$n=302$$n=302$

$s(302)$$s(302)$ — 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=302$$n=302$, $k=17$$k=17$ and interval 13 give $a(302)=17.433426084942247323703832494963\dots$$a(302)=17.433426084942247323703832494963\dots$, so the reported lower bound here is now $s(302)\ge 17.433426084942$$s(302)\ge 17.433426084942$, rounded down, above Nagamochi’s closed form, $17.40121946685$$17.40121946685$. 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, 16 October 2026. Polkovskyi’s tighter certificate of Couzo’s continuation of PingYou Ltd’s n = 302 arrangement (T-186) is now this case’s house: its exact certificate proves $s(302)\le 17.87001385040446435$$s(302)\le 17.87001385040446435$, the reported and the verified upper bound, rounded outward from its exact side. The exact certificate side improves the previous verified exact side by about 1.79×10−19$1.79\times {10}^{-19}$. The earlier ceiling, displayed outward as $17.8700138504044644$$17.8700138504044644$ from PingYou Ltd’s packing, improved by Couzo (T-181), applied until this adoption and stays retained as history (the certificate).

House adoption, 15 October 2026. Couzo’s continuation of PingYou Ltd’s n = 302 arrangement (T-181) was this case’s house until 16 October 2026: its exact certificate proves $s(302)\le 17.8700138504044644$$s(302)\le 17.8700138504044644$, the reported and the verified upper bound, 2.02×10−3$2.02\times {10}^{-3}$ below the ceiling the case held until then, $17.8720298490811800$$17.8720298490811800$ from Chaoweeraprasit’s packing, improved by Daniel (T-146), which stays retained as history (the certificate).

House adoption, 11 October 2026. Daniel’s display regularization of Chaoweeraprasit’s SQUISH packing (T-146) was this case’s house until 15 October 2026: its exact certificate proves $s(302)\le 17.8720298490811800$$s(302)\le 17.8720298490811800$, the reported and the verified upper bound, 9.28×10−3$9.28\times {10}^{-3}$ below the ceiling the case held until then, $17.8813062180958085$$17.8813062180958085$ from Chaoweeraprasit’s packing (T-116), which stays retained as history (the certificate).

The second SQUISH update

Chaoweeraprasit, using SQUISH, reported this packing on 7 October 2026: $s(302)\le 17.8813062180958085$$s(302)\le 17.8813062180958085$, with exact side $2516570125647313/140737488355328$$2516570125647313/140737488355328$. Two independently implemented exact rational SAT routes accepted the complete packing and rejected both complete-roster controls. Two separately prompted AI reviews accepted the mathematical inputs and full receipt bindings. The author reports the seed as SQUISH’s own s(88), grafted into n302’s record.

The revision-specific packet retains the source print $17.8813062180958084$$17.8813062180958084$ separately from the verified upward decimal ceiling. The exact side is unchanged; this certifies feasibility and an upper bound, without optimality, rigidity or human oversight claims. Earlier source geometry, certificates, credit and verification lanes remain historical evidence for their own packings.

Open. Previously, The best known published packing, Couzo’s at Daniel’s exact optimum (T-098), gives $s(302)\le 17.8859938925304106667\dots$$s(302)\le 17.8859938925304106667\dots$, and the strongest lower bound independently verified here is $17.389345$$17.389345$ from Karakuş’s general theorem, leaving a gap of $0.4966$$0.4966$. General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

The exact optimum

Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing at its exact optimum (T-098): the same 302 squares in a square of side $17.8859938925304106667\dots$$17.8859938925304106667\dots$, 6.3×10−12$6.3\times {10}^{-12}$ below the side Couzo prints. Square for square, its pose lies within 4.0×10−4$4.0\times {10}^{-4}$ of the binary64 pose the atlas pictures for this count. 140 squares move by more than 1×10−8$1\times {10}^{-8}$, 21 of them squares the source lists as carrying no force and the other 119 by at most 3.4×10−6$3.4\times {10}^{-6}$; every other square moves by at most 8.5×10−9$8.5\times {10}^{-9}$. The known-best witness this record lists is that binary64 pose, posed at its finder’s larger side; the side above is witnessed by the certificate itself. Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals, each square a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). Previously, That proves $s(302)\le 17.8859938925304106667\dots$$s(302)\le 17.8859938925304106667\dots$, the verified upper bound until 15 October 2026; it says nothing about optimality. The source reports the exact point as a KKT local minimum: multipliers that keep it in equilibrium, a reduced Hessian positive definite once its exact flat motions are set aside (its per-count report), and no first-order descent across corner-to-corner contacts, each computed numerically at that point; none of that is verified here, and it bears on this packing alone, not on $s(302)$$s(302)$. On jlevy/squares#375 its author wrote that the solver, its checkers and the batch “were written with Claude (Anthropic) as a coding and research agent, directed and reviewed by me.”

The packing

Couzo’s square-packing reports a packing of side $17.885993892536710$$17.885993892536710$ for this count, dated 26 September 2026 and unchanged when this record retained the repository on 27 September 2026 (T-056). Its first packing for this count, of side $17.886045193506881$$17.886045193506881$, is dated 25 September 2026. The repository names no method and no tolerance, and itself states no AI assistance; its author said on issue #227 that he found the 102 and 103 packings “with the help of Claude”.

This repository certifies it exactly. The retained decimal pose rounds to an exact rational packing at centre dilation 1, of side $17.8859938925367102500\dots$$17.8859938925367102500\dots$, 2.500×10−16$2.500\times {10}^{-16}$ above the printed side, and every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by the promotion’s exact separating-axis test and by an independent checker that shares no code with it (receipt). That proves $s(302)\le 17.885993892536711$$s(302)\le 17.885993892536711$, which was the verified upper bound until Daniel’s exact optimum above replaced it; it says nothing about optimality. Interval arithmetic on the printed pose itself, with each angle’s true cosine and sine and no rational rounding, decides every pair and wall again and gives the same bound (interval route).

The previous best known packing

Before this intake the best known packing was the Kingbird catalogue’s, of side $17.88674602860566$$17.88674602860566$.

Found by Ellsworth in 2024, via an unrecorded method. Its side length is algebraic of degree 4 over $\mathbb{Q}$$\mathbb{Q}$.

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-146)

Daniel’s display regularization of Chaoweeraprasit’s SQUISH packing, reported in his regularized record lists, on no issue, at exact side $558500932783786874181638432249/31250000000000000000000000000$$558500932783786874181638432249/31250000000000000000000000000$, displayed upward as $17.8720298490811800$$17.8720298490811800$ (T-146). The certificate gives a rational side and, for each of the 302 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 11 October 2026 and the final review of 11 October 2026 accepted the replay for finite feasibility. That proves $s(302)\le 17.8720298490811800$$s(302)\le 17.8720298490811800$, the verified upper bound until 15 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate regularizes for display the packing T-136 records, at the side the source reports as that packing’s exact optimum, so the packing stays with its finder and Daniel is credited with the regularization and the certificate. That the two are one packing is the source’s claim, measured here and not decided (T-146’s regularizations).

The source’s credits say the repository’s work was produced by Claude (Anthropic) under human direction.

The house it replaces, $17.8813062180958085$$17.8813062180958085$ from Chaoweeraprasit’s packing (T-116), stays retained with its own replay as history. The other certificates confirmed here at this count, T-136 at $17.8720298490811800$$17.8720298490811800$ and T-138 at $17.881306218091$$17.881306218091$, stay verified and are not selected.

Francisco Couzo’s certificate (T-181)

Couzo’s continuation of PingYou Ltd’s n = 302 arrangement, reported on jlevy/squares#583, at exact side $3574002770080892868927770707857/200000000000000000000000000000$$3574002770080892868927770707857/200000000000000000000000000000$, displayed upward as $17.8700138504044644$$17.8700138504044644$ (T-181). The certificate gives a rational side and, for each of the 302 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. One maintained route is the verifier the source ran; the independent rational corner checker and the third route are independently re-implemented. The review of 14 October 2026 accepted the replay for finite feasibility. That proves $s(302)\le 17.8700138504044644$$s(302)\le 17.8700138504044644$, the verified upper bound until 16 October 2026: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The author says OpenAI Codex assisted the search implementation, the verification orchestration and the requests, that earlier search work used Claude Code, and that he directed the work.

The house it replaces, $17.8720298490811800$$17.8720298490811800$ from Chaoweeraprasit’s packing, improved by Daniel (T-146), stays retained with its own replay as history. The other certificates confirmed here at this count, T-164 at $17.8702753827337598$$17.8702753827337598$, T-136 at $17.8720298490811800$$17.8720298490811800$, T-138 at $17.881306218091$$17.881306218091$ and T-116 at $17.8813062180958085$$17.8813062180958085$, stay verified and are not selected.

Mykhailo Polkovskyi’s certificate (T-186)

Polkovskyi’s tighter certificate of Couzo’s continuation of PingYou Ltd’s n = 302 arrangement, reported in his repository on 14 October 2026, with no issue, at an exact rational side, displayed upward as $17.87001385040446435$$17.87001385040446435$ (T-186). The certificate gives a rational side and, for each of the 302 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 16 October 2026 accepted the replay for finite feasibility. That proves $s(302)\le 17.87001385040446435$$s(302)\le 17.87001385040446435$, the verified upper bound: confirmed, independently re-implemented. It says nothing about optimality, rigidity or a local minimum.

The certificate is Polkovskyi’s tightening of the packing T-181 records, measured here pose by pose as that packing (T-186’s regularizations), so the packing keeps its finder’s credit and Polkovskyi is credited with the certificate.

The commit makes no statement about AI assistance; the author’s issue 593 discloses Gemini assistance with solver scripting, precision checks and formatting.

The house it replaces, $17.8700138504044644$$17.8700138504044644$ from PingYou Ltd’s packing, improved by Couzo (T-181), stays retained with its own replay as history. The other certificates confirmed here at this count, T-164 at $17.8702753827337598$$17.8702753827337598$, T-146 at $17.8720298490811800$$17.8720298490811800$, T-136 at $17.8720298490811800$$17.8720298490811800$, T-138 at $17.881306218091$$17.881306218091$ and T-116 at $17.8813062180958085$$17.8813062180958085$, stay verified and are not selected. T-193’s exact algebraic side at this count, a root of a degree-7 integer polynomial 1.79×10−39$1.79\times {10}^{-39}$ below this house, is confirmed here and accepted by the review of 16 October 2026. The owner’s house rule of 16 October 2026 adopts it: the case keeps this house until its adoption, staged under think-ot5c, moves it to T-193’s side, once the writer takes an algebraic side.