Case record

The best packing known for 44 squares, side

7

$n = 44$

=open

$\frac{2789}{400} \leq s (44) \leq 7$

6.977

678

6.6337.633

\sqrt{n}

\sqrt{n} + 1

Proven

$6.97250$ $\leq s (44) \leq$ $7$

exact

Citation record n-044

lowerHosono after Tokoharu, Levy et al. 2026, GitHub (confirmed T-090)

Open

optimality

Bounds

Best known packing

$7$

Construction: grid
Tilt angles: $0^{\circ}$
Source: [Kingbird]
Evidence: E-kingbird-upper-register

Verified upper bound

$7$

The reported value, verified here.

Evidence: E-basic-grid-upper

Reported lower bound

$\frac{2789}{400}$

Proved by: Hosono 2026
Kind: counting
Scope: Unrestricted unit-square packing with independent rotations and disjoint interiors.
Note: Hosono's square-packing-bounds (4 October 2026) reports $s (44) \geq 2789 / 400$ from a density of 399 uniform rectangles of total mass 4399999/100000, at core side 9977/10000 on 201 net half-angles, accepted there at every angle by its research copy of Tokoharu's verify.cpp at threshold one. It is above the 2777/400 this case reported, from the source's rectangle certificate rect_n44_L69425 (T-068, T-074). sqverify-fast decided it here on 6 October 2026 at all 201 net directions.
Source: [wand125 mixed bounds 2026-10-04]
Evidence: E-n044-wand125-mixed-69725-report

Verified lower bound

$\frac{2789}{400}$

The reported value, verified here.

Evidence: E-n044-wand125-mixed-69725-sqverify-fast-replay

Gap

$\frac{11}{400}$ = 0.0275

Verified upper minus verified lower.

Results in the register

T-007 V0 C1 Nagamochi · 2026-08-31 · 321 cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
T-046 V0 C0 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 48 cases
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
T-058 V3 C3 Hosono after Tokoharu, Daniel · 2026-09-29 · 100 cases
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
T-068 V3 C3 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-01 · 34 cases
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
T-070 V3 C3 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-02 · 25 cases
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
T-074 V3 C3 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-02 · 31 cases
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
T-083 V3 C3 Karakuş · 2026-10-02 · 301 cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
T-085 V3 C3 Karakuş; Vlasenko · 2026-10-02 · 315 cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
T-090 V3 C3 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-05 · 17 cases
Mixed rectangle-measure lower bounds verified at 17 counts in $n = 42 \dots 96$
T-124 V0 C0 Daniel after Couzo · 2026-10-07 · 178 cases
Reported non-strict local minima for 178 source configurations
T-177 V0 C1 Hosono after Tokoharu and Levy · 2026-10-14 · n = 44
Mixed rectangle-measure lower bound $s (44) \geq 2799 / 400$ reported, on a 2,073-direction declared net
Claim and records
Claim
A rectangle measure of wand125/square-packing, published on 13 October 2026 and reported on no issue, is reported to prove $s (44) \geq 2799 / 400$ = 6.9975.

The certificate, mixed_n44_L69975, is a density of 606 uniform rectangles in D4 orbits, and no point mass, of total mass 4399999/100000, on a net its candidate declares: core side 4999/5000 and 2,073 half-angle tangents of step 1/5002, the net of T-132, T-142 and T-143. Since 4999/5000 (1 + 1/5002) = 25009997/25010000 < 1, and the last tangent 1036/2501 is past tan(pi/8), every unit square contains a concentric core at a net angle strictly in its interior.

The source reports every such core capturing mass at least one at all 2,073 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, in two complete runs, and a control with every mass scaled by 197/200 refused at 32 sampled directions.

The value is above T-090's 2789/400 = 6.9725, the verified and reported lower bound at $n = 44$ , by 1/40, and 1/400 below the upper bound of 7.

Here the maintained readers accepted its exact mass, declared-net and check2 premises, and sqverify-fast, this repository's clean-room measure verifier, verified it at eight sampled directions with the source's node counts and least certified bounds: a diagnostic, not a replay. No complete replay has run here, so the value stands as reported.

Hosono after Tokoharu and Levy, square-packing. The source says parts of the work were produced with AI assistance under human direction.
Significance
Scored as a claim: if a complete replay passes, it is the strongest lower bound on record at $n = 44$ , 1/40 above T-090 and 1/400 below the upper bound of 7, by the certificate kind and declared net of T-132, T-142 and T-143, with no new technique, at S3 as they are.
Novelty
previously-published Present in an identified source
Records
n = 44 register evidence 1
n = 44 register evidence 1

Verification

upper: replayed here; lower: replayed here

—

Rigidity

not rigid, numerically checked, numerical multiprecision

Evidence: E-translation-escape-not-rigid

Scope

Square 37 of the retained witness (witness id 38) translates 1 along (0, 1) with the packing still valid, so the configuration admits a non-trivial feasible motion; 6 of its 44 squares do. Every constraint is exactly affine in the slide parameter, so the arithmetic carries no linearization error, but the coordinates are the witness's own finite-precision transcription: this settles the retained configuration, not the true optimum. Rigidity and optimality are independent, and this bears only on the former.

Evidence and sources

12 evidence entries

E-n044-wand125-mixed-69725-sqverify-fast-replay, E-n044-wand125-rect-69425-sqverify-fast-replay, E-wand125-rectangle-2026-10-01-source-replay, E-wand125-rectangle-2026-09-28-source-replay, E-wand125-rectangle-2026-10-01-report, E-wand125-rectangle-2026-09-28-report, E-wand125-rectangle-report, E-kingbird-upper-register, E-nagamochi-lower, E-basic-grid-upper, E-karakus-strip-lower, E-nagamochi-lemma1-counterexample

[wand125 mixed bounds 2026-10-04] lower bound proof
[wand125 rectangle bounds 2026-10-01] lower bound proof
[wand125 rectangle bounds 2026-09-28] lower bound proof
[wand125 rectangle bounds 2026] lower bound proof
[Kingbird] record catalogue
[Nagamochi 2005] lower bound proof
[Friedman DS7] survey
[Karakuş 2026] lower bound proof

In the frontier survey On GitHub

$s (44)$ — open

External intake, 2026-10-05. Hosono’s mixed-certificate source reports $s (44) \geq 2789 / 400 = 6.9725$ (T-090), from a density of 399 rectangles of total mass $4399999 / 100000 < 44$ , accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the $2777 / 400$ this case reported, by $0.03$ . It is the source’s first mixed certificate at this count. sqverify-fast, this repository’s clean-room verifier, decided it here on 6 October 2026 at all 201 net directions, so it is also the verified lower bound: confirmed, independently re-implemented. Hosono’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-01. Hosono’s rectangle-density source reports $s (44) \geq 2777 / 400 = 6.9425$ , since superseded (T-090), with total mass $4399 / 100 = 43.99 < 44$ , accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again, after this repository’s exact audit checked that the regenerated checker input is the published one and checked the mass and net premises, so it is also verified. 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 $1387 / 200 = 6.935$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass $4399 / 100 = 43.99 < 44$ , 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 $277 / 40 = 6.925$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass $4399 / 100 = 43.99 < 44$ , 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 $s (44) \leq 7$ , and the verified lower bound is $s (44) \geq 2789 / 400 = 6.9725$ , from Hosono’s mixed rectangle-density certificate of 4 October (T-090), leaving a gap of $0.0275$ .

The packing

The record catalogue does not picture $n = 44$ : no arrangement has ever been found that beats the trivial $⌈ \sqrt{44} ⌉ = 7$ grid, so the grid is still the best known packing. That is a statement about what has been searched, not a proof.

The lower bound

The verified field rests on a complete decision here (E-n044-wand125-mixed-69725-sqverify-fast-replay, V3/C3) of Hosono’s mixed certificate of 4 October by sqverify-fast, this repository’s clean-room measure verifier: all 201 net directions of the retained candidate on 6 October 2026, with two mutants scaled below coverage one refused. It shares no code with the source’s checker, so it decides coverage a second way with the same method; the source’s own checker was not run here. Until this decision the verified lower bound was $2777 / 400$ , Hosono’s rectangle-density certificate of 1 October (E-wand125-rectangle-2026-10-01-source-replay).

Nagamochi’s general closed form, the verified lower bound before this certificate, is now a reported bound (see the correction below). Karakuş’s general bound, independently verified here and below this certificate at this $n$ , applies to every nonsquare $N \geq 8$ :

s (N) \geq \frac{1}{2} + \sqrt{N - ⌊ \sqrt{N} ⌋ + \frac{1}{4}}

Correction, 2 October 2026. Until that date the verified lower bound here was Nagamochi’s general closed form, $s (N) \geq min (⌈ \sqrt{N} ⌉, \sqrt{N - 2 ⌊ \sqrt{N} ⌋ + 1} + 1)$ , which is stronger at this $n$ 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.

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.

bound	evidence	run	code	programs
verified lower	`E-n044-wand125-mixed-69725-sqverify-fast-replay`	replayed here	independent	`V-sqverify-fast` (first-party)
verified upper	`E-basic-grid-upper`	replayed here	independent	`V-check-basic-bounds` (first-party)