$n=37$$n=37$

$s(37)$$s(37)$ — open

Exact regularized certificate, 2026-10-12. This project’s regularization pass, Daniel’s exact solver and regularizer run here on this register’s witness, produced an exact rational packing of this arrangement, decided here over $\mathbb{Q}$$\mathbb{Q}$ by two exact checkers: it proves $s(37)\le 6.59861960924436011617837046268$$s(37)\le 6.59861960924436011617837046268$, the verified upper bound (T-155), which the record still prints as $6.59861960924437$$6.59861960924437$. Until then the verified upper bound was exactly $6.59861960924437$$6.59861960924437$, the side of Daniel’s exact certificate of this packing rounded up at the printed precision (T-101).

Exact certificate, 2026-10-06. Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing, decided here over $\mathbb{Q}$$\mathbb{Q}$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves $s(37)\le 6.59861960924437$$s(37)\le 6.59861960924437$, the verified upper bound from 6 to 12 October 2026 (T-101). Until then the verified upper bound was the trivial grid bound $7$$7$.

External intake, 2026-10-01. Hosono’s mixed-certificate source reports $s(37)\ge 161/25=6.44$$s(37)\ge 161/25=6.44$ (T-069), from a density of 350 rectangles of total mass $3699999/100000<37$$3699999/100000<37$, accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. Its complete replay here on 2 October 2026 returned the certificate’s own record at all 201 directions, so it is also the verified lower bound. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. Hosono’s rectangle-density source reports a direct $257/40=6.425$$257/40=6.425$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass $3699/100=36.99<37$$3699/100=36.99<37$, 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. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-27. Hosono’s rectangle-density source reports a direct $32/5=6.4$$32/5=6.4$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass $3699/100=36.99<37$$3699/100=36.99<37$, 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.

Open. The best known packing gives $s(37)\le 6.59861961$$s(37)\le 6.59861961$, and the verified lower bound is $s(37)\ge 161/25=6.44$$s(37)\ge 161/25=6.44$, from Hosono’s mixed rectangle-density certificate, leaving a gap of $0.1586$$0.1586$.

The regularized exact certificate

This project’s regularization pass of 11 and 12 October 2026 ran Daniel’s exact solver and regularizer (evand/square-packing, MIT, vendored here) on this register’s witness for this count, with the conservative angle policy and coincide placement, and this project’s same-side re-solve then fixed the result’s tight contacts over $\mathbb{Q}$$\mathbb{Q}$ at the witness’s own side. The result is an exact rational packing of the same 37 squares in a square of side $6.59861960924436011617837046268$$6.59861960924436011617837046268$, 9.9×10−15$9.9\times {10}^{-15}$ below the verified upper bound the record held before, and 6.6×10−20$6.6\times {10}^{-20}$ below the side of Daniel’s certificate of this packing (T-101): it is the witness’s own side. Measured against the witness under the box’s eight symmetries, its squares pair one to one with the witness’s; 7 move by more than 1×10−3$1\times {10}^{-3}$, the largest by 1.7×10−1$1.7\times {10}^{-1}$, and none turns by more than 1×10−6$1\times {10}^{-6}$ as a sine. It redraws the packing visibly, and is drawn only once the regularized layer admits it under the regularization policy.

Each pair and each wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and both accept it; both refuse it with its tightest pair pushed into overlap and with its box shrunk past its least clearance (certificate). The same two checkers accepted it in the run that made it, so this is a reproduction with the producer’s code. That proves $s(37)\le 6.59861960924436011617837046268$$s(37)\le 6.59861960924436011617837046268$, the verified upper bound (T-155), the exact form of the case’s verified_upper_bound, which prints it as $6.59861960924437$$6.59861960924437$, the decimal it printed before and an upper display of it. It says nothing about optimality, and the packing stays its finders’.

The earlier exact certificate

Daniel’s square-packing published on 5 October 2026 exact rational certificates of the packings this register lists as best known, solved from its own witnesses, and offered those that do not lower a printed side as a replay of the existing bounds, asking for nothing to be registered from them. The certificate for this count holds the same 37 squares, each a rational centre and a rational $t=\tan (\theta /2)$$t=\tan (\theta /2)$, in a square of side $6.5986196092443601163\dots$$6.5986196092443601163\dots$, 1.2×10−16$1.2\times {10}^{-16}$ above the side the Kingbird catalogue prints, $6.59861960924436$$6.59861960924436$. Square for square, its pose lies within 2.9×10−4$2.9\times {10}^{-4}$ of the binary64 pose the atlas pictures for this count. 4 squares move by more than 1×10−8$1\times {10}^{-8}$, all of them squares the source lists as carrying no force; every other square’s centre rounds to the witness’s. 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.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $\mathbb{Q}$$\mathbb{Q}$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves $s(37)\le 6.59861960924437$$s(37)\le 6.59861960924437$, the verified upper bound until 12 October 2026: the certificate’s side rounded up at the fourteen decimals the catalogue prints, as the record writes any certificate of a printed side. It says nothing about optimality. 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.”

Its rounded-up side and the printed side agree to one unit of the printed side’s last place, the precision at which the record compares them. The printed side itself is not certified here: the certificate’s side lies above it.

The packing

Found by Cantrell in 2002, via a hand construction. Its side length is algebraic of degree 8 over $\mathbb{Q}$$\mathbb{Q}$. The degree-8 rational polynomial is not the sharpest characterization available. Over Q(2​)$\mathbb{Q}(\sqrt{2})$ it factors, and the archived catalogue records the quartic factor:

This was checked here: the quartic divides the degree-8 polynomial with remainder identically zero, and its real root $6.5986196092443601161783$$6.5986196092443601161783$ matches the recorded value. Because the factor has degree 4 it is solvable in radicals, so a genuine closed form exists for this case, unlike $n=28$$n=28$ and $n=39$$n=39$.

The lower bound

The earlier external report, [Friedman DS7], gives the lower-bound expression 103​/37+22​+113/37$10\sqrt{3}/37+2\sqrt{2}+113/37$ for $s(37)$$s(37)$ (approximately $6.350603018684$$6.350603018684$). Friedman’s DS7 survey, Theorem 9, k=6, reports this bound at n=37; reference [8] is Green’s private communication (2000). The source proof has not been recovered. The unavoidable-set argument DS7’s Figure 34 illustrates does not prove it: at $k=6$$k=6$ that point pattern leaves a unit square empty (review). Hosono’s rectangle-density certificate above has since replaced it in the reported field; the verified field now holds Hosono’s replayed mixed certificate. The source audit compares the exact theorem expressions separately from opaque table decimals.

The verified field rests on a complete replay here (E-n037-wand125-mixed-644-source-replay, V3/C3) of Hosono’s mixed certificate: the source’s unchanged checker and per-angle functions on the pinned bundle, all 201 directions on 2 October 2026, each returning the certificate’s own record. It runs the source’s own algorithm, so it confirms the source’s run rather than deciding coverage a second way.

Until 2 October 2026 the verified lower bound was Nagamochi’s general closed form, now a reported bound (see the correction below). Karakuş’s general bound, independently verified here and below this certificate at this $n$$n$, applies to every nonsquare $N\ge 8$$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.

Update, 3 October 2026. Bašić and Slivková’s piercing bound is now registered (T-087), 53​/2+22​−1≈6.158554$5\sqrt{3}/2+2\sqrt{2}-1\approx 6.158554$, above both Karakuş’s $6.090169$$6.090169$ and Nagamochi’s $6.099019$$6.099019$. At sides just below it their Theorem 7 counts a piercing set of 36 points, so no 37 unit squares fit; the paper states the theorem and applies it at $n=61$$n=61$, and this case is its evaluation here by check_piercing_lower_bounds. The proof was read and re-derived, not machine-checked, and it uses nothing of Nagamochi 2005. It was the verified lower bound here on its own line of the register until that line merged, the same day, with the replay of Hosono’s mixed certificate above, $161/25=6.44$$161/25=6.44$ (T-069), which superseded it; the reported lower bound is that same certificate.

See the Frontier corpus summary for the current aggregate count; source-reported bounds are recorded separately.

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.