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s(261) ≤ 16.6787798753429098149372205115301919485639: further certificate refinement #567
Description
Updated 13 October 2026 — further refinement of this existing submission. Please register the improved feasible upper bound
s(261) ≤ 16.6787798753429098149372205115301919485639.
The decimal is rounded upward from the complete exact rational witness. Compared with the witness in our previous submission, the side is reduced by approximately 2.0196044479e-13. The current files below supersede that version.
Method. The inherited coupled-contact realization is first made feasible by an exact common-centre/host separation repair. The final centre refinement retains all 261 owners and all frames of that repaired intermediate witness, using the complete physical support-inequality system. Exact rational repair and cropping produce the final certificate. The reported bound includes both repairs; it is not presented as an exact solution of every original contact equation.
Current files.
- Exact rational certificate
- Squares Project Witness/v2
- Portable package with method, provenance and verification receipts
Certificate SHA-256: f850cb35c31e018d5f6fdde90940f3d55ad36ecdb88341b805af47639c7e16c2.
Verification and reproduction. Extract the package, enter its top-level directory, and run python3 VERIFY.py using Python 3.10 or later, without -O. All 261 unit squares, all wall conditions and all 33,930 unordered pairs pass exact checks. Separate rational polygon-intersection checks agree. Both unchanged Squares Project native verifiers at commit 074ac28d4ceb9a1897303f936d794771cfc6a7bd accept the native witness. The package retains the relevant receipts and replay instructions. Boundary contacts are permitted; no overlap tolerance is used.
This strengthens the upper bound for the existing record; it does not claim a new value of N or prove local or global optimality. Maintainer review is pending.
Credit and lineage. Ryan Xu: original layout. Robert Barnett: preceding refinement. PingYou Ltd: this further refinement. Evan Daniel: contact-solver source. Upstream software and schema licenses and credits are retained in the package.
Activity
[-]s(261) ≤ 16.678779875343112: exact certificate refinement[/-][+]s(261) ≤ 16.6787798753429098149372205115301919485639: further certificate refinement[/+]BuddenPing commentedon Oct 12, 2026
Updated this submission to s(261) ≤ 16.6787798753429098149372205115301919485639, improving the previously submitted exact witness by approximately 2.0196044479e-13. The issue body now links the replacement certificate, native witness and verified portable package, pinned to publication commit
e70342d149f9363b32ef6043ecfa8d0e0c4b51aa.The final bound includes the inherited-contact repair and subsequent centre refinement with exact repair and cropping. All exact geometric checks, all 34,974 declared support inequalities and both pinned native verifiers pass. Further refinement credit: PingYou Ltd, with the prior construction and software lineage retained. Please use this version for registration.
jlevy commentedon Oct 14, 2026
@BuddenPing, thank you. We have the n261 certificate from your update of 12 October, pinned at e70342d (n261.cert.json, SHA-256 f850cb35…). We retain it as derived exact facts; its size needed our importer's limits raised. It replayed and passed both of this project's maintained exact checkers, and both refused the two controls. As your release says it permits, the certificate has exact contacts: some square touches the wall and some pair of squares touch, and both checkers admit them. Its side is 2.979e-13 below the current register entry at 261, which keeps its own claim. Registration comes next. PR #574 states a further n261 certificate; it is a separate import.
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jlevy commentedon Oct 14, 2026
@BuddenPing, thank you; this is the report-order evaluation of every open report at the counts #567 claims.
At each count, a report is recorded when its side is decidably below every side reported before it (and below the record on main); a tie stays with the earlier report. Sides are rounded up at 12 places.
The registrations are in the records pull request #602, and a follow-up with the result ids and rungs will come once it is on main.
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jlevy commentedon Oct 14, 2026
@BuddenPing, thank you. With #602 merged, your n261 refinement is registered on
main:Evidence:
E-pingyou-567-certificate-report(your report, recorded). Onmain: RESULTS.md, the packet.Generated by Claude Code
intake: retain and replay PingYou Ltd's seven certificates of 12 October
jlevy commentedon Oct 15, 2026
@BuddenPing, thank you. Where this request stands in the record on
main.Corrections to earlier replies.
What is registered.
On the case page. Our reply of 14 October said adoption would follow the review. On 15 October 2026 T-169 became the house at 261; Ryan Xu's packing as Robert Barnett improved it (T-150) stays retained as history.
On
main.packing/frontier/RESULTS.md,docs/project/reviews/review-2026-10-14-upper-bound-imports-561-593.md,packing/frontier/n-261.md.Nothing you asked for is queued, so this issue is closed with this comment.
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test: hosted CI's findings on #718 (a3ff722)