T-219: Daniel's smaller n271 packing, continuing Couzo's arrangement, certified exactly

V3 C3 Upper bound confirmed

2026-10-17 published · Daniel after Couzo, Chaoweeraprasit, Levy · $n = 271$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s (271) \leq 16.948947733713657510723401112423$ .

The certificate and its duplicate-square and outside-container controls were decided here on 19 October 2026 by both maintained exact routes, and the frozen comparison rebuilt; no route here is the source's code, so the replay is independently re-implemented. The 19 October review found no blocking defect.

Issue 664 reports a packing found by Daniel's basin explorer from the register's witness for 271 with one tilt-swap move and a re-polish, calls it a refinement of Couzo's #561 packing, not a new construction, and asks that #561 and Chaoweeraprasit's SQUISH packing it refines (#500) be credited; it also calls it a different local minimum, its load-bearing tilt angles matching #561's within 0.12 degrees except that four squares near 31.7 to 32.0 degrees are three. The author reports that his two exact checkers accept the certificate and its 21-group regularized view at the same side; KKT and local-minimum statements are numerical evidence, not proofs. The issue says the work was done with Claude (Claude Code) under Daniel's direction.

His comment on the issue of 20 October says the two wordings contradicted each other and he meant a different local minimum, found from #561's packing, and asks to be named its finder with #561's packing and the SQUISH packing it continues as its start.

Significance, composition and next rung

Significance: A continuation of #561's arrangement to a packing the source calls a different local minimum, 4.46e-05 below the case, as T-119's new arrangements and T-130's refinements are; above T-188's 9.29e-06 (S2).
Next rung: None for the bound. V4 and C4 need a second accepted adversarial review by a distinct reviewer and a retained human oversight record; one AI review, of 19 October, is retained.
Regularized drawings: n = 271: a regularization of T-219’s packing by Levy (Daniel's search/regularize (MIT), evand/square-packing@675a5ea, conservative, coincide, through devtools.regularize_evand; the project’s modifications: same-side exact re-solve (devtools.regularize_same_side); at its own side; 2 free squares straightened, 2 free angles moved; identity: measured here; nearest centres one to one; 19 squares moved, at most 0.0339327; turns up to 0.143387°).
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, Daniel 2026 [Daniel n271 2026-10-17]

16.4716.949

161718

16.46217.462

\sqrt{n}

\sqrt{n} + 1

Proven

$16.47654$ $\leq s (271) \leq$ $16.948948$

exact

Citation record n-271

lowerKarakuş 2026, arXiv corrects Nagamochi 2005 (confirmed T-083)

upperDaniel 2026, GitHub (confirmed T-219)

Open

optimality

The case record

LowerUpper

Verified16.47654530…16.94894773…

Reported16.52918032…16.94894773…

Gap0.47240242…

Results on the case

11 results in the register on $n = 271$ , 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-083 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-219
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
Karakuş · Karakuş 2026 · source · register
2026-10-05 published T-098
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
V3 C3 Upper bound confirmed on this case, superseded by T-219
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-11 published T-151
Nine exact rational packings from SQUISH's fourth request
V3 C3 Upper bound confirmed superseded by T-219
Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register
2026-10-12 published T-161
Exact rational certificates at 156, 176, 237 and 271 from Couzo
V3 C3 Upper bound confirmed superseded by T-219
Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 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 on this case, superseded by T-219
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
2026-10-14 published T-186
Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there
V3 C3 Upper bound confirmed on this case, superseded by T-219
Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register
2026-10-17 published T-219 this result
Daniel's smaller n271 packing, continuing Couzo's arrangement, certified exactly
V3 C3 Upper bound confirmed
Daniel after Couzo, Chaoweeraprasit, Levy · Daniel n271 2026-10-17 · packet · register

T-219: Daniel's smaller n271 packing, continuing Couzo's arrangement, certified exactly

The case

Results on the case

Links