T-124: Reported non-strict local minima for 178 source configurations

V0 C0 Restricted optimality recorded

2026-10-07 published · Daniel after Couzo · 178 cases, n=1 to 324

Daniel reports that 178 configurations are non-strict local minima under the source matched-index perturbation convention. The retained catalogue names 176 integer-band cases and the separate n11 and n28 cases, with angles taken modulo 90 degrees and plateaux allowed. This assertion is restricted to those source configurations and their local neighborhoods; no global bound or rigidity is claimed here.

Significance, composition and next rung
Significance
Reported local structure for a named configuration roster; no global optimum or new feasible bound. A non-strict local minimum under a stated perturbation convention is a property of each packing short of rigidity: T-012 to T-014 (S3) prove rigidity and its failure at single famous cases with machinery the corpus lacked, and T-036 and T-112 (S3) are optimality statements; this is weaker than either at every count, and reported, so S2. Confirmed on 11 October.
Next rung
Audit the retained theorem, embedded configuration and perturbation-definition bindings and pinned dependencies, acquire any inputs that audit finds missing, and independently build the pinned Lean proof before raising either rung.
Unfinished confirmations
C3: an audit of the retained theorem, embedded configuration and perturbation-definition bindings and pinned dependencies, and an independent build here of the pinned Lean proof. Not priced.
Novelty
previously-published Present in an identified source

The cases

This result concerns 178 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
1110provedO=Rfrontier n-001.md
222provedO=frontier n-002.md
3frontier n-003.md
4provedO=Rfrontier n-004.md
633provedO=frontier n-006.md
73provedO=frontier n-007.md
8frontier n-008.md
93provedO=Rfrontier n-009.md
113.8770843.877084provedO=Rfrontier n-011.md
123.97150040.0285open=frontier n-012.md
1340provedO=frontier n-013.md
144provedO=frontier n-014.md
15frontier n-015.md
164provedO=Rfrontier n-016.md
204.90500050.095open=frontier n-020.md
2150provedO=frontier n-021.md
225provedO=frontier n-022.md
235provedO=frontier n-023.md
24frontier n-024.md
255provedO=Rfrontier n-025.md
285.7350005.8244450.08944461…open=Rfrontier n-028.md
305.88350060.1165open=frontier n-030.md
315.9525000.0475frontier n-031.md
3260provedO=frontier n-032.md
336provedO=frontier n-033.md
346provedO=frontier n-034.md
35frontier n-035.md
366provedO=Rfrontier n-036.md
426.84750070.1525open=frontier n-042.md
436.9075000.0925frontier n-043.md
446.9725000.0275frontier n-044.md
4570provedO=frontier n-045.md
467provedO=frontier n-046.md
477provedO=frontier n-047.md
48frontier n-048.md
497provedO=Rfrontier n-049.md
567.80250080.1975open=frontier n-056.md
577.8725000.1275frontier n-057.md
587.9350000.065frontier n-058.md
5980provedO=frontier n-059.md
60frontier n-060.md
61frontier n-061.md
62frontier n-062.md
63frontier n-063.md
648provedO=Rfrontier n-064.md
728.76000090.24open=frontier n-072.md
738.8130000.187frontier n-073.md
748.8675000.1325frontier n-074.md
758.9400000.06frontier n-075.md
768.9650000.035frontier n-076.md
7790provedO=frontier n-077.md
78frontier n-078.md
79frontier n-079.md
80frontier n-080.md
819provedO=Rfrontier n-081.md
909.730000100.27open=frontier n-090.md
919.7625000.2375frontier n-091.md
929.7700000.23frontier n-092.md
939.8800000.12frontier n-093.md
949.9500000.05frontier n-094.md
959.9650000.035frontier n-095.md
969.9700000.03frontier n-096.md
97100provedO=frontier n-097.md
98frontier n-098.md
99frontier n-099.md
10010provedO=Rfrontier n-100.md
11110.562305110.43769410…open=frontier n-111.md
11210.6118740.38812579…frontier n-112.md
11310.6612000.33879928…frontier n-113.md
11410.7102880.28971107…frontier n-114.md
11510.7591420.24085773…frontier n-115.md
11610.8077640.19223593…frontier n-116.md
11710.8561570.14384241…frontier n-117.md
118110provedO=frontier n-118.md
119frontier n-119.md
120frontier n-120.md
12111provedO=Rfrontier n-121.md
13311.556672120.44332780…open=frontier n-133.md
13411.6018010.39819834…frontier n-134.md
13511.6467480.35325159…frontier n-135.md
13611.6915140.30848535…frontier n-136.md
13711.7361020.26389747…frontier n-137.md
13811.7805140.21948582…frontier n-138.md
13911.8247510.17524834…frontier n-139.md
14011.8688170.13118299…frontier n-140.md
141120provedO=frontier n-141.md
142frontier n-142.md
143frontier n-143.md
14412provedO=Rfrontier n-144.md
15712.551970130.44802920…open=frontier n-157.md
15812.5933860.40661337…frontier n-158.md
15912.6346610.36533890…frontier n-159.md
16012.6757950.32420433…frontier n-160.md
16112.7167910.28320827…frontier n-161.md
16212.7576500.24234932…frontier n-162.md
16312.7983730.20162612…frontier n-163.md
16412.8389620.16103732…frontier n-164.md
16512.8794180.12058159…frontier n-165.md
166130provedO=frontier n-166.md
167frontier n-167.md
168frontier n-168.md
16913provedO=Rfrontier n-169.md
18313.54798813.9995810.45159182…open=frontier n-183.md
18413.586252140.41374767…frontier n-184.md
18513.6244040.37559525…frontier n-185.md
18613.6624460.33755341…frontier n-186.md
18713.7003780.29962121…frontier n-187.md
18813.7382020.26179770…frontier n-188.md
18913.7759180.22408195…frontier n-189.md
19013.8135260.18647304…frontier n-190.md
19113.8510290.14897007…frontier n-191.md
19213.8884270.11157216…frontier n-192.md
193140provedO=frontier n-193.md
194frontier n-194.md
195frontier n-195.md
19614provedO=Rfrontier n-196.md
21214.580127150.41987215…open=frontier n-212.md
21314.6155940.38440578…frontier n-213.md
21414.6509710.34902830…frontier n-214.md
21514.6862600.31373904…frontier n-215.md
21614.7214620.27853734…frontier n-216.md
21714.7565770.24342257…frontier n-217.md
21814.7916050.20839407…frontier n-218.md
21914.8265480.17345121…frontier n-219.md
22014.8614060.13859338…frontier n-220.md
22114.8961800.10381995…frontier n-221.md
222150provedO=frontier n-222.md
223frontier n-223.md
224frontier n-224.md
22515provedO=Rfrontier n-225.md
24215.574813160.42518656…open=frontier n-242.md
24315.6079440.39205507…frontier n-243.md
24415.6410030.35899607…frontier n-244.md
24515.6739900.32600909…frontier n-245.md
24615.7069060.29309367…frontier n-246.md
24715.7397500.26024934…frontier n-247.md
24815.7725240.22747565…frontier n-248.md
24915.8052270.19477213…frontier n-249.md
25015.8378610.16213834…frontier n-250.md
25115.8704260.12957385…frontier n-251.md
25215.9029210.09707819…frontier n-252.md
253160provedO=frontier n-253.md
254frontier n-254.md
255frontier n-255.md
25616provedO=Rfrontier n-256.md
27416.570158170.42984132…open=frontier n-274.md
27516.6012420.39875781…frontier n-275.md
27616.6322650.36773419…frontier n-276.md
27716.6632290.33677012…frontier n-277.md
27816.6941340.30586525…frontier n-278.md
27916.7249800.27501926…frontier n-279.md
28016.7557680.24423179…frontier n-280.md
28116.7864970.21350252…frontier n-281.md
28216.8171680.18283112…frontier n-282.md
28316.8477820.15221727…frontier n-283.md
28416.8783390.12166064…frontier n-284.md
28516.9088390.09116091…frontier n-285.md
286170provedO=frontier n-286.md
287frontier n-287.md
288frontier n-288.md
28917provedO=Rfrontier n-289.md
30817.56604817.9971500.43110095…open=frontier n-308.md
30917.595320180.40467900…frontier n-309.md
31017.6245430.37545621…frontier n-310.md
31117.6537160.34628320…frontier n-311.md
31217.6828400.31715972…frontier n-312.md
31317.7119140.28808552…frontier n-313.md
31417.7409390.25906035…frontier n-314.md
31517.7699160.23008396…frontier n-315.md
31617.7988430.20115610…frontier n-316.md
31717.8277230.17227654…frontier n-317.md
31817.8565540.14344504…frontier n-318.md
31917.8853380.11466134…frontier n-319.md
32017.9140740.08592523…frontier n-320.md
321180provedO=frontier n-321.md
322frontier n-322.md
323frontier n-323.md
32418provedO=Rfrontier n-324.md

Results on these cases

84 results in the register on these cases, oldest first
  1. 1979 published T-011 · case 11

    Trump's 1979 packing is exactly valid, so s(11)≤3.877083590022814…

    Upper bound confirmed

    Trump · Trump 2023 · register

  2. 2005 published T-007 · 175 of these cases

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

    Lower bound incomplete

    Nagamochi · Nagamochi 2005 · source · register

  3. 2010-09-13 published T-004 · case 46

    Bentz 2010, Theorem 8 (s(46)≥7) is correct as printed, machine-audited in full

    Audit confirmed

    Bentz · Bentz 2010 · source · register

  4. 2010-09-13 published T-006 · case 13

    s(13)=4

    Optimality confirmed

    Bentz; Daniel after Burns, Massaccesi · Bentz 2010 · evand square-packing 2026-09-28 · packet · source 1 · source 2 · review · register

  5. 2010-09-13 published T-008 · case 46

    s(46)=7

    Optimality confirmed

    Bentz · Bentz 2010 · source · register

  6. 2018 published T-087 · case 61

    s(37)≥53/2+22−1 and s(61)≥73/2+22−1, by optimal piercing

    Lower bound reviewed superseded by T-063

    Bašić, Slivková · Basic-Slivkova 2018 · source · register

  7. 2026-08-24 established T-010 · case 11

    s(11)≥2+4/5, by a repair of Stromquist 2003's Figure 14 point set

    Lower bound confirmed superseded by T-060

    Levy after Stromquist · register

  8. 2026-08-25 published T-049 · case 12

    s(12)≥15680/3951=3.9686155…

    Lower bound confirmed superseded by T-095

    Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · source 1 · source 2 · review 1 · review 2 · register

  9. 2026-08-31 established T-005 · case 13

    Bentz 2010, Lemma 10 is false as printed and true as corrected to (1.74,1)

    Correction confirmed

    Levy after Bentz · source · register

  10. 2026-09-04 established T-017 · case 12

    s(12)≥99/25=3.96

    Lower bound confirmed superseded by T-095

    Levy after Burns, Massaccesi · register

  11. 2026-09-04 established T-018 · case 11

    s(11)≥381/100=3.81

    Lower bound confirmed superseded by T-060

    Levy after Burns, Massaccesi · register

  12. 2026-09-04 established T-020 · cases 20, 21

    s(n)≥24/5=4.80 for n=19,20,21

    Lower bound confirmed superseded by T-052 and T-104

    Levy after Burns, Massaccesi · register

  13. 2026-09-04 published T-085 · 170 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

  14. 2026-09-04 published T-086 · 16 of these cases

    s(k2−2)=k for every integer k≥2; k=3…18 are the cases held here

    Optimality confirmed

    Vlasenko · chelokot Nagamochi counterexample 2026 · packet · source · register

  15. 2026-09-05 established T-021 · cases 20, 21

    s(n)≥97/20=4.85 for n=20,21

    Lower bound confirmed superseded by T-052 and T-104

    Levy after Burns, Massaccesi · register

  16. 2026-09-06 established T-022 · case 11

    s(11)≥381008100042893309449/899996306539=3.8100257…

    Lower bound confirmed superseded by T-060

    Levy after Burns, Massaccesi · source · register

  17. 2026-09-08 established T-023 · case 11

    Conditional exclusion: no eleven-square packing in the four-owner branch at q=96/25

    Case exclusion confirmed superseded in part by T-060

    Levy · source · register

  18. 2026-09-09 established T-024 · case 11

    s(11)≥3175000518400042893309449/598960960743657=3.8166095…

    Lower bound confirmed superseded by T-060

    Levy after Burns, Massaccesi · source · register

  19. 2026-09-09 established T-025 · case 11

    s(11)≥191/50=3.82, by a threshold certificate

    Lower bound confirmed superseded by T-060

    Levy · source · register

  20. 2026-09-09 established T-026 · case 11

    s(11)≥955000518400042893309449/179696714646249=3.8264474…

    Lower bound confirmed superseded by T-060

    Levy · source · register

  21. 2026-09-20 established T-031 · case 11

    The octagon corner class (threshold 1/2) holds no eleven-square packing at side 96/25

    Case exclusion confirmed superseded by T-060

    Levy · register

  22. 2026-09-22 established T-033 · case 11

    s(11)≥9550002073600042893309449/359341754646249=3.8269975…

    Lower bound confirmed superseded by T-060

    Levy · source · register

  23. 2026-09-22 published T-037 · case 11

    s(11)>31/8=3.875

    Lower bound confirmed superseded by T-060

    Overbeck after Levy, Guzhou0806, Mira-acc · Kleddamag n11 2026 · packet · source 1 · source 2 · review 1 · review 2 · register

  24. 2026-09-22 published T-044 · cases 56, 57, 72, 73

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

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

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

  25. 2026-09-22 published T-047 · cases 11, 28, 30, 31

    s(11)≥381/100; s(n)≥1377/250 for n=26…28; s(n)≥571/100 for n=29…31

    Lower bound confirmed superseded by T-060, T-068, T-074, T-107 and T-109

    Tokoharu after Levy, Hosono, Stromquist, Nagamochi, Burns, Massaccesi · Tokoharu density 2026 · packet · source · review · register

  26. 2026-09-23 established T-034 · case 21

    s(21)≥122/25=4.88

    Lower bound confirmed superseded by T-052

    Levy after Burns, Massaccesi · register

  27. 2026-09-23 published T-050 · case 21

    s(21)≥5000/1001=4.995004995…

    Lower bound confirmed superseded by T-052

    Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · source · review · register

  28. 2026-09-24 established T-035 · case 11

    Six-plus-five packings near Trump's tilt with side ≤Uhi lie within rho of his pose

    Case exclusion confirmed

    Levy · register

  29. 2026-09-24 established T-036 · case 11

    Trump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetry

    Restricted optimality confirmed superseded in part by T-060 and T-112

    Levy · register

  30. 2026-09-26 published T-051 · case 32

    s(32)=6

    Optimality confirmed

    Daniel after Burns, Massaccesi · evand square-packing 2026 · packet · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · review 3 · register

  31. 2026-09-27 published T-045 · 10 of these cases

    Rectangle-density lower bounds replayed at 15 counts in n=18…78

    Lower bound confirmed superseded by T-051, T-063, T-064, T-067, T-068, T-074, T-090, T-091, T-104, T-107 and T-109

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

  32. 2026-09-27 published T-046 · 24 of these cases

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

    Lower bound recorded superseded by T-062, T-063, T-064, T-066, T-067, T-068, T-074, T-082, T-090, T-091, T-104, T-107 and T-109

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

  33. 2026-09-27 published T-052 · case 21

    s(21)=5, by a mixed cover of points and grid-line segments

    Optimality confirmed

    Daniel after Burns, Massaccesi · evand square-packing 2026-09-28 · packet · source · review · register

  34. 2026-09-27 published T-053 · case 45

    s(45)=7, by a mixed cover of points and grid-line segments

    Optimality confirmed

    Daniel after Burns, Massaccesi · evand square-packing 2026-09-28 · packet · source · review · register

  35. 2026-09-28 published T-054 · case 45

    s(45)=7 by a second, point-only route

    Simplification confirmed

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

  36. 2026-09-28 published T-055 · case 21

    s(21)=5 by a point-only route

    Simplification confirmed

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

  37. 2026-09-28 published T-062 · case 60

    s(60)=8, by a mixed cover of points and grid-line segments

    Optimality confirmed

    Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · source · review · register

  38. 2026-09-28 published T-063 · case 61

    s(61)=8, by monotonicity from T-062

    Optimality confirmed

    Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register

  39. 2026-09-28 published T-070 · 10 of these cases

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

    Lower bound confirmed superseded by T-062, T-066, T-082, T-090 and T-091

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

  40. 2026-09-29 published T-058 · 66 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

  41. 2026-09-29 published T-059 · case 11

    Reported equality of 12028 n11 row minima reproduced by a complete bound replay

    Audit confirmed

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

  42. 2026-09-29 published T-060 · case 11

    Trump's eleven-square packing is globally optimal

    Optimality confirmed

    Ahmed after Levy, Overbeck · Ahmed n11 optimality 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

  43. 2026-09-29 published T-061 · case 11

    s(11)>3875000000/999999999=3.875000003875…, 3.9e-9 above 31/8

    Lower bound confirmed superseded by T-060

    Wang, Li after Overbeck, Levy · Wang Li n11 2026 · packet · source 1 · source 2 · review · register

  44. 2026-09-29 published T-083 · 156 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

  45. 2026-09-29 published T-084 · 16 of these cases

    s(k2−1)=k for every integer k≥3; k=3…18 are the cases held here

    Optimality confirmed

    Karakuş · Karakuş 2026 · source · register

  46. 2026-09-30 published T-064 · 13 of these cases

    s(k2−3)=k for every integer k from 6 up; k=6…18 are the cases held here

    Optimality confirmed

    Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register

  47. 2026-10-01 published T-066 · case 59

    s(59)=8, by a mixed cover of points and grid-line segments

    Optimality confirmed

    Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register

  48. 2026-10-01 published T-067 · cases 77, 78

    s(77)=9, by a mixed cover extended from Daniel's s(60) cover

    Optimality confirmed

    Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register

  49. 2026-10-01 published T-068 · 15 of these cases

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

    Lower bound confirmed

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

  50. 2026-10-01 published T-069 · cases 90, 92

    Mixed rectangle-measure lower bounds replayed at n=37,65,66,90,92

    Lower bound confirmed on these cases, superseded by T-082 and T-091

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

  51. 2026-10-01 published T-074 · 15 of these cases

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

    Lower bound confirmed

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

  52. 2026-10-02 published T-072 · case 76

    Mixed rectangle-measure lower bound replayed at n=76

    Lower bound confirmed superseded by T-091

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds n76 2026-10-02 · packet · packet · source · review · register

  53. 2026-10-02 published T-075 · cases 91, 92, 93, 96

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

    Lower bound confirmed on these cases, superseded by T-081 (reported), T-082, 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

  54. 2026-10-02 published T-077 · cases 20, 42

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

    Lower bound confirmed superseded by T-090 and T-104

    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

  55. 2026-10-02 published T-078 · case 12

    s(12)≥31360/7901=3.9691178…, Daniel's certificate rescaled by 7902/7901

    Lower bound confirmed superseded by T-095

    Ryu after Daniel · squarepacker s12 2026 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

  56. 2026-10-02 established T-079 · case 12

    s(12)≥15680000/3949423=3.9702002…, Daniel's points re-weighted

    Lower bound confirmed superseded by T-095

    Levy after Daniel · source 1 · source 2 · review · register

  57. 2026-10-03 published T-081 · 14 of these cases

    s(k2−4)=k for every integer k from 5 up; k=5…18 are the cases held here

    Optimality reviewed

    Daniel after Burns, Massaccesi · evand square-packing 2026-10-03 · packet · packet · source · review · register

  58. 2026-10-03 published T-082 · 12 of these cases

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

    Lower bound confirmed

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

  59. 2026-10-04 published T-090 · 11 of these cases

    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

  60. 2026-10-04 published T-091 · 10 of these cases

    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

  61. 2026-10-05 published T-095 · case 12

    s(12)≥7943/2000=3.9715, Daniel's points dilated and re-weighted

    Lower bound confirmed

    Ryu after Daniel, Levy · squarepacker s12 2026-10-05 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

  62. 2026-10-05 published T-101 · case 28

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

    Upper bound confirmed

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

  63. 2026-10-06 published T-104 · case 20

    Mixed rectangle-measure lower bound verified at n=20, on a declared net

    Lower bound confirmed

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register

  64. 2026-10-06 published T-107 · case 28

    Mixed rectangle-measure lower bound verified at n=28, on a declared net

    Lower bound confirmed

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register

  65. 2026-10-06 published T-109 · case 30

    Mixed rectangle-measure lower bound verified at n=30, on a declared net

    Lower bound confirmed

    Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds check2 2026-10-06 · packet · source · review · register

  66. 2026-10-06 established T-112 · case 11

    Trump's packing is the only optimal packing of eleven squares, up to symmetry

    Uniqueness confirmed

    Levy after Ahmed · packet · register

  67. 2026-10-07 published T-124 this result · 178 of these cases

    Reported non-strict local minima for 178 source configurations

    Restricted optimality recorded

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

  68. 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

  69. 2026-10-10 published T-141 · case 308

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

    Upper bound confirmed superseded by T-221

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

  70. 2026-10-10 published T-142 · case 28

    Mixed rectangle-measure lower bound s(28)≥2297/400 reported, on a 2,073-direction declared net

    Lower bound recorded

    Hosono after Tokoharu and Levy · wand125 fine-net n28 n30 2026-10-10 · packet · register

  71. 2026-10-10 published T-143 · case 30

    Mixed rectangle-measure lower bound s(30)≥2357/400 reported, on a 2,073-direction declared net

    Lower bound recorded

    Hosono after Tokoharu and Levy · wand125 fine-net n28 n30 2026-10-10 · packet · register

  72. 2026-10-12 established T-155 · case 28

    Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace

    Upper bound confirmed

    Levy after Daniel · register

  73. 2026-10-13 published T-174 · case 308

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

    Upper bound confirmed on these cases, superseded by T-221

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

  74. 2026-10-13 published T-177 · case 44

    Mixed rectangle-measure lower bound s(44)≥2799/400 reported, on a 2,073-direction declared net

    Lower bound reviewed

    Hosono after Tokoharu and Levy · wand125 fine-net n37 n39 n44 2026-10-13 · packet · register

  75. 2026-10-13 published T-180 · 131 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

  76. 2026-10-14 published T-184 · case 76

    s(76)=9, by a mixed cover of points and grid-line segments

    Optimality reviewed

    Greff after Daniel, Hosono · bengreff s76 certificate 2026-10-14 · packet · source · review · register

  77. 2026-10-15 published T-191 · case 95

    s(95)=10, by a mixed cover of points and grid-line segments

    Optimality reviewed

    Greff after Daniel, Hosono · bengreff s95 certificate 2026-10-15 · packet · source · review · register

  78. 2026-10-15 published T-192 · case 28

    Continuous-pose lower bound s(28)>1149/200 reported, by two of the source's programs

    Lower bound recorded

    Hosono after Guzhou0806, Tokoharu, Levy · wand125 continuous n28 2026-10-15 · packet · register

  79. 2026-10-16 published T-203 · case 43

    Mixed rectangle-measure lower bound s(43)≥2777/400 reported, on a 2,073-direction declared net

    Lower bound recorded

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

  80. 2026-10-16 published T-211 · case 91

    Mixed rectangle-measure lower bound s(91)≥783/80 reported, on the 201-direction net

    Lower bound recorded

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

  81. 2026-10-16 published T-212 · case 90

    Mixed rectangle-measure lower bound s(90)≥489/50 reported, on the 201-direction net

    Lower bound recorded

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

  82. 2026-10-16 published T-213 · case 92

    Mixed rectangle-measure lower bound s(92)≥493/50 reported, on the 201-direction net

    Lower bound recorded

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

  83. 2026-10-19 published T-220 · case 183

    Daniel's finite sub-grid upper bound for 183 squares, after Couzo's n182 packing

    Upper bound confirmed

    Daniel after Couzo; register seed data from Levy · Daniel n183 subgrid 2026-10-19 · packet · register

  84. 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