\(s (76)\)$s(76)$ — open

External intake, 2026-10-05. wand125’s mixed-certificate source reports \(s(76) \ge 1793/200 = 8.965\)$s(76)\ge 1793/200=8.965$ (T-091), from a density of 312 rectangles of total mass \(7599999/100000 < 76\)$7599999/100000<76$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(224/25\)$224/25$ this case reported, by \(0.005\)$0.005$. It supersedes the source’s mixed_n76_L896. sqverify-fast, this repository’s clean-room verifier, decided it here on 5 October 2026 at all 201 net directions, so it is also the verified lower bound: confirmed, independently re-implemented. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-03. wand125’s mixed-certificate source reports the mixed certificate mixed_n76_L896 at side \(224/25 = 8.96\)$224/25=8.96$ (T-082), since superseded (T-091), from a density of 341 rectangles of total mass \(7599999/100000 < 76\)$7599999/100000<76$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(447/50\)$447/50$ this case reported, by \(0.02\)$0.02$, and supersedes the source’s mixed_n76_L894. sqverify-fast, this repository’s clean-room verifier, decided it here on 6 October 2026 at all 201 net directions: confirmed, independently re-implemented. The verified lower bound is unchanged, the certificate above being higher. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-02. wand125’s mixed-certificate source reports \(s(76) \ge 447/50 = 8.94\)$s(76)\ge 447/50=8.94$ (T-072), since superseded (T-091), from a density of 317 rectangles of total mass \(7599999/100000 < 76\)$7599999/100000<76$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It supersedes the rectangle certificate of the 2026-10-01 intake below. Its complete replay here on 2 October 2026, of the source’s unchanged checker on the pinned bundle, returned the certificate’s own record at all 201 directions, so it is also the verified lower bound. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-01. wand125’s rectangle-density source reports a direct \(357/40 = 8.925\)$357/40=8.925$ certificate for this case, whose reported bound the 2026-10-02 intake above raises, with total mass \(7599/100 = 75.99 < 76\)$7599/100=75.99<76$, accepted by Tokoharu’s unchanged interval checker. The replayed n75 certificate carried \(889/100 = 8.89\)$889/100=8.89$ here by monotonicity, the verified lower bound until the 2026-10-02 intake above. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. wand125’s rectangle-density source reports a direct \(223/25 = 8.92\)$223/25=8.92$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(7599/100 = 75.99 < 76\)$7599/100=75.99<76$, accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay has not yet run here, so the verified lower bound is unchanged. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-27. wand125’s rectangle-density source reports a direct \(89/10 = 8.9\)$89/10=8.9$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(7599/100 = 75.99 < 76\)$7599/100=75.99<76$, accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay has not yet run here, so the verified lower bound is unchanged.

Open. The best known packing gives \(s(76) \le 9\)$s(76)\le 9$, and the verified lower bound is \(s(76) \ge 1793/200 = 8.965\)$s(76)\ge 1793/200=8.965$, from wand125’s mixed rectangle-density certificate of 4 October, leaving a gap of \(0.035\)$0.035$.

The packing

The record catalogue does not picture \(n = 76\)$n=76$: no arrangement has ever been found that beats the trivial \(\lceil\sqrt{76}\rceil = 9\)$\lceil \sqrt{76}\rceil =9$ grid, so the grid is still the best known packing. That is a statement about what has been searched, not a proof.

The lower bound

The verified field rests on a complete decision here (E-n076-wand125-mixed-8965-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 4 October by sqverify-fast, this repository’s clean-room measure verifier: all 201 net directions of the retained candidate on 5 October 2026, with two mutants scaled below coverage one refused. It shares no code with the source’s checker, so it decides coverage a second way with the same method; the source’s own checker was not run here.

The certificate that held the field before it, mixed_n76_L894, is a density of 317 uniform rectangles with no point mass, of total mass \(7599999/100000 < 76\)$7599999/100000<76$, at core side \(9977/10000\)$9977/10000$ on 201 net half-angles of step \(83/40000\)$83/40000$, such that the shrunken core of every rotated unit square at each net direction captures mass at least \(1\)$1$. The source’s research copy of Tokoharu’s interval checker, which accepts coverage at least \(1\)$1$, decides the oblique directions, and exact integer tables the axis direction.

Until 5 October 2026 the verified field rested on a complete replay here (E-n076-wand125-mixed-894-source-replay, V3/C3): the pinned bundle fetched and checked by digest, and all 201 directions replayed with the source’s unchanged checker and per-angle functions on 2 October, each returning the certificate’s own record (3.64 CPU-hours). It runs the source’s own algorithm, so it confirms the source’s run rather than deciding coverage a second way. The checker refuses two mutated copies of the source’s \(n = 37\)$n=37$ certificate of the same kind, and the review of 2 October found no defect in this one.

Earlier lower bounds

wand125’s n75 rectangle-density certificate, replayed here, gave \(s (76) \ge 889 / 100 = 8.89\)$s(76)\ge 889/100=8.89$ by monotonicity, which was the verified lower bound until 2 October 2026. Before 2026 it was Nagamochi’s general closed form, now a reported bound (see the correction below). Karakuş’s general bound, independently verified here and below this certificate at this \(n\)$n$, applies to every nonsquare \(N \ge 8\)$N\ge 8$:

Correction, 2 October 2026. Until that date the verified lower bound here was Nagamochi’s general closed form, \(s(N) \ge \min(\lceil\sqrt{N}\rceil, \sqrt{N - 2\lfloor\sqrt{N}\rfloor + 1} + 1)\)$s(N)\ge min(\lceil \sqrt{N}\rceil ,\sqrt{N-2\lfloor \sqrt{N}\rfloor +1}+1)$, which is stronger at this \(n\)$n$ and is now recorded as a reported bound. Its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false; nothing is disproved, and no packing beating it is known (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516.

See the Frontier corpus summary for the current aggregate count; source-reported bounds are recorded separately.

Verification Code

The programs behind this case’s verified bounds, by their evidence. The code column says how the code that ran stands to the code its producer used. VERIFIERS.md says what each program is and whose it is.