T-174: Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own

V3 C3 Upper bound confirmed

2026-10-13 published · PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · 68 cases, n=68 to 308

PingYou Ltd's pull request 574 reports finite upper-bound improvements at 68 counts: sixty-four complete rational source certificates, at 68, 84, 86, 88, 102, 103, 105, 108, 110, 123, 126, 127, 129, 130, 131, 132, 146, 155, 156, 172, 175, 176, 179, 180, 181, 182, 199, 204, 206, 208, 209, 210, 211, 228, 232, 236, 237, 238, 239, 240, 241, 258, 259, 261, 263, 264, 266, 267, 268, 269, 270, 271, 272, 273, 292, 295, 297, 301, 303, 304, 305, 306, 307 and 308, and algebraic ones at 70, 153, 207 and 302. Each rational side is a fraction whose numerator runs to 92 to 1,787 digits; the packet's claims record (acquisition/claims.json in pingyou-pr574-certificates-2026-10-13) states each exactly, and the notes compare each with its case and the earlier reports.

Its algebraic supplement states s(70)≤8.880960371555579594696691574923…, s(153)≤12.870013850404464344460153400780…, s(207)≤14.880960371555579594696691574923… and s(302)≤17.870275382555059669572692573398…, roots of integer polynomials of degree 5, 7, 5 and 3, as exact witnesses over their fields.

The author reports that a standard-library exact checker of its own, by separating axes and by rational polygon clipping, accepts all 147 v2 certificates and their pre-batch comparators with 2,678,449 unordered pairs per checker, rejects duplicate-square and outside-container controls, and that this repository's packing-witness verify and check_rational_witness_independent, run unchanged, accept every native corners Witness/v2 of the v2 release; and, for the supplement, that its checker isolates each declared real root by rational Sturm counts and accepts the 60 exact certificates with 917,302 pairs, and that the native verifier accepts their native witnesses. Those statements are the author's; no source program ran here.

Here both maintained exact routes decided all 64 rational certificates with their two controls each, and sqpack's exact witness verifier decided the four algebraic ones over their fields; no second route over a number field exists here. The third exact route decided the 64 rational certificates again with six controls each, and no route here is the source's code; the source's runs of this repository's two maintained routes are its own and decide nothing here, so the replay is independently re-implemented. The 14 October review of the pull request found no blocking defect.

Credit PingYou Ltd (https://github.com/BuddenPing/square-packing-certificates) with the certificate refinements, as the submission asks: smaller padding, an exact common-centre crop or an exact re-solve of each packing's own contacts. The packings are others': the submission names per count the original construction authors and preceding refiners it starts from, among them Friedman's and Ellsworth's catalogue constructions as Daniel's exact certificates and regularized lists state them, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit (SQUISH), Gupta, Fang and Polkovskyi (Mishapolk), and it says that padding reduction does not transfer authorship of an existing packing.

The pull request names at each count but 176 the registered packing whose certificate it tightens; at 176 it states Ostashev's packing of issue 575, which the record registers at no count there, every square within 9.6e-13 of his and the side 1.4e-19 below his certificate. Measured pose by pose by the 14 October review, the certificate is the named packing with no square moved beyond 1e-03 at 27 of those counts, pairs with it one to one with squares moved beyond 1e-03 at 33, and at 199, 292 and 303 pairs with it, or with any registered packing there, nowhere one to one; each count's regularizations item records which. At 199 the poses are those issue 593 states later.

The source makes no statement about AI assistance in this pull request; for its #503 and #504 certificates the author said on 12 October 2026 that they were AI assisted by 'PY', an unreleased agentic AI offering by PingYou Ltd.

The per-count ledger of 14 October (jlevy/squares#601) records it at these 68 counts, each below every earlier report there, and at no other: at 59 counts it ties or trails an earlier report, and at 21 the printed precision did not decide it.

Seventeen of those 21 are now decided exactly, by devtools.decide_undecided_comparisons. At 14 counts the pull request's side is above the standing one: the case records' reported lanes at 17, 28, 37, 39, 41, 69, 87, 128, 205, 235 and 300, and the entries pending adoption at 106, 152 and 177, T-121, T-122 and T-123. At 54, 107 and 178 it equals the reported lane. It is recorded at none of the 17. At 29, 55, 71 and 83 the comparison stays undecided, because the record holds only a print of the standing side.

The 14 October review recommends this entry as the house at the 60 rational counts where it was then the smallest exact side. On 15 October 2026 the coordinator of the 14 October intake held those houses for the owner (devtools.select_houses.LIFTED keeps the hold's words): their witnesses would add about 18.5 MB to a checkout and their custody inputs about 5.8 MB to every worker's snapshot. The owner's house rule of 16 October 2026 lifted the hold: size is met by OR-18's means, never by holding a house, and the houses were staged for adoption (devtools.select_houses.PENDING_ADOPTION).

The hold skipped this entry only: at 105, 129, 146, 156, 176, 204, 261, 271, 272 and 303 the next certificate below the case that a replay here decided and a review accepted held it from that day (T-170, T-162, T-168, T-161, T-165, T-162, T-169, T-161, T-171, T-162), this entry being smaller by 1.7e-99 to 1.18e-02, and at the other 50 counts the cases kept their houses. The adoption of 16 October 2026 (PR #657) then took ten of the 60 below this entry, 146 to T-187 and 129, 156, 175, 176, 204, 232, 264, 271 and 303 to T-186.

The adoption of 17 October 2026 (think-3mry), the first under the house rule, moved 37 of the other 50 to this entry and regularized each in the same pull request. At twelve of the thirteen left, 88, 102, 123, 126, 199, 258, 259, 266, 267, 292, 295 and 301, T-193's exact algebraic sides, at V3/C3 since its stage 4 review of 16 October (PR #658), are smaller and staged under think-ot5c. At 236 Daniel's issue 659 reports a side 2.02e-3 below this entry's, whose import adopts it; this entry stays staged there under think-3mry until then.

At 70, 153 and 207 the algebraic certificates are the smallest sides known but rest on one route; the rule of 15 October left them out for the owner (RP-3), and the house rule of 16 October 2026 adopts them, staged under think-ot5c until the writer takes an algebraic side; meanwhile the case at 70 keeps its house and Polkovskyi's issue 573 (T-173) holds 153 and 207. At 211, 237, 241, 302 and 304 smaller certificates hold the cases (T-188, T-190, T-182, T-186, T-186).

Significance, composition and next rung
Significance
Tighter certificates of packings the record credits to others, by padding or an exact re-solve, 1e-99 to 1e-19 below the case ceilings at most counts, as T-155; the 1.70e-02 drop below the case at 176 belongs to Ostashev's packing of issue 575; a citable detail of each case that moves no theorem; 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. The house rule of 16 October 2026 makes it the house at its 41 smallest counts: the adoption of 17 October 2026 (think-3mry) moved 37 rational ones to it, 236 stays staged under think-3mry until Daniel's smaller issue 659 side is adopted, and the algebraic ones at 70, 153 and 207 wait under think-ot5c, staged in devtools.select_houses.PENDING_ADOPTION until the writer takes an algebraic side. It was the smallest at 63 counts until the adoption of 16 October 2026 (PR #657) took ten of them below it, with T-186 and T-187, and T-193's stage 4 exit (PR #658) twelve more.
Regularizations
n = 68: a regularization of T-118’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 22 squares moved, at most 0.05544; turns up to 0.1293°).

n = 70: a regularization of T-146’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim).

n = 84: a regularization of T-130’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 45 squares moved, at most 0.07207; turns up to 1.066°).

n = 86: a regularization of T-130’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 64 squares moved, at most 0.2786; turns up to 2.347°).

n = 88: a regularization of T-127’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 35 squares moved, at most 0.1504; turns up to 4.756°).

n = 102: a regularization of T-146’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.429e-19; turns up to 1.105e-19°).

n = 103: a regularization of T-146’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.44e-19; turns up to 1.994e-18°).

n = 105: a regularization of T-130’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.455e-19; turns up to 1.126e-21°).

n = 108: a regularization of T-128’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 13 squares moved, at most 0.1213; turns up to 0.5055°).

n = 110: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 5 squares moved, at most 0.05405; turns up to 0.3583°).

n = 123: a regularization of T-146’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.566e-19; turns up to 3.767e-34°).

n = 126: a regularization of T-139’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 59 squares moved, at most 0.1372; turns up to 28.45°).

n = 127: a regularization of T-128’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 53 squares moved, at most 0.1916; turns up to 0.06535°).

n = 129: a regularization of T-162’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 2 squares moved, at most 0.1394; no square turned).

n = 130: a regularization of T-127’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 15 squares moved, at most 0.004243; turns up to 3.703e-34°).

n = 131: a regularization of T-136’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.479e-31; turns up to 2.233e-53°).

n = 132: a regularization of T-141’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 18 squares moved, at most 0.02147; turns up to 0.3469°).

n = 146: a regularization of T-168’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.414e-07; no square turned).

n = 153: a regularization of T-148’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim).

n = 155: a regularization of T-149’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 3 squares moved, at most 0.00824; turns up to 7.161e-05°).

n = 156: a regularization of T-161’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.765e-19; no square turned).

n = 172: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 69 squares moved, at most 0.1897; turns up to 1.325°).

n = 175: a regularization of T-158’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 4 squares moved, at most 0.4533; turns up to 0.00573°).

n = 179: a regularization of T-127’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 42 squares moved, at most 0.3164; turns up to 0.3196°).

n = 180: a regularization of T-128’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 25 squares moved, at most 0.04797; turns up to 0.00573°).

n = 181: a regularization of T-151’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1e-06; turns up to 4.902e-54°).

n = 182: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 27 squares moved, at most 0.08739; turns up to 1.004°).

n = 199: a regularization of T-127’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres not one to one).

n = 204: a regularization of T-162’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 9 squares moved, at most 0.8087; turns up to 0.1719°).

n = 206: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 8.276e-16; turns up to 1.2e-14°).

n = 207: a regularization of T-147’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim).

n = 208: a regularization of T-127’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 20 squares moved, at most 0.08253; turns up to 2.292°).

n = 209: a regularization of T-140’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 23 squares moved, at most 0.05204; turns up to 0.1512°).

n = 210: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 33 squares moved, at most 0.343; turns up to 0.4941°).

n = 211: a regularization of T-149’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.414e-06; no square turned).

n = 228: a regularization of T-128’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 46 squares moved, at most 0.3392; turns up to 0.09865°).

n = 232: a regularization of T-158’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 5 squares moved, at most 0.4533; turns up to 4.848e-15°).

n = 236: a regularization of T-146’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.173e-19; turns up to 4.844e-34°).

n = 237: a regularization of T-161’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.144e-19; no square turned).

n = 238: a regularization of T-151’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.004e-31; turns up to 2.327e-52°).

n = 239: a regularization of T-151’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 0.0001414; turns up to 0.00573°).

n = 240: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 28 squares moved, at most 0.3846; turns up to 0.4668°).

n = 241: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 33 squares moved, at most 0.05803; turns up to 5.575°).

n = 258: a regularization of T-146’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.272e-19; turns up to 2.276e-34°).

n = 259: a regularization of T-136’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.135e-31; turns up to 2.656e-30°).

n = 261: a regularization of T-169’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.414e-12; no square turned).

n = 263: a regularization of T-146’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.296e-19; turns up to 2.648e-18°).

n = 264: a regularization of T-158’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 2 squares moved, at most 0.4449; turns up to 3.565e-16°).

n = 266: a regularization of T-119’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 27 squares moved, at most 0.1194; turns up to 3.137°).

n = 267: a regularization of T-134’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.194e-19; turns up to 2.191e-18°).

n = 268: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 117 squares moved, at most 0.3617; turns up to 2.005°).

n = 269: a regularization of T-146’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.32e-19; turns up to 1.923e-18°).

n = 270: a regularization of T-151’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.183e-31; turns up to 9.689e-53°).

n = 271: a regularization of T-161’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.326e-19; no square turned).

n = 272: a regularization of T-151’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.414e-06; turns up to 1.423e-05°).

n = 273: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 49 squares moved, at most 0.3545; turns up to 3.884°).

n = 292: a regularization of T-146’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres not one to one).

n = 295: a regularization of T-146’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.433e-19; turns up to 1.399e-34°).

n = 297: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 119 squares moved, at most 0.2158; turns up to 2.271°).

n = 301: a regularization of T-098’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 189 squares moved, at most 0.4405; turns up to 1.792°).

n = 302: a regularization of T-164’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim).

n = 303: a regularization of T-162’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres not one to one).

n = 304: a regularization of T-152’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.465e-19; no square turned).

n = 305: a regularization of T-152’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.468e-19; no square turned).

n = 306: a regularization of T-128’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 34 squares moved, at most 0.368; turns up to 1.784°).

n = 307: a regularization of T-136’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.414e-06; turns up to 4.108e-51°).

n = 308: a regularization of T-141’s packing by PingYou Ltd (certificate refinement: smaller padding, an exact common-centre crop or an exact re-solve of the packing's own contacts; solved to its side; identity: the source’s claim; nearest centres one to one; 25 squares moved, at most 0.2385; turns up to 0.003911°).
Regularized drawings
n = 68: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 1 square moved, at most 0.0624451; no square turned).

n = 84: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 5.57737e-95; no square turned).

n = 86: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.10125e-94; no square turned).

n = 103: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 6.2913e-95; no square turned).

n = 105: a regularization of T-174’s packing by Levy (regularize_grid_slack/1: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/1) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 31 squares moved, at most 0.163115; turns up to 0.327601°).

n = 108: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 6.38887e-95; no square turned).

n = 110: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 2 squares moved, at most 0.0140541; no square turned).

n = 127: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.01486e-94; no square turned).

n = 130: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.23613e-94; no square turned).

n = 131: a regularization of T-174’s packing by Levy (regularize_grid_slack/1: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/1) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 3 squares moved, at most 0.266296; no square turned).

n = 132: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 8.676e-95; no square turned).

n = 172: a regularization of T-174’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; identity: measured here; nearest centres one to one; 3 squares moved, at most 0.00401552; turns up to 4.54359e-19°).

n = 179: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.87893e-94; no square turned).

n = 180: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.32582e-94; no square turned).

n = 181: a regularization of T-174’s packing by Levy (regularize_grid_slack/1: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/1) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 3 squares moved, at most 0.018627; no square turned).

n = 182: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 9.20759e-95; no square turned).

n = 206: a regularization of T-174’s packing by Levy (regularize_grid_slack/1: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/1) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 80 squares moved, at most 0.365425; turns up to 0.392072°).

n = 208: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.1935e-84; no square turned).

n = 209: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.21578e-94; no square turned).

n = 210: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.35755e-84; no square turned).

n = 228: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.63314e-94; no square turned).

n = 238: a regularization of T-174’s packing by Levy (regularize_grid_slack/1: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/1) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 11 squares moved, at most 0.0816735; turns up to 4.33173°).

n = 239: a regularization of T-174’s packing by Levy (regularize_grid_slack/1: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/1) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 14 squares moved, at most 0.135299; turns up to 0.00572958°).

n = 240: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.65217e-94; no square turned).

n = 261: a regularization of T-174’s packing by Levy (regularize_grid_slack/1: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/1) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 23 squares moved, at most 0.0777529; turns up to 4.92095e-10°).

n = 263: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.66446e-94; no square turned).

n = 268: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.26238e-94; no square turned).

n = 269: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.27066e-94; no square turned).

n = 272: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 22 squares moved, at most 0.347911; no square turned).

n = 273: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 16 squares moved, at most 0.359318; no square turned).

n = 297: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 2.37962e-90; turns up to 2.96678e-90°).

n = 305: a regularization of T-174’s packing by Levy (regularize_grid_slack/1: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/1) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 19 squares moved, at most 0.13004; turns up to 4.46319°).

n = 306: a regularization of T-174’s packing by Levy (regularize_axis_components/2; at its own side; identity: measured here; nearest centres one to one; 0 squares moved, at most 1.90821e-94; no square turned).

n = 307: a regularization of T-174’s packing by Levy (regularize_grid_slack/1: Daniel's coincidence placement (evand/square-packing search/regularize reg.coincidences and reg.place, MIT), ported to start from the witness's exact frame and to keep its contacts, then the same-side exact re-solve, through devtools.regularize_grid_slack; the project’s modifications: port of his coincidence placement (regularize_grid_slack/1) rather than his code, same-side exact re-solve (devtools.regularize_same_side); at its own side; identity: measured here; nearest centres one to one; 6 squares moved, at most 0.0630177; no square turned).
Novelty
previously-published Present in an identified source

The cases

This result concerns 68 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.

nProved lowerBest knownGapStatusRecords
688.5300008.7987960.26879523…open=frontier n-068.md
708.6575008.8809610.22346037…frontier n-070.md
849.4225009.6979350.27543479…frontier n-084.md
869.5030009.8205360.31753540…frontier n-086.md
889.6200009.8824520.26245103…frontier n-088.md
10210.28000010.6058290.32582869…frontier n-102.md
10310.6792330.39923204…frontier n-103.md
10510.7893040.50930378…frontier n-105.md
10810.62000010.9048220.28482101…frontier n-108.md
11010.51249210.9967840.48429119…frontier n-110.md
12311.09481011.5913790.49656809…frontier n-123.md
12611.23545511.7426410.50718541…frontier n-126.md
12711.28192911.8108790.52894946…frontier n-127.md
12911.37428111.8700140.49573226…frontier n-129.md
13011.42016411.9044840.48431819…frontier n-130.md
13111.46585611.9496600.48380348…frontier n-131.md
13211.51135711.9856810.47432242…frontier n-132.md
14612.08663012.5837830.49715196…frontier n-146.md
15312.38486412.8700140.48514952…frontier n-153.md
15512.46870912.9522610.48355174…frontier n-155.md
15612.51041212.9791660.46875370…frontier n-156.md
17213.11942913.6189890.49955949…frontier n-172.md
17513.23773913.7641750.52643511…frontier n-175.md
17613.27693213.7958730.51893997…frontier n-176.md
17913.39379613.8837960.48999853…frontier n-179.md
18013.43251713.9169940.48447636…frontier n-180.md
18113.47112113.9508200.47969775…frontier n-181.md
18213.50961113.9740910.46447888…frontier n-182.md
19914.11065714.6175730.50691458…frontier n-199.md
20414.29311414.8086430.51552861…frontier n-204.md
20614.36542414.8601590.49473403…frontier n-206.md
20714.40143814.8809610.47952159…frontier n-207.md
20814.43735914.9245190.48715890…frontier n-208.md
20914.47318814.9462240.47303461…frontier n-209.md
21014.50892514.9730020.46407539…frontier n-210.md
21114.54457114.9935500.44897772…frontier n-211.md
22815.10308115.6046020.50151960…frontier n-228.md
23215.23940215.7641750.52477134…frontier n-232.md
23615.37447415.8619350.48745960…frontier n-236.md
23715.40805115.9026040.49455212…frontier n-237.md
23815.44155215.9241070.48255398…frontier n-238.md
23915.47497915.9484440.47346419…frontier n-239.md
24015.50833115.9696860.46135431…frontier n-240.md
24115.54160815.9871180.44550813…frontier n-241.md
25816.06438216.5634490.49906558…frontier n-258.md
25916.09647316.5913790.49490418…frontier n-259.md
26116.16045916.6787800.51832011…frontier n-261.md
26316.22418516.7331670.50898081…frontier n-263.md
26416.25595116.7747360.51878387…frontier n-264.md
26616.31929216.8230290.50373673…frontier n-266.md
26716.35086716.8388160.48794778…frontier n-267.md
26816.38238016.8788150.49643464…frontier n-268.md
26916.41383016.9015140.48768311…frontier n-269.md
27016.44521816.9281630.48294353…frontier n-270.md
27116.47654516.9489480.47240242…frontier n-271.md
27216.50781016.9655870.45777588…frontier n-272.md
27316.53901416.9839260.44491103…frontier n-273.md
29217.09066017.5913790.50071812…frontier n-292.md
29517.18082717.7042330.52340547…frontier n-295.md
29717.24066917.7404180.49974824…frontier n-297.md
30117.35971517.8466680.48695189…open≈frontier n-301.md
30217.38934517.8700140.48066809…open=frontier n-302.md
30317.41892417.9099590.49103397…frontier n-303.md
30417.44845117.9294280.48097650…frontier n-304.md
30517.47792617.9504670.47253963…frontier n-305.md
30617.50735117.9634340.45608236…frontier n-306.md
30717.53672517.9808630.44413774…frontier n-307.md
30817.56604817.9971500.43110095…frontier n-308.md

Results on these cases

99 results in the register on these cases, oldest first

  1. 2005 published T-007 · 68 of these cases

    s(n)≥min(⌈n⌉,n−2⌊n⌋+1+1) for 4≤n≤324

    Lower bound incomplete on these cases, superseded by T-073, T-080, T-083, T-090, T-091, T-180 (reported), T-205, T-206 and T-216

    Nagamochi · Nagamochi 2005 · source · register

  2. 2026-09-04 published T-085 · 68 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

  3. 2026-09-16 published T-057 · case 211

    s(211)≤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

  4. 2026-09-22 published T-044 · cases 68, 70

    Weighted point lower bounds for ten counts in n=26…72, plus seven from the same files

    Lower bound confirmed superseded by T-091 and T-205

    Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register

  5. 2026-09-27 published T-046 · cases 68, 70, 86, 88

    Rectangle-density lower bounds reported for 48 counts in n=18…95

    Lower bound recorded superseded by T-090, T-091 and T-205

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register

  6. 2026-09-27 published T-056 · 45 of these cases

    Smaller packings for 49 counts from n=68 to 307, each certified two independent ways

    Upper bound confirmed superseded by T-098, T-127, T-136, T-146, T-173, T-174, T-182, T-186, T-190, T-197, T-218, T-219 and T-224

    Couzo · franciscouzo square-packing 2026-09-27 · packet · register

  7. 2026-09-28 published T-070 · cases 68, 70, 86

    Rectangle-density lower bounds replayed at 25 counts in n=29…95

    Lower bound confirmed superseded by T-090, T-091 and T-205

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-09-28 · packet · packet · source · review · register

  8. 2026-09-29 published T-058 · 5 of these cases

    Rectangle-certificate ceiling α·UB(n) proved for n=1..100; B·UB(n) on 64 grid rows

    Method limit confirmed

    Hosono after Tokoharu, Daniel · wand125 tools 2026 · packet · register

  9. 2026-09-29 published T-083 · 68 of these cases

    s(n)≥1/2+n−⌊n⌋+1/4 for every nonsquare 8≤n≤324

    Lower bound confirmed

    Karakuş · Karakuş 2026 · source · register

  10. 2026-10-01 published T-068 · cases 68, 70, 86, 88

    Rectangle-density lower bounds verified at 34 counts in n=19…95

    Lower bound confirmed on these cases, superseded by T-090, T-091 and T-205

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register

  11. 2026-10-01 published T-074 · cases 68, 70, 88

    Rectangle-density lower bounds replayed at 31 counts in n=19…95

    Lower bound confirmed on these cases, superseded by T-091 and T-205

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register

  12. 2026-10-02 published T-071 · cases 84, 86

    Mixed rectangle-measure lower bounds replayed at n=84…87

    Lower bound confirmed superseded by T-090 and T-206

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-02 · packet · source · review · register

  13. 2026-10-02 published T-073 · cases 102, 103, 105

    Linear-measure lower bounds replayed at n=83 and n=101…105

    Lower bound confirmed

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register

  14. 2026-10-02 published T-075 · cases 86, 88

    Mixed rectangle-measure lower bounds replayed at nine counts in n=83…96

    Lower bound confirmed on these cases, superseded by T-090 and T-091

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds afternoon 2026-10-02 · packet · source 1 · source 2 · review · register

  15. 2026-10-02 published T-077 · case 70

    Rectangle-density lower bounds replayed at n=20, 42 and 70

    Lower bound confirmed superseded by T-091

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-02 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

  16. 2026-10-02 published T-080 · cases 102, 103, 105

    Linear-measure lower bound replayed at n=101…105

    Lower bound confirmed

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register

  17. 2026-10-03 published T-082 · cases 70, 86, 88

    Mixed rectangle-measure lower bounds verified at 22 counts in n=51…96

    Lower bound confirmed on these cases, superseded by T-090 and T-091

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register

  18. 2026-10-03 published T-092 · 7 of these cases

    Smaller packings again at seven counts from n=208 to 306, each certified two independent ways

    Upper bound confirmed superseded by T-174 and T-186

    Couzo · franciscouzo square-packing 2026-10-03 · packet · register

  19. 2026-10-04 published T-090 · cases 84, 86, 88

    Mixed rectangle-measure lower bounds verified at 17 counts in n=42…96

    Lower bound confirmed

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-04 · packet · source · review · register

  20. 2026-10-04 published T-091 · cases 70, 88

    Mixed rectangle-measure lower bounds verified at 17 counts in n=53…95

    Lower bound confirmed

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds evening 2026-10-04 · packet · source · review · register

  21. 2026-10-05 published T-094 · case 84

    Mixed rectangle-measure lower bounds verified at n=67 and 84

    Lower bound confirmed on these cases, superseded by T-206

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-05 · packet · source · review · register

  22. 2026-10-05 published T-098 · 44 of these cases

    Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side

    Upper bound confirmed

    Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register

  23. 2026-10-05 published T-101 · 18 of these cases

    Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

    Upper bound confirmed on these cases, superseded by T-119, T-127, T-134, T-146, T-173, T-174, T-186, T-187 and T-225

    Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register

  24. 2026-10-07 published T-113 · 9 of these cases

    s(n)≤Sn at ten counts, n=108 to 303, from new SQUISH rational packings

    Upper bound confirmed superseded by T-139, T-174, T-186 and T-197

    Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register

  25. 2026-10-07 published T-114 · case 153

    s(153)≤7250614903299225/562949953421312, from a SQUISH rational packing

    Upper bound confirmed superseded by T-173

    Chaoweeraprasit after Ellsworth · SQUISH n153 2026-10-07 · packet · packet · register

  26. 2026-10-07 published T-115 · 11 of these cases

    s(n)≤Sn at twelve counts, from the SQUISH update’s new and smaller rational packings

    Upper bound confirmed superseded by T-139, T-146, T-174, T-186, T-190 and T-197

    Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register

  27. 2026-10-07 published T-116 · 9 of these cases

    s(n)≤Sn at nine counts, from the second SQUISH update’s new rational packings

    Upper bound confirmed superseded by T-127, T-173, T-174, T-186 and T-218

    Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register

  28. 2026-10-07 published T-117 · cases 105, 292

    Exact rational ceiling refinements at n=105, 292

    Upper bound confirmed superseded by T-146 and T-174

    Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register

  29. 2026-10-07 published T-118 · case 68

    Exact rational ceiling refinements at n=68

    Upper bound confirmed superseded by T-174

    Rehwaldt after Couzo and earlier contributors · Rehwaldt n68 refinement 2026-10-07 · packet · register

  30. 2026-10-07 published T-119 · cases 266, 270, 272

    Three exact feasible upper bounds at n=266,270,272 from new source arrangements

    Upper bound confirmed

    Daniel after Levy, Ellsworth, Couzo, Stead · Daniel new arrangements 2026-10-07 · packet · register

  31. 2026-10-07 published T-120 · case 102

    Reported s(102) ≤ alpha_102 with a degree-8 exact side form

    Upper bound recorded superseded by T-146

    Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

  32. 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

  33. 2026-10-07 published T-129 · cases 105, 130, 292

    Three dated Daniel certificate reports with historical source custody

    Upper bound recorded superseded by T-146 and T-174

    Daniel after Couzo, Levy · Daniel dated certificates 105 and 130 2026-10-07 · Daniel dated certificate 292 2026-10-07 · packet · packet · packet · register

  34. 2026-10-08 published T-125 · 24 of these cases

    Complete rational construction reports at 25 counts

    Upper bound confirmed superseded by T-127, T-134, T-139, T-146, T-173, T-174, T-186, T-187 and T-218

    Xu · ry-xu square packing 2026 · packet · register

  35. 2026-10-08 published T-127 · 16 of these cases

    Fourteen rational refinements, seventeen complete source cases

    Upper bound confirmed

    Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register

  36. 2026-10-08 published T-128 · 8 of these cases

    Eight complete rational refinements from Couzo

    Upper bound confirmed superseded by T-174 and T-197

    Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register

  37. 2026-10-08 published T-130 · 5 of these cases

    Five follow-up rational refinements from Couzo

    Upper bound confirmed superseded by T-174, T-186 and T-224

    Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register

  38. 2026-10-09 published T-131 · case 132

    s(132)≤11.987099332245063… from Daniel's record hunt

    Upper bound confirmed superseded by T-174

    Daniel after Couzo, Chaoweeraprasit, Levy · Daniel record hunt 2026-10-09 · packet · register

  39. 2026-10-09 published T-134 · 6 of these cases

    Six exact rational certificates from Couzo

    Upper bound confirmed

    Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register

  40. 2026-10-09 published T-136 · 16 of these cases

    Seventeen exact rational packings from SQUISH's third request

    Upper bound confirmed

    Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register

  41. 2026-10-09 published T-138 · 16 of these cases

    Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered

    Upper bound confirmed superseded by T-134, T-146, T-174, T-186 and T-224

    Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register

  42. 2026-10-09 published T-139 · case 126

    A more precise certificate of Xu's 126-square arrangement, from Daniel

    Upper bound confirmed

    Daniel after Xu, Chaoweeraprasit · Daniel trio126 2026-10-09 · packet · register

  43. 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

  44. 2026-10-10 published T-137 · case 70

    An exact rational refinement of Xu's packing of 70 squares

    Upper bound confirmed superseded by T-146

    Deleeuw after Xu, Levy · Deleeuw n70 refinement 2026-10-10 · packet · register

  45. 2026-10-10 published T-140 · 6 of these cases

    Six more exact rational certificates from Couzo

    Upper bound confirmed superseded by T-174, T-186, T-190 and T-224

    Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register

  46. 2026-10-10 published T-141 · cases 132, 308

    Exact rational certificates at 132 and 308 from Daniel's third record hunt

    Upper bound confirmed superseded by T-174 and T-221

    Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · Daniel record hunt 3 2026-10-10 · packet · register

  47. 2026-10-10 published T-144 · cases 123, 126

    Linear-measure lower bound s(n)≥563/50=11.26 for n=122…126, reported

    Lower bound recorded

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n122 2026-10-10 · packet · register

  48. 2026-10-10 published T-146 · 15 of these cases

    Exact rational certificates at fifteen counts from Daniel's regularized record lists

    Upper bound confirmed

    Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register

  49. 2026-10-11 published T-147 · case 207

    A packing of 207 squares from two placed patches of Xu's packing of 70

    Upper bound confirmed superseded by T-173

    PingYou Ltd after Xu, Daniel, Levy · PingYou n207 certificate 2026-10-11 · packet · register

  50. 2026-10-11 published T-148 · case 153

    A packing of 153 squares built from Xu's packing of 70, its contacts reclosed at a new angle

    Upper bound confirmed superseded by T-173

    PingYou Ltd after Xu, Daniel, Levy · PingYou n153 certificate 2026-10-11 · packet · register

  51. 2026-10-11 published T-149 · cases 155, 156, 211

    New packings of 155, 156 and 211 squares, each with an exact rational certificate

    Upper bound confirmed superseded by T-186, T-188 and T-197

    Ostashev after Nakajima, Levy, Xu · Ostashev records 2026-10-11 · packet · register

  52. 2026-10-11 published T-150 · cases 146, 261

    Exact rational refinements of Xu's packings of 146 and 261 squares

    Upper bound confirmed superseded by T-174 and T-187

    Barnett after Xu, Daniel, Levy · Barnett refinements 2026-10-11 · packet · register

  53. 2026-10-11 published T-151 · 9 of these cases

    Nine exact rational packings from SQUISH's fourth request

    Upper bound confirmed superseded by T-174, T-186, T-219 and T-224

    Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register

  54. 2026-10-11 published T-152 · cases 304, 305

    Exact rational refinements of SQUISH's Mondrian packings of 304 and 305 squares

    Upper bound confirmed superseded by T-174 and T-186

    Couzo after Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-11 · packet · register

  55. 2026-10-11 published T-159 · case 102

    Another arrangement of Xu's design for 102 squares, certified exactly from its decimal layout

    Upper bound confirmed superseded by T-146

    Calle after Xu, Levy · Calle n102 2026-10-12 · packet · register

  56. 2026-10-11 published T-185 · case 175

    A tighter packing of 175 squares from Couzo, superseded by Ostashev's composition

    Upper bound confirmed superseded by T-186

    Couzo after Xu, Daniel, Ellsworth, Levy · Couzo n175 2026-10-11 · packet · register

  57. 2026-10-12 established T-155 · cases 204, 264

    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-186 and T-225

    Levy after Daniel · register

  58. 2026-10-12 published T-158 · cases 175, 232, 264

    New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactly

    Upper bound confirmed superseded by T-186

    Ostashev after Nakajima, Ellsworth, Schadt, Friedman, de Winter, Daniel, Levy, Xu · Ostashev records 2026-10-12 · packet · register

  59. 2026-10-12 published T-161 · cases 156, 176, 237, 271

    Exact rational certificates at 156, 176, 237 and 271 from Couzo

    Upper bound confirmed superseded by T-186, T-190 and T-219

    Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-12 · packet · register

  60. 2026-10-12 published T-162 · cases 129, 204, 303

    New packings of 129, 204 and 303 squares, a cut-out and two compositions, each certified exactly

    Upper bound confirmed superseded by T-186 and T-225

    Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · Ostashev compositions 2026-10-12 · packet · register

  61. 2026-10-12 published T-163 · cases 211, 241, 273, 307

    s(n² − n + 1) < n for every n ≥ 15, and n² − n + F(n) squares below side n, from Fang

    Upper bound reviewed superseded by T-174, T-182 and T-188

    Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo · Fang excess theorem 2026-10-13 · packet · register

  62. 2026-10-12 published T-164 · case 302

    A packing of 302 squares from four boundary patches of Xu's packing of 70 and two upright banks

    Upper bound confirmed superseded by T-186

    PingYou Ltd after Xu, Daniel, Levy · PingYou n302 certificate 2026-10-12 · packet · register

  63. 2026-10-12 published T-165 · case 176

    A tighter certificate at 176 from Couzo, published with no issue

    Upper bound confirmed superseded by T-186

    Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo certificates 2026-10-13 · packet · register

  64. 2026-10-12 published T-166 · case 153

    A tighter rational realization of PingYou Ltd's packing of 153 squares

    Upper bound confirmed superseded by T-173

    PingYou Ltd after Xu, Daniel, Levy · PingYou n153 refinement 2026-10-12 · packet · register

  65. 2026-10-12 published T-167 · case 207

    A tighter rational realization of PingYou Ltd's packing of 207 squares

    Upper bound confirmed superseded by T-173

    PingYou Ltd after Xu, Daniel, Levy · PingYou n207 refinement 2026-10-12 · packet · register

  66. 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

  67. 2026-10-12 published T-169 · case 261

    A further exact refinement of Xu's packing of 261 squares, after Barnett's

    Upper bound confirmed superseded by T-174

    PingYou Ltd after Xu, Barnett, Daniel, Levy · PingYou n261 refinement 2026-10-12 · packet · register

  68. 2026-10-12 published T-170 · case 105

    An exact contact refinement of Xu's packing of 105 squares, after Couzo's

    Upper bound confirmed superseded by T-174

    PingYou Ltd after Xu, Couzo, Daniel, Levy · PingYou n105 refinement 2026-10-12 · packet · register

  69. 2026-10-12 published T-171 · case 272

    An exact contact refinement of SQUISH's packing of 272 squares

    Upper bound confirmed superseded by T-174

    PingYou Ltd after Chaoweeraprasit, Daniel, Levy · PingYou n272 refinement 2026-10-12 · packet · register

  70. 2026-10-12 published T-173 · cases 153, 207

    Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements there

    Upper bound confirmed

    Polkovskyi after Xu, Chaoweeraprasit, Levy · Mishapolk exact certificates 2026-10-12 · packet · register

  71. 2026-10-13 published T-174 this result · 68 of these cases

    Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own

    Upper bound confirmed

    PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · PingYou PR 574 certificates 2026-10-13 · packet · register

  72. 2026-10-13 published T-175 · cases 211, 241, 304

    New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactly

    Upper bound confirmed superseded by T-182, T-186 and T-188

    Ostashev after Nakajima, Daniel, de Winter, Couzo, Levy, Xu · Ostashev merged bands 2026-10-13 · packet · register

  73. 2026-10-13 published T-179 · case 146

    Linear-measure lower bound s(n)≥306/25=12.24 for n=145…149, reported

    Lower bound reviewed

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n145 2026-10-13 · packet · register

  74. 2026-10-13 published T-180 · 68 of these cases

    Closed-form lower bound s(N)≥a(N) for every non-square N≥49, reported

    Lower bound reviewed

    Hosono · wand125 closed form 2026-10-14 · packet · packet · register

  75. 2026-10-13 published T-181 · cases 211, 241, 302

    Exact rational certificates at 211, 241 and 302 from Couzo

    Upper bound confirmed superseded by T-182, T-186 and T-188

    Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy; Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo issue-583 certificates 2026-10-13 · Couzo certificates 2026-10-13 · packet · packet · register

  76. 2026-10-13 published T-182 · cases 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

  77. 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

  78. 2026-10-14 published T-186 · 11 of these cases

    Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there

    Upper bound confirmed

    Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register

  79. 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

  80. 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

  81. 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

  82. 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

  83. 2026-10-16 published T-193 · 16 of these cases

    Exact algebraic sides at sixteen counts from Daniel, of packings the record credits

    Upper bound confirmed

    Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register

  84. 2026-10-16 published T-205 · case 68

    Mixed rectangle-measure lower bound s(68)≥853/100 verified, on the 201-direction net

    Lower bound confirmed

    Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register

  85. 2026-10-16 published T-206 · case 84

    Mixed rectangle-measure lower bound s(84)≥3769/400 verified, on the 201-direction net

    Lower bound confirmed

    Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register

  86. 2026-10-16 published T-208 · case 86

    Mixed rectangle-measure lower bound s(86)≥3811/400 reported, on the 201-direction net

    Lower bound recorded

    Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

  87. 2026-10-16 published T-209 · case 88

    Mixed rectangle-measure lower bound s(88)≥3853/400 reported, on the 201-direction net

    Lower bound recorded

    Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

  88. 2026-10-16 published T-216 · case 108

    Mixed rectangle-measure lower bound s(108)≥531/50 verified, on the 201-direction net

    Lower bound confirmed

    Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register

  89. 2026-10-16 published T-217 · case 110

    Mixed rectangle-measure lower bound s(110)≥1073/100 reported, on the 201-direction net

    Lower bound recorded

    Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

  90. 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

  91. 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

  92. 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

  93. 2026-10-17 published T-226 · case 204

    Couzo's 204 of #670, below Polkovskyi's T-186, since superseded by his own #675 and #678

    Upper bound recorded superseded by T-225

    Couzo after Polkovskyi, Daniel, Levy · Couzo issue 670 2026-10-17 · packet · packet · register

  94. 2026-10-18 published T-223 · case 204

    Couzo's smaller packings at 204 and 260, the two in the corpus of issue 678's 85 bounds

    Upper bound confirmed superseded by T-225

    Couzo after de Winter, Polkovskyi, Daniel, Fang, PingYou Ltd, Levy · Couzo issue 678 2026-10-18 · packet · register

  95. 2026-10-18 published T-228 · case 204

    Couzo's 204 of #675, below his #670, since superseded by his #678

    Upper bound recorded superseded by T-225

    Couzo after Polkovskyi, Daniel, Levy · Couzo issue 675 2026-10-18 · packet · packet · register

  96. 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

  97. 2026-10-20 published T-224 · case 270

    Daniel's smaller n270 packing, another arrangement found from Chaoweeraprasit's, certified exactly

    Upper bound confirmed

    Daniel after Chaoweeraprasit, Levy · Daniel n270 sweep 2026-10-20 · packet · register

  98. 2026-10-20 published T-225 · case 204

    Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there

    Upper bound confirmed

    Daniel after Couzo, de Winter, Polkovskyi, Levy · Daniel regularized lists 2026-10-20 · packet · register

  99. 2026-10-22 published T-229 · cases 105, 204

    Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225

    Upper bound recorded

    Polkovskyi after Couzo, Muglich, Daniel, Xu, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register

Links

On this site
The frontier survey · T-174 in the results table

On GitHub, at main

Register
T-174 in results.yaml, line 22090
Evidence
E-pingyou-574-certificate-report · E-pingyou-574-algebraic-report · E-pingyou-574-exact-feasibility · E-pingyou-574-algebraic-feasibility
Proofs and certificates
certificate facts
Sources
PingYou PR 574 certificates 2026-10-13 (its own site, retained copy)
Source packet
resources/web/pingyou-pr574-certificates-2026-10-13/README.md
Artifacts
14 artifacts and controlsresources/web/pingyou-pr574-certificates-2026-10-13/README.md resources/web/pingyou-pr574-certificates-2026-10-13/acquisition/claims.json resources/web/pingyou-pr574-certificates-2026-10-13/acquisition/algebraic-claims.json campaign/issue-intake/upper-bounds-20261014/undecided-resolved.json devtools/algebraic_witness_reports.py devtools/upper_bound_reports.py resources/web/pingyou-pr574-certificates-2026-10-13/receipts/algebraic-route1-n070.json.xz resources/web/pingyou-pr574-certificates-2026-10-13/receipts/algebraic-route1-n153.json.xz resources/web/pingyou-pr574-certificates-2026-10-13/receipts/algebraic-route1-n207.json.xz resources/web/pingyou-pr574-certificates-2026-10-13/receipts/algebraic-route1-n302.json.xz devtools/check_half_angle_area.py docs/project/reviews/review-2026-10-14-upper-bound-imports-574.md tests/test_upper_bound_reports.py tests/test_check_half_angle_area.py
Case file
each case’s file is linked from its row above