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Problem 43 (Steiner ratio): certificate-layer lower bound 0.860 - #159
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Independently verified re-run of the [KHSHGW2026] certificate layer plus regeneration at a higher target: 1,104,177,103 regions verified at the exact rational 43/50, zero failures. Conditional on the same lemma layer. DOI 10.5281/zenodo.22223485
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- 87a: the depth criterion is [HMR2019, Proposition 1.3(iii)], not 1.8(iii). Also softened the reference to the question opening §5, whose number I could not confirm: ar5iv renders it as an unexpanded \the\smf@thm macro, and the paper's own §3.3.4 says only "see the beginning of §5". - 43a and README: asterisk the 0.860 Steiner ratio bound. The source is a self-archived preprint and the bound is, as the PR says plainly, conditional on the [KHSHGW2026] lemma layer, so the README keeps 0.8559 as the certified value and shows 0.860* alongside it. Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
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This updates the Gilbert-Pollak lower bound from 0.8559 to 0.860 (exact rational 43/50), a certificate-layer result built directly on the machinery of the current record holder, [KHSHGW2026] (arXiv:2601.22365).
What was done. Their lemma set and case decomposition are used unchanged. Their published partition generator was re-run with only its target-rho constant modified (a one-line diff against the pinned upstream commit
709673a8), and every one of the 1,104,177,103 regenerated regions was verified non-positive at the exact rational 43/50 by an independently implemented verifier — Arb ball arithmetic with outward rounding, exact-rational rho, all-2^n-vertices semantics; 244,138,895 regions were additionally certified over their entire box. Zero failures, zero invalid records, zero inconclusive enclosures.The same verifier first independently re-verified the published 0.8559 certificate layer in full (330,193,755 records, zero failures) — to my knowledge its first independent replication. That audit also established, by an exact per-record critical-rho computation, that the published partition's supremal certifiable bound is 0.8559000606: the generator subdivides only until each certificate clears zero by 1e-6, so the partition was pinned to its target rather than to the mathematics. That is what makes the higher target reachable by regeneration alone.
Claim scope, stated plainly. Like the incumbent entry, this is a computer-assisted certificate result resting on the lemma layer of [KHSHGW2026] (trapped-point bounds, case-decomposition topology, monotonicity/unimodality claims). Those claims concern geometry and never rho, so they transfer from 0.8559 to 0.860 unchanged; the 244.1M regions closed over their whole box do not need the unimodality claim at all. Partition coverage is the generator's construction, exactly as in the published work. The comment column and the constants page are worded accordingly rather than presenting this as an unconditional theorem.
Reproduce it. https://github.com/aimathpapers/Steiner_Ratio (MIT), archived at https://doi.org/10.5281/zenodo.22223485. Four commands: fetch the pinned upstream sources (hash-verified — nothing unlicensed is redistributed), regenerate a partition at 0.860, verify it at exact 43/50, run the margin analysis. Certificate SHA-256 hashes and per-run records are in the bundle; the full paper is in
paper/.Disclosure. The verification software, the analysis and the paper were produced by an AI system (Claude Fable 5, Anthropic) working under my direction; every mathematical claim rests on certified interval arithmetic and is reproducible from the archived artifact. The authors of [KHSHGW2026] were notified before this PR was opened. All credit for the underlying method is theirs; this contribution is the independent verification stack, the exact-rational analysis that identified the subdivision tolerance as the binding constraint, and the computation.