T-260: Polkovskyi's sub-print re-certifications at 105, 146, 156, 229 and 308, 1e-41 to 1e-21 below each

V0 C1 Upper bound reviewed

2026-10-30 published · Polkovskyi after Couzo, Xu, Daniel, Ostashev, Levy · 5 cases, $n = 105$ to 308

Five complete rational source certificates of issue 846 report finite upper-bound improvements at the exact sides they state: $s (105) \leq 10.789303783742975392483567321744$ , $s (146) \leq 12.583782190151860342291508597077$ , $s (156) \leq 648958292691713822691804519015415761935308555406679 / 50000000000000000000000000000000000000000000000000$ , $s (229) \leq 15.650161513876431064008548865321$ and $s (308) \leq 17.997149121270908735152160752315$ . They lie 1e-21 to 1e-41 below the houses' certificates, T-229 at 105 and T-186 at 156, both Polkovskyi's own, and T-187, T-225 and T-221, Daniel's, at 146, 229 and 308: at 105, 146, 229 and 308 the house certificate with every square unchanged and its side lowered by exactly 1e-21, and at 156 by exactly 1e-41 (RV-2, RV-3).

Each is a sub-print re-certification of the house certificate at its count: the 31 October review measured the margin as the house certificate's padding, the contact side unchanged, and no new arrangement. The owner's ruling of 23 October (think-fwa9) makes such re-certifications restatements, and the record applies it beyond the corpus only; whether it reaches counts inside the corpus is the owner's question (think-ikku, RV-1 of the review), and it is recorded as a question, not a verdict. Until the answer the entry stays below the stage 4 exit and no count is adopted; the review accepted every certificate for finite feasibility. These counts were registered on #852's branch within T-259, as the review read it, and are split from it so that the question holds only them.

The issue says every certificate yields valid and strictly valid, with positive wall and pair margin signs, under verify_bound_certificate.py --strict, Daniel's MIT separating-axis verifier with Couzo's modifications, byte-identical to the copy the issue-678 packet retains, and states the least wall and pair clearances of each: 4e-21 and 1e-20 at 105, 146, 229 and 308, and 4e-41 and 1e-40 at 156. Its table states larger margins at 156, 229 and 308 than the record's standing sides allow (RV-7). Those statements are the author's; no source program ran here.

Credit each packing to its finder, as the register credits the house it re-certifies, and Polkovskyi (Mishapolk, https://github.com/Mishapolk/square-packing-records), with what the issue calls the Antigravity Autonomous Research Lab, for the re-certification, as a regularization. The issue reports eight more counts that the ledger of 31 October does not record: 182, 268 and 304 lie above each case's house, 204 and 260 above issue 845's certificates, 299 above Daniel's update to issue 824, and 211 and 241 restate the source's own certificates first committed on 23 October.

Significance, composition and next rung

Significance: Sub-print re-certifications of the house certificates at $n = 105$ , 146, 156, 229 and 308, 1.00e-41 to 1.00e-21 below them: the side lowered with every square unchanged, the house certificate's padding, as T-170's (S1) is. No lower bound or optimum.
Next rung: Held for the owner's question (think-ikku): whether the 23 October restatement ruling reaches counts inside the corpus. If it does, the entry is withdrawn and no house moves; if it does not, the replay and review evidence are entered, both rungs rise to V3 and C3, and each count is adopted under the house rule with both strong generators' regularization. The 31 October review accepted every certificate for finite feasibility and recommends reading them as restatements.
Unfinished confirmations: C3 (think-ikku): The owner's answer to whether the 23 October restatement ruling reaches counts inside the corpus (RV-1 of the 31 October review); then, if the counts stay recorded, the replay and review evidence already run, entered as the stage 4 exit. About 0.01 CPU-hours, up to about 0.02 (priced 2026-10-31).
Regularizations: n = 105: a regularization of T-229’s packing by Polkovskyi (the house certificate with every square unchanged and its side lowered by exactly 1e-21, as the 31 October review measured it (the source states no method); solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 0; no square turned).

n = 146: a regularization of T-187’s packing by Polkovskyi (the house certificate with every square unchanged and its side lowered by exactly 1e-21, as the 31 October review measured it (the source states no method); solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 0; no square turned).

n = 156: a regularization of T-186’s packing by Polkovskyi (the house certificate with every square unchanged and its side lowered by exactly 1e-41, as the 31 October review measured it (the source states no method); solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 0; no square turned).

n = 229: a regularization of T-225’s packing by Polkovskyi (the house certificate with every square unchanged and its side lowered by exactly 1e-21, as the 31 October review measured it (the source states no method); solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 0; no square turned).

n = 308: a regularization of T-221’s packing by Polkovskyi (the house certificate with every square unchanged and its side lowered by exactly 1e-21, as the 31 October review measured it (the source states no method); solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 0; no square turned).
Novelty: previously-published Present in an identified source

The cases

This result concerns 5 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
105	10.280000	10.789304	0.50930378…	open=	frontier n-105.md
146	12.086630	12.583783	0.49715196…		frontier n-146.md
156	12.510412	12.979166	0.46875370…		frontier n-156.md
229	15.137281	15.650162	0.51288034…		frontier n-229.md
308	17.566048	17.997150	0.43110095…		frontier n-308.md

Results on these cases

38 results in the register on these cases, oldest first

2005 published T-007 · 5 of these cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
Lower bound incomplete on these cases, superseded by T-073, T-080, T-083 and T-180 (reported)
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 5 of these cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
Correction confirmed
Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-27 published T-056 · cases 105, 156
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
Upper bound confirmed superseded by T-186 and T-229
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · 5 of these cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
Lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-02 published T-073 · case 105
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-080 · case 105
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
2026-10-05 published T-098 · case 156
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
Upper bound confirmed on these cases, superseded by T-186
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · cases 146, 229
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side
Upper bound confirmed on these cases, superseded by T-187 and T-225
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-117 · case 105
Exact rational ceiling refinements at $n = 105$ , 292
Upper bound confirmed superseded by T-229
Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register
2026-10-07 published T-124 · case 308
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 · case 105
Three dated Daniel certificate reports with historical source custody
Upper bound recorded superseded by T-229
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 · cases 105, 146
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-187 and T-229
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-128 · case 105
Eight complete rational refinements from Couzo
Upper bound confirmed superseded by T-229
Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register
2026-10-08 published T-130 · case 105
Five follow-up rational refinements from Couzo
Upper bound confirmed superseded by T-229
Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register
2026-10-09 published T-138 · case 105
Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered
Upper bound confirmed superseded by T-229
Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Gupta, Daniel, Levy · Mishapolk decimal poses 2026-10-09 · packet · register
2026-10-10 published T-135 · case 308
A packing of 308 squares below the grid, from Fang
Upper bound confirmed superseded by T-221
Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead · Fang two-wedge certificates 2026-10-10 · packet · register
2026-10-10 published T-141 · case 308
Exact rational certificates at 132 and 308 from Daniel's third record hunt
Upper bound confirmed superseded by T-221
Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · Daniel record hunt 3 2026-10-10 · packet · register
2026-10-10 published T-146 · case 146
Exact rational certificates at fifteen counts from Daniel's regularized record lists
Upper bound confirmed on these cases, superseded by T-187
Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register
2026-10-11 published T-149 · case 156
New packings of 155, 156 and 211 squares, each with an exact rational certificate
Upper bound confirmed superseded by T-186
Ostashev after Nakajima, Levy, Xu · Ostashev records 2026-10-11 · packet · register
2026-10-11 published T-150 · case 146
Exact rational refinements of Xu's packings of 146 and 261 squares
Upper bound confirmed superseded by T-187
Barnett after Xu, Daniel, Levy · Barnett refinements 2026-10-11 · packet · register
2026-10-12 established T-155 · case 229
Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace
Upper bound confirmed on these cases, superseded by T-225
Levy after Daniel · register
2026-10-12 published T-161 · case 156
Exact rational certificates at 156, 176, 237 and 271 from Couzo
Upper bound confirmed superseded by T-186
Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 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-170 · case 105
An exact contact refinement of Xu's packing of 105 squares, after Couzo's
Upper bound confirmed superseded by T-229
PingYou Ltd after Xu, Couzo, Daniel, Levy · PingYou n105 refinement 2026-10-12 · packet · register
2026-10-12 published T-261 · case 229
Reported grade-C local minima for 233 source configurations of field degree at most 4
Restricted optimality incomplete
Daniel after Schadt, Ellsworth, Levy · Daniel exact and local reports 2026-10-12 · packet · packet · register
2026-10-13 published T-174 · cases 105, 146, 156, 308
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
Upper bound confirmed on these cases, superseded by T-186, T-187, T-221 and T-229
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-179 · case 146
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 · 5 of these cases
Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported
Lower bound reviewed
Hosono · wand125 closed form 2026-10-14 · packet · packet · register
2026-10-14 published T-186 · case 156
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
2026-10-17 published T-196 · case 229
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-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-227 · case 229
Couzo's refinements of de Winter's 227 and 229, below T-196 at both
Upper bound confirmed superseded by T-225
Couzo after de Winter, Daniel, Levy · Couzo issue 672 2026-10-17 · packet · packet · register
2026-10-18 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
2026-10-19 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-225 · case 229
Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there
Upper bound confirmed
Daniel after Couzo, de Winter, Polkovskyi, Ostashev, PingYou Ltd, Levy · Daniel regularized lists 2026-10-20 · packet · register
2026-10-22 published T-229 · case 105
Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225
Upper bound confirmed
Polkovskyi after Couzo, Muglich, Daniel, Xu, Ostashev, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register
2026-10-30 published T-260 this result · 5 of these cases
Polkovskyi's sub-print re-certifications at 105, 146, 156, 229 and 308, 1e-41 to 1e-21 below each
Upper bound reviewed
Polkovskyi after Couzo, Xu, Daniel, Ostashev, Levy · Mishapolk issue 846 2026-10-30 · packet · register

Links

On this site: Frontier Survey · T-260 in the results table
Issues: Issue #846

On GitHub, at main

Register: T-260 in results.yaml, line 37616
Evidence: E-mishapolk-846-certificate-report
Proofs and certificates: certificate 0000-0999
Sources: Mishapolk issue 846 2026-10-30
Source packet: resources/web/mishapolk-issue846-2026-10-30/README.md
Artifacts: 31 artifacts and controls in 11 directories. Complete artifact list in the result entry. docs/project/reviews packing/campaign/issue-intake/upper-bounds-843-846-20261031 packing/devtools packing/resources/web/mishapolk-issue846-2026-10-30 packing/resources/web/mishapolk-issue846-2026-10-30/acquisition packing/resources/web/mishapolk-issue846-2026-10-30/measure packing/resources/web/mishapolk-issue846-2026-10-30/receipts packing/resources/web/mishapolk-issue846-2026-10-30/review/review-2026-10-31 packing/resources/web/mishapolk-issue846-2026-10-30/review/review-2026-10-31/out packing/resources/web/mishapolk-issue846-2026-10-30/source/certificates/0000-0999 packing/tests
Case file: each case’s file is linked from its row above