steiner-ratio
Gilbert–Pollak conjecture · August 2026
A certificate-layer bound of ρ ≥ 0.860 for the Steiner ratio
An independent verification of the 2026 Peking certificate, an exact analysis of what it actually proved, and a new certified record — built on their mathematics, checked by independent software, reproducible from a public archive.
The Steiner ratio measures how much shorter a network can be when it is allowed to invent new junctions. Gilbert and Pollak conjectured in 1968 that the answer is √3 / 2 ≈ 0.8660254 — the saving an equilateral triangle already achieves. A celebrated 1990 proof collapsed on inspection; the certified record stood at 0.824 for four decades, until a 2026 preprint by Ke, Huang, Shu, He, Gai and Wang reached 0.8559 with a computer-checked argument over 330 million parameter regions.
This work does three things. It independently re-verifies that certificate — with clean-room software, stricter checking rules, and zero failures across all 330,193,755 records. It computes exactly what the published certificate could have proved, and finds the answer to be 0.8559000606: the bound was pinned by the generator's subdivision tolerance, not by the mathematics. And it turns that dial, regenerating the partitions at a higher target and verifying all 1,104,177,103 resulting regions at the exact rational 43/50 — establishing ρ ≥ 0.860 with no new mathematics, inheriting the same lemma-layer obligations as the original proof.
Soap Films, a Broken Proof, and a New Record
The whole story for a general reader: what the conjecture says, why the 1990 proof failed, how a billion machine checks work, what the audit found, and the single line of code that had been holding the record in place.
An Independent Verification … and a Bound of ρ ≥ 0.860
The full write-up: certificate framework, the verifier's semantics, the audit and its findings, the critical-ρ proposition with proof, the record, and an explicit statement of scope and threats to validity. PDF
Verification Note
The re-verification of the published 0.8559 certificate layer in detail: method, per-case results, the five findings, reproduction instructions and limitations.
Record Note
The 0.860 claim document: what is claimed and what is not, the regeneration method, per-case verification counts, certificate hashes, and the inherited obligations spelled out.
Reproducing it
Everything needed is public. The upstream artifacts are not redistributed — they carry no license — so reproduction fetches them from their original sources and checks them against pinned SHA-256 hashes.
uv sync
uv run python -m steiner_audit.acquire # fetch + hash-verify upstream
uv run python -m steiner_audit.regen --family d_steiner --subcase f_ge_d --rho 0.860
uv run python -m steiner_audit.verify --family d_steiner --subcase f_ge_d \
--cert <certificate> --rho 0.860 --workers 16Code, tests, manifests and per-run records: github.com/aimathpapers/Steiner_Ratio. Permanent archive: doi.org/10.5281/zenodo.22223485.
Credit where it belongs
The lemma set, the case decomposition and the partition generator are the work of Yisi Ke, Tianyu Huang, Yankai Shu, Di He, Jingchu Gai and Liwei Wang (arXiv:2601.22365, 2026). This project contributes an independent verification stack, the exact-rational analysis that identified the binding constraint, and the computation. Nothing here would exist without their paper.