\(s (26)\)$s(26)$ — open

External intake, 2026-10-06. wand125’s check2 source reports \(s(26) \ge 1109/200 = 5.545\)$s(26)\ge 1109/200=5.545$ (T-105), from a density of 477 rectangles of total mass \(2599999/100000 < 26\)$2599999/100000<26$, on a net the certificate declares: core side \(1999/2000\)$1999/2000$ and 832 half-angle tangents of step \(1/2006\)$1/2006$. Since \(1999/2000 \cdot (1 + 1/2006) < 1\)$1999/2000·(1+1/2006)<1$, every unit square contains a core at a net angle strictly in its interior. The source ships no record of its C++ checker for it: it accepts it at every direction with its copy of this repository’s sqverify_fast, changed to read a declared net. It is above the reported \(2213/400\)$2213/400$ (T-068) by \(0.0125\)$0.0125$. This repository’s clean-room verifier sqverify-fast decided it here on 6 October 2026 at all 832 directions of its net, and refused two mutants scaled below coverage one: confirmed, re-implemented sharing the producer’s components (the source’s check is a copy of the same crate), 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 wand125’s rectangle-density source reports \(s (26) \ge 2213 / 400 = 5.5325\)$s(26)\ge 2213/400=5.5325$, since superseded (T-105), with total mass \(2599/100 = 25.99 < 26\)$2599/100=25.99<26$, accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again, 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one and checked the mass and net premises, so it was also verified until 2026-10-06, when the certificate above superseded it in both lanes. 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 \(553/100 = 5.53\)$553/100=5.53$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass \(2599/100 = 25.99 < 26\)$2599/100=25.99<26$, 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.

The Tokoharu density source, retained on 2026-09-22, reports \(s (26) \ge 1377 / 250 = 5.508\)$s(26)\ge 1377/250=5.508$, which superseded wand125’s \(109/20 = 5.45\)$109/20=5.45$ point certificate and was the verified lower bound until 2026-10-02. The mathematical audit records full local interval replay and the independent premise checks, with their limits. That evidence verifies the retained fixed certificate at \(1377/250\)$1377/250$. Tokoharu’s README says parts of the work were produced with AI assistance under human direction.

Open. The verified interval is \(1109 / 200 \le s (26) \le (7 + 3 \sqrt{2}) / 2\)$1109/200\le s(26)\le (7+3\sqrt{2})/2$, with endpoints \(5.545\)$5.545$ and approximately \(5.621320344\)$5.621320344$, leaving a gap of about \(0.0763\)$0.0763$. Its lower end is wand125’s check2 certificate on a declared net (T-105, 2026-10-06, V3/C3), which superseded its rectangle-density certificate of 1 October at \(2213/400\)$2213/400$ (T-074, replayed here on 2026-10-02), that one its replayed \(553/100\)$553/100$ of 27 September (T-045), and that one Tokoharu’s replayed \(1377/250\)$1377/250$. Friedman’s historical Green report, approximately \(5.3918\)$5.3918$, is weaker than all of them; Green’s proof has not been recovered.

The packing

Found by Erich Friedman in 1997, via the off-centre variant of Göbel’s family — the rule of [Friedman DS7] section 3 at \((a, b) = (2, 3)\)$(a,b)=(2,3)$, a rectangle packing with the tilted block centred in the rectangle plus a column of \(2a + 1\)$2a+1$ unit squares against its tall side. cases/gobel_offcentre builds it and verifies it exactly over Q(sqrt 2), which is what moved verified_upper_bound from the grid ceiling onto the exact side. The retained witness declares that side rounded up at its own digits, so the construction fits inside the witness’s declared value (D-398’s comfortable direction), but its layout is the witness’s own; the certificate carries the construction’s coordinates.

Stromquist’s Memo III, Table 1 and Figure 4(b), gives a different construction at approximately \(5.65062919143938\)$5.65062919143938$. It improved earlier packings and is already credited in Ellsworth’s historical catalogue, but it does not improve the current upper bound. The dedicated verification records the author’s clarification and exact geometry checks.

The September 7 current-source audit found no smaller n26 packing in the checked catalogues, literature, or recent public solver and proof projects. Exact normalization shows that the MinMax Arena submission and a recent solver cache are slightly worse than Friedman’s side. The equal-side Ellsworth rearrangements also leave the bound unchanged. This supports the dated best-known claim within the searched sources; optimality remains open.

The lower bound

The verified lower bound is \(1109/200 = 5.545\)$1109/200=5.545$, from wand125’s check2 certificate of 6 October on a declared net (T-105), decided here by sqverify-fast at every direction of its net. Its n26 rectangle-density certificate of 1 October, \(2213/400 = 5.5325\)$2213/400=5.5325$, replayed in full here (T-074), held the field until then, and its certificate of 27 September, \(553/100 = 5.53\)$553/100=5.53$ (T-045), held the field earlier on 2 October. Before it, the complete interval replay and exact premise checks verified Tokoharu’s lower bound \(1377/250 = 5.508\)$1377/250=5.508$, which was the verified lower bound until 2026-10-02. The replay uses the retained source coverage implementation; a method-distinct coverage implementation would add confidence and improve native support for future rectangle-density certificates, but it is not a missing premise of this fixed bound.

Corrected 2 October 2026: Nagamochi’s Lemma 1 is false (Karakuş 2026), so his closed form below is now a reported bound whose published proof is incomplete (review). This register had recorded that proof as verified, its own error, logged as defect D-516. Nagamochi’s earlier general closed form remains part of the evidence history and applies to every \(N \ge 4\)$N\ge 4$:

Friedman’s survey, Theorem 9 at \(k = 5\)$k=5$ and Table 2, reports Green’s \(2 \sqrt{2} + (27 + 2 \sqrt{10}) / 13 \approx 5.3918\)$2\sqrt{2}+(27+2\sqrt{10})/13\approx 5.3918$ for \(n = 26,27\)$n=26,27$. Reference [8] is a private communication from 2000; the survey supplies neither a proof nor the twenty-six-square point configuration. The unavoidable-set argument DS7’s Figure 34 illustrates does not prove it: at \(k = 5\)$k=5$ that point pattern leaves a unit square empty (review). This historical report is stronger than Nagamochi’s bound but weaker than Tokoharu’s replayed \(1377/250\)$1377/250$ certificate and wand125’s \(553/100\)$553/100$. Recovering Green’s proof remains a separate historical provenance question. The follow-up review separates proof recovery from a new independently certified covering argument.

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.