Theorem 5.3 at a fixed pair, stationary form

Davie--Stothers Theorem 5.3 at a fixed pair, in stationary form.

Let $\tau$ satisfy $2\le 3\tau\le 3$, let $a\in Z$ be a strictly positive admissible profile, and let $b\in\mathcal N$ be a strictly positive stationary profile on the same marginal fibre, i.e. $a-b\in Y$. Then for every $V\ge 0$ with

the fourth power $CW_6^{\otimes 4}$ has $\tau$-value at least $V$.

This is the same statement as the slice-infimum form, with the minimality hypothesis on $b$ replaced by membership in $\mathcal N$. The two are interchangeable for the purposes of Theorem 5.3 -- Lemma 5.2 derives minimality from stationarity -- but stationarity is the more useful hypothesis to carry into the extraction: it is exactly what makes $\log b$ an affine function of the nine grade statistics, hence what identifies $b$'s histogram as the maximum-entropy one on the fibre and so what controls the completion-star degree. The factor $\mathcal E(b)/\mathcal E(a)\le1$ is the combination loss of Equation (3.4), and it degenerates to $1$ exactly on the diagonal $a=b$, which is the case the published numerical endpoint uses.