Results

Added in the Last Seven Days

Every result that reached this site over 15–22 October 2026 (its newest day and the seven before it), at any significance, by the day it arrived, newest first. Each id leads to its row in the table below; S4 and S5 are marked, and a muted entry says it has since been superseded. Browse these arrivals in the table, or open each day’s link for that day; what arrived in the last day counts whole days on the same calendar as these dates.

22 October 2026 · 5 results

T-225 Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported theren = 204, 227, 229, 260 · V3 C3 S2 · Daniel
T-226 Couzo's 204 of #670, below Polkovskyi's T-186, since superseded by his own #675 and #678n = 204 · V0 C0 S2 · Couzo · superseded
T-227 Couzo's refinements of de Winter's 227 and 229, below T-196 at bothn = 227, 229 · V0 C0 S2 · Couzo · superseded
T-228 Couzo's 204 of #675, below his #670, since superseded by his #678n = 204 · V0 C0 S2 · Couzo · superseded
T-229 Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225n = 105, 204 · V0 C0 S2 · Polkovskyi

Show these 5 in the table

21 October 2026 · 2 results

T-223 Couzo's smaller packings at 204 and 260, the two in the corpus of issue 678's 85 boundsn = 204, 260 · V3 C3 S3 · Couzo · superseded
T-224 Daniel's smaller n270 packing, another arrangement found from Chaoweeraprasit's, certified exactlyn = 270 · V3 C3 S3 · Daniel

Show these 2 in the table

20 October 2026 · 5 results

T-218 Daniel's smaller n236 packing, after Chaoweeraprasit's composition, certified exactlyn = 236 · V3 C3 S3 · Daniel
T-219 Daniel's smaller n271 packing, continuing Couzo's arrangement, certified exactlyn = 271 · V3 C3 S3 · Daniel
T-220 Daniel's finite sub-grid upper bound for 183 squares, after Couzo's n182 packingn = 183 · V3 C3 S3 · Daniel
T-221 Daniel's smaller n308 packing, started from Chaoweeraprasit's 307 packing, certified exactlyn = 308 · V3 C3 S3 · Daniel
T-222 Daniel's smaller n154 packing, continuing Couzo's #586 arrangement, certified exactlyn = 154 · V3 C3 S3 · Daniel

Show these 5 in the table

19 October 2026 · 23 results

T-196 de Winter reports three smaller rational upper bounds at 227, 229 and 260n = 227, 229, 260 · V3 C3 S3 · de Winter · superseded
T-197 Daniel's smaller n155 packing, started from Ostashev's packingn = 155 · V3 C3 S3 · Daniel
T-200 Mixed rectangle-measure lower bound s(38) ≥ 2631/400 reported, on a 2,073-direction declared netn = 38 · V0 C0 S3 · Hosono
T-201 Mixed rectangle-measure lower bound s(40) ≥ 1347/200 reported, on a 2,073-direction declared netn = 40 · V0 C0 S3 · Hosono
T-202 Mixed rectangle-measure lower bound s(41) ≥ 2721/400 reported, on a 2,073-direction declared netn = 41 · V0 C0 S3 · Hosono
T-203 Mixed rectangle-measure lower bound s(43) ≥ 2777/400 reported, on a 2,073-direction declared netn = 43 · V0 C0 S3 · Hosono
T-204 Mixed rectangle-measure lower bound s(51) ≥ 3011/400 reported, on a 2,073-direction declared netn = 51 · V0 C0 S3 · Hosono
T-205 Mixed rectangle-measure lower bound s(68) ≥ 853/100 verified, on the 201-direction netn = 68 · V3 C3 S3 · Hosono
T-206 Mixed rectangle-measure lower bound s(84) ≥ 3769/400 verified, on the 201-direction netn = 84 · V3 C3 S3 · Hosono
T-207 Mixed rectangle-measure lower bound s(85) ≥ 379/40 verified, on the 201-direction netn = 85 · V3 C3 S3 · Hosono
T-208 Mixed rectangle-measure lower bound s(86) ≥ 3811/400 reported, on the 201-direction netn = 86 · V0 C0 S3 · Hosono
T-209 Mixed rectangle-measure lower bound s(88) ≥ 3853/400 reported, on the 201-direction netn = 88 · V0 C0 S3 · Hosono
T-210 Mixed rectangle-measure lower bound s(89) ≥ 1937/200 reported, on the 201-direction netn = 89 · V0 C0 S3 · Hosono
T-211 Mixed rectangle-measure lower bound s(91) ≥ 783/80 reported, on the 201-direction netn = 91 · V0 C0 S3 · Hosono
T-212 Mixed rectangle-measure lower bound s(90) ≥ 489/50 reported, on the 201-direction netn = 90 · V0 C0 S3 · Hosono
T-213 Mixed rectangle-measure lower bound s(92) ≥ 493/50 reported, on the 201-direction netn = 92 · V0 C0 S3 · Hosono
T-214 Mixed rectangle-measure lower bound s(106) ≥ 10503/1000 reported, on the 201-direction netn = 106 · V0 C0 S3 · Hosono
T-215 Mixed rectangle-measure lower bound s(107) ≥ 529/50 reported, on the 201-direction netn = 107 · V0 C0 S3 · Hosono
T-216 Mixed rectangle-measure lower bound s(108) ≥ 531/50 verified, on the 201-direction netn = 108 · V3 C3 S3 · Hosono
T-217 Mixed rectangle-measure lower bound s(110) ≥ 1073/100 reported, on the 201-direction netn = 110 · V0 C0 S3 · Hosono
T-198 Daniel's opening n229 dilation certificate, after Couzo and de Wintern = 229 · V3 C3 S2 · Daniel · superseded
T-199 Daniel's smaller n229 update, replacing the opening certificaten = 229 · V3 C3 S2 · Daniel · superseded
T-195 A tighter rational certificate of Schadt and Ellsworth's 29-square packing, 1.57e-36 below T-155n = 29 · V3 C3 S1 · PingYou Ltd

Show these 23 in the table

16 October 2026 · 7 results

T-194 Fitzharding reports a Lean proof that Wainwright's packing is optimal, s(19) = 3 + 4√2/3n = 19 · V0 C1 S5 · Fitzharding
T-189 Couzo's refinement of Polkovskyi's 237, and fifteen beyond-horizon boundsn = 237 · V3 C3 S3 · Couzo · superseded
T-191 s(95) = 10, by a mixed cover of points and grid-line segmentsn = 95 · V3 C1 S3 · Greff
T-192 Continuous-pose lower bound s(28) > 1149/200 reported, by two of the source's programsn = 28 · V0 C0 S3 · Hosono
T-188 A tighter exact certificate for 211 squares from Couzo, of Ostashev's arrangementn = 211 · V3 C3 S2 · Couzo
T-190 A tighter exact certificate for 237 squares from Couzo, below his issue-607 certificaten = 237 · V3 C3 S2 · Couzo
T-193 Exact algebraic sides at sixteen counts from Daniel, of packings the record credits16 counts from 88 to 302 · V3 C3 S2 · Daniel

Show these 7 in the table

15 October 2026 · 3 results

T-185 A tighter packing of 175 squares from Couzo, superseded by Ostashev's compositionn = 175 · V3 C3 S2 · Couzo · superseded
T-187 A smaller packing of 146 squares from Daniel's chain search, started from Xu's packingn = 146 · V3 C3 S2 · Daniel
T-186 Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there12 counts from 129 to 304 · V3 C3 S1 · Polkovskyi

Show these 3 in the table

Earlier Publications

Results at S3 and above from the four weeks before 15 October 2026, by the week each was published (or established, for this project’s own), with who found them. A result added above that was published earlier is listed here too. A muted, underlined id has since been superseded.

12–14 October 2026 · 20 results

Fitzharding 1 formalization result: T-154S5
Overbeck 1 optimality result: T-153S5
Hosono 5 lower bounds: T-176, T-177, T-178, T-179, T-180S4
Fang 1 upper bound: T-163 (superseded)S4
Couzo 5 upper bounds: T-161 (superseded), T-172 (superseded), T-181 (superseded), T-182, T-189 (superseded)
Ostashev 3 upper bounds: T-158 (superseded), T-162 (superseded), T-175 (superseded)
Greff 1 optimality result: T-184
Polkovskyi 1 upper bound: T-183 (superseded)
Daniel 1 upper bound: T-160 (superseded)
PingYou Ltd 1 upper bound: T-164 (superseded)

5–11 October 2026 · 46 results

Hosono 19 lower bounds: T-094, T-096 (superseded), T-097, T-099 (superseded), T-100 (superseded), T-102, T-103, T-104, T-105, T-106, T-107, T-108, T-109, T-110, T-111, T-132, T-142, T-143, T-144
Chaoweeraprasit 6 upper bounds: T-113 (superseded), T-114 (superseded), T-115 (superseded), T-116 (superseded), T-136, T-151 (superseded)
Couzo 5 upper bounds: T-128 (superseded), T-130 (superseded), T-134, T-140 (superseded), T-152 (superseded)
Guzhou0806 3 lower bounds: T-133 (superseded), T-145 (superseded), T-157
Daniel 3 upper bounds: T-119, T-131 (superseded), T-141 (superseded)
PingYou Ltd 2 upper bounds: T-147 (superseded), T-148 (superseded)
Xu 2 upper bounds: T-125 (superseded), T-126
Ostashev 1 upper bound: T-149 (superseded)
Fang 1 upper bound: T-135 (superseded)
Polkovskyi 1 upper bound: T-138 (superseded)
Gupta 1 upper bound: T-127
Levy 1 uniqueness result: T-112
Ryu 1 lower bound: T-095

28 September – 4 October 2026 · 27 results

Ahmed 1 optimality result: T-060S5
Daniel 4 optimality results: T-062, T-063, T-064S4, T-081S4
Hosono 15 lower bounds and 2 optimality results: T-048, T-070 (superseded), T-066, T-067, T-068, T-069, T-074, T-071 (superseded), T-072 (superseded), T-073, T-075, T-076, T-077 (superseded), T-080, T-082, T-090, T-091
Karakuş 1 lower bound and 1 optimality result: T-083, T-084
Couzo 1 upper bound: T-092 (superseded)
Levy 1 lower bound: T-079 (superseded)
Guzhou0806 1 lower bound: T-043 (superseded)

21–27 September 2026 · 18 results

Overbeck 4 lower bounds: T-038 (superseded), T-037 (superseded)S5, T-040 (superseded), T-041 (superseded)
Daniel 1 lower bound and 3 optimality results: T-050 (superseded), T-051S4, T-052S4, T-053
Tokoharu 1 lower bound: T-047 (superseded)S4
Levy 2 lower bounds, 1 case exclusion result and 1 restricted optimality result: T-033 (superseded), T-034 (superseded), T-035, T-036
Hosono 3 lower bounds: T-044 (superseded), T-045 (superseded), T-046 (superseded)
Couzo 1 upper bound: T-056 (superseded)
Guzhou0806 1 lower bound: T-039 (superseded)

Verification Ladders

Each result has an identifier, a claim, and three ratings, defined in epistemics.md: S, its significance; V, the strongest verification its original evidence supports; and C, what this repository has confirmed independently. For results by others, this repository assigns V and C under its review policy. Each ladder links to its rubric; each rung has a tooltip with its full meaning.

Level

Significance How significant is the result?

Verification How was it originally verified?

Confirmation How has it been confirmed?

Level 5

Moves a central open case, or broad adoption

V5Formal, expert-reviewed

C5Formal, replayed here, open, two experts

Level 4

Reusable technique, bound family, or settled value

V4Mechanized, AI-reviewed, human-overseen

C4Mechanized confirmation, AI-reviewed, overseen

Level 3

A substantive case result or machine audit

V3Checkable; review record pending

C3Machine-replayed; review record pending

Level 2

A citable detail that changes no theorem

V2Proof asserted, not recoverable

C2Replayed, no machine certificate

Level 1

Bookkeeping or a routine consequence

V1Numerically checked

C1Read

Level 0

V0Claimed or recorded only

C0Recorded

Evidence and Exactness

The ladders grade a result. Its evidence carries an assurance label: reported, a named source’s claim not checked here; numerically checked, a finite-precision calculation with its precision, rounding and tolerance recorded; or verified, an exact check, rigorous interval certificate or complete proof covering the claim and its preconditions. A verified packing proves an upper bound; optimality also needs a matching verified lower bound.

Finite precision is not enough where squares touch exactly. A tolerance that accepts a true zero-gap contact also accepts a small overlap, so a contact-heavy packing requires exact algebraic signs or outward-rounded intervals (why). Schadt’s $n = 29$ packing passes its 300-digit numerical check, while the verified interval witness proves a slightly weaker side. Trump’s $n = 11$ packing is verified exactly over a degree-eight number field, including fourteen zero-gap contacts.

These checks also audit published work: T-004 and T-008 check Bentz’s 2010 Theorem 8, including both halves of $s (46) = 7$ , and T-011 checks Trump’s 1979 packing for eleven squares. The theorem remains the source’s; this repository adds an exact machine check.

Result Types and Credits

A result’s kind is lower bound, upper bound, optimality (settling an exact value), or a kind that bounds no case, such as rigidity, case exclusion, simplification (a simpler proof of a value another result establishes) or formalization (a proof in a proof assistant of a value another result establishes).

Credits name the source’s authors; this project’s results are credited to Levy as Levy. X after Y means X’s result rests directly on Y’s proof, method or tool.

Status and Supersession

A result’s status is recorded (registered from its source without review or replay here), reviewed (its argument read here), confirmed (its certificate replayed successfully) or incomplete (a known defect remains open). Of the 229 results, 185 are confirmed; the rest are 30 recorded, 13 reviewed and 1 incomplete. Status follows the confirmation rung and is never set by hand; the status policy defines it. In analysis marks a review or replay under way here; waiting on marks a request with the source or another party.

A bound is superseded when no current case bound rests on it and the bounds recorded for its cases match or beat it. A better bound not yet adopted by its case records remains unmarked. For other kinds, superseded means the register records a later result implying the whole claim; superseded in part means it implies only part, so the original remains current. The table names the most recent replacement, with “and others” when needed; the full record lists every replacement.

The overview highlights recent major results.

Browse Results

Significance Verification Confirmation Kind Status Hide supersededSource Case n Max age daysAdded within days229 results

Date	Result	n	Credit	Rungs	Status	ID
2026-10-22 published	Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225	105, 204	Polkovskyi after Couzo, Muglich, Daniel, Xu, PingYou Ltd, Levy	V0 C0Upper bound	recorded	T-229
2026-10-20 published	Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there	204, 227, 229, 260	Daniel after Couzo, de Winter, Polkovskyi, Levy	V3 C3Upper bound	confirmed	T-225
2026-10-20 published	Daniel's smaller n270 packing, another arrangement found from Chaoweeraprasit's, certified exactly	270	Daniel after Chaoweeraprasit, Levy	V3 C3Upper bound	confirmed	T-224
2026-10-20 published	Daniel's smaller n154 packing, continuing Couzo's #586 arrangement, certified exactly	154	Daniel after Couzo, Ostashev, Levy	V3 C3Upper bound	confirmed	T-222
2026-10-19 published	Daniel's smaller n308 packing, started from Chaoweeraprasit's 307 packing, certified exactly	308	Daniel after Chaoweeraprasit, Levy	V3 C3Upper bound	confirmed	T-221
2026-10-19 published	Daniel's finite sub-grid upper bound for 183 squares, after Couzo's n182 packing	183	Daniel after Couzo; register seed data from Levy	V3 C3Upper bound	confirmed	T-220
2026-10-18 published	Couzo's 204 of #675, below his #670, since superseded by his #678	204	Couzo after Polkovskyi, Daniel, Levy	V0 C0Upper bound	recorded superseded by T-225	T-228
2026-10-18 published	Couzo's smaller packings at 204 and 260, the two in the corpus of issue 678's 85 bounds	204, 260	Couzo after de Winter, Polkovskyi, Daniel, Fang, PingYou Ltd, Levy	V3 C3Upper bound	confirmed superseded by T-225	T-223
2026-10-18 published	Daniel's smaller n229 update, replacing the opening certificate	229	Daniel after Couzo, de Winter, Levy	V3 C3Upper bound	confirmed superseded by T-225	T-199
2026-10-17 published	Couzo's refinements of de Winter's 227 and 229, below T-196 at both	227, 229	Couzo after de Winter, Daniel, Levy	V0 C0Upper bound	recorded superseded by T-225	T-227
2026-10-17 published	Couzo's 204 of #670, below Polkovskyi's T-186, since superseded by his own #675 and #678	204	Couzo after Polkovskyi, Daniel, Levy	V0 C0Upper bound	recorded superseded by T-225	T-226
2026-10-17 published	Daniel's smaller n271 packing, continuing Couzo's arrangement, certified exactly	271	Daniel after Couzo, Chaoweeraprasit, Levy	V3 C3Upper bound	confirmed	T-219
2026-10-17 published	Daniel's opening n229 dilation certificate, after Couzo and de Winter	229	Daniel after Couzo, de Winter, Levy	V3 C3Upper bound	confirmed superseded by T-225	T-198
2026-10-17 published	Daniel's smaller n155 packing, started from Ostashev's packing	155	Daniel after Ostashev, Levy	V3 C3Upper bound	confirmed	T-197
2026-10-17 published	de Winter reports three smaller rational upper bounds at 227, 229 and 260	227, 229, 260	de Winter after Levy	V3 C3Upper bound	confirmed superseded by T-225	T-196
2026-10-16 published	Daniel's smaller n236 packing, after Chaoweeraprasit's composition, certified exactly	236	Daniel after Chaoweeraprasit, Levy	V3 C3Upper bound	confirmed	T-218
2026-10-16 published	Mixed rectangle-measure lower bound $s (110) \geq 1073 / 100$ reported, on the 201-direction net	110	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-217
2026-10-16 published	Mixed rectangle-measure lower bound $s (108) \geq 531 / 50$ verified, on the 201-direction net	108	Hosono after Tokoharu and Levy	V3 C3Lower bound	confirmed	T-216
2026-10-16 published	Mixed rectangle-measure lower bound $s (107) \geq 529 / 50$ reported, on the 201-direction net	107	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-215
2026-10-16 published	Mixed rectangle-measure lower bound $s (106) \geq 10503 / 1000$ reported, on the 201-direction net	106	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-214
2026-10-16 published	Mixed rectangle-measure lower bound $s (92) \geq 493 / 50$ reported, on the 201-direction net	92	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-213
2026-10-16 published	Mixed rectangle-measure lower bound $s (90) \geq 489 / 50$ reported, on the 201-direction net	90	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-212
2026-10-16 published	Mixed rectangle-measure lower bound $s (91) \geq 783 / 80$ reported, on the 201-direction net	91	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-211
2026-10-16 published	Mixed rectangle-measure lower bound $s (89) \geq 1937 / 200$ reported, on the 201-direction net	89	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-210
2026-10-16 published	Mixed rectangle-measure lower bound $s (88) \geq 3853 / 400$ reported, on the 201-direction net	88	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-209
2026-10-16 published	Mixed rectangle-measure lower bound $s (86) \geq 3811 / 400$ reported, on the 201-direction net	86	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-208
2026-10-16 published	Mixed rectangle-measure lower bound $s (85) \geq 379 / 40$ verified, on the 201-direction net	85	Hosono after Tokoharu and Levy	V3 C3Lower bound	confirmed	T-207
2026-10-16 published	Mixed rectangle-measure lower bound $s (84) \geq 3769 / 400$ verified, on the 201-direction net	84	Hosono after Tokoharu and Levy	V3 C3Lower bound	confirmed	T-206
2026-10-16 published	Mixed rectangle-measure lower bound $s (68) \geq 853 / 100$ verified, on the 201-direction net	68	Hosono after Tokoharu and Levy	V3 C3Lower bound	confirmed	T-205
2026-10-16 published	Mixed rectangle-measure lower bound $s (51) \geq 3011 / 400$ reported, on a 2,073-direction declared net	51	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-204
2026-10-16 published	Mixed rectangle-measure lower bound $s (43) \geq 2777 / 400$ reported, on a 2,073-direction declared net	43	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-203
2026-10-16 published	Mixed rectangle-measure lower bound $s (41) \geq 2721 / 400$ reported, on a 2,073-direction declared net	41	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-202
2026-10-16 published	Mixed rectangle-measure lower bound $s (40) \geq 1347 / 200$ reported, on a 2,073-direction declared net	40	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-201
2026-10-16 published	Mixed rectangle-measure lower bound $s (38) \geq 2631 / 400$ reported, on a 2,073-direction declared net	38	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-200
2026-10-16 published	Fitzharding reports a Lean proof that Wainwright's packing is optimal, $s (19) = 3 + 4 \sqrt{2} / 3$	19	Fitzharding after Levy, Ahmed	V0 C1Optimality	reviewed	T-194
2026-10-16 published	Exact algebraic sides at sixteen counts from Evan Daniel, of packings the record credits	88, 102, 123, 126, 175, 199, 204, 232, 258, 259, 266, 267, 292, 295, 301, 302	Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy	V3 C3Upper bound	confirmed	T-193
2026-10-15 published	A tighter rational certificate of Schadt and Ellsworth's 29-square packing, `1.57e-36` below T-155	29	PingYou Ltd after Schadt, Ellsworth, Daniel, Levy	V3 C3Upper bound	confirmed	T-195
2026-10-15 published	Continuous-pose lower bound $s (28) > 1149 / 200$ reported, by two of the source's programs	28	Hosono after Guzhou0806, Tokoharu, Levy	V0 C0Lower bound	recorded	T-192
2026-10-15 published	$s (95) = 10$ , by a mixed cover of points and grid-line segments	95	Greff after Daniel, Hosono	V3 C1Optimality	reviewed	T-191
2026-10-15 published	A tighter exact certificate for 237 squares from Francisco Couzo, below his issue-607 certificate	237	Couzo after Polkovskyi, Chaoweeraprasit, Daniel, Levy	V3 C3Upper bound	confirmed	T-190
2026-10-14 published	Francisco Couzo's refinement of Mykhailo Polkovskyi's 237, and fifteen beyond-horizon bounds	237	Couzo after Polkovskyi, Fang, Daniel, Ellsworth, Levy	V3 C3Upper bound	confirmed superseded by T-190	T-189
2026-10-14 published	A tighter exact certificate for 211 squares from Francisco Couzo, of Daniil Ostashev's arrangement	211	Couzo after Ostashev, Daniel, Levy	V3 C3Upper bound	confirmed	T-188
2026-10-14 published	A smaller packing of 146 squares from Daniel's chain search, started from Xu's packing	146	Daniel after Xu	V3 C3Upper bound	confirmed	T-187
2026-10-14 published	Exact certificates at twelve counts from Mykhailo Polkovskyi, tightening the smallest reported there	129, 154, 156, 175, 176, 204, 232, 264, 271, 302–304	Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy	V3 C3Upper bound	confirmed	T-186
2026-10-14 published	$s (76) = 9$ , by a mixed cover of points and grid-line segments	76	Greff after Daniel, Hosono	V3 C1Optimality	reviewed	T-184
2026-10-13 published	Mykhailo Polkovskyi's certificate for 237 squares, the smallest known side there when registered	237	Polkovskyi after Friedman, Daniel, Xu, Chaoweeraprasit, Barnett, Couzo, Ostashev, Ellsworth, Fang, Levy	V3 C3Upper bound	confirmed superseded by T-190	T-183
2026-10-13 published	Exact rational certificates at 154, 211 and 241 from Couzo	154, 211, 241	Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy	V3 C3Upper bound	confirmed	T-182
2026-10-13 published	Exact rational certificates at 211, 241 and 302 from Couzo	211, 241, 302	Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy; Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy	V3 C3Upper bound	confirmed superseded by T-188 and others	T-181
2026-10-13 published	Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported	50–63, 65–80, 82–99, 101–120, 122–143, 145–168, 170–195, 197–224, 226–255, 257–288, 290–323	Hosono	V0 C1Lower bound	reviewed	T-180
2026-10-13 published	Linear-measure lower bound $s (n) \geq 306 / 25 = 12.24$ for $n = 145 \dots 149$ , reported	145–149	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V0 C1Lower bound	reviewed	T-179
2026-10-13 published	Mixed rectangle-measure lower bound $s (37) \geq 2599 / 400$ reported, on a 2,073-direction declared net	37	Hosono after Tokoharu and Levy	V0 C1Lower bound	reviewed	T-178
2026-10-13 published	Mixed rectangle-measure lower bound $s (44) \geq 2799 / 400$ reported, on a 2,073-direction declared net	44	Hosono after Tokoharu and Levy	V0 C1Lower bound	reviewed	T-177
2026-10-13 published	Mixed rectangle-measure lower bound $s (39) \geq 533 / 80$ reported, on a 2,073-direction declared net	39	Hosono after Tokoharu and Levy	V0 C1Lower bound	reviewed	T-176
2026-10-13 published	New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactly	211, 241, 304	Ostashev after Nakajima, Daniel, de Winter, Couzo, Levy, Xu	V3 C3Upper bound	confirmed superseded by T-188 and others	T-175
2026-10-13 published	Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own	68, 70, 84, 86, 88, 102, 103, 105, 108, 110, 123, 126, 127, 129–132, 146, 153, 155, 156, 172, 175, 176, 179–182, 199, 204, 206–211, 228, 232, 236–241, 258, 259, 261, 263, 264, 266–273, 292, 295, 297, 301–308	PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy	V3 C3Upper bound	confirmed	T-174
2026-10-13 established	$s (290) \leq 17.5355339059327376 \dots$ , by monotonicity from T-155, with its own exact certificate	290	Levy after Daniel	V3 C3Upper bound	confirmed	T-156
2026-10-12 published	Mykhailo Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements there	153, 207	Polkovskyi after Xu, Chaoweeraprasit, Levy	V3 C3Upper bound	confirmed	T-173
2026-10-12 published	A packing of 154 squares from Couzo, continuing Daniel's arrangement	154	Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy	V3 C3Upper bound	confirmed superseded by T-222	T-172
2026-10-12 published	An exact contact refinement of SQUISH's packing of 272 squares	272	PingYou Ltd after Chaoweeraprasit, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-174	T-171
2026-10-12 published	An exact contact refinement of Xu's packing of 105 squares, after Couzo's	105	PingYou Ltd after Xu, Couzo, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-174	T-170
2026-10-12 published	A further exact refinement of Xu's packing of 261 squares, after Barnett's	261	PingYou Ltd after Xu, Barnett, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-174	T-169
2026-10-12 published	A further exact refinement of Xu's packing of 146 squares, after Barnett's	146	PingYou Ltd after Xu, Barnett, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-187	T-168
2026-10-12 published	A tighter rational realization of PingYou Ltd's packing of 207 squares	207	PingYou Ltd after Xu, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-173	T-167
2026-10-12 published	A tighter rational realization of PingYou Ltd's packing of 153 squares	153	PingYou Ltd after Xu, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-173	T-166
2026-10-12 published	A tighter certificate at 176 from Couzo, published with no issue	176	Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy	V3 C3Upper bound	confirmed superseded by T-186	T-165
2026-10-12 published	A packing of 302 squares from four boundary patches of Xu's packing of 70 and two upright banks	302	PingYou Ltd after Xu, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-186	T-164
2026-10-12 published	s(n² − n + 1) < n for every n ≥ 15, and n² − n + F(n) squares below side n, from Kevin Fang	211, 241, 273, 307	Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo	V2 C1Upper bound	reviewed superseded by T-188 and others	T-163
2026-10-12 published	New packings of 129, 204 and 303 squares, a cut-out and two compositions, each certified exactly	129, 204, 303	Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy	V3 C3Upper bound	confirmed superseded by T-225 and others	T-162
2026-10-12 published	Exact rational certificates at 156, 176, 237 and 271 from Couzo	156, 176, 237, 271	Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy	V3 C3Upper bound	confirmed superseded by T-219 and others	T-161
2026-10-12 published	A smaller packing of 154 squares from Daniel's size-chaining explorer, certified exactly	154	Daniel after Couzo, Levy	V3 C3Upper bound	confirmed superseded by T-222	T-160
2026-10-12 published	New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactly	175, 232, 264	Ostashev after Nakajima, Ellsworth, Schadt, Friedman, de Winter, Daniel, Levy, Xu	V3 C3Upper bound	confirmed superseded by T-186	T-158
2026-10-12 established	Exact regularized sides at 59 counts, `5.4e-20` to `9.9e-15` below the ceilings they replace	28, 29, 37, 39, 41, 50, 53–55, 69, 71, 83, 87, 101, 104, 107, 109, 122, 124, 125, 128, 145, 147–149, 151, 170, 171, 173, 174, 178, 197, 198, 200–205, 226, 227, 229–231, 233–235, 257, 260, 262, 264, 265, 291, 293, 294, 296, 298–300	Levy after Daniel	V3 C3Upper bound	confirmed	T-155
2026-10-12 published	Fitzharding reports a Lean proof that $s (17) = 4.67553009360455 \dots$	17	Fitzharding after Levy, Ahmed, MacIver	V3 C1Formalization	reviewed	T-154
2026-10-12 published	Candidate proof that Bidwell's 1998 packing is optimal, $s (17) = 4.67553009360455 \dots$	17	Overbeck after Levy, Ahmed, Mira-acc, Guzhou0806, Maaz	V0 C1Optimality	reviewed	T-153
2026-10-11 published	A tighter packing of 175 squares from Couzo, superseded by Ostashev's composition	175	Couzo after Xu, Daniel, Ellsworth, Levy	V3 C3Upper bound	confirmed superseded by T-186	T-185
2026-10-11 published	Another arrangement of Xu's design for 102 squares, certified exactly from its decimal layout	102	Calle after Xu, Levy	V3 C3Upper bound	confirmed superseded by T-146	T-159
2026-10-11 published	$s (40) > 3359 / 500 = 6.718$ , by T-145's continuous-pose kernel on a density with 163 new supports	40	Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-157
2026-10-11 published	Exact rational refinements of SQUISH's Mondrian packings of 304 and 305 squares	304, 305	Couzo after Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy	V3 C3Upper bound	confirmed superseded by T-186 and others	T-152
2026-10-11 published	Nine exact rational packings from SQUISH's fourth request	176, 181, 238, 239, 270–272, 304, 305	Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy	V3 C3Upper bound	confirmed superseded by T-224 and others	T-151
2026-10-11 published	Exact rational refinements of Xu's packings of 146 and 261 squares	146, 261	Barnett after Xu, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-187 and others	T-150
2026-10-11 published	New packings of 155, 156 and 211 squares, each with an exact rational certificate	155, 156, 211	Ostashev after Nakajima, Levy, Xu	V3 C3Upper bound	confirmed superseded by T-197 and others	T-149
2026-10-11 published	A packing of 153 squares built from Xu's packing of 70, its contacts reclosed at a new angle	153	PingYou Ltd after Xu, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-173	T-148
2026-10-11 published	A packing of 207 squares from two placed patches of Xu's packing of 70	207	PingYou Ltd after Xu, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-173	T-147
2026-10-10 published	Exact rational certificates at fifteen counts from Daniel's regularized record lists	70, 102, 103, 123, 129, 146, 153, 236, 258, 263, 269, 292, 295, 302, 303	Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy	V3 C3Upper bound	confirmed	T-146
2026-10-10 published	$s (40) > 1340000 / 199529 = 6.7158157 \dots$ , by a continuous-pose interval kernel	40	Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-157	T-145
2026-10-10 published	Linear-measure lower bound $s (n) \geq 563 / 50 = 11.26$ for $n = 122 \dots 126$ , reported	122–126	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V0 C0Lower bound	recorded	T-144
2026-10-10 published	Mixed rectangle-measure lower bound $s (30) \geq 2357 / 400$ reported, on a 2,073-direction declared net	30	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-143
2026-10-10 published	Mixed rectangle-measure lower bound $s (28) \geq 2297 / 400$ reported, on a 2,073-direction declared net	28	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-142
2026-10-10 published	Exact rational certificates at 132 and 308 from Daniel's third record hunt	132, 308	Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy	V3 C3Upper bound	confirmed superseded by T-221 and others	T-141
2026-10-10 published	Six more exact rational certificates from Couzo	132, 175, 209, 237, 270, 305	Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy	V3 C3Upper bound	confirmed superseded by T-224 and others	T-140
2026-10-10 published	An exact rational refinement of Xu's packing of 70 squares	70	Deleeuw after Xu, Levy	V3 C3Upper bound	confirmed superseded by T-146	T-137
2026-10-10 published	A packing of 308 squares below the grid, from Fang	308	Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead	V3 C3Upper bound	confirmed superseded by T-221	T-135
2026-10-10 published	$s (40) > 335427 / 50000 = 6.70854$ , by a clipped-corner transfer on a 401-direction net	40	Guzhou0806 after Hosono, Tokoharu, Levy, Stromquist, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-157	T-133
2026-10-10 published	Mixed rectangle-measure lower bound $s (29) \geq 291 / 50$ reported, on a 2,073-direction declared net	29	Hosono after Tokoharu and Levy	V0 C0Lower bound	recorded	T-132
2026-10-09 published	A more precise certificate of Xu's 126-square arrangement, from Daniel	126	Daniel after Xu, Chaoweeraprasit	V3 C3Upper bound	confirmed	T-139
2026-10-09 published	Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered	84, 86, 103, 105, 108, 127, 131, 132, 175, 180, 258, 267, 270, 302, 303, 306	Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy	V3 C3Upper bound	confirmed superseded by T-224 and others	T-138
2026-10-09 published	Seventeen exact rational packings from SQUISH's third request	131, 153, 154, 207, 209, 232, 236, 237, 259, 263, 269, 270, 292, 302, 303, 305, 307	Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy	V3 C3Upper bound	confirmed	T-136
2026-10-09 published	Six exact rational certificates from Couzo	132, 237, 263, 267, 270, 303	Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy	V3 C3Upper bound	confirmed	T-134
2026-10-09 published	$s (132) \leq 11.987099332245063 \dots$ from Daniel's record hunt	132	Daniel after Couzo, Chaoweeraprasit, Levy	V3 C3Upper bound	confirmed superseded by T-174	T-131
2026-10-08 published	Five follow-up rational refinements from Couzo	84, 86, 105, 175, 270	Couzo after Xu, Daniel, Ellsworth, Levy	V3 C3Upper bound	confirmed superseded by T-224 and others	T-130
2026-10-08 published	Eight complete rational refinements from Couzo	105, 108, 127, 131, 155, 180, 228, 306	Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy	V3 C3Upper bound	confirmed superseded by T-197 and others	T-128
2026-10-08 published	Fourteen rational refinements, seventeen complete source cases	88, 108, 123, 129, 130, 153, 154, 179, 180, 199, 207–209, 236–239	Gupta after Chaoweeraprasit, Daniel	V3 C3Upper bound	confirmed	T-127
2026-10-08 published	Undilated n51 construction over Q(sqrt2)	51	Xu	V3 C3Upper bound	confirmed	T-126
2026-10-08 published	Complete rational construction reports at 25 counts	51, 70, 84, 86, 88, 102, 103, 105, 108, 123, 126, 127, 129–131, 146, 153, 175, 179, 236, 258, 261, 263, 267, 295	Xu	V3 C3Upper bound	confirmed superseded by T-218 and others	T-125
2026-10-07 published	Three dated Daniel certificate reports with historical source custody	105, 130, 292	Daniel after Couzo, Levy	V0 C0Upper bound	recorded superseded by T-174 and others	T-129
2026-10-07 published	Reported non-strict local minima for 178 source configurations	1–4, 6–9, 11–16, 20–25, 28, 30–36, 42–49, 56–64, 72–81, 90–100, 111–121, 133–144, 157–169, 183–196, 212–225, 242–256, 274–289, 308–324	Daniel after Couzo	V0 C0Restricted optimality	recorded	T-124
2026-10-07 published	Reported `s(177) ≤ alpha_177` with a degree-32 exact side form	177	Daniel after Couzo	V0 C0Upper bound	recorded	T-123
2026-10-07 published	Reported `s(152) ≤ alpha_152` with a degree-40 exact side form	152	Daniel after Couzo	V0 C0Upper bound	recorded	T-122
2026-10-07 published	Reported `s(106) ≤ alpha_106` with a degree-32 exact side form	106	Daniel after Couzo	V0 C0Upper bound	recorded	T-121
2026-10-07 published	Reported `s(102) ≤ alpha_102` with a degree-8 exact side form	102	Daniel after Couzo	V0 C0Upper bound	recorded superseded by T-146	T-120
2026-10-07 published	Three exact feasible upper bounds at n=266,270,272 from new source arrangements	266, 270, 272	Daniel after Levy, Ellsworth, Couzo, Stead	V3 C3Upper bound	confirmed	T-119
2026-10-07 published	Exact rational ceiling refinements at $n = 68$	68	Rehwaldt after Couzo and earlier contributors	V3 C3Upper bound	confirmed superseded by T-174	T-118
2026-10-07 published	Exact rational ceiling refinements at $n = 105$ , 292	105, 292	Rehwaldt after Couzo and earlier contributors	V3 C3Upper bound	confirmed superseded by T-174 and others	T-117
2026-10-07 published	$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings	88, 108, 179, 180, 199, 207, 236, 263, 302	Chaoweeraprasit after Ellsworth, Couzo and SQUISH	V3 C3Upper bound	confirmed superseded by T-218 and others	T-116
2026-10-07 published	$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings	123, 126, 129, 154, 155, 179, 208, 237–239, 258, 263	Chaoweeraprasit after Couzo and Ellsworth	V3 C3Upper bound	confirmed superseded by T-222 and others	T-115
2026-10-07 published	$s (153) \leq 7250614903299225 / 562949953421312$ , from a SQUISH rational packing	153	Chaoweeraprasit after Ellsworth	V3 C3Upper bound	confirmed superseded by T-173	T-114
2026-10-07 published	$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings	108, 126, 129, 130, 154, 155, 180, 209, 238, 303	Chaoweeraprasit after Ellsworth	V3 C3Upper bound	confirmed superseded by T-222 and others	T-113
2026-10-06 established	Trump's packing is the only optimal packing of eleven squares, up to symmetry	11	Levy after Ahmed	V3 C2Uniqueness	confirmed	T-112
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 41$ , on a declared net	41	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-111
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 39$ , on a declared net	39	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-110
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 30$ , on a declared net	30	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-109
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 29$ , on a declared net	29	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-108
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 28$ , on a declared net	28	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-107
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 27$ , on a declared net	27	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-106
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 26$ , on a declared net	26	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-105
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 20$ , on a declared net	20	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-104
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 19$ , on the finest declared net yet	19	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-103
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 18$ , on the finest declared net yet	18	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-102
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 19$ , past the best 18-square packing	19	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-103	T-100
2026-10-06 published	Mixed rectangle-measure lower bound verified at $n = 18$ , on a finer declared net	18	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-102	T-099
2026-10-05 published	Exact certificates of 77 catalogue packings: `s(n) ≤ S'`, `1.2e-16` to `9.8e-15` above each side	28, 37, 39, 41, 50, 51, 53–55, 69–71, 83, 87, 88, 101, 104, 107–109, 122, 124, 125, 127–129, 145–151, 153, 170, 171, 173–176, 178, 179, 197, 198, 200–205, 226, 227, 229–235, 257, 258, 260–262, 264–267, 290, 291, 293–296, 298–300	Daniel after Couzo, de Winter, Ellsworth, Levy	V3 C3Upper bound	confirmed	T-101
2026-10-05 published	Exact optima of 48 known-best packings: `s(n) ≤ S'`, `3.5e-13` to `5.0e-11` below each printed side	68, 102, 103, 106, 110, 123, 126, 131, 132, 152, 154–156, 172, 177, 180–182, 199, 206–211, 228, 236–241, 259, 263, 268–273, 297, 301–307	Daniel after Couzo, de Winter, Ellsworth, Levy	V3 C3Upper bound	confirmed	T-098
2026-10-05 published	Mixed rectangle-measure lower bound verified at $n = 66$	66	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-097
2026-10-05 published	Mixed rectangle-measure lower bound replayed at $n = 18$ , on a net the certificate declares	18	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-102	T-096
2026-10-05 published	$s (12) \geq 7943 / 2000 = 3.9715$ , Daniel's points dilated and re-weighted	12	Ryu after Daniel, Levy	V3 C3Lower bound	confirmed	T-095
2026-10-05 published	Mixed rectangle-measure lower bounds verified at $n = 67$ and $84$	67, 84	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-094
2026-10-04 published	Mixed rectangle-measure lower bounds verified at 17 counts in $n = 53 \dots 95$	53, 54, 58, 70–73, 76, 87–95	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-091
2026-10-04 published	Mixed rectangle-measure lower bounds verified at 17 counts in $n = 42 \dots 96$	42–44, 51, 56, 57, 67, 69, 72, 75, 84, 86, 88, 93–96	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-090
2026-10-03 published	Smaller packings again at seven counts from $n = 208$ to $306$ , each certified two independent ways	208, 209, 228, 263, 272, 303, 306	Couzo	V3 C3Upper bound	confirmed superseded by T-186 and others	T-092
2026-10-03 published	Mixed rectangle-measure lower bounds verified at 22 counts in $n = 51 \dots 96$	51, 52, 55, 58, 69–71, 73–76, 86–96	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-082
2026-10-03 published	$s (k^{2} - 4) = k$ for every integer $k$ from 5 up; $k = 5 \dots 18$ are the cases held here	21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320	Daniel after Burns, Massaccesi	V0 C1Optimality	reviewed	T-081
2026-10-02 published	Linear-measure lower bound replayed at $n = 101 \dots 105$	101–105	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-080
2026-10-02 established	$s (12) \geq 15680000 / 3949423 = 3.9702002 \dots$ , Daniel's points re-weighted	12	Levy after Daniel	V3 C3Lower bound	confirmed superseded by T-095	T-079
2026-10-02 published	$s (12) \geq 31360 / 7901 = 3.9691178 \dots$ , Daniel's certificate rescaled by $7902 / 7901$	12	Ryu after Daniel	V3 C3Lower bound	confirmed superseded by T-095	T-078
2026-10-02 published	Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70	20, 42, 70	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-104 and others	T-077
2026-10-02 published	Linear-measure lower bound replayed at $n = 82$	82	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-076
2026-10-02 published	Mixed rectangle-measure lower bounds replayed at nine counts in $n = 83 \dots 96$	83, 85–88, 91–93, 96	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-075
2026-10-02 published	Linear-measure lower bounds replayed at $n = 83$ and $n = 101 \dots 105$	83, 101–105	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-073
2026-10-02 published	Mixed rectangle-measure lower bound replayed at $n = 76$	76	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-091	T-072
2026-10-02 published	Mixed rectangle-measure lower bounds replayed at $n = 84 \dots 87$	84–87	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-207 and others	T-071
2026-10-01 published	Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$	19, 20, 26–31, 38–44, 53–58, 68–70, 74, 75, 88, 89, 93–95	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-074
2026-10-01 published	Mixed rectangle-measure lower bounds replayed at $n = 37, 65, 66, 90, 92$	37, 65, 66, 90, 92	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-069
2026-10-01 published	Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$	19, 20, 26–31, 38–44, 53–56, 66, 68–70, 74–76, 86–90, 93–95	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-068
2026-10-01 published	$s (77) = 9$ , by a mixed cover extended from Daniel's $s (60)$ cover	77, 78	Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Optimality	confirmed	T-067
2026-10-01 published	$s (59) = 8$ , by a mixed cover of points and grid-line segments	59	Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Optimality	confirmed	T-066
2026-09-30 published	$s (17) > 18641771 / 4000000 = 4.66044275$	17	Guzhou0806 after Overbeck, Mira-acc, Levy	V3 C3Lower bound	confirmed	T-093
2026-09-30 published	$s (k^{2} - 3) = k$ for every integer $k$ from 6 up; $k = 6 \dots 18$ are the cases held here	33, 46, 61, 78, 97, 118, 141, 166, 193, 222, 253, 286, 321	Daniel after Burns, Massaccesi	V3 C3Optimality	confirmed	T-064
2026-09-29 published	$s (k^{2} - 1) = k$ for every integer $k \geq 3$ ; $k = 3 \dots 18$ are the cases held here	8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323	Karakuş	V3 C3Optimality	confirmed	T-084
2026-09-29 published	$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$	8, 10–15, 17–24, 26–35, 37–48, 50–63, 65–80, 82–99, 101–120, 122–143, 145–168, 170–195, 197–224, 226–255, 257–288, 290–323	Karakuş	V3 C3Lower bound	confirmed	T-083
2026-09-29 published	$s (11) > 3875000000 / 999999999 = 3.875000003875 \dots$ , 3.9e-9 above $31 / 8$	11	Wang, Li after Overbeck, Levy	V3 C3Lower bound	confirmed superseded by T-060	T-061
2026-09-29 published	Trump's eleven-square packing is globally optimal	11	Ahmed after Levy, Overbeck	V3 C3Optimality	confirmed	T-060
2026-09-29 published	Reported equality of 12028 n11 row minima reproduced by a complete bound replay	11	Hosono after Tokoharu, Daniel	V3 C3Audit	confirmed	T-059
2026-09-29 published	Rectangle-certificate ceiling `α·UB(n)` proved for $n = 1$ ..100; `B·UB(n)` on 64 grid rows	1–100	Hosono after Tokoharu, Daniel	V3 C3Method limit	confirmed	T-058
2026-09-28 published	Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$	29, 38, 39, 41–44, 51–55, 57, 59, 60, 67–74, 86, 95	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-205 and others	T-070
2026-09-28 published	$s (61) = 8$ , by monotonicity from T-062	61	Daniel after Burns, Massaccesi	V3 C3Optimality	confirmed	T-063
2026-09-28 published	$s (60) = 8$ , by a mixed cover of points and grid-line segments	60	Daniel after Burns, Massaccesi	V3 C3Optimality	confirmed	T-062
2026-09-28 published	$s (21) = 5$ by a point-only route	21	Hosono after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Simplification	confirmed	T-055
2026-09-28 published	$s (45) = 7$ by a second, point-only route	45	Hosono after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Simplification	confirmed	T-054
2026-09-28 published	$s (50) \geq 37 / 5 = 7.4$	50, 51	Hosono after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed	T-048
2026-09-28 published	$s (17) > 116511 / 25000 = 4.66044$	17	Guzhou0806 after Overbeck, Mira-acc, Levy	V3 C3Lower bound	confirmed superseded by T-153 (reported) and others	T-043
2026-09-28 published	$s (17) > 233009 / 50000 = 4.66018$	17	Guzhou0806 after Overbeck, Mira-acc, Levy	V3 C3Lower bound	confirmed superseded by T-153 (reported) and others	T-042
2026-09-27 published	Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways	68, 102, 103, 105, 106, 110, 123, 130–132, 152, 154–156, 172, 177, 180–182, 199, 206–210, 228, 236–241, 259, 263, 268–273, 292, 297, 301–307	Couzo	V3 C3Upper bound	confirmed superseded by T-224 and others	T-056
2026-09-27 published	$s (45) = 7$ , by a mixed cover of points and grid-line segments	45	Daniel after Burns, Massaccesi	V3 C3Optimality	confirmed	T-053
2026-09-27 published	$s (21) = 5$ , by a mixed cover of points and grid-line segments	21	Daniel after Burns, Massaccesi	V3 C3Optimality	confirmed	T-052
2026-09-27 published	Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$	18–20, 26–31, 37–44, 51–61, 66–78, 86, 88–91, 94, 95	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V0 C0Lower bound	recorded superseded by T-205 and others	T-046
2026-09-27 published	Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$	18–20, 26–28, 30–32, 40, 61, 75–78	Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-157 and others	T-045
2026-09-27 published	$s (17) > 466001 / 100000 = 4.66001$	17	Overbeck after Levy, Mira-acc, Guzhou0806	V3 C3Lower bound	confirmed superseded by T-153 (reported) and others	T-041
2026-09-26 published	$s (32) = 6$	32	Daniel after Burns, Massaccesi	V3 C3Optimality	confirmed	T-051
2026-09-26 published	$s (17) > 232001 / 50000 = 4.64002$	17	Overbeck after Levy, Mira-acc, Guzhou0806	V3 C3Lower bound	confirmed superseded by T-153 (reported) and others	T-040
2026-09-25 published	$s (17) > 231001 / 50000 = 4.62002$	17	Guzhou0806 after Overbeck, Mira-acc, Levy	V3 C3Lower bound	confirmed superseded by T-153 (reported) and others	T-039
2026-09-24 published	$s (83) \leq 9.63475764863195$ and $s (87) \leq 9.83881526994915$ , Chang's packings, certified exactly	83, 87	Chang, Ellsworth after Cantrell, Hajba, Stenlund, Bidwell; Chang after Ellsworth, Cantrell, Schadt, Hajba, DeVincentis	V3 C3Upper bound	confirmed superseded by T-155 and others	T-089
2026-09-24 published	$s (69) \leq 8.82719465572975$ , Ellsworth's degree-38 packing, certified exactly from its picture	69	Ellsworth after hmbelvedere, Cantrell, Schadt, Morandi	V3 C3Upper bound	confirmed superseded by T-155 and others	T-088
2026-09-24 established	Trump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetry	11	Levy	V3 C2Restricted optimality	confirmed superseded in part by T-112 and others	T-036
2026-09-24 established	Six-plus-five packings near Trump's tilt with side $\leq U_{h i}$ lie within `rho` of his pose	11	Levy	V3 C3Case exclusion	confirmed	T-035
2026-09-23 published	$s (21) \geq 5000 / 1001 = 4.995004995 \dots$	21	Daniel after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-052	T-050
2026-09-23 established	$s (21) \geq 122 / 25 = 4.88$	21	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-052	T-034
2026-09-22 published	$s (11) \geq 381 / 100$ ; $s (n) \geq 1377 / 250$ for $n = 26 \dots 28$ ; $s (n) \geq 571 / 100$ for $n = 29 \dots 31$	11, 26–31	Tokoharu after Levy, Hosono, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-109 and others	T-047
2026-09-22 published	Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files	26, 29, 39–41, 52–57, 68–73	Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-205 and others	T-044
2026-09-22 published	$s (11) > 31 / 8 = 3.875$	11	Overbeck after Levy, Guzhou0806, Mira-acc	V3 C3Lower bound	confirmed superseded by T-060	T-037
2026-09-22 established	$s (11) \geq 955000 \sqrt{2073600042893309449} / 359341754646249 = 3.8269975 \dots$	11	Levy	V3 C3Lower bound	confirmed superseded by T-060	T-033
2026-09-21 published	$s (17) \leq 4.6755300936045509 \dots$ , Bidwell's packing certified exactly	17	Overbeck after Levy, Mira-acc, Guzhou0806	V3 C3Upper bound	confirmed	T-065
2026-09-21 published	$s (17) > 461300 / 99853 = 4.6197910929 \dots$	17	Overbeck after Levy, Mira-acc, Guzhou0806	V3 C3Lower bound	confirmed superseded by T-153 (reported) and others	T-038
2026-09-20 published	$s (17) \geq 461300 / 99999 = 4.61304613 \dots$ , and beneath it Mira's $s (17) \geq 4613 / 1000$	17	Guzhou0806, Mira-acc after Levy, Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-153 (reported) and others	T-032
2026-09-20 established	The octagon corner class (threshold $1 / 2$ ) holds no eleven-square packing at side $96 / 25$	11	Levy	V3 C3Case exclusion	confirmed superseded by T-060	T-031
2026-09-19 established	$s (18) \geq 4679 / 1000 = 4.679$	18	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-102	T-030
2026-09-19 established	$s (18) \geq 1871 / 400 = 4.6775$	18	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-102	T-029
2026-09-19 established	$s (18) \geq 187 / 40 = 4.675$	18	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-102	T-028
2026-09-18 established	$s (18) \geq 467 / 100 = 4.67$	18	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-102	T-027
2026-09-16 published	$s (211) \leq 14.99796070496771500150 < 15$ , the first packing of 211 squares below the grid on record	211	de Winter	V3 C3Upper bound	confirmed superseded by T-188	T-057
2026-09-09 established	$s (11) \geq 955000 \sqrt{518400042893309449} / 179696714646249 = 3.8264474 \dots$	11	Levy	V3 C3Lower bound	confirmed superseded by T-060	T-026
2026-09-09 established	$s (11) \geq 191 / 50 = 3.82$ , by a threshold certificate	11	Levy	V3 C3Lower bound	confirmed superseded by T-060	T-025
2026-09-09 established	$s (11) \geq 3175000 \sqrt{518400042893309449} / 598960960743657 = 3.8166095 \dots$	11	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-060	T-024
2026-09-08 established	Conditional exclusion: no eleven-square packing in the four-owner branch at $q = 96 / 25$	11	Levy	V3 C3Case exclusion	confirmed superseded in part by T-060	T-023
2026-09-06 established	$s (11) \geq 38100 \sqrt{8100042893309449} / 899996306539 = 3.8100257 \dots$	11	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-060	T-022
2026-09-05 established	$s (n) \geq 97 / 20 = 4.85$ for $n = 20, 21$	20, 21	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-104 and others	T-021
2026-09-04 published	$s (k^{2} - 2) = k$ for every integer $k \geq 2$ ; $k = 3 \dots 18$ are the cases held here	7, 14, 23, 34, 47, 62, 79, 98, 119, 142, 167, 194, 223, 254, 287, 322	Vlasenko	V3 C3Optimality	confirmed	T-086
2026-09-04 published	Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$	10–324	Karakuş; Vlasenko	V3 C3Correction	confirmed	T-085
2026-09-04 established	$s (n) \geq 24 / 5 = 4.80$ for $n = 19, 20, 21$	19–21	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-104 and others	T-020
2026-09-04 established	$s (n) \geq 459 / 100 = 4.59$ for $n = 17, 18, 19$	17–19	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-153 (reported) and others	T-019
2026-09-04 established	$s (11) \geq 381 / 100 = 3.81$	11	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-060	T-018
2026-09-04 established	$s (12) \geq 99 / 25 = 3.96$	12	Levy after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-095	T-017
2026-09-03 established	Goebel's $n = 5$ optimum is rigid at fixed side: its pose is an isolated feasible point	5	Levy	V3 C3Rigidity	confirmed	T-014
2026-08-31 established	Bentz 2010, Lemma 10 is false as printed and true as corrected to $(1.74, 1)$	13	Levy after Bentz	V3 C3Correction	confirmed	T-005
2026-08-31 established	The sixteen-point set's unavoidability ceiling lies in $[4426213 / 1000000, 4427 / 1000)$	17, 18	Levy after Bentz	V3 C3Method limit	confirmed	T-003
2026-08-31 established	$s (18) \geq 4426213 / 1000000$ , by monotonicity from T-001	18	Levy after Bentz	V3 C3Lower bound	confirmed superseded by T-102	T-002
2026-08-31 established	$s (17) \geq 4426213 / 1000000 = 4.426213$ , from a sixteen-point unavoidable set	17	Levy after Bentz	V3 C3Lower bound	confirmed superseded by T-153 (reported) and others	T-001
2026-08-30 established	Goebel's $n = 40$ packing: seven verified first-order flexes, each refused at second order	40	Levy	V3 C3Rigidity	confirmed	T-013
2026-08-30 established	Goebel's $n = 5$ packing is second-order rigid at fixed side	5	Levy	V3 C3Rigidity	confirmed	T-012
2026-08-29 established	$s (29) \leq 5.93383346267692918974 \dots$ , by a Krawczyk interval certificate	29	Levy	V3 C3Upper bound	confirmed superseded by T-195	T-009
2026-08-25 published	$s (12) \geq 15680 / 3951 = 3.9686155 \dots$	12	Daniel after Burns, Massaccesi	V3 C3Lower bound	confirmed superseded by T-095	T-049
2026-08-24 established	$s (11) \geq 2 + 4 / \sqrt{5}$ , by a repair of Stromquist 2003's Figure 14 point set	11	Levy after Stromquist	V3 C3Lower bound	confirmed superseded by T-060	T-010
2026-08-21 published	$s (n) \geq 22529 / 5000$ for $n = 18, 19$ , by monotonicity from T-015	18, 19	Massaccesi after Burns	V3 C3Lower bound	confirmed superseded by T-103 and others	T-016
2026-08-21 published	$s (17) \geq 22529 / 5000 = 4.5058$	17	Massaccesi after Burns	V3 C3Lower bound	confirmed superseded by T-153 (reported) and others	T-015
2018 published	$s (37) \geq 5 \sqrt{3} / 2 + 2 \sqrt{2} - 1$ and $s (61) \geq 7 \sqrt{3} / 2 + 2 \sqrt{2} - 1$ , by optimal piercing	37, 61	Bašić, Slivková	V3 C1Lower bound	reviewed superseded by T-069 and others	T-087
2010-09-13 published	$s (46) = 7$	46	Bentz	V3 C3Optimality	confirmed	T-008
2010-09-13 published	$s (13) = 4$	13	Bentz; Daniel after Burns, Massaccesi	V3 C3Optimality	confirmed	T-006
2010-09-13 published	Bentz 2010, Theorem 8 ( $s (46) \geq 7$ ) is correct as printed, machine-audited in full	46	Bentz	V3 C3Audit	confirmed	T-004
2005 published	$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$	4–324	Nagamochi	V0 C1Lower bound	incomplete	T-007
1979 published	Trump's 1979 packing is exactly valid, so $s (11) \leq 3.877083590022814 \dots$	11	Trump	V3 C3Upper bound	confirmed	T-011