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

2005 published · Nagamochi · 321 cases, $n = 4$ to 324

For every integer $4 \leq N \leq 324$ , Nagamochi 2005, Theorem 2 gives $s (N)$ >= min(ceil(sqrt(N)), sqrt(N - 2*floor(sqrt(N)) + 1) + 1).

Significance, composition and next rung

Significance: The workhorse lower bound until 2026-10-02, and no less important for being unproved: 95 of the hundred case records cite this result, and since that date it supplies the operative verified lower bound in 0 of them. It supplied 88 until 2026-08-31, when this project's certificates began displacing it; replayed external certificates displaced 18 more case values on 2026-09-22, one more, $n = 32$ , on 2026-09-27, and another, $n = 45$ , on 2026-09-29, leaving 63 at n <= 100 and 287 in all when its proof was found incomplete. Replayed certificates recorded in parallel and merged on 2026-10-03 have since raised 51 of those floors, 38 of them at n <= 100, above the ones that replaced it. Its values remain recorded as reported bounds.
Next rung: A checkable proof of Theorem 1, or of Theorem 2 by another route, restores a rung: for the m^2 - 2 family a measure on [0,m]^2 of total at most m^2 - 2 that gives every square of side in (1, 1.01] more than one, or a replay of chelokot's Lean compensation proof with its axiom receipt; for the general floor, a proof at full strength. The read that found the defect is what stands, and the transcribed values are still re-derived in the gate.
Novelty: previously-published Present in an identified source

The cases

This result concerns 321 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
4	2	2	0	provedO=R	frontier n-004.md
5	2.707107	2.707107		provedO=R	frontier n-005.md
6	3	3		provedO=	frontier n-006.md
7	3			provedO=	frontier n-007.md
8	3			provedO=	frontier n-008.md
9	3			provedO=R	frontier n-009.md
10	3.707107	3.707107		provedO=	frontier n-010.md
11	3.877084	3.877084		provedO=R	frontier n-011.md
12	3.971500	4	0.0285	open=	frontier n-012.md
13	4		0	provedO=	frontier n-013.md
14	4			provedO=	frontier n-014.md
15	4			provedO=	frontier n-015.md
16	4			provedO=R	frontier n-016.md
17	4.660442	4.675531	0.01508734…	open=	frontier n-017.md
18	4.705000	4.822876	0.11787565…		frontier n-018.md
19	4.825000	4.885619	0.06061808…		frontier n-019.md
20	4.905000	5	0.095		frontier n-020.md
21	5		0	provedO=	frontier n-021.md
22	5			provedO=	frontier n-022.md
23	5			provedO=	frontier n-023.md
24	5			provedO=	frontier n-024.md
25	5			provedO=R	frontier n-025.md
26	5.545000	5.621321	0.07632034…	open=	frontier n-026.md
27	5.643500	5.707107	0.06360678…	open=	frontier n-027.md
28	5.735000	5.824445	0.08944461…	open=R	frontier n-028.md
29	5.810000	5.933834	0.12383346…	open=	frontier n-029.md
30	5.883500	6	0.1165		frontier n-030.md
31	5.952500		0.0475		frontier n-031.md
32	6		0	provedO=	frontier n-032.md
33	6			provedO=	frontier n-033.md
34	6			provedO=	frontier n-034.md
35	6			provedO=	frontier n-035.md
36	6			provedO=R	frontier n-036.md
37	6.440000	6.598620	0.15861960…	open=	frontier n-037.md
38	6.545000	6.707107	0.16210678…		frontier n-038.md
39	6.650000	6.810723	0.16072208…		frontier n-039.md
40	6.718000	6.828428	0.11042712…	open=R	frontier n-040.md
41	6.775000	6.926694	0.15169309…	open=	frontier n-041.md
42	6.847500	7	0.1525		frontier n-042.md
43	6.907500		0.0925		frontier n-043.md
44	6.972500		0.0275		frontier n-044.md
45	7		0	provedO=	frontier n-045.md
46	7			provedO=	frontier n-046.md
47	7			provedO=	frontier n-047.md
48	7			provedO=	frontier n-048.md
49	7			provedO=R	frontier n-049.md
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
64	8			provedO=R	frontier n-064.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	8.965000		0.035		frontier n-076.md
77	9		0	provedO=	frontier n-077.md
78					frontier n-078.md
79					frontier n-079.md
80					frontier n-080.md
81	9			provedO=R	frontier n-081.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	10	0.27		frontier n-090.md
91	9.762500		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	9.950000		0.05		frontier n-094.md
95	9.965000		0.035		frontier n-095.md
96	9.970000		0.03		frontier n-096.md
97	10		0	provedO=	frontier n-097.md
98					frontier n-098.md
99					frontier n-099.md
100	10			provedO=R	frontier n-100.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
121	11			provedO=R	frontier n-121.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	11.904484	0.48431819…		frontier n-130.md
131	11.465856	11.949660	0.48380348…		frontier n-131.md
132	11.511357	11.985681	0.47432242…		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
144	12			provedO=R	frontier n-144.md
145	12.043396	12.535534	0.49213752…	open=	frontier n-145.md
146	12.086630	12.583783	0.49715196…		frontier n-146.md
147	12.129703	12.656855	0.52715089…		frontier n-147.md
148	12.172617	12.656855	0.48423671…		frontier n-148.md
149	12.215374	12.707107	0.49173226…		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	12.979166	0.46875370…		frontier n-156.md
157	12.551970	13	0.44802920…		frontier n-157.md
158	12.593386		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
169	13			provedO=R	frontier n-169.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	13.618989	0.49955949…		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	13.764175	0.52643511…		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	13.883796	0.48999853…		frontier n-179.md
180	13.432517	13.916994	0.48447636…		frontier n-180.md
181	13.471121	13.950820	0.47969775…		frontier n-181.md
182	13.509611	13.974091	0.46447888…		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
196	14			provedO=R	frontier n-196.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	14.617573	0.50691458…		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	14.808643	0.51552861…		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
225	15			provedO=R	frontier n-225.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	15.604602	0.50151960…		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	15.764175	0.52477134…		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	15.902604	0.49455212…		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
256	16			provedO=R	frontier n-256.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	16.645754	0.51725364…		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	16.878815	0.49643464…		frontier n-268.md
269	16.413830	16.901514	0.48768311…		frontier n-269.md
270	16.445218	16.928163	0.48294353…		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
289	17			provedO=R	frontier n-289.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	17.634147	0.51337620…		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	17.740418	0.49974824…		frontier n-297.md
298	17.270509	17.778175	0.50766476…		frontier n-298.md
299	17.300297	17.822876	0.52257803…		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	17.909959	0.49103397…		frontier n-303.md
304	17.448451	17.929428	0.48097650…		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	17.980863	0.44413774…		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
324	18			provedO=R	frontier n-324.md

Results on these cases

229 results in the register on these cases, oldest first

1979 published T-011 · case 11
Trump's 1979 packing is exactly valid, so $s (11) \leq 3.877083590022814 \dots$
Upper bound confirmed
Trump · Trump 2023 · register
2005 published T-007 this result · 321 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
2010-09-13 published T-004 · case 46
Bentz 2010, Theorem 8 ( $s (46) \geq 7$ ) is correct as printed, machine-audited in full
Audit confirmed
Bentz · Bentz 2010 · source · register
2010-09-13 published T-006 · case 13
$s (13) = 4$
Optimality confirmed
Bentz; Daniel after Burns, Massaccesi · Bentz 2010 · evand square-packing 2026-09-28 · packet · source 1 · source 2 · review · register
2010-09-13 published T-008 · case 46
$s (46) = 7$
Optimality confirmed
Bentz · Bentz 2010 · source · register
2018 published T-087 · cases 37, 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 and T-069
Bašić, Slivková · Basic-Slivkova 2018 · source · register
2026-08-21 published T-015 · case 17
$s (17) \geq 22529 / 5000 = 4.5058$
Lower bound confirmed superseded by T-093 and T-153 (reported)
Massaccesi after Burns · Burns–Massaccesi n17 · packet · register
2026-08-21 published T-016 · cases 18, 19
$s (n) \geq 22529 / 5000$ for $n = 18, 19$ , by monotonicity from T-015
Lower bound confirmed superseded by T-102 and T-103
Massaccesi after Burns · Burns–Massaccesi n17 · packet · register
2026-08-24 established T-010 · case 11
$s (11) \geq 2 + 4 / \sqrt{5}$ , by a repair of Stromquist 2003's Figure 14 point set
Lower bound confirmed superseded by T-060
Levy after Stromquist · register
2026-08-25 published T-049 · case 12
$s (12) \geq 15680 / 3951 = 3.9686155 \dots$
Lower bound confirmed superseded by T-095
Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · source 1 · source 2 · review 1 · review 2 · register
2026-08-29 established T-009 · case 29
$s (29) \leq 5.93383346267692918974 \dots$ , by a Krawczyk interval certificate
Upper bound confirmed superseded by T-195
Levy · register
2026-08-30 established T-012 · case 5
Goebel's $n = 5$ packing is second-order rigid at fixed side
Rigidity confirmed
Levy · register
2026-08-30 established T-013 · case 40
Goebel's $n = 40$ packing: seven verified first-order flexes, each refused at second order
Rigidity confirmed
Levy · register
2026-08-31 established T-001 · case 17
$s (17) \geq 4426213 / 1000000 = 4.426213$ , from a sixteen-point unavoidable set
Lower bound confirmed superseded by T-093 and T-153 (reported)
Levy after Bentz · register
2026-08-31 established T-002 · case 18
$s (18) \geq 4426213 / 1000000$ , by monotonicity from T-001
Lower bound confirmed superseded by T-102
Levy after Bentz · register
2026-08-31 established T-003 · cases 17, 18
The sixteen-point set's unavoidability ceiling lies in $[4426213 / 1000000, 4427 / 1000)$
Method limit confirmed
Levy after Bentz · register
2026-08-31 established T-005 · case 13
Bentz 2010, Lemma 10 is false as printed and true as corrected to $(1.74, 1)$
Correction confirmed
Levy after Bentz · source · register
2026-09-03 established T-014 · case 5
Goebel's $n = 5$ optimum is rigid at fixed side: its pose is an isolated feasible point
Rigidity confirmed
Levy · register
2026-09-04 established T-017 · case 12
$s (12) \geq 99 / 25 = 3.96$
Lower bound confirmed superseded by T-095
Levy after Burns, Massaccesi · register
2026-09-04 established T-018 · case 11
$s (11) \geq 381 / 100 = 3.81$
Lower bound confirmed superseded by T-060
Levy after Burns, Massaccesi · register
2026-09-04 established T-019 · cases 17, 18, 19
$s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$
Lower bound confirmed superseded by T-093, T-102, T-103 and T-153 (reported)
Levy after Burns, Massaccesi · register
2026-09-04 established T-020 · cases 19, 20, 21
$s (n) \geq 24 / 5 = 4.80$ for $n = 19, 20, 21$
Lower bound confirmed superseded by T-052, T-103 and T-104
Levy after Burns, Massaccesi · register
2026-09-04 published T-085 · 315 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 · 16 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
2026-09-05 established T-021 · cases 20, 21
$s (n) \geq 97 / 20 = 4.85$ for $n = 20, 21$
Lower bound confirmed superseded by T-052 and T-104
Levy after Burns, Massaccesi · register
2026-09-06 established T-022 · case 11
$s (11) \geq 38100 \sqrt{8100042893309449} / 899996306539 = 3.8100257 \dots$
Lower bound confirmed superseded by T-060
Levy after Burns, Massaccesi · source · register
2026-09-08 established T-023 · case 11
Conditional exclusion: no eleven-square packing in the four-owner branch at $q = 96 / 25$
Case exclusion confirmed superseded in part by T-060
Levy · source · register
2026-09-09 established T-024 · case 11
$s (11) \geq 3175000 \sqrt{518400042893309449} / 598960960743657 = 3.8166095 \dots$
Lower bound confirmed superseded by T-060
Levy after Burns, Massaccesi · source · register
2026-09-09 established T-025 · case 11
$s (11) \geq 191 / 50 = 3.82$ , by a threshold certificate
Lower bound confirmed superseded by T-060
Levy · source · register
2026-09-09 established T-026 · case 11
$s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$
Lower bound confirmed superseded by T-060
Levy · 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
2026-09-18 established T-027 · case 18
$s (18) \geq 467 / 100 = 4.67$
Lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-028 · case 18
$s (18) \geq 187 / 40 = 4.675$
Lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-029 · case 18
$s (18) \geq 1871 / 400 = 4.6775$
Lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-19 established T-030 · case 18
$s (18) \geq 4679 / 1000 = 4.679$
Lower bound confirmed superseded by T-102
Levy after Burns, Massaccesi · register
2026-09-20 established T-031 · case 11
The octagon corner class (threshold $1 / 2$ ) holds no eleven-square packing at side $96 / 25$
Case exclusion confirmed superseded by T-060
Levy · register
2026-09-20 published T-032 · case 17
$s (17) \geq 461300 / 99999 = 4.61304613 \dots$ , and beneath it Mira's $s (17) \geq 4613 / 1000$
Lower bound confirmed superseded by T-093 and T-153 (reported)
Guzhou0806, Mira-acc after Levy, Burns, Massaccesi · n17 weighted certificates 2026-09-20 · packet · register
2026-09-21 published T-038 · case 17
$s (17) > 461300 / 99853 = 4.6197910929 \dots$
Lower bound confirmed superseded by T-093 and T-153 (reported)
Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 certified bound · packet · source · review · register
2026-09-21 published T-065 · case 17
$s (17) \leq 4.6755300936045509 \dots$ , Bidwell's packing certified exactly
Upper bound confirmed
Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 certified bound · packet · source · review · register
2026-09-22 established T-033 · case 11
$s (11) \geq 955000 \sqrt{2073600042893309449} / 359341754646249 = 3.8269975 \dots$
Lower bound confirmed superseded by T-060
Levy · source · register
2026-09-22 published T-037 · case 11
$s (11) > 31 / 8 = 3.875$
Lower bound confirmed superseded by T-060
Overbeck after Levy, Guzhou0806, Mira-acc · Kleddamag n11 2026 · packet · source 1 · source 2 · review 1 · review 2 · register
2026-09-22 published T-044 · 17 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, T-105, T-108, T-110, T-111, T-157 and T-205
Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register
2026-09-22 published T-047 · 7 of these cases
$s (11) \geq 381 / 100$ ; $s (n) \geq 1377 / 250$ for $n = 26 \dots 28$ ; $s (n) \geq 571 / 100$ for $n = 29 \dots 31$
Lower bound confirmed superseded by T-060, T-068, T-074, T-105, T-106, T-107, T-108 and T-109
Tokoharu after Levy, Hosono, Stromquist, Nagamochi, Burns, Massaccesi · Tokoharu density 2026 · packet · source · review · register
2026-09-23 established T-034 · case 21
$s (21) \geq 122 / 25 = 4.88$
Lower bound confirmed superseded by T-052
Levy after Burns, Massaccesi · register
2026-09-23 published T-050 · case 21
$s (21) \geq 5000 / 1001 = 4.995004995 \dots$
Lower bound confirmed superseded by T-052
Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · source · review · register
2026-09-24 established T-035 · case 11
Six-plus-five packings near Trump's tilt with side $\leq U_{h i}$ lie within rho of his pose
Case exclusion confirmed
Levy · register
2026-09-24 established T-036 · case 11
Trump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetry
Restricted optimality confirmed superseded in part by T-060 and T-112
Levy · 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
2026-09-25 published T-039 · case 17
$s (17) > 231001 / 50000 = 4.62002$
Lower bound confirmed superseded by T-093 and T-153 (reported)
Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R052 · packet · source · review · register
2026-09-26 published T-040 · case 17
$s (17) > 232001 / 50000 = 4.64002$
Lower bound confirmed superseded by T-093 and T-153 (reported)
Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 4.640020 · packet · source · review · register
2026-09-26 published T-051 · case 32
$s (32) = 6$
Optimality confirmed
Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · review 3 · register
2026-09-27 published T-041 · case 17
$s (17) > 466001 / 100000 = 4.66001$
Lower bound confirmed superseded by T-093 and T-153 (reported)
Overbeck after Levy, Mira-acc, Guzhou0806 · Kleddamag n17 4.66001 · packet · source · review · register
2026-09-27 published T-045 · 15 of these cases
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
Lower bound confirmed superseded by T-051, T-063, T-064, T-067, T-068, T-074, T-090, T-091, T-102, T-103, T-104, T-105, T-106, T-107, T-109 and T-157
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register
2026-09-27 published T-046 · 48 of these cases
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
Lower bound recorded superseded by T-062, T-063, T-064, T-066, T-067, T-068, T-069, T-074, T-082, T-090, T-091, T-094, T-097, T-102, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110, T-111, T-157 and T-205
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register
2026-09-27 published T-052 · case 21
$s (21) = 5$ , by a mixed cover of points and grid-line segments
Optimality confirmed
Daniel after Burns, Massaccesi · evand square-packing 2026-09-28 · packet · source · review · register
2026-09-27 published T-053 · case 45
$s (45) = 7$ , by a mixed cover of points and grid-line segments
Optimality confirmed
Daniel after Burns, Massaccesi · evand square-packing 2026-09-28 · 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-127, T-136, T-146, T-173, T-174, T-182, T-186, T-190, T-197, T-218, T-219, T-222 and T-224
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-28 published T-042 · case 17
$s (17) > 233009 / 50000 = 4.66018$
Lower bound confirmed superseded by T-093 and T-153 (reported)
Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R067 · packet · source · review · register
2026-09-28 published T-043 · case 17
$s (17) > 116511 / 25000 = 4.66044$
Lower bound confirmed superseded by T-093 and T-153 (reported)
Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R068 · packet · source · review · 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-054 · case 45
$s (45) = 7$ by a second, point-only route
Simplification 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-055 · case 21
$s (21) = 5$ by a point-only route
Simplification 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 · 25 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-068, T-074, T-082, T-090, T-091, T-094, T-108, T-110, T-111 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 · 97 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-059 · case 11
Reported equality of 12028 n11 row minima reproduced by a complete bound replay
Audit confirmed
Hosono after Tokoharu, Daniel · wand125 tools 2026 · packet · register
2026-09-29 published T-060 · case 11
Trump's eleven-square packing is globally optimal
Optimality confirmed
Ahmed after Levy, Overbeck · Ahmed n11 optimality 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-09-29 published T-061 · case 11
$s (11) > 3875000000 / 999999999 = 3.875000003875 \dots$ , 3.9e-9 above $31 / 8$
Lower bound confirmed superseded by T-060
Wang, Li after Overbeck, Levy · Wang Li n11 2026 · packet · source 1 · source 2 · review · register
2026-09-29 published T-083 · 301 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 · 16 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 · 13 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
2026-09-30 published T-093 · case 17
$s (17) > 18641771 / 4000000 = 4.66044275$
Lower bound confirmed
Guzhou0806 after Overbeck, Mira-acc, Levy · Guzhou0806 n17 R071 · packet · source · review · 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 · 34 of these cases
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register
2026-10-01 published T-069 · 5 of these cases
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 · 31 of these cases
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
Lower bound confirmed
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-091
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 · cases 20, 42, 70
Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70
Lower bound confirmed superseded by T-090, T-091 and T-104
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-078 · case 12
$s (12) \geq 31360 / 7901 = 3.9691178 \dots$ , Daniel's certificate rescaled by $7902 / 7901$
Lower bound confirmed superseded by T-095
Ryu after Daniel · squarepacker s12 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-10-02 established T-079 · case 12
$s (12) \geq 15680000 / 3949423 = 3.9702002 \dots$ , Daniel's points re-weighted
Lower bound confirmed superseded by T-095
Levy after Daniel · source 1 · source 2 · review · 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 · 14 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 and T-186
Couzo · franciscouzo square-packing 2026-10-03 · packet · register
2026-10-04 published T-090 · 17 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-095 · case 12
$s (12) \geq 7943 / 2000 = 3.9715$ , Daniel's points dilated and re-weighted
Lower bound confirmed
Ryu after Daniel, Levy · squarepacker s12 2026-10-05 · packet · packet · source 1 · source 2 · review 1 · review 2 · register
2026-10-05 published T-096 · case 18
Mixed rectangle-measure lower bound replayed at $n = 18$ , on a net the certificate declares
Lower bound confirmed superseded by T-102
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-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 · 77 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
2026-10-06 published T-099 · case 18
Mixed rectangle-measure lower bound verified at $n = 18$ , on a finer declared net
Lower bound confirmed superseded by T-102
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-06 · packet · source 1 · source 2 · review · register
2026-10-06 published T-100 · case 19
Mixed rectangle-measure lower bound verified at $n = 19$ , past the best 18-square packing
Lower bound confirmed superseded by T-103
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-06 · packet · source · review · register
2026-10-06 published T-102 · case 18
Mixed rectangle-measure lower bound verified at $n = 18$ , on the finest declared net yet
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-103 · case 19
Mixed rectangle-measure lower bound verified at $n = 19$ , on the finest declared net yet
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-104 · case 20
Mixed rectangle-measure lower bound verified at $n = 20$ , on a declared net
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-105 · case 26
Mixed rectangle-measure lower bound verified at $n = 26$ , on a declared net
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-106 · case 27
Mixed rectangle-measure lower bound verified at $n = 27$ , on a declared net
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-107 · case 28
Mixed rectangle-measure lower bound verified at $n = 28$ , on a declared net
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-108 · case 29
Mixed rectangle-measure lower bound verified at $n = 29$ , on a declared net
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-109 · case 30
Mixed rectangle-measure lower bound verified at $n = 30$ , on a declared net
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-110 · case 39
Mixed rectangle-measure lower bound verified at $n = 39$ , on a declared net
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 published T-111 · case 41
Mixed rectangle-measure lower bound verified at $n = 41$ , on a declared net
Lower bound confirmed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register
2026-10-06 established T-112 · case 11
Trump's packing is the only optimal packing of eleven squares, up to symmetry
Uniqueness confirmed
Levy after Ahmed · 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 and T-222
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-173
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-146, T-174, T-186, T-190, T-197 and T-222
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-173, T-174, T-186 and T-218
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-174
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-146
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 · 175 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 and T-174
Daniel after Couzo, Levy · Daniel dated certificates 105 and 130 2026-10-07 · Daniel dated certificate 292 2026-10-07 · packet · packet · packet · register
2026-10-08 published T-125 · 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-146, T-173, T-174, T-186, T-187 and T-218
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 and T-197
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-186 and T-224
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-174
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
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-146, T-174, T-186 and T-224
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, 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-132 · case 29
Mixed rectangle-measure lower bound $s (29) \geq 291 / 50$ reported, on a 2,073-direction declared net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net n29 2026-10-10 · packet · register
2026-10-10 published T-133 · case 40
$s (40) > 335427 / 50000 = 6.70854$ , by a clipped-corner transfer on a 401-direction net
Lower bound confirmed superseded by T-157
Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 clipped corner 2026-10-10 · packet · packet · source 1 · source 2 · review · 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-146
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-186, T-190 and T-224
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-174 and T-221
Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · Daniel record hunt 3 2026-10-10 · packet · register
2026-10-10 published T-142 · case 28
Mixed rectangle-measure lower bound $s (28) \geq 2297 / 400$ reported, on a 2,073-direction declared net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net n28 n30 2026-10-10 · packet · register
2026-10-10 published T-143 · case 30
Mixed rectangle-measure lower bound $s (30) \geq 2357 / 400$ reported, on a 2,073-direction declared net
Lower bound recorded
Hosono after Tokoharu and Levy · wand125 fine-net n28 n30 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-145 · case 40
$s (40) > 1340000 / 199529 = 6.7158157 \dots$ , by a continuous-pose interval kernel
Lower bound confirmed superseded by T-157
Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 continuous pose 2026-10-10 · packet · packet · source 1 · source 2 · review · 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-173
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-173
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-186, T-188 and T-197
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 and T-224
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-186
Couzo after Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-11 · packet · register
2026-10-11 published T-157 · case 40
$s (40) > 3359 / 500 = 6.718$ , by T-145's continuous-pose kernel on a density with 163 new supports
Lower bound confirmed
Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi · Guzhou0806 n40 new supports 2026-10-11 · packet · source 1 · source 2 · review · 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-146
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-186
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo n175 2026-10-11 · packet · register
2026-10-12 published T-153 · case 17
Candidate proof that Bidwell's 1998 packing is optimal, $s (17) = 4.67553009360455 \dots$
Optimality reviewed
Overbeck after Levy, Ahmed, Mira-acc, Guzhou0806, Maaz · Overbeck n17 candidate 2026 · packet · register
2026-10-12 published T-154 · case 17
Fitzharding reports a Lean proof that $s (17) = 4.67553009360455 \dots$
Formalization reviewed
Fitzharding after Levy, Ahmed, MacIver · Fitzharding n17 Lean 2026 · packet · source · source build receipt · review 1 · review 2 · register
2026-10-12 established T-155 · 59 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
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-190 and T-219
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 and T-225
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 and T-188
Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo · Fang excess theorem 2026-10-13 · packet · 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-186
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-173
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-173
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-174
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
Polkovskyi after Xu, Chaoweeraprasit, Levy · Mishapolk exact certificates 2026-10-12 · 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-186 and T-188
Ostashev after Nakajima, Daniel, de Winter, Couzo, Levy, Xu · Ostashev merged bands 2026-10-13 · packet · register
2026-10-13 published T-176 · case 39
Mixed rectangle-measure lower bound $s (39) \geq 533 / 80$ reported, on a 2,073-direction declared net
Lower bound reviewed
Hosono after Tokoharu and Levy · wand125 fine-net n37 n39 n44 2026-10-13 · packet · register
2026-10-13 published T-177 · case 44
Mixed rectangle-measure lower bound $s (44) \geq 2799 / 400$ reported, on a 2,073-direction declared net
Lower bound reviewed
Hosono after Tokoharu and Levy · wand125 fine-net n37 n39 n44 2026-10-13 · packet · register
2026-10-13 published T-178 · case 37
Mixed rectangle-measure lower bound $s (37) \geq 2599 / 400$ reported, on a 2,073-direction declared net
Lower bound reviewed
Hosono after Tokoharu and Levy · wand125 fine-net n37 n39 n44 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 $n = 145 \dots 149$ , reported
Lower bound reviewed
Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n145 2026-10-13 · packet · register
2026-10-13 published T-180 · 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-186 and T-188
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-190
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 reviewed
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-190
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
Couzo after Polkovskyi, Chaoweeraprasit, Daniel, Levy · Couzo n237 2026-10-15 · packet · register
2026-10-15 published T-191 · case 95
$s (95) = 10$ , by a mixed cover of points and grid-line segments
Optimality reviewed
Greff after Daniel, Hosono · bengreff s95 certificate 2026-10-15 · packet · source · review · register
2026-10-15 published T-192 · case 28
Continuous-pose lower bound $s (28) > 1149 / 200$ reported, by two of the source's programs
Lower bound recorded
Hosono after Guzhou0806, Tokoharu, Levy · wand125 continuous n28 2026-10-15 · packet · register
2026-10-15 published T-195 · case 29
A tighter rational certificate of Schadt and Ellsworth's 29-square packing, 1.57e-36 below T-155
Upper bound confirmed
PingYou Ltd after Schadt, Ellsworth, Daniel, Levy · PingYou PR 622 certificate 2026-10-15 · packet · 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-194 · case 19
Fitzharding reports a Lean proof that Wainwright's packing is optimal, $s (19) = 3 + 4 \sqrt{2} / 3$
Optimality reviewed
Fitzharding after Levy, Ahmed · Fitzharding n19 Lean 2026 · packet · source · review · register
2026-10-16 published T-200 · case 38
Mixed rectangle-measure lower bound $s (38) \geq 2631 / 400$ 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-201 · case 40
Mixed rectangle-measure lower bound $s (40) \geq 1347 / 200$ 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-202 · case 41
Mixed rectangle-measure lower bound $s (41) \geq 2721 / 400$ 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-203 · case 43
Mixed rectangle-measure lower bound $s (43) \geq 2777 / 400$ 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-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
2026-10-17 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
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 recorded superseded by T-225
Couzo after Polkovskyi, 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 recorded superseded by T-225
Couzo after de Winter, Daniel, Levy · Couzo issue 672 2026-10-17 · packet · packet · 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-225
Couzo after de Winter, Polkovskyi, 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 recorded superseded by T-225
Couzo after Polkovskyi, Daniel, Levy · Couzo issue 675 2026-10-18 · packet · packet · 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
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, 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 recorded
Polkovskyi after Couzo, Muglich, Daniel, Xu, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register

T-007: s(n)≥min(⌈n​⌉,n−2⌊n​⌋+1​+1)s(n)≥min(⌈n⌉,n−2⌊n⌋+1+1) for 4≤n≤3244≤n≤324

The cases

Results on these cases

Links

T-007: $s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$