T-253: Polkovskyi's sub-print re-certifications of nine house packings, 1.3e-19 to 1.7e-19 below each

V0 C1 Upper bound reviewed

2026-10-23 published · Polkovskyi after Daniel, Couzo, Ostashev, de Winter, Chaoweeraprasit, Levy · 9 cases, $n = 154$ to 271

Nine complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s (154) \leq 12.9211009407616805497273710781811018832596$ , $s (155) \leq 12.9522609446505640351376570514307748530773$ , $s (183) \leq 13.9995801787362525019414750957971146726967$ , $s (211) \leq 14.993549626299421038785594964633603835513$ , $s (227) \leq 15.5636256523251490846368074671935498402163$ , $s (236) \leq 15.8619343906073586170851792275034202808198$ , $s (237) \leq 15.9026036382595968800201182179071384887302$ , $s (241) \leq 15.9871170951502387581618869582526723489714$ and $s (271) \leq 16.9489477337136575105539116351026963841491$ .

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.

The certificates lie 1.3e-19 to 1.7e-19 below the houses' certificates at their counts (T-222, T-197, T-220, T-188, T-225, T-218, T-190, T-182 and T-219), each a re-certification of the house's packing. At six counts every square stays within 1.6e-6 of the house's; at 154, 183 and 227 47, 24 and 16 squares move beyond 1e-3, by at most 0.078, 0.358 and 0.168, in a region that deforms without changing the side, which equals the house's contact side to within 1e-30 (RV-4).

Each commit's title calls its certificate a world record and states how far below the earlier certificate it lies, and the index certified_bounds.json records each as valid and strictly valid under the source's verifier, Daniel's MIT separating-axis checker with Couzo's modifications, byte-identical to the copy the issue-678 packet retains. 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) for the re-certification, as a regularization: the index says 'packing refined from Daniel' at 154, 155, 236 and 271 and 'from Daniel & Couzo' at 183, and credits Polkovskyi alone at 211, 227, 237 and 241, whose houses are Ostashev's (211, 241), de Winter's (227) and Chaoweeraprasit's (237) packings. A publication in a watched repository that no issue reports, recorded from its commits by the owner's rule of 14 October 2026.

The same commits' certificates at 88 and 126 lie below the house but above an exact side a register entry states pending adoption (T-193's), 152 is undecided against T-122's exact side, and those at 204 and 270 tied the houses when published; the ledger records none of them.

Significance, composition and next rung

Significance: Sub-print re-certifications of the house certificates at $n = 154$ , 155, 183, 211, 227, 236, 237, 241 and 271, 1.29e-19 to 1.69e-19 below them: the house certificates' padding, the contact side unchanged, 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 = 154: a regularization of T-222’s packing by Polkovskyi (the house certificate's contact side re-dilated from 1 + 1e-20 to about 1 + 1e-30, as the 31 October review measured it (the source states no method); solved to its side; identity: measured here; nearest centres one to one; 47 squares moved, at most 0.07793; turns up to 0.8514°).

n = 155: a regularization of T-197’s packing by Polkovskyi (the house certificate's contact side re-dilated from 1 + 1e-20 to about 1 + 1e-30, 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 8.601e-16; turns up to 5e-15°).

n = 183: a regularization of T-220’s packing by Polkovskyi (the house certificate's contact side re-dilated from 1 + 1e-20 to about 1 + 1e-30, as the 31 October review measured it (the source states no method); solved to its side; identity: measured here; nearest centres one to one; 24 squares moved, at most 0.3578; turns up to 6.237°).

n = 211: a regularization of T-188’s packing by Polkovskyi (the house certificate's contact side re-dilated from 1 + 1e-20 to about 1 + 1e-30, 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 8.262e-16; turns up to 1e-14°).

n = 227: a regularization of T-225’s packing by Polkovskyi (the house certificate's contact side re-dilated from 1 + 1e-20 to about 1 + 1e-30, as the 31 October review measured it (the source states no method); solved to its side; identity: measured here; nearest centres one to one; 16 squares moved, at most 0.1682; turns up to 2.542°).

n = 236: a regularization of T-218’s packing by Polkovskyi (the house certificate's contact side re-dilated from 1 + 1e-20 to about 1 + 1e-30, 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 9.795e-16; turns up to 5.717e-34°).

n = 237: a regularization of T-190’s packing by Polkovskyi (the house certificate's contact side re-dilated from 1 + 1e-20 to about 1 + 1e-30, 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 1.341e-15; turns up to 7e-15°).

n = 241: a regularization of T-182’s packing by Polkovskyi (the house certificate's contact side re-dilated from 1 + 1e-20 to about 1 + 1e-30, 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 1.563e-06; turns up to 5e-15°).

n = 271: a regularization of T-219’s packing by Polkovskyi (the house certificate's contact side re-dilated from 1 + 1e-20 to about 1 + 1e-30, 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 2.326e-19; turns up to 5.184e-34°).
Novelty: previously-published Present in an identified source

The cases

This result concerns 9 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
154	12.426860	12.921101	0.49424049…	open=	frontier n-154.md
155	12.468709	12.952261	0.48355174…		frontier n-155.md
183	13.547988	13.999581	0.45159182…		frontier n-183.md
211	14.544571	14.993550	0.44897772…		frontier n-211.md
227	15.068802	15.563626	0.49482336…		frontier n-227.md
236	15.374474	15.861935	0.48745960…		frontier n-236.md
237	15.408051	15.902604	0.49455212…		frontier n-237.md
241	15.541608	15.987118	0.44550813…		frontier n-241.md
271	16.476545	16.948948	0.47240242…		frontier n-271.md

Results on these cases

46 results in the register on these cases, oldest first

2005 published T-007 · 9 of these cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
Lower bound incomplete on these cases, superseded by T-083 and T-180 (reported)
Nagamochi · Nagamochi 2005 · source · register
2026-09-04 published T-085 · 9 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
2026-09-16 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
2026-09-27 published T-056 · 6 of these cases
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
Upper bound confirmed superseded by T-182, T-190, T-197, T-218, T-219 and T-222
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · 9 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
2026-10-05 published T-098 · 7 of these cases
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
Upper bound confirmed on these cases, superseded by T-182, T-188, T-190, T-197, T-218, T-219 and T-222
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · case 227
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-225
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-113 · cases 154, 155
$s (n) \leq S_{n}$ at ten counts, $n = 108$ to 303, from new SQUISH rational packings
Upper bound confirmed superseded by T-197 and T-222
Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register
2026-10-07 published T-115 · cases 154, 155, 237
$s (n) \leq S_{n}$ at twelve counts, from the SQUISH update’s new and smaller rational packings
Upper bound confirmed superseded by 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 · case 236
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
Upper bound confirmed superseded by T-218
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register
2026-10-07 published T-124 · case 183
Reported non-strict local minima for 178 source configurations
Restricted optimality recorded
Daniel after Couzo · Daniel exact and local reports 2026 · packet · register
2026-10-08 published T-125 · case 236
Complete rational construction reports at 25 counts
Upper bound confirmed superseded by T-218
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · cases 154, 236, 237
Fourteen rational refinements, seventeen complete source cases
Upper bound confirmed on these cases, superseded by T-190, T-218 and T-222
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-08 published T-128 · case 155
Eight complete rational refinements from Couzo
Upper bound confirmed superseded by T-197
Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register
2026-10-09 published T-134 · case 237
Six exact rational certificates from Couzo
Upper bound confirmed on these cases, superseded by T-190
Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register
2026-10-09 published T-136 · cases 154, 236, 237
Seventeen exact rational packings from SQUISH's third request
Upper bound confirmed superseded by T-190, T-218 and T-222
Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-10 published T-140 · case 237
Six more exact rational certificates from Couzo
Upper bound confirmed superseded by T-190
Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register
2026-10-10 published T-146 · case 236
Exact rational certificates at fifteen counts from Daniel's regularized record lists
Upper bound confirmed on these cases, superseded by T-218
Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register
2026-10-11 published T-149 · cases 155, 211
New packings of 155, 156 and 211 squares, each with an exact rational certificate
Upper bound confirmed superseded by T-188 and T-197
Ostashev after Nakajima, Levy, Xu · Ostashev records 2026-10-11 · packet · register
2026-10-11 published T-151 · case 271
Nine exact rational packings from SQUISH's fourth request
Upper bound confirmed superseded by T-219
Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register
2026-10-12 established T-155 · case 227
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-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 237, 271
Exact rational certificates at 156, 176, 237 and 271 from Couzo
Upper bound confirmed superseded by 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-163 · cases 211, 241
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-182 and T-188
Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo · Fang excess theorem 2026-10-13 · packet · packet · packet · source 1 · source 2 · register
2026-10-12 published T-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-261 · cases 183, 227
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 · 6 of these cases
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
Upper bound confirmed on these cases, superseded by T-182, T-188, T-190, T-197, T-218 and T-219
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
New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactly
Upper bound confirmed superseded by T-182 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-180 · 9 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
Exact rational certificates at 211, 241 and 302 from Couzo
Upper bound confirmed superseded by T-182 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-186 · cases 154, 271
Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there
Upper bound confirmed on these cases, superseded by T-219 and T-222
Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 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
2026-10-16 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 · case 227
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-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-227 · case 227
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-236 · cases 211, 241
Daniel reports Lean proofs of Fang's Theorems 1 and 2, $s (n^{2} - n + 1) < n$ for $n \geq 15$
Formalization recorded
Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Levy · Daniel Fang and Ryu Lean 2026-10-18 · packet · source · register
2026-10-19 published T-220 · case 183
Daniel's finite sub-grid upper bound for 183 squares, after Couzo's n182 packing
Upper bound confirmed
Daniel after Couzo; register seed data from Levy · Daniel n183 subgrid 2026-10-19 · packet · register
2026-10-20 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-225 · case 227
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
2026-10-23 published T-253 this result · 9 of these cases
Polkovskyi's sub-print re-certifications of nine house packings, 1.3e-19 to 1.7e-19 below each
Upper bound reviewed
Polkovskyi after Daniel, Couzo, Ostashev, de Winter, Chaoweeraprasit, Levy · Mishapolk refinements 2026-10-23 · packet · packet · register

Links

On this site: Frontier Survey · T-253 in the results table

On GitHub, at main

Register: T-253 in results.yaml, line 36204
Evidence: E-mishapolk-1023-certificate-report
Proofs and certificates: certificate 0000-0999
Sources: Mishapolk refinements 2026-10-23
Source packet: resources/web/mishapolk-1023-refinements-2026-10-30/README.md · resources/web/mishapolk-issue846-2026-10-30/README.md
Artifacts: 47 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-1023-refinements-2026-10-30 packing/resources/web/mishapolk-1023-refinements-2026-10-30/acquisition packing/resources/web/mishapolk-1023-refinements-2026-10-30/measure packing/resources/web/mishapolk-1023-refinements-2026-10-30/receipts packing/resources/web/mishapolk-1023-refinements-2026-10-30/source/certificates/0000-0999 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/tests
Case file: each case’s file is linked from its row above