finite Schur matrix.

The same architecture supplies endpoint floors for the preceding atom chambers. An exact cutoff-subspace dilation embeds every smaller support into the endpoint space, so the min-max principle transports each endpoint eigenvalue floor across its entire chamber.

\[
(J_{q,Q}f)(x)=\sqrt{Q/q}\,f(Qx/q)\mathbf1_{(0,q/Q)}(x)
\]

\[
\mathfrak A_{4/q}^{\rm full}(f)=\mathfrak A_{4/Q}^{\rm full}(J_{q,Q}f),\qquad q\le Q
\]

\begin{enumerate}[leftmargin=*]
\item Represent each endpoint operator by exact action and moment data on the cells cut by support and reflection walls.
\item Lower-bound the omitted high modes and certify the finite Schur complement by directed Arb arithmetic.
\item Use the exact dilation identity and min-max to inherit positivity for every smaller support in the chamber.
\end{enumerate}
\section{Transfer to the finite matrices}

Each Euler atom in $W_n(r)$ is an exact half-integer cosine compression. After adjacent parity pairing, the alias branch is suppressed on the critical frequency scale, while logarithmic-frequency compactness sends the primary branch to the corresponding continuum translation.

Mosco compactness, aggregate control of later atoms, and a parity-minus Schur floor tending like $\tfrac12\log n-O(1)$ transfer the continuum gaps uniformly through the chambers ending at $q=\log6$. The argument proves existence of a threshold $N$ but extracts no numerical value for it.
\section{What the pinned certificate checks}

The public command runs the pinned outer-chamber certificate only. It covers $t\in[4/\log2,7.2056493]$ by 424 adjacent rational interval cells and proves, on every cell, a positive tail floor, positive Schur determinant, positive congruence pivots, and nonzero preconditioner determinants.

This certificate is one of six chamber certificates used by the canonical claim. The log-3 through log-7 endpoint certificates, nesting arguments, and finite-transfer proofs are separate named source files in the claim record; the pinned outer-chamber run is a reproducible anchor, not by itself a certificate of every assertion above.

\[
\tau_t>1.3514852913,\qquad \det S_t>1.1750152695\times10^{-3}
\]

\section{Pinned certificate}
The pinned 512-bit Arb program certifies the scalar-plus outer chamber uniformly on 424 exact interval cells. It supplies one load-bearing chamber anchor; the canonical theorem also relies on separately named endpoint, nesting, and finite-transfer certificates.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0007/certify_interval_operator.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0007/certify_interval_operator.py}
\item \path{canon/witnesses/C-0007/PIN.md}
\item \path{canon/witnesses/C-0007/interval_operator_certificate.json}
\item \path{canon/claims/C-0007-chamber-ladder-certified-range.md}
\end{itemize}

\section{Scope}
Continuum positivity is established through autocorrelation support $\log 7$; uniform eventual finite-matrix positivity is established through $\log 6$ with a non-effective threshold.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0007-chamber-ladder-certified-range.md}
\item Pinned certificate description: \path{canon/witnesses/C-0007/PIN.md}
\end{itemize}

\end{document}
