\(s (57)\)$s(57)$ — open

External intake, 2026-10-02. wand125’s rectangle-density source reports the rectangle certificate rect_n57_L7835 at side \(1567/200 = 7.835\)$1567/200=7.835$, since superseded (T-090), with total mass \(5699/100 = 56.99 < 57\)$5699/100=56.99<57$, accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again on 3 October, after this repository’s exact audit checked that the regenerated checker input is the published one and checked the mass and net premises, so it was the verified lower bound from its entry on 6 October (T-070) until T-090’s certificate was decided here later that day. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-05. wand125’s mixed-certificate source reports \(s(57) \ge 3149/400 = 7.8725\)$s(57)\ge 3149/400=7.8725$ (T-090), from a density of 532 rectangles of total mass \(5699999/100000 < 57\)$5699999/100000<57$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the \(1567/200\)$1567/200$ this case reported, by \(0.0375\)$0.0375$. It is the source’s first mixed certificate at this count. sqverify-fast, this repository’s clean-room verifier, decided it here on 6 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, 2026-10-01. wand125’s rectangle-density source reports the rectangle certificate rect_n57_L7835 at side \(1567/200 = 7.835\)$1567/200=7.835$, since superseded (T-090), with total mass \(5699/100 = 56.99 < 57\)$5699/100=56.99<57$, accepted by Tokoharu’s unchanged interval checker. The replayed n56 certificate carries \(s (57) \ge 3113 / 400 = 7.7825\)$s(57)\ge 3113/400=7.7825$ here by monotonicity, which was the verified lower bound until 6 October. 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source reports the rectangle certificate rect_n57_L7835 at side \(1567/200 = 7.835\)$1567/200=7.835$, since superseded (T-090), with total mass \(5699/100 = 56.99 < 57\)$5699/100=56.99<57$, 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 had not yet run here, so the verified lower bound was unchanged until that replay was entered on 6 October (T-070, above). 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 \(39/5 = 7.8\)$39/5=7.8$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass \(5699/100 = 56.99 < 57\)$5699/100=56.99<57$, 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 \(s(57) \le 8\)$s(57)\le 8$, and the verified lower bound is \(s(57) \ge 3149/400 = 7.8725\)$s(57)\ge 3149/400=7.8725$, from wand125’s mixed rectangle-density certificate of 4 October (T-090, decided here on 2026-10-06), leaving a gap of \(0.1275\)$0.1275$. Earlier that day it was wand125’s rectangle-density certificate of 28 September for this count, \(1567/200 = 7.835\)$1567/200=7.835$ (T-070, replayed here on 2026-10-03 and entered on 2026-10-06), and until then wand125’s n56 certificate, \(3113/400 = 7.7825\)$3113/400=7.7825$, by monotonicity (T-074).

The packing

The record catalogue does not picture \(n = 57\)$n=57$: no arrangement has ever been found that beats the trivial \(\lceil\sqrt{57}\rceil = 8\)$\lceil \sqrt{57}\rceil =8$ 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-n057-wand125-mixed-78725-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 6 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.

Earlier on 6 October the verified lower bound was wand125’s replayed rectangle-density certificate for this count, rect_n57_L7835 (T-070, E-wand125-rectangle-2026-09-28-source-replay), as the intakes above say. Before it, wand125’s replayed rectangle-density certificate for 56 squares was carried here by monotonicity (T-074). That had superseded wand125’s exact weighted point certificate for \(n = 56\)$n=56$, replayed here in full (T-044) and carried to \(n = 57\)$n=57$ the same way: deleting one square from a packing of 57 leaves a packing of 56, so \(s(57) \ge s(56) \ge 381/50 = 7.62\)$s(57)\ge s(56)\ge 381/50=7.62$, which was the strongest registered bound once Nagamochi’s formula left the verified lane. Karakuş’s general bound gives only \(7.588723\)$7.588723$ here.

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 (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516. Nagamochi’s value had masked the registered bound above. The review’s text would put Karakuş’s weaker bound here; the record keeps the strongest registered verified bound instead.

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.