Evan Daniel

Evan Daniel proves lower bounds for packings of unit squares in a square with exact weighted covers, beginning with s(12) ≥ 15680/3951 in August 2026. In September 2026 he added proofs of s(21) = 5, s(32) = 6 and s(45) = 7, and a proof of Bentz’s s(13) = 4 without case analysis; the proofs for 13 and 32 are also checked in full in Lean. His repository’s credits say this work was produced by Claude (Anthropic) in a single session under human direction and that none of it has been peer reviewed. The repository also holds the Square Packing Atlas, an explorer for the record packings, and in October 2026 he added regularized drawings of the record packings for 1 to 324 squares, with exact rational certificates.

Years active
2026

Resources

Repositorysquare-packing

By Daniel

github.com/evand/square-packingExact covers settling n=21, 32, 45 and 60, and s(k² − 3) = k for every k ≥ 6; s(k² − 4) = k awaits its replay here.

36 results (4 at S4, 17 at S3, 14 at S2, 1 at S1) · 24 records (4 at S4, 11 at S3, 9 at S2) · 1 credit (at S3)

WebsiteSquare Packing Atlas

By Daniel

evand.github.io/square-packingEvery record packing drawn beside the proven floor beneath it, with a page of open problems.

0 results

36 results (4 at S4, 17 at S3, 14 at S2, 1 at S1) · 24 records (4 at S4, 11 at S3, 9 at S2) · 51 credits (1 at S4, 26 at S3, 17 at S2, 7 at S1)

Results

Results: 36
  • T-006 (optimality, n = 13): source author, credited contributor — Daniel, in [evand square-packing 2026-09-28]
  • T-049 (lower bound, n = 12): source author, credited contributor — Daniel, in [evand square-packing 2026]
  • T-050 (lower bound, n = 21): source author, credited contributor — Daniel, in [evand square-packing 2026]
  • T-051 (optimality, n = 32): source author, credited contributor — Daniel, in [evand square-packing 2026]
  • T-052 (optimality, n = 21): source author, credited contributor — Daniel, in [evand square-packing 2026-09-28]
  • T-053 (optimality, n = 45): source author, credited contributor — Daniel, in [evand square-packing 2026-09-28] (Registered in: #563)
  • T-062 (optimality, n = 60): source author, credited contributor — Daniel, in [evand square-packing 2026-10-01] (Registered in: #508)
  • T-063 (optimality, n = 61): source author, credited contributor — Daniel, in [evand square-packing 2026-10-01] (Registered in: #563)
  • T-064 (optimality, 13 cases, n = 33 to 321): source author, credited contributor — Daniel, in [evand square-packing 2026-10-01] (Registered in: #508)
  • T-081 (optimality, 14 cases, n = 21 to 320): source author, credited contributor — Daniel, in [evand square-packing 2026-10-03] (Registered in: #508)
  • T-098 (upper bound, 48 cases, n = 68 to 307): source author, credited contributor — Daniel, in [evand exact optima 2026-10-05]
  • T-101 (upper bound, 77 cases, n = 28 to 300): source author, credited contributor — Daniel, in [evand exact optima 2026-10-05]
  • T-119 (upper bound, n = 266, 270, 272): source author, credited contributor — Daniel, in [Daniel new arrangements 2026-10-07] (Registered in: #441)
  • T-120 (upper bound, n = 102): source author, credited contributor — Daniel, in [Daniel exact and local reports 2026] (Registered in: #441)
  • T-121 (upper bound, n = 106): source author, credited contributor — Daniel, in [Daniel exact and local reports 2026] (Registered in: #441)
  • T-122 (upper bound, n = 152): source author, credited contributor — Daniel, in [Daniel exact and local reports 2026] (Registered in: #441)
  • T-123 (upper bound, n = 177): source author, credited contributor — Daniel, in [Daniel exact and local reports 2026] (Registered in: #441)
  • T-124 (restricted optimality, 178 cases, n = 1 to 324): source author, credited contributor — Daniel, in [Daniel exact and local reports 2026] (Registered in: #441)
  • T-129 (upper bound, n = 105, 130, 292): source author, credited contributor — Daniel, in [Daniel dated certificates 105 and 130 2026-10-07]; [Daniel dated certificate 292 2026-10-07] (Registered in: #487)
  • T-131 (upper bound, n = 132): source author, credited contributor — Daniel, in [Daniel record hunt 2026-10-09] (Registered in: #487 · #478)
  • T-139 (upper bound, n = 126): source author, credited contributor — Daniel, in [Daniel trio126 2026-10-09] (Registered in: #487)
  • T-141 (upper bound, n = 132, 308): source author, credited contributor — Daniel, in [Daniel record hunt 3 2026-10-10] (Registered in: #487)
  • T-146 (upper bound, 15 cases, n = 70 to 303): source author, credited contributor — Daniel, in [Daniel regularized lists 2026-10-10] (Registered in: #487)
  • T-160 (upper bound, n = 154): source author, credited contributor — Daniel, in [Daniel n154 2026-10-12] (Registered in: #598)
  • T-187 (upper bound, n = 146): source author, credited contributor — Daniel, in [Daniel n146 2026-10-14]
  • T-193 (upper bound, 16 cases, n = 88 to 302): source author, credited contributor — Daniel, in [Daniel exact current records 2026-10-16]
  • T-197 (upper bound, n = 155): source author, credited contributor — Daniel, in [Daniel n155 explorer 2026-10-17]
  • T-198 (upper bound, n = 229): source author, credited contributor — Daniel, in [Daniel n229 dilation 2026-10-17]
  • T-199 (upper bound, n = 229): source author, credited contributor — Daniel, in [Daniel n229 dilation update 2026-10-18]
  • T-218 (upper bound, n = 236): source author, credited contributor — Daniel, in [Daniel n236 2026-10-16]
  • T-219 (upper bound, n = 271): source author, credited contributor — Daniel, in [Daniel n271 2026-10-17]
  • T-220 (upper bound, n = 183): source author, credited contributor — Daniel, in [Daniel n183 subgrid 2026-10-19] (Registered in: #687; Supporting: #680)
  • T-221 (upper bound, n = 308): source author, credited contributor — Daniel, in [Daniel n308 2026-10-19]
  • T-222 (upper bound, n = 154): source author, credited contributor — Daniel, in [Daniel n154 sweep 2026-10-20]
  • T-224 (upper bound, n = 270): source author, credited contributor — Daniel, in [Daniel n270 sweep 2026-10-20]
  • T-225 (upper bound, n = 204, 227, 229, 260): source author, credited contributor — Daniel, in [Daniel regularized lists 2026-10-20]

Records

Current registered contributions in their recorded family and scope. A result counts once when any of its facets remains current.

Records: 24
  • T-006 (optimality, n = 13): source author, credited contributor — Daniel, in [evand square-packing 2026-09-28]
  • T-051 (optimality, n = 32): source author, credited contributor — Daniel, in [evand square-packing 2026]
  • T-052 (optimality, n = 21): source author, credited contributor — Daniel, in [evand square-packing 2026-09-28]
  • T-053 (optimality, n = 45): source author, credited contributor — Daniel, in [evand square-packing 2026-09-28] (Registered in: #563)
  • T-062 (optimality, n = 60): source author, credited contributor — Daniel, in [evand square-packing 2026-10-01] (Registered in: #508)
  • T-063 (optimality, n = 61): source author, credited contributor — Daniel, in [evand square-packing 2026-10-01] (Registered in: #563)
  • T-064 (optimality, 13 cases, n = 33 to 321): source author, credited contributor — Daniel, in [evand square-packing 2026-10-01] (Registered in: #508)
  • T-081 (optimality, 14 cases, n = 21 to 320): source author, credited contributor — Daniel, in [evand square-packing 2026-10-03] (Registered in: #508)
  • T-098 (upper bound, 48 cases, n = 68 to 307): source author, credited contributor — Daniel, in [evand exact optima 2026-10-05]
  • T-101 (upper bound, 77 cases, n = 28 to 300): source author, credited contributor — Daniel, in [evand exact optima 2026-10-05]
  • T-121 (upper bound, n = 106): source author, credited contributor — Daniel, in [Daniel exact and local reports 2026] (Registered in: #441)
  • T-122 (upper bound, n = 152): source author, credited contributor — Daniel, in [Daniel exact and local reports 2026] (Registered in: #441)
  • T-123 (upper bound, n = 177): source author, credited contributor — Daniel, in [Daniel exact and local reports 2026] (Registered in: #441)
  • T-124 (restricted optimality, 178 cases, n = 1 to 324): source author, credited contributor — Daniel, in [Daniel exact and local reports 2026] (Registered in: #441)
  • T-187 (upper bound, n = 146): source author, credited contributor — Daniel, in [Daniel n146 2026-10-14]
  • T-193 (upper bound, 16 cases, n = 88 to 302): source author, credited contributor — Daniel, in [Daniel exact current records 2026-10-16]
  • T-197 (upper bound, n = 155): source author, credited contributor — Daniel, in [Daniel n155 explorer 2026-10-17]
  • T-218 (upper bound, n = 236): source author, credited contributor — Daniel, in [Daniel n236 2026-10-16]
  • T-219 (upper bound, n = 271): source author, credited contributor — Daniel, in [Daniel n271 2026-10-17]
  • T-220 (upper bound, n = 183): source author, credited contributor — Daniel, in [Daniel n183 subgrid 2026-10-19] (Registered in: #687; Supporting: #680)
  • T-221 (upper bound, n = 308): source author, credited contributor — Daniel, in [Daniel n308 2026-10-19]
  • T-222 (upper bound, n = 154): source author, credited contributor — Daniel, in [Daniel n154 sweep 2026-10-20]
  • T-224 (upper bound, n = 270): source author, credited contributor — Daniel, in [Daniel n270 sweep 2026-10-20]
  • T-225 (upper bound, n = 204, 227, 229, 260): source author, credited contributor — Daniel, in [Daniel regularized lists 2026-10-20]

Credited work

Credits: 51
  • T-048 (lower bound, n = 50, 51): prerequisite — Daniel, in [wand125 point and mixed bounds 2026-09-28]
  • T-054 (simplification, n = 45): prerequisite — Daniel, in [wand125 point and mixed bounds 2026-09-28]
  • T-055 (simplification, n = 21): prerequisite — Daniel, in [wand125 point and mixed bounds 2026-09-28]
  • T-058 (method limit, 100 cases, n = 1 to 100): prerequisite — Daniel, in [wand125 tools 2026]
  • T-059 (audit, n = 11): prerequisite — Daniel, in [wand125 tools 2026]
  • T-066 (optimality, n = 59): prerequisite — Daniel, in [wand125 exact covers 2026-10-01]
  • T-067 (optimality, n = 77, 78): prerequisite — Daniel, in [wand125 exact covers 2026-10-01]
  • T-078 (lower bound, n = 12): prerequisite — Daniel, in [squarepacker s12 2026]
  • T-079 (lower bound, n = 12): prerequisite — Daniel
  • T-095 (lower bound, n = 12): prerequisite — Daniel, in [squarepacker s12 2026-10-05]
  • T-127 (upper bound, 17 cases, n = 88 to 239): prerequisite — Daniel, in [Gupta rational refinements 2026-10-08] (Registered in: #448 · #459)
  • T-128 (upper bound, 8 cases, n = 105 to 306): prerequisite — Daniel, in [Couzo exact refinements 2026-10-08] (Registered in: #487 · #478)
  • T-130 (upper bound, 5 cases, n = 84 to 270): prerequisite — Daniel, in [Couzo follow-up refinements 2026-10-08] (Registered in: #487)
  • T-134 (upper bound, 6 cases, n = 132 to 303): prerequisite — Daniel, in [Couzo exact certificates 2026-10-09] (Registered in: #487)
  • T-140 (upper bound, 6 cases, n = 132 to 305): prerequisite — Daniel, in [Couzo certificates 2026-10-10] (Registered in: #487)
  • T-147 (upper bound, n = 207): prerequisite — Daniel, in [PingYou n207 certificate 2026-10-11] (Registered in: #558 · #512 · #516 · #518)
  • T-148 (upper bound, n = 153): prerequisite — Daniel, in [PingYou n153 certificate 2026-10-11] (Registered in: #558 · #512 · #516 · #518)
  • T-150 (upper bound, n = 146, 261): prerequisite — Daniel, in [Barnett refinements 2026-10-11] (Registered in: #512 · #516 · #518)
  • T-152 (upper bound, n = 304, 305): prerequisite — Daniel, in [Couzo certificates 2026-10-11] (Registered in: #553)
  • T-155 (upper bound, 59 cases, n = 28 to 300): prerequisite — Daniel (Registered in: #511 · #530 · #548 · #549 · #559 · #580 · #585 · #587 · #591)
  • T-156 (upper bound, n = 290): prerequisite — Daniel (Registered in: #511 · #530 · #548 · #549 · #559 · #580 · #585 · #587 · #591)
  • T-158 (upper bound, n = 175, 232, 264): prerequisite — Daniel, in [Ostashev records 2026-10-12] (Registered in: #598)
  • T-161 (upper bound, n = 156, 176, 237, 271): prerequisite — Daniel, in [Couzo certificates 2026-10-12] (Registered in: #614 · #602)
  • T-162 (upper bound, n = 129, 204, 303): prerequisite — Daniel, in [Ostashev compositions 2026-10-12] (Registered in: #614 · #602)
  • T-163 (upper bound, n = 211, 241, 273, 307): prerequisite — Daniel, in [Fang excess theorem 2026-10-13] (Registered in: #602)
  • T-164 (upper bound, n = 302): prerequisite — Daniel, in [PingYou n302 certificate 2026-10-12] (Registered in: #614 · #602)
  • T-165 (upper bound, n = 176): prerequisite — Daniel, in [Couzo certificates 2026-10-13] (Registered in: #614 · #602)
  • T-166 (upper bound, n = 153): prerequisite — Daniel, in [PingYou n153 refinement 2026-10-12] (Registered in: #614 · #602)
  • T-167 (upper bound, n = 207): prerequisite — Daniel, in [PingYou n207 refinement 2026-10-12] (Registered in: #614 · #602)
  • T-168 (upper bound, n = 146): prerequisite — Daniel, in [PingYou n146 refinement 2026-10-12] (Registered in: #614 · #602)
  • T-169 (upper bound, n = 261): prerequisite — Daniel, in [PingYou n261 refinement 2026-10-12] (Registered in: #614 · #602)
  • T-170 (upper bound, n = 105): prerequisite — Daniel, in [PingYou n105 refinement 2026-10-12] (Registered in: #614 · #602)
  • T-171 (upper bound, n = 272): prerequisite — Daniel, in [PingYou n272 refinement 2026-10-12] (Registered in: #614 · #602)
  • T-172 (upper bound, n = 154): prerequisite — Daniel, in [Couzo issue-583 certificates 2026-10-13] (Registered in: #614 · #602)
  • T-174 (upper bound, 68 cases, n = 68 to 308): prerequisite — Daniel, in [PingYou PR 574 certificates 2026-10-13] (Registered in: #614 · #602)
  • T-175 (upper bound, n = 211, 241, 304): prerequisite — Daniel, in [Ostashev merged bands 2026-10-13] (Registered in: #614 · #602)
  • T-181 (upper bound, n = 211, 241, 302): prerequisite — Daniel, in [Couzo certificates 2026-10-13]; [Couzo issue-583 certificates 2026-10-13] (Registered in: #614 · #602)
  • T-182 (upper bound, n = 154, 211, 241): prerequisite — Daniel, in [Couzo certificates 2026-10-13] (Registered in: #614 · #602)
  • T-183 (upper bound, n = 237): prerequisite — Daniel, in [Mishapolk records 2026-10-14] (Registered in: #614 · #602)
  • T-184 (optimality, n = 76): prerequisite — Daniel, in [bengreff s76 certificate 2026-10-14] (Registered in: #602)
  • T-185 (upper bound, n = 175): prerequisite — Daniel, in [Couzo n175 2026-10-11]
  • T-188 (upper bound, n = 211): prerequisite — Daniel, in [Couzo verified bounds 2026-10-14]
  • T-189 (upper bound, n = 237): prerequisite — Daniel, in [Couzo certificates 2026-10-14]
  • T-190 (upper bound, n = 237): prerequisite — Daniel, in [Couzo n237 2026-10-15]
  • T-191 (optimality, n = 95): prerequisite — Daniel, in [bengreff s95 certificate 2026-10-15] (Registered in: #627; Supporting: #610)
  • T-195 (upper bound, n = 29): prerequisite — Daniel, in [PingYou PR 622 certificate 2026-10-15]
  • T-223 (upper bound, n = 204, 260): prerequisite — Daniel, in [Couzo issue 678 2026-10-18] (Registered in: #706; Supporting: #678)
  • T-226 (upper bound, n = 204): prerequisite — Daniel, in [Couzo issue 670 2026-10-17]
  • T-227 (upper bound, n = 227, 229): prerequisite — Daniel, in [Couzo issue 672 2026-10-17]
  • T-228 (upper bound, n = 204): prerequisite — Daniel, in [Couzo issue 675 2026-10-18]
  • T-229 (upper bound, n = 105, 204): prerequisite — Daniel, in [Mishapolk issue 719 2026-10-22]

Other reviewed records

Reviewed records: 82

Related News

Sources

Biography sources: intake-credit, repository-writeups, repository-credits, regularized-lists

  1. repository-license: Copyright (c) 2026 Evan Daniel.
  2. intake-credit: Title, Provenance table, and Credit, as the source states it.
  3. repository-writeups: Readable write-ups for s(11), s(12), s(13), s(21), s(32) and s(45) and the Square Packing Atlas link in the opening paragraphs; the Results list, with each certificate’s date and Lean status.
  4. repository-credits: Provenance of this work: “Produced by Claude (Anthropic) in a single session under human direction” and “Nothing here has been peer reviewed.”
  5. regularized-lists: Opening paragraph: display-regularized drawings of the record packings for n = 1 to 324, added in 346c6ed (2026-10-10T18:05:38Z), with exact rational certificates under search/regularize/lists/certs/.
  6. bibliography: credited_names maps Evan Daniel to Daniel.

Identity evidence