Jain's round-seven construction from its network savings

Jain's round seven, with its finite certificates factored out. Let $h^{\mathrm{bit}}$ be the child-width histogram of the round-seven bit network (witness 1: PR #62 producer, lifted frames, late copies, deferred readouts, V leaves, staircase entrance corner), of width $529$ with $W_b=108516254$ roles, and $h^{\mathrm{cx}}$ that of the round-six complex network, of width $576$ with $W_c=207387136$ roles and rank sum $s_c=\sum_w w\,n^{\mathrm{cx}}_w$. Put $\beta=1/1000$. Let $a_b,a_c,\varepsilon,x,C_1,\kappa$ be real numbers with $a_b,a_c\ge0$ such that

1. $\displaystyle\sum_w n^{\mathrm{bit}}_w\Big(\frac{w}{529}\Big)^{1-a_b}<W_b$;
2. $\displaystyle\sum_w n^{\mathrm{cx}}_w\Big(\frac{w}{576}\Big)^{1-a_c}<W_c$;
3. the crude-guard assembly $\mathrm{CrudeAssembly}(a_b,a_c,\beta,\varepsilon,x,C_1,\kappa,576,s_c)$ holds.

Then integer multiplication is possible in time

by one deterministic multitape Turing machine with fixed alphabet and tape count.

This is the machine-level content of the round: the OpenAI construction, Colkitt's framework and Jain's two networks turn certified network savings into a multiplication algorithm. The finite certificates and the real-analysis steps (rigorous bounds for $\log$ and $\exp$ applied to the frozen histograms) are separate. The source states the result as conditional on the OpenAI manuscript and on Colkitt's framework, and assumes that stage two is the complement time-reversal of stage one.

Formalization Note $\mathrm{KappaBound}(\kappa)$ is from IntMul_MultitapeModel: one deterministic multitape Turing machine with fixed alphabet and tape count, correct for every $n\ge1$, with running time at most $c\,n\,(\lg n)^{1-\kappa}$ for $n\ge n_0$. The moment $\mathrm{histMoment}$, rank sum and $\mathrm{CrudeAssembly}$ are from IntMul_KappaFramework; the histograms are from IntMul_KappaJainData. All hypotheses are real-number statements; the source checks rational majorants of them.