Jain's round-six construction from its network savings

Jain's round six, with its finite certificates factored out. Let $h^{\mathrm{bit}}$ be the child-width histogram of the round-six bit network, of width $529$ with $W_b=148225616$ 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 (copied retained centres, retained point totals and a two-stage complex interchange).

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.