$n=106$$n=106$

$s(106)$$s(106)$ — 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=106$$n=106$, $k=10$$k=10$ and interval 8 give $a(106)=10.403976649170868104031437025345\dots$$a(106)=10.403976649170868104031437025345\dots$, so the reported lower bound here is now $s(106)\ge 10.403976649170$$s(106)\ge 10.403976649170$, rounded down, above Nagamochi’s closed form, $10.32737905308$$10.32737905308$. 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.

Open. The best known published packing, Couzo’s at Daniel’s exact optimum (T-098), gives $s(106)\le 10.8229080441328475552\dots$$s(106)\le 10.8229080441328475552\dots$, and the strongest lower bound independently verified here is $10.310708$$10.310708$ from Karakuş’s general theorem, leaving a gap of $0.5122$$0.5122$. General bound: s(N)≥1/2+N−⌊N​⌋+1/4​$s(N)\ge 1/2+\sqrt{N-\lfloor \sqrt{N}\rfloor +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 106 squares in a square of side $10.8229080441328475552\dots$$10.8229080441328475552\dots$, 9.1×10−12$9.1\times {10}^{-12}$ below the side Couzo prints. Square for square, its pose lies within 4.4×10−4$4.4\times {10}^{-4}$ of the binary64 pose the atlas pictures for this count. 39 squares move by more than 1×10−8$1\times {10}^{-8}$, 9 of them squares the source lists as carrying no force and the other 30 by at most 6.3×10−6$6.3\times {10}^{-6}$; every other square moves by at most 2.4×10−11$2.4\times {10}^{-11}$. The known-best witness this record lists is that binary64 pose, posed at its finder’s larger side; the side above is witnessed by the certificate itself. Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals, each square a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves $s(106)\le 10.8229080441328475552\dots$$s(106)\le 10.8229080441328475552\dots$, the verified upper bound; 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(106)$$s(106)$. 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 $10.822908044141968$$10.822908044141968$ for this count, dated 27 September 2026 and unchanged when this record retained the repository on 27 September 2026 (T-056). Its first packing for this count, of side $10.822965780538063$$10.822965780538063$, is dated 26 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 $10.8229080441419683466\dots$$10.8229080441419683466\dots$, 3.466×10−16$3.466\times {10}^{-16}$ above the printed side, and every pair and every wall is decided over $\mathbb{Q}$$\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(106)\le 10.822908044141969$$s(106)\le 10.822908044141969$, 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).

Casson’s square-packing, dated 23 September 2026 and made with the help of Claude as its README says, reports a packing of side $10.82293693182615435$$10.82293693182615435$ for this count, larger than Couzo’s by $0.00002888768418635$$0.00002888768418635$. Casson’s is the earlier of the two by the timestamps this record’s priority notes keep: Couzo’s first packing for this count, of side $10.822965780538063$$10.822965780538063$, is dated 26 September 2026. The record states both dates and infers nothing about whether either packing derives from the other.

The previous best known packing

Before this intake the best known packing was the Kingbird catalogue’s, of side $10.82297973416944$$10.82297973416944$.

Found by Ellsworth in 2024, via an unrecorded method. Improved by Cantrell in December 2024. Its side length is algebraic of degree 32 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.