$n=183$$n=183$

$s(183)$$s(183)$ — open

Hosono’s closed form, 14 October 2026. Hosono’s closed form (wand125, 13 October 2026, T-180) states $s(N)\ge a(N)$$s(N)\ge a(N)$ for every non-square $N\ge 49$$N\ge 49$, from a measure of line segments on 34 intervals of the fractional part. At $n=183$$n=183$, $k=13$$k=13$ and interval 28 give $a(183)=13.613833531794956650722991072135\dots$$a(183)=13.613833531794956650722991072135\dots$, so the reported lower bound here is now $s(183)\ge 13.613833531794$$s(183)\ge 13.613833531794$, rounded down, above Nagamochi’s closed form, $13.56980508997$$13.56980508997$. The source replayed its certificate with its own checker; no record of it has been replayed here, so the verified lower bound is unchanged. The source states that its search, certificates, checkers and text were produced with AI assistance (Claude and Codex), under the direction of and reviewed by Hosono.

House adoption, 19 October 2026. Daniel’s basin-explorer packing after Couzo’s n = 182 packing (T-220) is now this case’s house: its exact certificate proves $s(183)\le 13.9995801787362526$$s(183)\le 13.9995801787362526$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 4.20×10−4$4.20\times {10}^{-4}$. The earlier ceiling, displayed outward as $14.0$$14.0$ from the trivial grid, applied until this adoption and stays retained as history (the certificate).

Original certificate registered, 19 October 2026. Daniel’s original rational certificate (T-220) has exact side $13.999580178736252502081470897571<14$$13.999580178736252502081470897571<14$, which issue 680 prints upward as $13.999580178737$$13.999580178737$. Its finite feasibility was independently re-verified by this project’s two maintained exact geometry implementations, with complete pair and wall coverage and both invalid controls refused, and accepted in the separately prompted mathematical review. The routes share parsing, half-angle conversion, Fraction, runtime and the separating-axis theorem; they establish neither a clean room nor author or model diversity. The source’s same-side regularized certificate is retained but was not executed here, and neither its identity with this packing nor a path between them is confirmed. The adoption above made the original certificate this case’s house the same day, and its regularization is recorded with the atlas’s regularized views.

Open. Daniel’s packing (T-220) gives $s(183)\le 13.9995801787362526$$s(183)\le 13.9995801787362526$, and the strongest lower bound independently verified here is $13.547988$$13.547988$ from Karakuş’s general theorem, leaving a gap of $0.4516$$0.4516$. Until 19 October 2026 the trivial grid gave the ceiling, $s(183)\le 14$$s(183)\le 14$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +1/4}$.

The packing

The record catalogue does not picture $n=183$$n=183$. Until 19 October 2026 this case’s house was the trivial ⌈183​⌉=14$\lceil \sqrt{183}\rceil =14$ grid. The house is now Daniel’s original certificate, whose witness the atlas draws (the certificate).

The lower bound

The strongest lower bound independently verified in this record is Karakuş’s general bound, which 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)≥min(⌈N​⌉,N−2⌊N​⌋+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.

Source-reported bounds are recorded separately from this independently verified theorem.

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.

Evan Daniel’s certificate (T-220)

Daniel’s basin-explorer packing after Couzo’s n = 182 packing, reported on jlevy/squares#680, at exact side $13999580178736252502081470897571/1000000000000000000000000000000$$13999580178736252502081470897571/1000000000000000000000000000000$, displayed upward as $13.9995801787362526$$13.9995801787362526$ (T-220). The certificate gives a rational side and, for each of the 183 unit squares, a rational centre and the tangent of its half-angle; the retained packet holds it with the complete native jobs that decided it.

Both maintained exact routes decided it here, sqpack’s exact separating-axis verifier and the independent rational corner checker, with its duplicate-square and outside-container controls; no third exact route has decided it. No route here is the source’s code, so the replay is independently re-implemented. The review of 19 October 2026 accepted the replay for finite feasibility. That proves $s(183)\le 13.9995801787362526$$s(183)\le 13.9995801787362526$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The source’s credits say its work was produced by Claude (Anthropic) under human direction, and the commit that publishes the certificate names Claude Opus 5.5 as co-author.

The author asks that the start be credited to Couzo’s n = 182 packing, this register’s witness of the time, and it is: his basin explorer ran 17 moves at n = 182 from that witness to a packing that was no record, a square was inserted into it, and a tilt-angle swap, two two-parent crossovers and refinement led to this certificate. The crossovers’ second parents, other packings of the same run, were not logged, so that lineage is the author’s account and is not measured here. The source’s 27 numerical candidate flat modes concern the 182-square force-bearing subsystem, excluding a free rattler, and prove no local optimality.

The house it replaces, $14.0$$14.0$ from the trivial grid, stays retained with its own replay as history.