Jain's round-eight construction from its network savings

Jain's round eight, with its finite certificates factored out. Let $h^{\mathrm{old}}$ be the child-width histogram of the round-seven bit network (witness 2, round-six entrance corner), $h^{\mathrm{bit}}$ that of the round-eight bit network (opposite bank orders on the same witness), and $h^{\mathrm{cx}}$ that of the round-eight complex network (signed reclaim and completed-core sharing), with rank sum $s_c=\sum_w w\,n^{\mathrm{cx}}_w$. Put $\theta=\beta=1/1000$. Let $a_{\mathrm{old}},a^*,a_b,a_c,\varepsilon,x,C_1,\kappa$ be real numbers with $a_{\mathrm{old}},a^*,a_c\ge0$ such that

1. $\displaystyle\sum_w n^{\mathrm{old}}_w\Big(\frac{w}{529}\Big)^{1-a_{\mathrm{old}}}<104719230$;
2. $\displaystyle\sum_w n^{\mathrm{bit}}_w\Big(\frac{w}{529}\Big)^{1-a^*}<104719230$;
3. $a_b=(1-\theta)a^*+\theta a_{\mathrm{old}}$ and $a_b<\theta$ (the stopped bit interchange);
4. $\displaystyle\sum_w n^{\mathrm{cx}}_w\Big(\frac{w}{576}\Big)^{1-a_c}<12869228$;
5. 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 Jain's round eight: the OpenAI construction, Colkitt's framework and Jain's 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 lists as assumed interfaces: stage two is the complement time-reversal of stage one; the normal form for alternating residuals (icekylinx, PRs #24 and #104); binary address adapters for the exterior children of the sharing; and the row stock of the stopped bit 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.