T-173: Mykhailo Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements there

V3 C3 Upper bound confirmed

2026-10-12 published · Polkovskyi after Xu, Chaoweeraprasit, Levy · $n = 153, 207$

Two complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s (153) \leq 12.87001385040448$ and $s (207) \leq 14.880960371558$ .

All 12 retained jobs, four certificates with their duplicate-square and outside-container controls, were decided again here on 14 October 2026 by both maintained exact routes and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided both certificates this entry cites with six controls each and held each derived fact to the pinned file it is derived from, fetched at the pin. No route here is the source's code; the author's runs of this repository's sqpack verifier, from copies outside the pinned tree, are his and decide nothing here, so the replay is independently re-implemented. The 14 October review found no blocking defect.

The author reports that this repository's sqpack verify.py and Daniel's verify_cert.py, at the revisions the issue names, accept all six certificates the issue requests in exact rational arithmetic. Those statements are the author's; no source program ran here.

Credit Polkovskyi (Mishapolk), as he asks on issue 593, with the certificates. The issue calls 153 and 207 new arrangements from basin hopping with tilted boundary corridors. Measured pose by pose by the 14 October review against PingYou Ltd's packings: at 207 every square pairs with one of T-147's within 2.8e-04, three turned by at most 0.012 degrees, a re-solve of PingYou's packing and not a new arrangement by pose; at 153 it is T-148's design with 100 squares moved beyond 1e-03 and six beyond a tenth, another arrangement of PingYou's design. Whether to credit them as new arrangements is the owner's decision 1 on pull request 602. It discloses that Gemini was used for the search implementation, verification orchestration and issue preparation, directed by the author.

On 15 October 2026 the case records adopted this entry as the house at 153 and 207, each case's reported and verified upper bounds moving together and PingYou Ltd's certificates (T-148, T-147) kept as history, with T-166 and T-167 verified and not selected; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json). PR 574's algebraic certificates (T-174) are smaller at both counts, by 1.6e-14 and 2.4e-12, and rest on one route; the rule of 15 October left them out for the owner's decision (RP-3 of the review of PR 574), and the house rule of 16 October 2026 adopts them instead, staged under think-ot5c until the writer takes an algebraic side (devtools.select_houses.PENDING_ADOPTION).

Significance, composition and next rung

Significance: Another arrangement of PingYou Ltd's design at 153 and a re-solve of its packing at 207, 2.56e-12 and 1.49e-09 below the case ceilings, as T-159 and T-150 are; no lower bound or optimum.
Next rung: V4 and C4 need a second accepted adversarial review by a distinct reviewer and a retained human oversight record; one AI review, of 14 October, is retained.
Regularized drawings: n = 153: a regularization of T-173’s packing by Levy (Daniel's search/regularize (MIT), evand/square-packing@675a5ea, conservative, coincide, through devtools.regularize_evand; the project’s modifications: same-side exact re-solve (devtools.regularize_same_side); at its own side; 21 free squares straightened, 5 free angles moved; identity: measured here; nearest centres one to one; 99 squares moved, at most 0.0839527; turns up to 2.16769°).

n = 207: a regularization of T-173’s packing by Levy (Daniel's search/regularize (MIT), evand/square-packing@675a5ea, conservative, coincide, through devtools.regularize_evand; the project’s modifications: same-side exact re-solve (devtools.regularize_same_side); at its own side; 2 free squares straightened, 6 free angles moved; identity: measured here; nearest centres one to one; 6 squares moved, at most 0.0176234; turns up to 0.0114592°).
Novelty: previously-published Present in an identified source

The cases

Case record

Regularized by the Squares Project with Daniel's regularizer (evand/square-packing, MIT)

As published

The packing as published, PingYou Ltd 2026 [Mishapolk exact certificates 2026-10-12]

12.3812.870

121314

12.36913.369

\sqrt{n}

\sqrt{n} + 1

Proven

$12.38486$ $\leq s (153) \leq$ $12.870014$

exact

Citation record n-153

lowerKarakuş 2026, arXiv corrects Nagamochi 2005 (confirmed T-083)

upperPingYou Ltd & Mishapolk, GitHub (confirmed T-173)

Open

optimality

The case record

LowerUpper

Verified12.38486432…12.87001385…

Reported12.44683233…12.87001385…

Gap0.48514952…

Case record

Regularized by the Squares Project with Daniel's regularizer (evand/square-packing, MIT)

As published

The packing as published, PingYou Ltd 2026 [Mishapolk exact certificates 2026-10-12]

14.4014.881

141516

14.38715.387

\sqrt{n}

\sqrt{n} + 1

Proven

$14.40143$ $\leq s (207) \leq$ $14.880961$

exact

Citation record n-207

lowerKarakuş 2026, arXiv corrects Nagamochi 2005 (confirmed T-083)

upperPingYou Ltd & Mishapolk, GitHub (confirmed T-173)

Open

optimality

The case record

LowerUpper

Verified14.40143877…14.88096037…

Reported14.45408302…14.88096037…

Gap0.47952159…

Results on these cases

19 results in the register on $n = 153, 207$ , oldest first, each with what it established and how it stands now.

2005 published T-007 · $n = 153, 207$
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
V0 C1 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 · $n = 153, 207$
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
V3 C3 Correction confirmed
Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register
2026-09-27 published T-056 · $n = 207$
Smaller packings for 49 counts from $n = 68$ to $307$ , each certified two independent ways
V3 C3 Upper bound confirmed superseded by T-173
Couzo · franciscouzo square-packing 2026-09-27 · packet · register
2026-09-29 published T-083 · $n = 153, 207$
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
V3 C3 Lower bound confirmed
Karakuş · Karakuş 2026 · source · register
2026-10-05 published T-098 · $n = 207$
Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side
V3 C3 Upper bound confirmed on these cases, superseded by T-173
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-05 published T-101 · $n = 153$
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side
V3 C3 Upper bound confirmed on these cases, superseded by T-173
Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register
2026-10-07 published T-114 · $n = 153$
$s (153) \leq 7250614903299225 / 562949953421312$ , from a SQUISH rational packing
V3 C3 Upper bound confirmed superseded by T-173
Chaoweeraprasit after Ellsworth · SQUISH n153 2026-10-07 · packet · packet · register
2026-10-07 published T-116 · $n = 207$
$s (n) \leq S_{n}$ at nine counts, from the second SQUISH update’s new rational packings
V3 C3 Upper bound confirmed superseded by T-173
Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register
2026-10-08 published T-125 · $n = 153$
Complete rational construction reports at 25 counts
V3 C3 Upper bound confirmed superseded by T-173
Xu · ry-xu square packing 2026 · packet · register
2026-10-08 published T-127 · $n = 153, 207$
Fourteen rational refinements, seventeen complete source cases
V3 C3 Upper bound confirmed on these cases, superseded by T-173
Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register
2026-10-09 published T-136 · $n = 153, 207$
Seventeen exact rational packings from SQUISH's third request
V3 C3 Upper bound confirmed on these cases, superseded by T-173
Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register
2026-10-10 published T-146 · $n = 153$
Exact rational certificates at fifteen counts from Daniel's regularized record lists
V3 C3 Upper bound confirmed on these cases, superseded by T-173
Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register
2026-10-11 published T-147 · $n = 207$
A packing of 207 squares from two placed patches of Xu's packing of 70
V3 C3 Upper bound confirmed superseded by T-173
PingYou Ltd after Xu, Daniel, Levy · PingYou n207 certificate 2026-10-11 · packet · register
2026-10-11 published T-148 · $n = 153$
A packing of 153 squares built from Xu's packing of 70, its contacts reclosed at a new angle
V3 C3 Upper bound confirmed superseded by T-173
PingYou Ltd after Xu, Daniel, Levy · PingYou n153 certificate 2026-10-11 · packet · register
2026-10-12 published T-166 · $n = 153$
A tighter rational realization of PingYou Ltd's packing of 153 squares
V3 C3 Upper bound confirmed superseded by T-173
PingYou Ltd after Xu, Daniel, Levy · PingYou n153 refinement 2026-10-12 · packet · register
2026-10-12 published T-167 · $n = 207$
A tighter rational realization of PingYou Ltd's packing of 207 squares
V3 C3 Upper bound confirmed superseded by T-173
PingYou Ltd after Xu, Daniel, Levy · PingYou n207 refinement 2026-10-12 · packet · register
2026-10-12 published T-173 this result · $n = 153, 207$
Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements there
V3 C3 Upper bound confirmed
Polkovskyi after Xu, Chaoweeraprasit, Levy · Mishapolk exact certificates 2026-10-12 · packet · register
2026-10-13 published T-174 · $n = 153, 207$
Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own
V3 C3 Upper bound confirmed on these cases, pending adoption
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-180 · $n = 153, 207$
Closed-form lower bound $s (N) \geq a (N)$ for every non-square $N \geq 49$ , reported
V0 C1 Lower bound reviewed
Hosono · wand125 closed form 2026-10-14 · packet · packet · register

Links

On this site: Case record, $n = 153$ · Frontier row, $n = 153$ · Case record, $n = 207$ · Frontier row, $n = 207$ · T-173 in the results table

On GitHub, at main

Register: T-173 in results.yaml, line 21935
Evidence: E-mishapolk-573-certificate-report · E-mishapolk-573-exact-feasibility
Proofs and certificates: certificate facts
Sources: Mishapolk exact certificates 2026-10-12 (its own site, retained copy)
Source packet: resources/web/mishapolk-exact-certificates-2026-10-12/README.md
Artifacts: 9 artifacts and controls
resources/web/mishapolk-exact-certificates-2026-10-12/README.md resources/web/mishapolk-exact-certificates-2026-10-12/acquisition/claims.json resources/web/mishapolk-exact-certificates-2026-10-12/facts/n-153.yaml.gz resources/web/mishapolk-exact-certificates-2026-10-12/facts/n-207.yaml.gz devtools/upper_bound_reports.py devtools/check_half_angle_area.py docs/project/reviews/review-2026-10-14-upper-bound-imports-561-593.md tests/test_upper_bound_reports.py tests/test_check_half_angle_area.py
Case file: frontier/n-153.md (verified lower, verified upper, reported lower, reported upper) · frontier/n-207.md (verified lower, verified upper, reported lower, reported upper)