T-170: An exact contact refinement of Xu's packing of 105 squares, after Couzo's

V3 C3 Upper bound confirmed superseded by T-174

2026-10-12 published · PingYou Ltd after Xu, Couzo, Daniel, Levy · $n = 105$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s (105) \leq 10789303783748158830926836660039617659736218711424421332884365754662904597598593330008796151 / 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$ .

All three retained jobs, the positive with its duplicate-square and outside-container controls, were decided again here on 14 October 2026 by both maintained exact routes and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided the certificate with six controls each and held the derived fact to the pinned file it is derived from, fetched at the pin. No route here is the source's code; the source's runs of this repository's two maintained routes on its corners witnesses are its own and decide nothing here, so the replay is independently re-implemented. The 14 October review found no blocking defect.

The author reports that exact separating-axis and rational polygon-intersection implementations accept all 105 squares, 420 wall conditions and 5,460 pairs, with duplicate and outside controls refused and the conversion to its corners Witness/v2 lossless; that an inverse-dilation control reproduces the source's side and its 97 nonfree rows; and that this repository's sqpack exact_verify and check_rational_witness_independent, run unchanged, accept that witness. Those statements are the author's; no source program ran here.

Credit PingYou Ltd (https://github.com/BuddenPing/square-packing-certificates) with this contact refinement, as the source asks, of Couzo's refinement (T-130) of Xu's layout (ry-xu, issue 432), with Daniel's exact contact solver. The issue states: 'Computation, checking and submission preparation were AI-assisted; the requested attribution is as stated above.' The pinned source makes no statement about AI assistance.

Measured pose by pose by that review, it is Couzo's packing (T-130) at less padding, no square moved beyond 2e-19. At 105 PR 574's certificate (T-174) is smaller, by 1.08e-79, and the house rule of 16 October 2026, which lifted the hold of 15 October, made T-174 the house there, and the adoption of 17 October 2026 (think-3mry) moved the case to it. This entry held the case until then, the smallest certificate there that the hold left in.

Significance, composition and next rung

Significance: Padding, 1.08e-19 below the case ceiling, as T-155; no lower bound or optimum.
Next rung: V4 and C4 need a second accepted adversarial review by a distinct reviewer and a retained human oversight record; one AI review, of 14 October, is retained.
Regularizations: n = 105: a regularization of T-130’s packing by PingYou Ltd (exact contact refinement with Daniel's exact contact solver; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.455e-19; turns up to 1.126e-21°).
Novelty: previously-published Present in an identified source

The case

Case record

Regularized by the Squares Project with Daniel's regularizer (evand/square-packing, MIT)

As published

The packing as published, Xu 2026 [PingYou PR 574 certificates 2026-10-13]

10.2810.789

101112

10.24711.247

\sqrt{n}

\sqrt{n} + 1

Proven

$10.28000$ $\leq s (105) \leq$ $10.789304$

exact

Citation record n-105

lowerHosono after Tokoharu, Levy et al. 2026 (confirmed T-073, T-080)

upperXu et al., GitHub (confirmed T-174)

Open

optimality

The case record

LowerUpper

Verified

\frac{257}{25}

10.78930378…

Reported10.35476005…10.78930378…

Gap0.50930378…

Results on the case

16 results in the register on $n = 105$ , oldest first, each with what it established and how it stands now.

2005 published T-007
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 Lower bound incomplete on this case, superseded by T-073, T-080 and T-180 (reported)
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
V3 C3 Correction confirmed
Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-27 published T-056
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
V3 C3 Upper bound confirmed superseded by T-174
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 Lower bound confirmed on this case, superseded by T-073, T-080 and T-180 (reported)
Karakuş · Karakuş 2026 · source · register
2026-10-02 published T-073
Linear-measure lower bounds replayed at $n = 83$ and $n = 101 \dots 105$
V3 C3 Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register
2026-10-02 published T-080
Linear-measure lower bound replayed at $n = 101 \dots 105$
V3 C3 Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register
2026-10-07 published T-117
Exact rational ceiling refinements at $n = 105$ , 292
V3 C3 Upper bound confirmed superseded by T-174
Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register
2026-10-07 published T-129
Three dated Daniel certificate reports with historical source custody
V0 C0 Upper bound recorded superseded by T-174
Daniel after Couzo, Levy · Daniel dated certificates 105 and 130 2026-10-07 · Daniel dated certificate 292 2026-10-07 · packet · packet · packet · register
2026-10-08 published T-125
Complete rational construction reports at 25 counts
V3 C3 Upper bound confirmed superseded by T-174
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-128
Eight complete rational refinements from Couzo
V3 C3 Upper bound confirmed superseded by T-174
Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register
2026-10-08 published T-130
Five follow-up rational refinements from Couzo
V3 C3 Upper bound confirmed superseded by T-174
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register
2026-10-09 published T-138
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
V3 C3 Upper bound confirmed superseded by T-174
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-12 published T-170 this result
An exact contact refinement of Xu's packing of 105 squares, after Couzo's
V3 C3 Upper bound confirmed superseded by T-174
PingYou Ltd after Xu, Couzo, Daniel, Levy · PingYou n105 refinement 2026-10-12 · packet · register
2026-10-13 published T-174
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
V3 C3 Upper bound confirmed
PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · PingYou PR 574 certificates 2026-10-13 · packet · register
2026-10-13 published T-180
Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported
V0 C1 Lower bound reviewed
Hosono · wand125 closed form 2026-10-14 · packet · packet · register
2026-10-22 published T-229
Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225
V0 C0 Upper bound recorded
Polkovskyi after Couzo, Muglich, Daniel, Xu, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register

T-170: An exact contact refinement of Xu's packing of 105 squares, after Couzo's

The case

Results on the case

Links