T-180: Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported

V0 C1 Lower bound reviewed

2026-10-13 published · Hosono · 264 cases, $n = 50$ to 323

For every non-square integer N >= 49, with k = floor(sqrt(N)), Hosono (wand125) reports $s (N)$ >= a(N) = k + sup{delta in (0, 1] : (k + delta)^2 - 4 S*(delta) < N}, where S*(delta) is the largest of 34 certified savings alpha_i delta + beta_i, exact rationals, over the intervals of delta that contain it. In closed form a(N) is the largest of min(2 alpha_i + sqrt(4 alpha_i^2 + N + 4 beta_i - 4 alpha_i k), k + d1_i) over the intervals whose root exceeds k + d0_i: one square root on each interval.

The certificate is a measure on the container of side k + delta: Lebesgue measure in the middle and, for each interval, a family of line segments along the walls and in the corners, quarter-turn symmetric, with total mass (k + delta)^2 - 4 S(delta) for every k >= 7 and mass at least 1 on every closed unit square. The source checks the last by 187,183,108 exact leaf tests in 136 records of four local regions, every one replayed by its Rust checker replay4d (4,294 CPU-hours), and the structure of all 68 families by an exact Python check.

At the counts held here it is above the reported lower bound at 156 counts, $n = 104$ to 116, 125 to 139, 148 to 164, 173 to 191, 201 to 220, 230 to 251, 261 to 284 and 294 to 319, mostly Nagamochi's closed form, and above the verified lower bound, Karakus's strip bound, at 189: $s (104) \geq 10.288527854902856 \dots$ , $s (230) \geq 15.212985634623918 \dots$ , $s (319) \geq 17.950279622148055 \dots$ . It is above no upper bound. It exceeds Nagamochi's capped formula at every other non-square N >= 49 than m^2 - 1 and m^2 - 2, without Nagamochi's Lemma 1 (T-085), and because S*(1) > 3/4 it gives s(m^2 - 1) = s(m^2 - 2) = s(m^2 - 3) = m for m >= 8, values this record already holds (T-084, T-086, T-064).

Here the source's structure check passed on all 68 families from the release, and an exact evaluation written here derives a(n) at every count from the retained intervals by two routes; both agree with the source's own a(N) at every count to 3600. Two of the 136 records, the smallest, were replayed here by replay4d as a calibration, REPLAY OK; a sample sets no rung, so the bound stands as reported. The 14 October review re-derived the argument from the certificate to the claim and found no blocking defect.

Hosono, square-packing, 13 October 2026, credited to Hosono (wand125) as issue 600 asks. Issue 600 states: 'The search, the certificates, the checkers and this text were produced with AI assistance (Claude and Codex), under the direction of and reviewed by Hosono (wand125).' The commit that adds it names Claude as co-author.

The case records take a(n) in the reported lane at 154 of the 156 counts, rounded down at 12 places. At 125 and 148 an earlier report is stronger and pending adoption, Hosono's 563/50 at 122 to 126 (T-144) and 306/25 at 145 to 149 (T-179), so those lanes are unchanged; at 149 this bound supersedes the latter. The verified lanes are unchanged until a replay.

Significance, composition and next rung

Significance: Scored as a claim: a bound family for every non-square N >= 49, by one new measure construction, that raises the reported lower bound at 156 of the counts held here and would raise the verified one at 189, and re-proves Nagamochi's values at N >= 49 without his Lemma 1. S4 by the anchor "a reusable technique, bound family, or resolved disputed value". The score is the theorem's, not ours.
Composition: One reported certificate on its reported entry. The theorem rests on the measure principle (SPEC.md 1.2), the total-mass identity (1.3), the reduction of every square to four local regions for k >= 7 (2.2, 2.3), the structure of each family, every leaf of every record, the coverage of (0, 1] by the intervals, and the chord of the affine saving on each interval. The value at each count is derived from the intervals by V-closed-form-evaluation, exactly. A replay here would run the source's replay4d unchanged, one implementation of one method; it has run here on 2 of the 136 records, as a calibration. The source's cfcheck is a second implementation of SPEC.md's procedures, sharing none of replay4d's code, run by the source on 8 of the 136 records and here on a sample of 2,909 leaves.
Next rung: C3 needs a complete replay of all 136 records by replay4d from the retained crates, built with the locked Cargo.lock, with the structure check, two mutated records refused and a review that reads the replay; the source prices it at 4,294 CPU-hours, and a calibration here at about 5,250 on this host. It is held over the owner's 40 CPU-hour cap for each validation item (think-wu73). The source's second checker, cfcheck, written from SPEC.md and sharing no code with replay4d (E-wand125-closed-form-cfcheck-report), has checked intervals 1 and 34 of the 34; a complete replay of all 136 records by it would be a second route at a higher level, priced at about 9,900 worker-hours.
Unfinished confirmations: C3 (think-tsox): a complete replay of all 136 records (187,183,108 leaves) by replay4d built from the retained crates with the locked dependencies, through replay_all.py, with the structure check and two mutated records refused by the same build. About 4294 CPU-hours (priced 2026-10-14).

C3 (think-3n1x): a complete replay of all 136 records by cfcheck built from the retained crate at 746e2c5 with its locked dependencies, from no resumed checkpoint, with the three re-sealed mutants refused by the same build. About 9900 CPU-hours (priced 2026-10-14).
Novelty: previously-published Present in an identified source

The cases

This result concerns 264 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
50	7.400000	7.571429	0.17142857…	open=	frontier n-050.md
51	7.470000	7.690356	0.22035593…		frontier n-051.md
52	7.550000	7.707107	0.15710678…		frontier n-052.md
53	7.627500	7.822876	0.19537565…		frontier n-053.md
54	7.685000	7.846668	0.16166719…		frontier n-054.md
55	7.728000	7.945772	0.21777100…	open≈	frontier n-055.md
56	7.802500	8	0.1975	open=	frontier n-056.md
57	7.872500		0.1275		frontier n-057.md
58	7.935000		0.065		frontier n-058.md
59	8		0	provedO=	frontier n-059.md
60					frontier n-060.md
61					frontier n-061.md
62					frontier n-062.md
63					frontier n-063.md
65	8.350000	8.535534	0.18553390…	open=	frontier n-065.md
66	8.430000	8.656855	0.22685424…		frontier n-066.md
67	8.480000	8.707107	0.22710678…		frontier n-067.md
68	8.530000	8.798796	0.26879523…		frontier n-068.md
69	8.620000	8.827195	0.20719465…		frontier n-069.md
70	8.657500	8.880961	0.22346037…		frontier n-070.md
71	8.721000	8.944072	0.22307155…	open≈	frontier n-071.md
72	8.760000	9	0.24	open=	frontier n-072.md
73	8.813000		0.187		frontier n-073.md
74	8.867500		0.1325		frontier n-074.md
75	8.940000		0.06		frontier n-075.md
76	9		0	provedO=	frontier n-076.md
77					frontier n-077.md
78					frontier n-078.md
79					frontier n-079.md
80					frontier n-080.md
82	9.320000	9.535534	0.21553390…	open=	frontier n-082.md
83	9.370000	9.634758	0.26475764…		frontier n-083.md
84	9.422500	9.697935	0.27543479…		frontier n-084.md
85	9.475000	9.742641	0.26764068…		frontier n-085.md
86	9.503000	9.820536	0.31753540…		frontier n-086.md
87	9.580000	9.838816	0.25881526…		frontier n-087.md
88	9.620000	9.882452	0.26245103…		frontier n-088.md
89	9.650000	9.949748	0.29974746…		frontier n-089.md
90	9.730000	9.998278	0.26827716…		frontier n-090.md
91	9.762500	10	0.2375		frontier n-091.md
92	9.770000		0.23		frontier n-092.md
93	9.880000		0.12		frontier n-093.md
94	9.972000		0.028		frontier n-094.md
95	10		0	provedO=	frontier n-095.md
96					frontier n-096.md
97					frontier n-097.md
98					frontier n-098.md
99					frontier n-099.md
101	10.280000	10.535534	0.25553390…	open=	frontier n-101.md
102		10.605829	0.32582869…		frontier n-102.md
103		10.679233	0.39923204…		frontier n-103.md
104		10.707107	0.42710678…		frontier n-104.md
105		10.789304	0.50930378…		frontier n-105.md
106	10.310708	10.822909	0.51219960…	open≈	frontier n-106.md
107	10.361541	10.846668	0.48512573…	open=	frontier n-107.md
108	10.620000	10.904822	0.28482101…		frontier n-108.md
109	10.462429	10.949748	0.48731804…		frontier n-109.md
110	10.512492	10.996784	0.48429119…		frontier n-110.md
111	10.562305	11	0.43769410…		frontier n-111.md
112	10.611874		0.38812579…		frontier n-112.md
113	10.661200		0.33879928…		frontier n-113.md
114	10.710288		0.28971107…		frontier n-114.md
115	10.759142		0.24085773…		frontier n-115.md
116	10.807764		0.19223593…		frontier n-116.md
117	10.856157		0.14384241…		frontier n-117.md
118	11		0	provedO=	frontier n-118.md
119					frontier n-119.md
120					frontier n-120.md
122	11.047511	11.535534	0.48802235…	open=	frontier n-122.md
123	11.094810	11.591379	0.49656809…		frontier n-123.md
124	11.141898	11.656855	0.51495592…		frontier n-124.md
125	11.188779	11.707107	0.51832761…		frontier n-125.md
126	11.235455	11.742641	0.50718541…		frontier n-126.md
127	11.281929	11.810879	0.52894946…		frontier n-127.md
128	11.328203	11.825092	0.49688805…		frontier n-128.md
129	11.374281	11.870014	0.49573226…		frontier n-129.md
130	11.420164	11.902431	0.48226581…		frontier n-130.md
131	11.465856	11.949660	0.48380348…		frontier n-131.md
132	11.511357	11.976049	0.46469061…		frontier n-132.md
133	11.556672	12	0.44332780…		frontier n-133.md
134	11.601801		0.39819834…		frontier n-134.md
135	11.646748		0.35325159…		frontier n-135.md
136	11.691514		0.30848535…		frontier n-136.md
137	11.736102		0.26389747…		frontier n-137.md
138	11.780514		0.21948582…		frontier n-138.md
139	11.824751		0.17524834…		frontier n-139.md
140	11.868817		0.13118299…		frontier n-140.md
141	12		0	provedO=	frontier n-141.md
142					frontier n-142.md
143					frontier n-143.md
145	12.240000	12.535534	0.29553390…	open=	frontier n-145.md
146		12.583783	0.34378219…		frontier n-146.md
147		12.651044	0.41104355…		frontier n-147.md
148		12.656855	0.41685424…		frontier n-148.md
149		12.707107	0.46710678…		frontier n-149.md
150	12.257976	12.778175	0.52019857…		frontier n-150.md
151	12.300423	12.822876	0.52245193…		frontier n-151.md
152	12.342719	12.830719	0.48799951…	open≈	frontier n-152.md
153	12.384864	12.870014	0.48514952…	open=	frontier n-153.md
154	12.426860	12.921101	0.49424049…		frontier n-154.md
155	12.468709	12.952261	0.48355174…		frontier n-155.md
156	12.510412	12.977054	0.46664093…		frontier n-156.md
157	12.551970	12.993303	0.44133181…		frontier n-157.md
158	12.593386	13	0.40661337…		frontier n-158.md
159	12.634661		0.36533890…		frontier n-159.md
160	12.675795		0.32420433…		frontier n-160.md
161	12.716791		0.28320827…		frontier n-161.md
162	12.757650		0.24234932…		frontier n-162.md
163	12.798373		0.20162612…		frontier n-163.md
164	12.838962		0.16103732…		frontier n-164.md
165	12.879418		0.12058159…		frontier n-165.md
166	13		0	provedO=	frontier n-166.md
167					frontier n-167.md
168					frontier n-168.md
170	13.039936	13.535534	0.49559770…	open=	frontier n-170.md
171	13.079745	13.571429	0.49168294…		frontier n-171.md
172	13.119429	13.614690	0.49526038…		frontier n-172.md
173	13.158988	13.656855	0.49786534…		frontier n-173.md
174	13.198425	13.707107	0.50868168…		frontier n-174.md
175	13.237739	13.764051	0.52631084…		frontier n-175.md
176	13.276932	13.795873	0.51893997…		frontier n-176.md
177	13.316005	13.822980	0.50697411…	open≈	frontier n-177.md
178	13.354960	13.846668	0.49170706…	open=	frontier n-178.md
179	13.393796	13.880961	0.48716341…		frontier n-179.md
180	13.432517	13.915619	0.48310134…		frontier n-180.md
181	13.471121	13.950820	0.47969775…		frontier n-181.md
182	13.509611	13.970917	0.46130460…		frontier n-182.md
183	13.547988	13.999581	0.45159182…		frontier n-183.md
184	13.586252	14	0.41374767…		frontier n-184.md
185	13.624404		0.37559525…		frontier n-185.md
186	13.662446		0.33755341…		frontier n-186.md
187	13.700378		0.29962121…		frontier n-187.md
188	13.738202		0.26179770…		frontier n-188.md
189	13.775918		0.22408195…		frontier n-189.md
190	13.813526		0.18647304…		frontier n-190.md
191	13.851029		0.14897007…		frontier n-191.md
192	13.888427		0.11157216…		frontier n-192.md
193	14		0	provedO=	frontier n-193.md
194					frontier n-194.md
195					frontier n-195.md
197	14.036986	14.535534	0.49854753…	open=	frontier n-197.md
198	14.073871	14.571429	0.49755661…		frontier n-198.md
199	14.110657	14.614690	0.50403226…		frontier n-199.md
200	14.147344	14.656855	0.50951018…		frontier n-200.md
201	14.183932	14.707107	0.52317459…		frontier n-201.md
202	14.220422	14.727923	0.50749932…		frontier n-202.md
203	14.256816	14.778175	0.52135810…		frontier n-203.md
204	14.293114	14.804362	0.51124715…		frontier n-204.md
205	14.329316	14.824452	0.49513446…		frontier n-205.md
206	14.365424	14.860159	0.49473403…		frontier n-206.md
207	14.401438	14.880961	0.47952159…		frontier n-207.md
208	14.437359	14.924519	0.48715890…		frontier n-208.md
209	14.473188	14.946224	0.47303461…		frontier n-209.md
210	14.508925	14.973002	0.46407539…		frontier n-210.md
211	14.544571	14.993550	0.44897772…		frontier n-211.md
212	14.580127	15	0.41987215…		frontier n-212.md
213	14.615594		0.38440578…		frontier n-213.md
214	14.650971		0.34902830…		frontier n-214.md
215	14.686260		0.31373904…		frontier n-215.md
216	14.721462		0.27853734…		frontier n-216.md
217	14.756577		0.24342257…		frontier n-217.md
218	14.791605		0.20839407…		frontier n-218.md
219	14.826548		0.17345121…		frontier n-219.md
220	14.861406		0.13859338…		frontier n-220.md
221	14.896180		0.10381995…		frontier n-221.md
222	15		0	provedO=	frontier n-222.md
223					frontier n-223.md
224					frontier n-224.md
226	15.034441	15.535534	0.50109205…	open=	frontier n-226.md
227	15.068802	15.563626	0.49482336…		frontier n-227.md
228	15.103081	15.596044	0.49296125…		frontier n-228.md
229	15.137281	15.650162	0.51288034…		frontier n-229.md
230	15.171400	15.682927	0.51152607…		frontier n-230.md
231	15.205441	15.707107	0.50166561…		frontier n-231.md
232	15.239402	15.761410	0.52200667…		frontier n-232.md
233	15.273286	15.778175	0.50488789…		frontier n-233.md
234	15.307092	15.822876	0.51578276…		frontier n-234.md
235	15.340822	15.826606	0.48578355…		frontier n-235.md
236	15.374474	15.861935	0.48745960…		frontier n-236.md
237	15.408051	15.893806	0.48575406…		frontier n-237.md
238	15.441552	15.924107	0.48255398…		frontier n-238.md
239	15.474979	15.948444	0.47346419…		frontier n-239.md
240	15.508331	15.969686	0.46135431…		frontier n-240.md
241	15.541608	15.987118	0.44550813…		frontier n-241.md
242	15.574813	16	0.42518656…		frontier n-242.md
243	15.607944		0.39205507…		frontier n-243.md
244	15.641003		0.35899607…		frontier n-244.md
245	15.673990		0.32600909…		frontier n-245.md
246	15.706906		0.29309367…		frontier n-246.md
247	15.739750		0.26024934…		frontier n-247.md
248	15.772524		0.22747565…		frontier n-248.md
249	15.805227		0.19477213…		frontier n-249.md
250	15.837861		0.16213834…		frontier n-250.md
251	15.870426		0.12957385…		frontier n-251.md
252	15.902921		0.09707819…		frontier n-252.md
253	16		0	provedO=	frontier n-253.md
254					frontier n-254.md
255					frontier n-255.md
257	16.032224	16.535534	0.50330933…	open=	frontier n-257.md
258	16.064382	16.563449	0.49906558…		frontier n-258.md
259	16.096473	16.591379	0.49490418…		frontier n-259.md
260	16.128499	16.633521	0.50502134…		frontier n-260.md
261	16.160459	16.678780	0.51832011…		frontier n-261.md
262	16.192354	16.707107	0.51475195…		frontier n-262.md
263	16.224185	16.733167	0.50898081…		frontier n-263.md
264	16.255951	16.774736	0.51878387…		frontier n-264.md
265	16.287653	16.778175	0.49052118…		frontier n-265.md
266	16.319292	16.823029	0.50373673…		frontier n-266.md
267	16.350867	16.838816	0.48794778…		frontier n-267.md
268	16.382380	16.872030	0.48964967…		frontier n-268.md
269	16.413830	16.889731	0.47589961…		frontier n-269.md
270	16.445218	16.928149	0.48292988…		frontier n-270.md
271	16.476545	16.948948	0.47240242…		frontier n-271.md
272	16.507810	16.965587	0.45777588…		frontier n-272.md
273	16.539014	16.983926	0.44491103…		frontier n-273.md
274	16.570158	17	0.42984132…		frontier n-274.md
275	16.601242		0.39875781…		frontier n-275.md
276	16.632265		0.36773419…		frontier n-276.md
277	16.663229		0.33677012…		frontier n-277.md
278	16.694134		0.30586525…		frontier n-278.md
279	16.724980		0.27501926…		frontier n-279.md
280	16.755768		0.24423179…		frontier n-280.md
281	16.786497		0.21350252…		frontier n-281.md
282	16.817168		0.18283112…		frontier n-282.md
283	16.847782		0.15221727…		frontier n-283.md
284	16.878339		0.12166064…		frontier n-284.md
285	16.908839		0.09116091…		frontier n-285.md
286	17		0	provedO=	frontier n-286.md
287					frontier n-287.md
288					frontier n-288.md
290	17.030275	17.535534	0.50525865…	open=	frontier n-290.md
291	17.060495	17.535534	0.47503874…		frontier n-291.md
292	17.090660	17.591379	0.50071812…		frontier n-292.md
293	17.120770	17.623066	0.50229507…		frontier n-293.md
294	17.150825	17.656855	0.50602844…		frontier n-294.md
295	17.180827	17.704233	0.52340547…		frontier n-295.md
296	17.210774	17.707107	0.49633181…		frontier n-296.md
297	17.240669	17.739456	0.49878651…		frontier n-297.md
298	17.270509	17.778175	0.50766476…		frontier n-298.md
299	17.300297	17.801005	0.50070681…		frontier n-299.md
300	17.330032	17.824124	0.49409070…		frontier n-300.md
301	17.359715	17.846668	0.48695189…	open≈	frontier n-301.md
302	17.389345	17.870014	0.48066809…	open=	frontier n-302.md
303	17.418924	17.901790	0.48286495…		frontier n-303.md
304	17.448451	17.920945	0.47249283…		frontier n-304.md
305	17.477926	17.950467	0.47253963…		frontier n-305.md
306	17.507351	17.963434	0.45608236…		frontier n-306.md
307	17.536725	17.979231	0.44250592…		frontier n-307.md
308	17.566048	17.997150	0.43110095…		frontier n-308.md
309	17.595320	18	0.40467900…		frontier n-309.md
310	17.624543		0.37545621…		frontier n-310.md
311	17.653716		0.34628320…		frontier n-311.md
312	17.682840		0.31715972…		frontier n-312.md
313	17.711914		0.28808552…		frontier n-313.md
314	17.740939		0.25906035…		frontier n-314.md
315	17.769916		0.23008396…		frontier n-315.md
316	17.798843		0.20115610…		frontier n-316.md
317	17.827723		0.17227654…		frontier n-317.md
318	17.856554		0.14344504…		frontier n-318.md
319	17.885338		0.11466134…		frontier n-319.md
320	17.914074		0.08592523…		frontier n-320.md
321	18		0	provedO=	frontier n-321.md
322					frontier n-322.md
323					frontier n-323.md

Results on these cases

203 results in the register on these cases, oldest first

2005 published T-007 · 264 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
Nagamochi · Nagamochi 2005 · source · register
2018 published T-087 · case 61
$s (37) \geq 5 \sqrt{3} / 2 + 2 \sqrt{2} - 1$ and $s (61) \geq 7 \sqrt{3} / 2 + 2 \sqrt{2} - 1$ , by optimal piercing
Lower bound reviewed superseded by T-063
Bašić, Slivková · Basic-Slivkova 2018 · source · register
2026-09-04 published T-085 · 264 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
2026-09-04 published T-086 · 11 of these cases
$s (k^{2} - 2) = k$ for every integer $k \geq 2$ ; $k = 3 \dots 18$ are the cases held here
Optimality confirmed
Vlasenko · chelokot Nagamochi counterexample 2026 · packet · source · register
2026-09-16 published T-057 · case 211
$s (211) \leq 14.99796070496771500150 < 15$ , the first packing of 211 squares below the grid on record
Upper bound confirmed superseded by T-188
de Winter · de Winter n211 2026-09-16 · packet · register
2026-09-22 published T-044 · 12 of these cases
Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files
Lower bound confirmed superseded by T-082, T-090, T-091 and T-205
Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register
2026-09-24 published T-088 · case 69
$s (69) \leq 8.82719465572975$ , Ellsworth's degree-38 packing, certified exactly from its picture
Upper bound confirmed superseded by T-101 and T-155
Ellsworth after hmbelvedere, Cantrell, Schadt, Morandi · Ellsworth n69 2026-09-24 · packet · packet · register
2026-09-24 published T-089 · cases 83, 87
$s (83) \leq 9.63475764863195$ and $s (87) \leq 9.83881526994915$ , Chang's packings, certified exactly
Upper bound confirmed superseded by T-101 and T-155
Chang, Ellsworth after Cantrell, Hajba, Stenlund, Bidwell; Chang after Ellsworth, Cantrell, Schadt, Hajba, DeVincentis · Chang n83 2026-09-24 · Chang n87 2026-09-24 · packet · packet · register
2026-09-27 published T-045 · 5 of these cases
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
Lower bound confirmed superseded by T-063, T-064, T-067, T-090 and T-184
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register
2026-09-27 published T-046 · 31 of these cases
Rectangle-density lower bounds verified at 48 counts in $n = 18 \dots 95$
Lower bound confirmed superseded by T-062, T-063, T-064, T-066, T-067, T-082, T-090, T-091, T-094, T-097, T-184, T-191, T-205 and T-266
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · source · review · register
2026-09-27 published T-056 · 49 of these cases
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
Upper bound confirmed superseded by T-098, T-146, T-174, T-182, T-193, T-197, T-218, T-219, T-222, T-229, T-243, T-259, T-273, T-278, T-279, T-282, T-283, T-284, T-287, T-288, T-289, T-292, T-295, T-297, T-299 and T-301
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-28 published T-048 · cases 50, 51
$s (50) \geq 37 / 5 = 7.4$
Lower bound confirmed
Hosono after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point and mixed bounds 2026-09-28 · packet · source · review · register
2026-09-28 published T-062 · case 60
$s (60) = 8$ , by a mixed cover of points and grid-line segments
Optimality confirmed
Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · source · review · register
2026-09-28 published T-063 · case 61
$s (61) = 8$ , by monotonicity from T-062
Optimality confirmed
Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-09-28 published T-070 · 18 of these cases
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
Lower bound confirmed superseded by T-062, T-066, T-082, T-090, T-091, T-094, T-191 and T-205
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-09-28 · packet · packet · source · review · register
2026-09-29 published T-058 · 48 of these cases
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
Method limit confirmed
Hosono after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-083 · 264 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
2026-09-29 published T-084 · 11 of these cases
$s (k^{2} - 1) = k$ for every integer $k \geq 3$ ; $k = 3 \dots 18$ are the cases held here
Optimality confirmed
Karakuş · Karakuş 2026 · source · register
2026-09-30 published T-064 · 11 of these cases
$s (k^{2} - 3) = k$ for every integer $k$ from 6 up; $k = 6 \dots 18$ are the cases held here
Optimality confirmed
Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-10-01 published T-066 · case 59
$s (59) = 8$ , by a mixed cover of points and grid-line segments
Optimality confirmed
Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register
2026-10-01 published T-067 · cases 77, 78
$s (77) = 9$ , by a mixed cover extended from Daniel's $s (60)$ cover
Optimality confirmed
Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register
2026-10-01 published T-068 · 19 of these cases
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
Lower bound confirmed on these cases, superseded by T-082, T-090, T-091, T-097, T-184, T-191, T-205 and T-266
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-069 · cases 65, 66, 90, 92
Mixed rectangle-measure lower bounds replayed at $n = 37, 65, 66, 90, 92$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-074 · 16 of these cases
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
Lower bound confirmed on these cases, superseded by T-082, T-090, T-091, T-191, T-205 and T-266
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register
2026-10-02 published T-071 · cases 84, 85, 86, 87
Mixed rectangle-measure lower bounds replayed at $n = 84 \dots 87$
Lower bound confirmed superseded by T-090, T-091, T-206 and T-207
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-02 · packet · source · review · register
2026-10-02 published T-072 · case 76
Mixed rectangle-measure lower bound replayed at $n = 76$
Lower bound confirmed superseded by T-184
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds n76 2026-10-02 · packet · packet · source · review · register
2026-10-02 published T-073 · 6 of these cases
Linear-measure lower bounds replayed at $n = 83$ and $n = 101 \dots 105$
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-075 · 9 of these cases
Mixed rectangle-measure lower bounds replayed at nine counts in $n = 83 \dots 96$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds afternoon 2026-10-02 · packet · source 1 · source 2 · review · register
2026-10-02 published T-076 · case 82
Linear-measure lower bound replayed at $n = 82$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n82 2026-10-02 · packet · source · review · register
2026-10-02 published T-077 · case 70
Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70
Lower bound confirmed superseded by T-091
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-02 · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-10-02 published T-080 · 5 of these cases
Linear-measure lower bound replayed at $n = 101 \dots 105$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register
2026-10-03 published T-081 · 11 of these cases
$s (k^{2} - 4) = k$ for every integer $k$ from 5 up; $k = 5 \dots 18$ are the cases held here
Optimality reviewed
Daniel after Burns, Massaccesi · evand square-packing 2026-10-03 · packet · packet · source · review · register
2026-10-03 published T-082 · 22 of these cases
Mixed rectangle-measure lower bounds verified at 22 counts in $n = 51 \dots 96$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register
2026-10-03 published T-092 · 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, T-278 and T-301
Couzo · franciscouzo square-packing 2026-10-03 · packet · register
2026-10-04 published T-090 · 14 of these cases
Mixed rectangle-measure lower bounds verified at 17 counts in $n = 42 \dots 96$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-04 · packet · source · review · register
2026-10-04 published T-091 · 17 of these cases
Mixed rectangle-measure lower bounds verified at 17 counts in $n = 53 \dots 95$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds evening 2026-10-04 · packet · source · review · register
2026-10-05 published T-094 · cases 67, 84
Mixed rectangle-measure lower bounds verified at $n = 67$ and $84$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-05 · packet · source · review · register
2026-10-05 published T-097 · case 66
Mixed rectangle-measure lower bound verified at $n = 66$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-05 · packet · source · review · register
2026-10-05 published T-098 · 48 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
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · 73 of these cases
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side
Upper bound confirmed
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · 10 of these cases
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
Upper bound confirmed superseded by T-139, T-174, T-186, T-197, T-222, T-259, T-278 and T-288
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-114 · case 153
$s (153) \leq 7250614903299225 / 562949953421312$ , from a SQUISH rational packing
Upper bound confirmed superseded by T-174
Chaoweeraprasit after Ellsworth · SQUISH n153 2026-10-07 · packet · packet · register
2026-10-07 published T-115 · 12 of these cases
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
Upper bound confirmed superseded by T-139, T-174, T-186, T-193, T-197, T-222, T-294 and T-295
Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register
2026-10-07 published T-116 · 9 of these cases
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
Upper bound confirmed superseded by T-127, T-174, T-193, T-218, T-284, T-288 and T-294
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register
2026-10-07 published T-117 · cases 105, 292
Exact rational ceiling refinements at $n = 105$ , 292
Upper bound confirmed superseded by T-146 and T-229
Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register
2026-10-07 published T-118 · case 68
Exact rational ceiling refinements at $n = 68$
Upper bound confirmed superseded by T-174
Rehwaldt after Couzo and earlier contributors · Rehwaldt n68 refinement 2026-10-07 · packet · register
2026-10-07 published T-119 · cases 266, 270, 272
Three exact feasible upper bounds at n=266,270,272 from new source arrangements
Upper bound confirmed
Daniel after Levy, Ellsworth, Couzo, Stead · Daniel new arrangements 2026-10-07 · packet · register
2026-10-07 published T-120 · case 102
Reported s(102) ≤ alpha_102 with a degree-8 exact side form
Upper bound recorded superseded by T-193
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-07 published T-121 · case 106
Reported s(106) ≤ alpha_106 with a degree-32 exact side form
Upper bound recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-07 published T-122 · case 152
Reported s(152) ≤ alpha_152 with a degree-40 exact side form
Upper bound recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-07 published T-123 · case 177
Reported s(177) ≤ alpha_177 with a degree-32 exact side form
Upper bound recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-07 published T-124 · 131 of these cases
Reported non-strict local minima for 178 source configurations
Restricted optimality recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-07 published T-129 · cases 105, 130, 292
Three dated Daniel certificate reports with historical source custody
Upper bound recorded superseded by T-146, T-229 and T-259
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 · 25 of these cases
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-126, T-127, T-134, T-139, T-174, T-186, T-187, T-193, T-218, T-229, T-259, T-267 and T-294
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-126 · case 51
Undilated n51 construction over Q(sqrt2)
Upper bound confirmed
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · 17 of these cases
Fourteen rational refinements, seventeen complete source cases
Upper bound confirmed
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-08 published T-128 · 8 of these cases
Eight complete rational refinements from Couzo
Upper bound confirmed superseded by T-174, T-197, T-229, T-288 and T-301
Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register
2026-10-08 published T-130 · 5 of these cases
Five follow-up rational refinements from Couzo
Upper bound confirmed superseded by T-174, T-229, T-267 and T-283
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register
2026-10-09 published T-131 · case 132
$s (132) \leq 11.987099332245063 \dots$ from Daniel's record hunt
Upper bound confirmed superseded by T-292
Daniel after Couzo, Chaoweeraprasit, Levy · Daniel record hunt 2026-10-09 · packet · register
2026-10-09 published T-134 · 6 of these cases
Six exact rational certificates from Couzo
Upper bound confirmed
Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register
2026-10-09 published T-136 · 17 of these cases
Seventeen exact rational packings from SQUISH's third request
Upper bound confirmed superseded by T-146, T-174, T-193, T-218, T-222, T-275, T-278, T-279, T-283, T-295 and T-297
Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-09 published T-138 · 16 of these cases
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
Upper bound confirmed superseded by T-134, T-174, T-193, T-229, T-267, T-278, T-283, T-288 and T-292
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Gupta, Daniel, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-09 published T-139 · case 126
A more precise certificate of Xu's 126-square arrangement, from Daniel
Upper bound confirmed
Daniel after Xu, Chaoweeraprasit · Daniel trio126 2026-10-09 · packet · register
2026-10-10 published T-135 · case 308
A packing of 308 squares below the grid, from Fang
Upper bound confirmed superseded by T-221
Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead · Fang two-wedge certificates 2026-10-10 · packet · register
2026-10-10 published T-137 · case 70
An exact rational refinement of Xu's packing of 70 squares
Upper bound confirmed superseded by T-174
Deleeuw after Xu, Levy · Deleeuw n70 refinement 2026-10-10 · packet · register
2026-10-10 published T-140 · 6 of these cases
Six more exact rational certificates from Couzo
Upper bound confirmed superseded by T-174, T-267, T-283, T-292 and T-295
Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register
2026-10-10 published T-141 · cases 132, 308
Exact rational certificates at 132 and 308 from Daniel's third record hunt
Upper bound confirmed superseded by T-221 and T-292
Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · Daniel record hunt 3 2026-10-10 · packet · register
2026-10-10 published T-144 · 5 of these cases
Linear-measure lower bound $s (n) \geq 563 / 50 = 11.26$ for $n = 122 \dots 126$ , reported
Lower bound recorded
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n122 2026-10-10 · packet · register
2026-10-10 published T-146 · 15 of these cases
Exact rational certificates at fifteen counts from Daniel's regularized record lists
Upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register
2026-10-11 published T-147 · case 207
A packing of 207 squares from two placed patches of Xu's packing of 70
Upper bound confirmed superseded by T-174
PingYou Ltd after Xu, Daniel, Levy · PingYou n207 certificate 2026-10-11 · packet · register
2026-10-11 published T-148 · case 153
A packing of 153 squares built from Xu's packing of 70, its contacts reclosed at a new angle
Upper bound confirmed superseded by T-174
PingYou Ltd after Xu, Daniel, Levy · PingYou n153 certificate 2026-10-11 · packet · register
2026-10-11 published T-149 · cases 155, 156, 211
New packings of 155, 156 and 211 squares, each with an exact rational certificate
Upper bound confirmed superseded by T-188, T-197 and T-287
Ostashev after Nakajima, Levy, Xu · Ostashev records 2026-10-11 · packet · register
2026-10-11 published T-150 · cases 146, 261
Exact rational refinements of Xu's packings of 146 and 261 squares
Upper bound confirmed superseded by T-174 and T-187
Barnett after Xu, Daniel, Levy · Barnett refinements 2026-10-11 · packet · register
2026-10-11 published T-151 · 9 of these cases
Nine exact rational packings from SQUISH's fourth request
Upper bound confirmed superseded by T-174, T-186, T-219, T-273 and T-283
Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register
2026-10-11 published T-152 · cases 304, 305
Exact rational refinements of SQUISH's Mondrian packings of 304 and 305 squares
Upper bound confirmed superseded by T-174 and T-273
Couzo after Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-11 · packet · register
2026-10-11 published T-159 · case 102
Another arrangement of Xu's design for 102 squares, certified exactly from its decimal layout
Upper bound confirmed superseded by T-193
Calle after Xu, Levy · Calle n102 2026-10-12 · packet · register
2026-10-11 published T-185 · case 175
A tighter packing of 175 squares from Couzo, superseded by Ostashev's composition
Upper bound confirmed superseded by T-267
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo n175 2026-10-11 · packet · register
2026-10-11 published T-235 · case 59
Hosono reports a Lean kernel proof that $s (59) = 8$ , on Daniel's mixed-measure checker
Formalization recorded
Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 n59 Lean kernel 2026-10-11 · packet · source · register
2026-10-12 established T-155 · 54 of these cases
Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace
Upper bound confirmed
Levy after Daniel · register
2026-10-12 published T-158 · cases 175, 232, 264
New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactly
Upper bound confirmed superseded by T-186, T-267 and T-275
Ostashev after Nakajima, Ellsworth, Schadt, Friedman, de Winter, Daniel, Levy, Xu · Ostashev records 2026-10-12 · packet · register
2026-10-12 published T-160 · case 154
A smaller packing of 154 squares from Daniel's size-chaining explorer, certified exactly
Upper bound confirmed superseded by T-222
Daniel after Couzo, Levy · Daniel n154 2026-10-12 · packet · register
2026-10-12 published T-161 · cases 156, 176, 237, 271
Exact rational certificates at 156, 176, 237 and 271 from Couzo
Upper bound confirmed superseded by T-186, T-219, T-287 and T-295
Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-12 · packet · register
2026-10-12 published T-162 · cases 129, 204, 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, T-268 and T-278
Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · Ostashev compositions 2026-10-12 · packet · register
2026-10-12 published T-163 · cases 211, 241, 273, 307
s(n² − n + 1) < n for every n ≥ 15, and n² − n + F(n) squares below side n, from Fang
Upper bound reviewed superseded by T-174, T-182, T-188 and T-279
Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo · Fang excess theorem 2026-10-13 · packet · packet · packet · source 1 · source 2 · register
2026-10-12 published T-164 · case 302
A packing of 302 squares from four boundary patches of Xu's packing of 70 and two upright banks
Upper bound confirmed superseded by T-193
PingYou Ltd after Xu, Daniel, Levy · PingYou n302 certificate 2026-10-12 · packet · register
2026-10-12 published T-165 · case 176
A tighter certificate at 176 from Couzo, published with no issue
Upper bound confirmed superseded by T-186
Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo certificates 2026-10-13 · packet · register
2026-10-12 published T-166 · case 153
A tighter rational realization of PingYou Ltd's packing of 153 squares
Upper bound confirmed superseded by T-174
PingYou Ltd after Xu, Daniel, Levy · PingYou n153 refinement 2026-10-12 · packet · register
2026-10-12 published T-167 · case 207
A tighter rational realization of PingYou Ltd's packing of 207 squares
Upper bound confirmed superseded by T-174
PingYou Ltd after Xu, Daniel, Levy · PingYou n207 refinement 2026-10-12 · packet · register
2026-10-12 published T-168 · case 146
A further exact refinement of Xu's packing of 146 squares, after Barnett's
Upper bound confirmed superseded by T-187
PingYou Ltd after Xu, Barnett, Daniel, Levy · PingYou n146 refinement 2026-10-12 · packet · register
2026-10-12 published T-169 · case 261
A further exact refinement of Xu's packing of 261 squares, after Barnett's
Upper bound confirmed superseded by T-174
PingYou Ltd after Xu, Barnett, Daniel, Levy · PingYou n261 refinement 2026-10-12 · packet · register
2026-10-12 published T-170 · case 105
An exact contact refinement of Xu's packing of 105 squares, after Couzo's
Upper bound confirmed superseded by T-229
PingYou Ltd after Xu, Couzo, Daniel, Levy · PingYou n105 refinement 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
2026-10-12 published T-172 · case 154
A packing of 154 squares from Couzo, continuing Daniel's arrangement
Upper bound confirmed superseded by T-222
Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy · Couzo issue-583 certificates 2026-10-13 · packet · register
2026-10-12 published T-173 · cases 153, 207
Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements there
Upper bound confirmed superseded by T-174
Polkovskyi after Xu, Chaoweeraprasit, Levy · Mishapolk exact certificates 2026-10-12 · packet · register
2026-10-12 published T-261 · 180 of these cases
Reported grade-C local minima for 233 source configurations of field degree at most 4
Restricted optimality incomplete
Daniel after Schadt, Ellsworth, Levy · Daniel exact and local reports 2026-10-12 · packet · packet · register
2026-10-13 established T-156 · case 290
$s (290) \leq 17.5355339059327376 \dots$ , by monotonicity from T-155, with its own exact certificate
Upper bound confirmed
Levy after Daniel · register
2026-10-13 published T-174 · 68 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-175 · cases 211, 241, 304
New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactly
Upper bound confirmed superseded by T-182, T-188 and T-273
Ostashev after Nakajima, Daniel, de Winter, Couzo, Levy, Xu · Ostashev merged bands 2026-10-13 · packet · register
2026-10-13 published T-179 · 5 of these cases
Linear-measure lower bound $s (n) \geq 306 / 25 = 12.24$ for $n = 145 \dots 149$ , replayed
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n145 2026-10-13 · packet · source · review · register
2026-10-13 published T-180 this result · 264 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
2026-10-13 published T-181 · cases 211, 241, 302
Exact rational certificates at 211, 241 and 302 from Couzo
Upper bound confirmed superseded by T-182, T-188 and T-193
Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy; Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo issue-583 certificates 2026-10-13 · Couzo certificates 2026-10-13 · packet · packet · register
2026-10-13 published T-182 · cases 154, 211, 241
Exact rational certificates at 154, 211 and 241 from Couzo
Upper bound confirmed
Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo certificates 2026-10-13 · packet · register
2026-10-13 published T-183 · case 237
Polkovskyi's certificate for 237 squares, the smallest known side there when registered
Upper bound confirmed superseded by T-295
Polkovskyi after Friedman, Daniel, Xu, Chaoweeraprasit, Barnett, Couzo, Ostashev, Ellsworth, Fang, Levy · Mishapolk records 2026-10-14 · packet · register
2026-10-14 published T-184 · case 76
$s (76) = 9$ , by a mixed cover of points and grid-line segments
Optimality confirmed
Greff after Daniel, Hosono · bengreff s76 certificate 2026-10-14 · packet · source · review · register
2026-10-14 published T-186 · 12 of these cases
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
2026-10-14 published T-187 · case 146
A smaller packing of 146 squares from Daniel's chain search, started from Xu's packing
Upper bound confirmed
Daniel after Xu · Daniel n146 2026-10-14 · packet · register
2026-10-14 published T-188 · case 211
A tighter exact certificate for 211 squares from Couzo, of Ostashev's arrangement
Upper bound confirmed
Couzo after Ostashev, Daniel, Levy · Couzo verified bounds 2026-10-14 · packet · register
2026-10-14 published T-189 · case 237
Couzo's refinement of Polkovskyi's 237, and fifteen beyond-horizon bounds
Upper bound confirmed superseded by T-295
Couzo after Polkovskyi, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-14 · packet · register
2026-10-15 published T-190 · case 237
A tighter exact certificate for 237 squares from Couzo, below his issue-607 certificate
Upper bound confirmed superseded by T-295
Couzo after Polkovskyi, Chaoweeraprasit, Daniel, Levy · Couzo n237 2026-10-15 · packet · register
2026-10-15 published T-191 · cases 95, 96
$s (95) = 10$ , by a mixed cover of points and grid-line segments
Optimality confirmed
Greff after Daniel, Hosono · bengreff s95 certificate 2026-10-15 · packet · source · review · register
2026-10-16 published T-193 · 16 of these cases
Exact algebraic sides at sixteen counts from Daniel, of packings the record credits
Upper bound confirmed
Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register
2026-10-16 published T-204 · case 51
Mixed rectangle-measure lower bound $s (51) \geq 3011 / 400$ reported, on a 2,073-direction declared net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-205 · case 68
Mixed rectangle-measure lower bound $s (68) \geq 853 / 100$ verified, on the 201-direction net
Lower bound confirmed
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register
2026-10-16 published T-206 · case 84
Mixed rectangle-measure lower bound $s (84) \geq 3769 / 400$ verified, on the 201-direction net
Lower bound confirmed
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register
2026-10-16 published T-207 · case 85
Mixed rectangle-measure lower bound $s (85) \geq 379 / 40$ verified, on the 201-direction net
Lower bound confirmed
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register
2026-10-16 published T-208 · case 86
Mixed rectangle-measure lower bound $s (86) \geq 3811 / 400$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-209 · case 88
Mixed rectangle-measure lower bound $s (88) \geq 3853 / 400$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-210 · case 89
Mixed rectangle-measure lower bound $s (89) \geq 1937 / 200$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-211 · case 91
Mixed rectangle-measure lower bound $s (91) \geq 783 / 80$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-212 · case 90
Mixed rectangle-measure lower bound $s (90) \geq 489 / 50$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-213 · case 92
Mixed rectangle-measure lower bound $s (92) \geq 493 / 50$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-214 · case 106
Mixed rectangle-measure lower bound $s (106) \geq 10503 / 1000$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-215 · case 107
Mixed rectangle-measure lower bound $s (107) \geq 529 / 50$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-216 · case 108
Mixed rectangle-measure lower bound $s (108) \geq 531 / 50$ verified, on the 201-direction net
Lower bound confirmed
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register
2026-10-16 published T-217 · case 110
Mixed rectangle-measure lower bound $s (110) \geq 1073 / 100$ reported, on the 201-direction net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register
2026-10-16 published T-218 · case 236
Daniel's smaller n236 packing, after Chaoweeraprasit's composition, certified exactly
Upper bound confirmed
Daniel after Chaoweeraprasit, Levy · Daniel n236 2026-10-16 · packet · register
2026-10-16 published T-251 · 11 of these cases
Hosono reports a Lean kernel proof of Daniel's ValidTilt9, so $s (k^{2} - 4) = k$ for every $k \geq 8$
Formalization recorded
Hosono after Daniel, Levy, Burns, Massaccesi · wand125 ValidTilt9 Lean 2026-10-16 · packet · source · register
2026-10-17 published T-196 · cases 227, 229, 260
de Winter reports three smaller rational upper bounds at 227, 229 and 260
Upper bound confirmed superseded by T-225 and T-269
de Winter after Levy · de Winter upper bounds 2026-10-17 · packet · register
2026-10-17 published T-197 · case 155
Daniel's smaller n155 packing, started from Ostashev's packing
Upper bound confirmed
Daniel after Ostashev, Levy · Daniel n155 explorer 2026-10-17 · packet · register
2026-10-17 published T-198 · case 229
Daniel's opening n229 dilation certificate, after Couzo and de Winter
Upper bound confirmed superseded by T-225
Daniel after Couzo, de Winter, Levy · Daniel n229 dilation 2026-10-17 · packet · register
2026-10-17 published T-219 · case 271
Daniel's smaller n271 packing, continuing Couzo's arrangement, certified exactly
Upper bound confirmed
Daniel after Couzo, Chaoweeraprasit, Levy · Daniel n271 2026-10-17 · packet · register
2026-10-17 published T-226 · case 204
Couzo's 204 of #670, below Polkovskyi's T-186, since superseded by his own #675 and #678
Upper bound confirmed superseded by T-268
Couzo after Polkovskyi, Ostashev, PingYou Ltd, Daniel, Levy · Couzo issue 670 2026-10-17 · packet · packet · register
2026-10-17 published T-227 · cases 227, 229
Couzo's refinements of de Winter's 227 and 229, below T-196 at both
Upper bound confirmed superseded by T-225
Couzo after de Winter, Daniel, Levy · Couzo issue 672 2026-10-17 · packet · packet · register
2026-10-17 published T-250 · 11 of these cases
Hosono reports a Lean kernel proof of Daniel's Valid7, and so of $s (k^{2} - 3) = k$ for every $k \geq 6$
Formalization recorded
Hosono after Daniel, Burns, Massaccesi; Hosono after Daniel, Levy, Burns, Massaccesi · wand125 Valid7 Lean 2026-10-11 · wand125 Valid7 Lean v2 2026-10-17 · packet · packet · source 1 · source 2 · register
2026-10-18 published T-199 · case 229
Daniel's smaller n229 update, replacing the opening certificate
Upper bound confirmed superseded by T-225
Daniel after Couzo, de Winter, Levy · Daniel n229 dilation update 2026-10-18 · packet · register
2026-10-18 published T-223 · cases 204, 260
Couzo's smaller packings at 204 and 260, the two in the corpus of issue 678's 85 bounds
Upper bound confirmed superseded by T-268 and T-269
Couzo after de Winter, Polkovskyi, Ostashev, Daniel, Fang, PingYou Ltd, Levy · Couzo issue 678 2026-10-18 · packet · register
2026-10-18 published T-228 · case 204
Couzo's 204 of #675, below his #670, since superseded by his #678
Upper bound confirmed superseded by T-268
Couzo after Polkovskyi, Ostashev, PingYou Ltd, Daniel, Levy · Couzo issue 675 2026-10-18 · packet · packet · register
2026-10-18 published T-236 · cases 211, 241, 273, 307
Daniel reports Lean proofs of Fang's Theorems 1 and 2, $s (n^{2} - n + 1) < n$ for $n \geq 15$
Formalization recorded
Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Levy · Daniel Fang and Ryu Lean 2026-10-18 · packet · source · register
2026-10-19 published T-220 · case 183
Daniel's finite sub-grid upper bound for 183 squares, after Couzo's n182 packing
Upper bound confirmed
Daniel after Couzo; register seed data from Levy · Daniel n183 subgrid 2026-10-19 · packet · register
2026-10-19 published T-221 · case 308
Daniel's smaller n308 packing, started from Chaoweeraprasit's 307 packing, certified exactly
Upper bound confirmed
Daniel after Chaoweeraprasit, Levy · Daniel n308 2026-10-19 · packet · register
2026-10-20 published T-222 · case 154
Daniel's smaller n154 packing, continuing Couzo's #586 arrangement, certified exactly
Upper bound confirmed
Daniel after Couzo, Ostashev, Levy · Daniel n154 sweep 2026-10-20 · packet · packet · register
2026-10-20 published T-224 · case 270
Daniel's smaller n270 packing, another arrangement found from Chaoweeraprasit's, certified exactly
Upper bound confirmed superseded by T-283
Daniel after Chaoweeraprasit, Levy · Daniel n270 sweep 2026-10-20 · packet · register
2026-10-20 published T-225 · cases 204, 227, 229, 260
Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there
Upper bound confirmed
Daniel after Couzo, de Winter, Polkovskyi, Ostashev, PingYou Ltd, Levy · Daniel regularized lists 2026-10-20 · packet · register
2026-10-22 published T-229 · cases 105, 204
Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225
Upper bound confirmed
Polkovskyi after Couzo, Muglich, Daniel, Xu, Ostashev, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register
2026-10-22 published T-230 · case 303
Daniel's smaller n303 packing, started from Ostashev's packing, certified exactly
Upper bound confirmed superseded by T-278
Daniel after Ostashev, Levy · Daniel n303 2026-10-22 · packet · register
2026-10-22 published T-231 · case 204
Polkovskyi's lower re-solve at 204 of Couzo's packing, 1.48e-11 below T-229
Upper bound confirmed superseded by T-268
Polkovskyi after Couzo, Ostashev, PingYou Ltd, Daniel, Levy · Mishapolk n204 2026-10-22 · packet · register
2026-10-23 published T-239 · case 303
Kristiansen's smaller n303 packing, found from Daniel's packing, certified exactly
Upper bound confirmed superseded by T-278
Kristiansen after Daniel, Ostashev, Levy · Kristiansen n303 2026-10-23 · packet · register
2026-10-23 published T-240 · cases 268, 303
Couzo's 268 and 303 of #763, below T-174 and T-239, from Kaggle notebook packings
Upper bound confirmed superseded by T-278 and T-289
Couzo after Lavrentev, Tymchishin, Muglich, Polkovskyi, Daniel, Fang, de Winter, Levy · Couzo issue 763 2026-10-23 · packet · register
2026-10-23 published T-241 · case 304
Couzo's certificate at 304 of Tymchishin's Kaggle notebook packing, 2.53e-3 below T-186
Upper bound confirmed superseded by T-273
Couzo after Tymchishin, Levy · Couzo n304 2026-10-23 · packet · packet · packet · register
2026-10-23 published T-253 · 9 of these cases
Polkovskyi's sub-print re-certifications of nine house packings, 1.3e-19 to 1.7e-19 below each
Upper bound reviewed
Polkovskyi after Daniel, Couzo, Ostashev, de Winter, Chaoweeraprasit, Levy · Mishapolk refinements 2026-10-23 · packet · packet · register
2026-10-24 published T-242 · cases 268, 304
Daniel's exact descents of public Kaggle packings at 268 and 304, below the reports before them
Upper bound confirmed superseded by T-273 and T-289
Daniel after Lavrentev, Tymchishin · Daniel Kaggle descents 2026-10-24 · packet · register
2026-10-25 published T-243 · case 182
Daniel's new arrangement at 182, 3.17e-3 below T-174
Upper bound confirmed
Daniel after Couzo, Ostashev, Levy · Daniel n182 2026-10-25 · packet · register
2026-10-25 published T-244 · cases 172, 263
Kim's exact algebraic sides at 172 and 263, as reported, of packings the record credits
Upper bound confirmed
Kim after Couzo, Chaoweeraprasit, Daniel, PingYou Ltd, Levy · Kim exact forms 2026-10-25 · packet · packet · register
2026-10-25 published T-247 · case 297
Kim's exact degree-21 side at 297, as reported, of the packing the record credits
Upper bound confirmed superseded by T-299
Kim after Couzo, Daniel, PingYou Ltd, Levy · Kim s297 2026-10-25 · packet · register
2026-10-26 published T-248 · case 299
Daniel's new arrangement at 299, 1.92e-2 below the catalogue's construction
Upper bound confirmed superseded by T-277
Daniel after Deotte · Daniel n299 2026-10-26 · packet · packet · register
2026-10-27 published T-254 · case 130
Couzo's new packing of 130 squares from PR #574's, 2.05e-3 below T-174
Upper bound confirmed superseded by T-259
Couzo after Chaoweeraprasit, Gupta, PingYou Ltd, Levy · Couzo n130 2026-10-27 · packet · packet · register
2026-10-28 published T-255 · cases 260, 307
Couzo's certificates of ntumlnoob's Kaggle packings at 260 and 307, below T-225 and T-174
Upper bound confirmed superseded by T-269 and T-279
Couzo after ntumlnoob, Daniel, Levy · Couzo a5a5d76 2026-10-28 · packet · packet · register
2026-10-28 published T-256 · cases 232, 260, 307
Couzo's packings from Kaggle notebook packings at 232, 260 and 307, below T-186, T-225 and T-174
Upper bound confirmed superseded by T-269, T-275 and T-279
Couzo after Chilton, ntumlnoob, Lavrentev, Ellsworth, Daniel, Levy · Couzo issue 844 2026-10-28 · packet · register
2026-10-29 published T-257 · cases 172, 204, 260, 293
Daniel's new packings of 172, 204, 260 and 293 squares, 1.64e-4 to 1.18e-2 below each house
Upper bound confirmed superseded by T-268, T-269, T-271 and T-282
Daniel after Chaoweeraprasit, Gupta, Ostashev, kirilltym, Friedman · Daniel issue 845 2026-10-29 · packet · register
2026-10-29 published T-258 · case 299
Daniel's lower packing of 299 squares from his T-248 packing, 2.46e-3 below it
Upper bound confirmed superseded by T-277
Daniel after Deotte · Daniel n299 update 2026-10-29 · packet · packet · register
2026-10-30 published T-259 · case 130
Polkovskyi's re-solve of Couzo's packing of 130 squares, 2.08e-12 below T-254
Upper bound confirmed
Polkovskyi after Couzo, Xu, Daniel, Ostashev, Levy · Mishapolk issue 846 2026-10-30 · packet · register
2026-10-30 published T-260 · 5 of these cases
Polkovskyi's sub-print re-certifications at 105, 146, 156, 229 and 308, 1e-41 to 1e-21 below each
Upper bound reviewed
Polkovskyi after Couzo, Xu, Daniel, Ostashev, Levy · Mishapolk issue 846 2026-10-30 · packet · register
2026-11-03 published T-262 · case 94
$s (94) \geq 1993 / 200$ verified, T-091's measure dilated onto a 600-direction net
Lower bound confirmed superseded by T-266
Ozden after Hosono, Tokoharu, Levy · Özden s94 dilation 2026-11-03 · packet · source · review · register
2026-11-03 published T-263 · case 299
Özden's further arrangement at 299 of Daniel's T-258 packing, one to one, 1.54e-8 below it
Upper bound confirmed superseded by T-277
Ozden after Daniel, Deotte · Özden s299 upper bound 2026-11-03 · packet · register
2026-11-03 published T-264 · case 204
Özden's further arrangement at 204 of Daniel's T-257 packing, one to one, 6.48e-6 below it
Upper bound confirmed superseded by T-268
Ozden after Daniel, Ostashev · Özden s204 upper bound 2026-11-03 · packet · register
2026-11-04 published T-265 · case 260
Yim's further arrangement at 260 of Daniel's T-257 packing, one to one, 1.99e-10 below it
Upper bound confirmed superseded by T-269
Yim after Daniel, Tymchishin · Yim s260 upper bound 2026-11-05 · packet · register
2026-11-05 published T-266 · case 94
$s (94) \geq 2493 / 250$ verified, T-091's measure dilated onto a 4,972-direction net
Lower bound confirmed
Ozden after Hosono, Tokoharu, Levy · Özden s94 lower bound 2026-11-05 · packet · source · review · register
2026-11-05 published T-267 · case 175
Daniel's new packing of 175 squares, from gadgets of T-186's, 1.24e-4 below it
Upper bound confirmed
Daniel after Deotte, Ostashev, PingYou Ltd, Polkovskyi · Daniel issue 914 n175 2026-11-05 · packet · packet · register
2026-11-05 published T-268 · case 204
Daniel's new packing of 204 squares, 3.26e-3 below T-264
Upper bound confirmed
Daniel after Deotte · Daniel issue 914 2026-11-05 · packet · packet · register
2026-11-05 published T-269 · case 260
Daniel's new packing of 260 squares, T-257's tilted island recomposed, 4.62e-4 below T-265
Upper bound confirmed
Daniel after Deotte, Tymchishin · Daniel issue 914 n260 2026-11-05 · packet · register
2026-11-05 published T-270 · case 268
Daniel's new packing of 268 squares, 1.86e-3 below T-242
Upper bound confirmed superseded by T-289
Daniel after Deotte · Daniel issue 914 2026-11-05 · packet · packet · register
2026-11-05 published T-271 · case 293
Daniel's new packing of 293 squares, 1.09e-2 below his T-257
Upper bound confirmed
Daniel after Deotte · Daniel issue 914 2026-11-05 · packet · packet · register
2026-11-05 published T-272 · case 303
Daniel's new packing of 303 squares, 5.15e-3 below T-240
Upper bound confirmed superseded by T-278
Daniel after Deotte · Daniel issue 914 2026-11-05 · packet · packet · register
2026-11-05 published T-273 · case 304
Daniel's new packing of 304 squares, 5.95e-3 below T-242
Upper bound confirmed
Daniel after Deotte · Daniel issue 914 2026-11-05 · packet · packet · register
2026-11-09 published T-275 · case 232
Özden's further arrangement at 232 of Couzo's T-256 packing, one to one, 2.52e-4 below it
Upper bound confirmed
Ozden after Chilton, Couzo, Daniel, Levy · Özden s232 upper bound 2026-11-09 · packet · register
2026-11-09 published T-276 · case 268
Özden's further arrangement at 268 of Daniel's T-270 packing, one to one, 2.96e-5 below it
Upper bound confirmed superseded by T-289
Ozden after Daniel, Deotte, Levy · Özden s268 upper bound 2026-11-09 · packet · packet · register
2026-11-09 published T-277 · case 299
Özden's further arrangement at 299 of his own T-263 packing, after Daniel, 2.34e-4 below it
Upper bound confirmed
Ozden after Daniel, Deotte, Levy · Özden s299 upper bound 2026-11-09 · packet · packet · register
2026-11-09 published T-278 · case 303
Özden's further arrangement at 303 of Daniel's T-272 packing, one to one, 4.35e-5 below it
Upper bound confirmed
Ozden after Daniel, Deotte, Levy · Özden s303 upper bound 2026-11-09 · packet · packet · register
2026-11-09 published T-279 · case 307
Özden's further arrangement at 307 of Couzo's T-256 packing, one to one, 2.03e-5 below it
Upper bound confirmed
Ozden after ntumlnoob, Couzo, Daniel, Levy · Özden s307 upper bound 2026-11-09 · packet · packet · register
2026-11-09 published T-280 · cases 299, 303
Yim's re-solves of Özden's T-277 and T-278 at 299 and 303, within their padding; held
Upper bound reviewed
Yim after Ozden, Daniel, Deotte, Casson, Couzo, Lavrentev, alexchilton, Polkovskyi, Levy · Yim certificates 2026-11-09 · packet · packet · register
2026-11-09 published T-281 · case 130
Yim's translation of T-259's certificate at 130, 1.96e-14 below it at the same print; held
Upper bound reviewed
Yim after Ozden, Daniel, Deotte, Casson, Couzo, Lavrentev, alexchilton, Polkovskyi, Levy · Yim certificates 2026-11-09 · packet · register
2026-11-09 published T-282 · case 172
Özden's further arrangement at 172 of Daniel's T-257 packing, 3.25e-3 below it
Upper bound confirmed
Ozden after Daniel, Levy · Özden s172 upper bound 2026-11-09 · packet · register
2026-11-09 published T-283 · case 270
Özden's further arrangement at 270 of Daniel's T-224 packing, one to one, 1.36e-5 below it
Upper bound confirmed
Ozden after Daniel, Chaoweeraprasit, Levy · Özden s270 upper bound 2026-11-09 · packet · packet · register
2026-11-09 published T-284 · case 199
Özden's T-282 packing plus an L of 27 squares at 199, 2.88e-3 below the house
Upper bound confirmed
Ozden after Daniel, Levy · Özden s199 upper bound 2026-11-09 · packet · register
2026-11-09 published T-285 · case 199
De Winter's re-solve of T-284's packing at 199, 1.0399 of its margin through the L step; held
Upper bound reviewed
de Winter after Ozden, Daniel, Levy · de Winter s199 upper bound 2026-11-09 · packet · register
2026-11-10 published T-286 · case 172
De Winter's re-solve of T-282's packing at 172, T-285's core, 1.0399 of T-282's margin; held
Upper bound reviewed
de Winter after Ozden, Daniel, Levy · de Winter s172 upper bound 2026-11-10 · packet · register
2026-11-10 published T-287 · case 156
Özden's packing at 156, seeded from the registry's 182, 2.11e-03 below the house, T-186
Upper bound confirmed
Ozden after Daniel, Levy · Özden s156 upper bound 2026-11-10 · packet · register
2026-11-10 published T-288 · case 180
Özden's packing at 180, seeded from the registry's 154, 1.38e-03 below the house, T-174
Upper bound confirmed
Ozden after Daniel, Couzo, Levy · Özden s180 upper bound 2026-11-10 · packet · packet · register
2026-11-10 published T-289 · case 268
Özden's packing at 268, seeded from the registry's 302, 4.40e-04 below his own T-276
Upper bound confirmed
Ozden after PingYou, Couzo, Mishapolk, Daniel, Levy · Özden s268 upper bound 2026-11-10 · packet · packet · register
2026-11-11 published T-290 · case 269
Özden's packing at 269, seeded from the registry's 304, 5.19e-04 below the house, T-174
Upper bound confirmed superseded by T-297
Ozden after Daniel, Levy · Özden s269 upper bound 2026-11-11 · packet · packet · register
2026-11-11 published T-291 · case 179
Özden's packing at 179, a band built from the registry's 70, 2.43e-03 below the house, T-174
Upper bound confirmed superseded by T-294
Ozden after Xu, Daniel, PingYou, Levy · Özden s179 upper bound 2026-11-11 · packet · packet · register
2026-11-11 published T-292 · case 132
De Winter's pinwheel at 132 from Samanyu-dev's 109, a new arrangement 9.63e-03 below T-174
Upper bound confirmed
de Winter after Samanyu-dev · de Winter s132 upper bound 2026-11-11 · packet · packet · register
2026-11-11 published T-293 · case 157
De Winter's pinwheel at 157 from Samanyu-dev's 109, the first below the grid's 13, by 6.70e-03
Upper bound confirmed
de Winter after Samanyu-dev · de Winter s157 upper bound 2026-11-11 · packet · packet · packet · register
2026-11-11 published T-294 · case 179
Özden's further arrangement of his T-291 band at 179, 4.07e-04 below it, 2.645e-12 above $s (70)$ + 5
Upper bound confirmed
Ozden after Xu, Daniel, PingYou, Levy · Özden s179 issue 959 2026-11-11 · packet · packet · register
2026-11-11 published T-295 · case 237
Özden's packing at 237, a band from two copies of the registry's 70, 8.80e-03 below T-190
Upper bound confirmed
Ozden after Xu, Daniel, PingYou, Levy · Özden s237 upper bound 2026-11-11 · packet · packet · register
2026-11-11 published T-296 · case 269
Özden's band at 269 from tilted squares of the 207, the 70 and his 179, 1.53e-03 below his T-290
Upper bound confirmed superseded by T-297
Ozden after PingYou, Xu, Daniel, Levy · Özden s269 issue 962 2026-11-11 · packet · packet · register
2026-11-11 published T-301 · case 228
Ono's packing at 228, five band periods cut from Couzo's 404, 4.76e-03 below T-300
Upper bound confirmed
Ono after Couzo, Levy · Ono issue 984 2026-11-14 · packet · packet · register
2026-11-11 published T-302 · case 147
Ono's packing at 147, one band period cut from Daniel's 229, 5.81e-03 below 7 + 4*sqrt(2)
Upper bound confirmed
Ono after Daniel, Levy · Ono issue 985 2026-11-14 · packet · packet · register
2026-11-12 published T-297 · case 269
Özden's column-band packing at 269, 9.74e-03 below his T-296 and 1.13e-02 below his T-290
Upper bound confirmed
Ozden after PingYou, Xu, Daniel, Levy · Özden s269 upper bound 2026-11-12 · packet · packet · register
2026-11-13 published T-298 · 11 of these cases
$s (k^{2} - 5) = k$ for every integer k from 7 up; k = 7…18 are the cases held here
Optimality reviewed
Khalili, Gharabaghi after Daniel · Khalili Gharabaghi k2m5 family 2026-11-13 · packet · packet · packet · register
2026-11-14 published T-299 · case 297
De Winter's double-step staircases in T-174's layout at 297, a new arrangement 9.62e-04 below it
Upper bound confirmed
de Winter after Couzo, Daniel, PingYou, Levy · de Winter s297 upper bound 2026-11-14 · packet · packet · register
2026-11-14 published T-300 · case 228
Özden's column-band packing at 228, a new arrangement 3.80e-03 below T-174
Upper bound confirmed superseded by T-301
Ozden after Daniel, Levy · Özden s228 upper bound 2026-11-14 · packet · packet · register
2026-11-15 published T-303 · case 90
jazivxt's packing at 90 from a crop of de Winter's 157, the first below the grid's 10, by 1.72e-03
Upper bound confirmed
jazivxt after de Winter · jazivxt s90 upper bound 2026-11-15 · packet · packet · register

Links

On this site: Frontier Survey · T-180 in the results table
Issues: Issue #600

On GitHub, at main

Register: T-180 in results.yaml, line 24995
Evidence: E-wand125-closed-form-report · E-wand125-closed-form-cfcheck-report · E-wand125-closed-form-cfcheck-sample · E-wand125-closed-form-replay4d-sample
Proofs and certificates: certificate intervals.json.gz · certificate interval-1-premises-and-kind-sample.json · certificate calibration.json
Sources: wand125 closed form 2026-10-14 (its own site, retained copy) · wand125 cfcheck 2026-10-14
Source packet: resources/web/wand125-closed-form-2026-10-14/README.md · resources/web/wand125-closed-form-source-hash-2026-10-14/README.md
Artifacts: 7 artifacts and controls
resources/web/wand125-closed-form-2026-10-14/README.md resources/web/wand125-closed-form-source-hash-2026-10-14/README.md resources/web/wand125-closed-form-2026-10-14/receipts/release-custody.json resources/web/wand125-closed-form-2026-10-14/receipts/structure-check/run.json resources/web/wand125-closed-form-2026-10-14/evaluation/closed_form_bounds.py resources/web/wand125-closed-form-2026-10-14/evaluation/outputs/bounds.json docs/project/reviews/review-2026-10-14-wand125-closed-form-lower-bound.md
Case file: each case’s file is linked from its row above

T-180: Closed-form lower bound s(N)≥a(N)s(N)≥a(N) for every non-square N≥49N≥49, reported

The cases

Results on these cases

Links

T-180: Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported