The FK record and the missing verification step

For the FK model in the supplied preprint, set

and define

The decoration-summed FK weight is $tw(M)$; the extra factor is constant across maps. Joining two maps at one vertex adds their exponents in this formula, so $w$ is multiplicative. The one-link and one-loop block weights are each $1+t$, giving $b_1=2(1+t)>0$.

The supplied preprint reports:

1. the exact word coefficient identity;

2. a Catalan upper bound giving $T(u)<\infty$;

3. strict inventory drift for $p>1/2$;

4. ladder-cut matching giving (eq:lowerinput);

5. the exact corner-substitution representation.

These are attributed source claims OpenAI211 (Proposition 2.2, Proposition 2.4, Proposition 3.1, Lemma 4.1, and Proposition 4.3). They are not independently verified by recording them.