Corrected Theorem 5.3 for an integral profile and a stationary partner

The fourth-power value of a general integral profile, corrected by the combination loss.

Work over a field $K$ at the Coppersmith--Winograd parameter $q=6$ and fix an exponent $\tau$. Let $\beta$ and $\beta^{*}$ be strictly positive integral ten-class profiles with the same nine-grade marginals, $Q\beta^{*} = Q\beta$, and suppose the normalised partner $b = \beta^{*}/D$ is stationary, $b \in \mathcal N$. Write $a = \beta/D$. Assume the ten class constituents have their Lemma 5.1 values, i.e. the cyclically symmetrized constituent of class $r$ has tau-value at least $V$ for every $0 \le V < v_r(\tau)$, and that the fourth-power grading is supported in total degree eight.

Then for every

the literal fourth power $CW_6^{\otimes 4}$ has tau-value at least $V$, where $G(\tau,a)$ is the Equation (5.3) global rate of $a$ against itself.

This is Theorem 5.3 of Davie--Stothers in the form the source actually supports. The published statement of the platform's mme_stothers_theorem53_global_value carries the reciprocal factor $E(a)/E(b) \ge 1$ and is false; the correct factor is $E(b)/E(a) \le 1$, and it is the combination loss recorded as an infimum over the marginal fibre in Equation (3.4). Its mechanism is visible here: the exact-profile targets of $\beta$ number $V_N \Delta_\beta$, but the completion star whose degree Behrend's construction must beat is governed by the maximum-entropy profile on the same marginal fibre, so the surviving family retains only the fraction $\Delta_\beta/\Delta_{\beta^{*}} = (E(b)/E(a))^N$ of them.

On the diagonal $\beta = \beta^{*}$ the correction is $1$ and the statement reduces to the published fixed-profile value; that is why the fixed chain, which lives only on the diagonal, never had to carry it, and why the $\omega < 2.3737$ endpoint it supports is unaffected. What the general form adds is the freedom to vary $\beta$, which is exactly what an optimisation over profiles needs.

Formalization note. Stationarity of $b$ is used only through the Gibbs argument that makes $\beta^{*}$ entropy-maximal on its marginal fibre; no symmetrisation of the histogram is required. The polynomial slack in the star-degree comparison is absorbed by choosing a strict intermediate endpoint, so no additional error constant appears in the conclusion.