T-092: Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways

V3 C3 Upper bound confirmed superseded by T-174 and T-186

2026-10-03 published · Couzo · 7 cases, $n = 208$ to 306

For seven counts, $s (n)$ is at most the bound certified here from Couzo's packing as published on 3 October 2026: $s (208) \leq 14.926534459703512$ , $s (209) \leq 14.953939011860643$ , $s (228) \leq 15.604602454552252$ , $s (263) \leq 16.742270262031792$ , $s (272) \leq 16.968165867864400$ , $s (303) \leq 17.924341009860250$ and $s (306) \leq 17.963438139777141$ .

Each bound is the side the source prints, or where the printed pose's own side rounds up past it, that side plus one unit of its fifteenth decimal at $n = 208$ , 209 and 263 and two units at $n = 306$ . Each printed side is below the one Couzo published at the same count on 26 or 27 September (T-056), by 8.3e-6 at $n = 303$ up to 0.0105 at $n = 208$ , and below the Kingbird catalogue as captured on 30 September 2026.

Each bound is certified here twice, by T-056's two methods, which share no geometry code. The first is an exact rational packing, the source's decimal pose rounded to rationals at centre dilation 1, decided pair by pair and wall by wall over Q by two checkers that share no code. The second is outward-rounded interval arithmetic on the source's own pose as printed, with the true cosine and sine of each angle, which decides every pair and wall at 40 digits.

Couzo, franciscouzo/square-packing. The author said on jlevy/squares#227 that he found the 102 and 103 packings "with the help of Claude", and the repository itself states no AI assistance.

Significance, composition and next rung

Significance: Seven substantive case results: each moves both the best known side and the verified ceiling at its count, by 8.3e-6 to 0.0105, below the packings that held them (T-056). S3 by the anchor "a substantive case result", as T-056; the source publishes no method, so nothing here is a reusable technique. Kept by the review of 2026-10-05, beside T-056 (S3, 49 such results) and T-088 and T-089 (S2, which move a best known side and no verified bound).
Composition: Both machine entries decide all seven bounds, each at its case's verified value, by different methods (exact-algebraic on a rational rounding of each pose; interval-certified on the printed pose), so every load-bearing part is C3, with two machine methods shown beside the rung. Their common mode is stated in E-franciscouzo-2026-10-03-interval-replay. The comparisons with Couzo's earlier sides and the retained catalogue are recorded values and set no rung.
Next rung: V4 and C4 need a second adversarial AI review of the complete claim by a reviewer distinct from the one of 2026-10-05, and a human oversight record. At $n = 306$ both routes refute the printed side for the printed pose, so the printed side itself needs a pose refined beyond the source's binary64 digits, or coordinates the source prints at higher precision; that would be a new packing, not a replay.
Novelty: previously-published Present in an identified source

The cases

This result concerns 7 cases, too many to draw one by one. Each is listed with the film’s bounds, the proved lower bound and the best known side, and links to its case record, where its packing and number line are drawn.

n	Proved lower	Best known	Gap	Status	Records
208	14.437359	14.924519	0.48715890…	open=	frontier n-208.md
209	14.473188	14.946224	0.47303461…		frontier n-209.md
228	15.103081	15.604602	0.50151960…		frontier n-228.md
263	16.224185	16.733167	0.50898081…		frontier n-263.md
272	16.507810	16.965587	0.45777588…		frontier n-272.md
303	17.418924	17.909959	0.49103397…		frontier n-303.md
306	17.507351	17.963434	0.45608236…		frontier n-306.md

Results on these cases

24 results in the register on these cases, oldest first

2005 published T-007 · 7 of these cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
Lower bound incomplete on these cases, superseded by T-083 and T-180 (reported)
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 7 of these cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
Correction confirmed
Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-27 published T-056 · 7 of these cases
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
Upper bound confirmed superseded by T-174 and T-186
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · 7 of these cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
Lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-03 published T-092 this result · 7 of these cases
Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways
Upper bound confirmed superseded by T-174 and T-186
Couzo · franciscouzo square-packing 2026-10-03 · packet · register
2026-10-05 published T-098 · 7 of these cases
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
Upper bound confirmed on these cases, superseded by T-174 and T-186
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · cases 209, 303
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
Upper bound confirmed superseded by T-174 and T-186
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-115 · cases 208, 263
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
Upper bound confirmed superseded by T-174
Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register
2026-10-07 published T-116 · case 263
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
Upper bound confirmed superseded by T-174
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register
2026-10-07 published T-119 · case 272
Three exact feasible upper bounds at n=266,270,272 from new source arrangements
Upper bound confirmed on these cases, superseded by T-174
Daniel after Levy, Ellsworth, Couzo, Stead · Daniel new arrangements 2026-10-07 · packet · register
2026-10-08 published T-125 · case 263
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-174
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · cases 208, 209
Fourteen rational refinements, seventeen complete source cases
Upper bound confirmed on these cases, superseded by T-174
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-08 published T-128 · cases 228, 306
Eight complete rational refinements from Couzo
Upper bound confirmed superseded by T-174
Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register
2026-10-09 published T-134 · cases 263, 303
Six exact rational certificates from Couzo
Upper bound confirmed on these cases, superseded by T-174 and T-186
Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register
2026-10-09 published T-136 · cases 209, 263, 303
Seventeen exact rational packings from SQUISH's third request
Upper bound confirmed on these cases, superseded by T-174 and T-186
Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-09 published T-138 · cases 303, 306
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
Upper bound confirmed superseded by T-174 and T-186
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-10 published T-140 · case 209
Six more exact rational certificates from Couzo
Upper bound confirmed superseded by T-174
Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register
2026-10-10 published T-146 · cases 263, 303
Exact rational certificates at fifteen counts from Daniel's regularized record lists
Upper bound confirmed on these cases, superseded by T-174 and T-186
Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register
2026-10-11 published T-151 · case 272
Nine exact rational packings from SQUISH's fourth request
Upper bound confirmed superseded by T-174
Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register
2026-10-12 published T-162 · case 303
New packings of 129, 204 and 303 squares, a cut-out and two compositions, each certified exactly
Upper bound confirmed superseded by T-186
Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · Ostashev compositions 2026-10-12 · packet · register
2026-10-12 published T-171 · case 272
An exact contact refinement of SQUISH's packing of 272 squares
Upper bound confirmed superseded by T-174
PingYou Ltd after Chaoweeraprasit, Daniel, Levy · PingYou n272 refinement 2026-10-12 · packet · register
2026-10-13 published T-174 · 7 of these cases
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
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 · 7 of these cases
Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported
Lower bound reviewed
Hosono · wand125 closed form 2026-10-14 · packet · packet · register
2026-10-14 published T-186 · case 303
Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there
Upper bound confirmed
Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register

Links

On this site: The frontier survey · T-092 in the results table

On GitHub, at main

Register: T-092 in results.yaml, line 9360
Evidence: E-franciscouzo-2026-10-03-report · E-franciscouzo-2026-10-03-exact-replay · E-franciscouzo-2026-10-03-interval-replay
Proofs and certificates: certificate franciscouzo-2026-10-03 · certificate facts
Sources: franciscouzo square-packing 2026-10-03 (its own site, retained copy)
Source packet: resources/web/franciscouzo-square-packing-2026-10-03/README.md
Artifacts: 13 artifacts and controls
resources/web/franciscouzo-square-packing-2026-10-03/README.md resources/web/franciscouzo-square-packing-2026-10-03/acquisition/sources.json resources/web/franciscouzo-square-packing-2026-10-03/receipts/certification.json resources/web/franciscouzo-square-packing-2026-10-03/receipts/interval-certification.json devtools/upper_bound_packets.py devtools/check_rational_witness_independent.py devtools/upper_bound_intervals.py devtools/apply_upper_bound_packets.py docs/project/reviews/review-2026-10-05-couzo-6042c56.md resources/web/franciscouzo-square-packing-2026-10-03/receipts/negative-controls.json resources/web/franciscouzo-square-packing-2026-10-03/receipts/interval-negative-controls.json tests/test_upper_bound_packets.py tests/test_upper_bound_intervals.py
Case file: each case’s file is linked from its row above

T-092: Smaller packings again at seven counts from n=208n=208 to 306306, each certified two independent ways

The cases

Results on these cases

Links

T-092: Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways