$n=308$$n=308$

$s(308)$$s(308)$ — 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=308$$n=308$, $k=17$$k=17$ and interval 28 give $a(308)=17.616620106628016467339740238257\dots$$a(308)=17.616620106628016467339740238257\dots$, so the reported lower bound here is now $s(308)\ge 17.616620106628$$s(308)\ge 17.616620106628$, rounded down, above Nagamochi’s closed form, $17.58312395177$$17.58312395177$. 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, 20 October 2026. Daniel’s arrangement of Fang’s two-wedge design in a search started from Chaoweeraprasit’s n = 307 packing (T-221) is now this case’s house: its exact certificate proves $s(308)\le 17.9971491212709088$$s(308)\le 17.9971491212709088$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.12×10−3$1.12\times {10}^{-3}$. The earlier ceiling, displayed outward as $17.9982698795262558753$$17.9982698795262558753$ from Daniel’s packing, improved by PingYou Ltd (T-174), applied until this adoption and stays retained as history (the certificate).

Reported certificate, 2026-10-19. Daniel reported a 308-square certificate at container side $17.997149121271$$17.997149121271$, rounded upward from the retained exact certificate. The exact rational side is registered in the result register. Its V0/C0 registration left both case upper lanes, the house witness and its drawing unchanged until the house adoption of 20 October 2026, and until then the rigidity and arrangement notes below described that retained house.

The source calls this a new arrangement, found by inserting one square into Chaoweeraprasit’s 307-square packing of #481 and descending with its explorer, and says it neither refines nor descends from Fang’s #484 or Daniel’s #489, whose packing the house refined until 20 October 2026. Its original and regularized display certificates have the same side and yield one result. The registration asserted no measured packing identity or lineage; the review of 20 October 2026 measured it pose by pose: under a quarter turn, 15 of the 308 squares lie beyond a tenth of #489’s packing and 293 within one, so the packing is another arrangement of the two-wedge design, as its certificate’s section says.

House adoption, 17 October 2026. PingYou Ltd’s refinement of Daniel’s packing based on Fang’s two-wedge construction (T-174) was this case’s house until 20 October 2026: its exact certificate proves $s(308)\le 17.9982698795262558753$$s(308)\le 17.9982698795262558753$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.80×10−19$1.80\times {10}^{-19}$. The earlier exact ceiling, $17998269879526255875387992744507/1000000000000000000000000000000$$17998269879526255875387992744507/1000000000000000000000000000000$ from Fang’s packing, improved by Daniel (T-141), applied until this adoption and stays retained as history; the reported upper bound it replaces, $17.998269879526255875387992744507$$17.998269879526255875387992744507$ from [Daniel record hunt 3 2026-10-10], lies 1.80×10−19$1.80\times {10}^{-19}$ above the new side (the certificate).

House adoption, 11 October 2026. Daniel’s move of Fang’s two-wedge construction (T-141) was this case’s house until 17 October 2026: its exact certificate proves $s(308)\le 17.9982698795262559$$s(308)\le 17.9982698795262559$, the reported and the verified upper bound, 1.73×10−3$1.73\times {10}^{-3}$ below the ceiling the case held until then, $18.0$$18.0$ from the trivial grid, which stays retained as history (the certificate).

Open. Previously, the best known packing gave $s(308)\le 18$$s(308)\le 18$, and the strongest lower bound independently verified here is $17.566048$$17.566048$ from Karakuş’s general theorem, leaving a gap of $0.434$$0.434$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +1/4}$.

The packing

The record catalogue does not picture $n=308$$n=308$: until October 2026 no arrangement was known that beats the trivial ⌈308​⌉=18$\lceil \sqrt{308}\rceil =18$ grid, and the catalogue’s conjecture that the grid is optimal here was a statement about what had been searched, not a proof. Fang’s two-wedge construction (T-135, issue 484) beat it first, and Daniel’s move of that construction (T-141, issue 489) is smaller again, and his arrangement of the same design (T-221, issue 689), the house from 20 October 2026, smaller still; the grid stays retained as history.

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-141)

Daniel’s move of Fang’s two-wedge construction, reported on jlevy/squares#489, at exact side $17998269879526255875387992744507/1000000000000000000000000000000$$17998269879526255875387992744507/1000000000000000000000000000000$, displayed upward as $17.9982698795262559$$17.9982698795262559$ (T-141). The certificate gives a rational side and, for each of the 308 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, and a third exact route, half-extent containment and exact intersection area, decided it again. No route here is the source’s code, so the replay is independently re-implemented. The review of 10 October 2026 and the final review of 11 October 2026 accepted the replay for finite feasibility. That proves $s(308)\le 17.9982698795262559$$s(308)\le 17.9982698795262559$, the verified upper bound until 17 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The author says the search code and the request were written with Claude under his direction.

The house it replaces, $18.0$$18.0$ from the trivial grid, stays retained with its own replay as history. The other certificates confirmed here at this count, T-135 at $17.9993098551627480$$17.9993098551627480$, stay verified and are not selected. T-174 at $17.9982698795262559$$17.9982698795262559$, 1.80×10−19$1.80\times {10}^{-19}$ below this house, is confirmed here and accepted by the review of 14 October 2026. From 15 October 2026 until the owner’s house rule of 16 October 2026 lifted the hold, PR #574’s houses were held for the owner, since their witnesses would add about 18.5 MB to a checkout and their custody inputs about 5.8 MB to every worker’s snapshot.

PingYou Ltd’s certificate (T-174)

PingYou Ltd’s refinement of Daniel’s packing based on Fang’s two-wedge construction, reported in pull request #574, at an exact rational side, displayed upward as $17.9982698795262558753$$17.9982698795262558753$ (T-174). The certificate gives a rational side and, for each of the 308 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, and a third exact route, half-extent containment and exact intersection area, decided it again. The source ran both maintained routes itself on its corners witnesses; those runs are its own, and the replay here is independently re-implemented. The review of 14 October 2026 accepted the replay for finite feasibility. That proves $s(308)\le 17.9982698795262558753$$s(308)\le 17.9982698795262558753$, the verified upper bound until 20 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate is PingYou Ltd’s tightening of the packing T-141 records, that packing by the source’s statement, not decided here (T-174’s regularizations), so the packing keeps its finder’s credit and PingYou Ltd is credited with the certificate.

The source makes no statement about AI assistance in this pull request; for its #503 and #504 certificates the author said on 12 October 2026 that they were AI assisted by ‘PY’, an unreleased agentic AI offering by PingYou Ltd.

The house it replaces, $17998269879526255875387992744507/1000000000000000000000000000000$$17998269879526255875387992744507/1000000000000000000000000000000$ from Fang’s packing, improved by Daniel (T-141), stays retained with its own replay as history. The other certificates confirmed here at this count, T-135 at $17.9993098551627480$$17.9993098551627480$, stay verified and are not selected.

Evan Daniel’s certificate (T-221)

Daniel’s arrangement of Fang’s two-wedge design in a search started from Chaoweeraprasit’s n = 307 packing, reported on jlevy/squares#689, at exact side $3599429824254181747030632150463/200000000000000000000000000000$$3599429824254181747030632150463/200000000000000000000000000000$, displayed upward as $17.9971491212709088$$17.9971491212709088$ (T-221). The certificate gives a rational side and, for each of the 308 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, and a third exact route, half-extent containment and exact intersection area, decided it again. No route here is the source’s code, so the replay is independently re-implemented. The review of 20 October 2026 accepted the replay for finite feasibility. That proves $s(308)\le 17.9971491212709088$$s(308)\le 17.9971491212709088$, 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 done with Claude (Anthropic) under human direction, and the commit that publishes the certificate names Claude Opus 5.5 as co-author and says Daniel approved it.

The author’s chained explorer inserted one square into Chaoweeraprasit’s n = 307 packing of #481, the register’s witness for 307, and his basin explorer descended from there to this packing; the issue asks that the 307 packing be credited as the ultimate start, and says the packing neither refines nor descends from Fang’s #484 or Daniel’s #489. Measured by the 20 October review, square by square, the best assignment against #489’s packing under a quarter turn moves 15 squares beyond a tenth, at most 0.708, and keeps 293 within one, and 263 of the 307 squares of #481’s packing lie within a tenth of this one: another arrangement of the two-wedge design that #484, #489 and #481’s 307 share, found from #481’s 307 by the source’s account (T-221).

The house it replaces, $17.9982698795262558753$$17.9982698795262558753$ from Daniel’s packing, improved by PingYou Ltd (T-174), stays retained with its own replay as history. The other certificates confirmed here at this count, T-141 at $17.9982698795262559$$17.9982698795262559$ and T-135 at $17.9993098551627480$$17.9993098551627480$, stay verified and are not selected.