External-theorem dependency and reproducibility

The circuit reduction, synthesis correctness, branch-and-bound certificates and nonlinear cost frontier
are proved here without any ETH-refutation premise. The conditional counting-time corollary invokes
the fixed-density part of the supplied October25 OpenAI preprint, Theorem1.1[7]. Its rational tensor
adapter invokes that preprint’s TheoremD.1. These are external theorem assumptions, not conclusions
of the prototype checks.
The cited official Lean scope description reports the weaker exponent
Ok(n(log log n)2/√log n) and distinguishes the sharper printed bounds[8]. The official release describes
ongoing external examination[9]. The supplied Anthropic manuscript reports formal checking but
explicitly leaves statement correspondence and mathematical interpretation to readers[3], AppendixE.
The supplied OpenAI manuscript is dated 17 October 2026, revised 25 October 2026; the supplied
Anthropic manuscript is dated 26 October 2026. Their exact input hashes and inspected theorem/page
locators are recorded in the supplement. We inspected the relevant printed interfaces and primary-source
scope descriptions; we did not build those formalizations, establish identity between current repository
files and every pinned version, or independently reconstruct their entire proofs. No sharper theorem is
labelled machine-verified here solely because a related Lean statement exists. The supplement supplies
inspected source code and exact integer/rational tests, source-pair witnesses, menu instances, covering-
tree certificates, weighted examples and independent enumeration checks. Primary implementations are
conventional DPLL, branch-and-bound, enumeration and variable elimination. Their finite execution
counts and benchmark timings do not implement the preprints’ asymptotic algorithms or establish a
practical general SAT speedup. Exact rational arithmetic, asserted checks and actual oracle access
remain part of the reproducibility contract.