Completed Weil positivity through support log 7

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.

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.

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.

1. Represent each endpoint operator by exact action and moment data on the cells cut by support and reflection walls.
2. Lower-bound the omitted high modes and certify the finite Schur complement by directed Arb arithmetic.
3. Use the exact dilation identity and min-max to inherit positivity for every smaller support in the chamber.

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.

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.