BREAKTHROUGH REVIEW

Breakthrough review.

A numerical lead is evidence to review, not a conclusion. Pending entries pass the arena verifier; certified entries additionally rule out source, normalisation and published-precision artefacts.

Verified breakthroughs39
Pending candidates47
Problem families9
CLAIM POLICY

One certification standard

  1. 01A traceable external benchmarkThere must be a public source and an explicit comparison value. A baseline found by this site is not an external record.
  2. 02A complete construction verifiesThe complete answer is rescored by the submission verifier. The verified entry is pinned to that exact score and contributor; it does not transfer with the current record.
  3. 03Rule out precision artefactsTruncated and rounded values are expanded to their most conservative interval, together with source-coordinate precision, solver tolerance and reconstruction error. A last-digit floating-point tweak without a rigorous error bound remains only a candidate; verification requires either a proof or a margin beyond every identified uncertainty.

A verified breakthrough is not a proof of global optimality. It means a uniformly reviewed construction genuinely improves on a traceable external best-known construction. A player can earn the 3-point award only once per sub-problem, and keeps it after the record changes hands.

CURRENT VERIFIED RESULTS

Verified breakthroughs

One best verified result per sub-problem, ranked by conservative relative improvement. Historical certifications and personal awards are retained.

Current verifiable construction for Lighting a unit square, n = 21This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
01 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 21

Published best known172.658+
Verified here184.439329
Relative improvement+6.822887%
Verified contributor
Gaozx1#151
Solution method
AI · Z.ai: GLM 5.3 Flash
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (5)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

NUE_13 #31

172.658+184.439282+6.82286%

Human

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
今天 #84

172.658+184.009074+6.573694%

Human

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Lighting a unit square, n = 28This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
02 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 28

Published best known252.447+
Verified here268.113242
Relative improvement+6.205334%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (3)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

邱仲普 #72

252.447+267.937821+6.135846%

AI · MoonshotAI: Kimi K3

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Lighting a unit square, n = 20This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
03 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 20

Published best known165.272+
Verified here174.495297
Relative improvement+5.580038%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The credited historical answer was fetched and independently checked by integer interval branch-and-bound over the whole square. Its certified intensity lower bound clears the complete three-decimal truncated source interval; this is not a finite-sampling estimate.

The external P51 benchmark and publication interval reviewed on 2026-09-05

Earlier certifications (3)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

Current verifiable construction for Lighting a unit square, n = 27This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
04 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 27

Published best known242.055+
Verified here252.044246
Relative improvement+4.12642%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (3)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

邱仲普 #72

242.055+251.883491+4.060007%

AI · MoonshotAI: Kimi K3

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Lighting a unit square, n = 18This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
05 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 18

Published best known141.556+
Verified here147.340063
Relative improvement+4.085325%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The credited historical answer was fetched and independently checked by integer interval branch-and-bound over the whole square. Its certified intensity lower bound clears the complete three-decimal truncated source interval; this is not a finite-sampling estimate.

The external P51 benchmark and publication interval reviewed on 2026-09-05

Earlier certifications (4)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

Current verifiable construction for Lighting a unit square, n = 19This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
06 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 19

Published best known152.854+
Verified here158.868263
Relative improvement+3.933966%
Verified contributor
邱仲普#72
Solution method
AI · MoonshotAI: Kimi K3
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (3)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

Current verifiable construction for The smallest triangle in a disc, n = 11This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
07 · P22VERIFIED BREAKTHROUGH · +3

The smallest triangle in a disc · n = 11

Published best known0.1097733212804615 (rounded)
Verified here0.1139381173301516175
Relative improvement+3.793997%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The original historical coordinates pass exact integer containment and every triangle-area check. After converting the unit-area disc literature value by multiplying by pi, the gain is approximately 3.79%, far above conversion, output and figure-reconstruction precision.

The external P22 benchmark and publication interval reviewed on 2026-09-05

Current verifiable construction for Lighting a unit square, n = 30This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
08 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 30

Published best known285.345+
Verified here294.494986
Relative improvement+3.206278%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (4)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

邱仲普 #72

285.345+294.382627+3.166902%

AI · MoonshotAI: Kimi K3

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Lighting a unit square, n = 35This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
09 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 35

Published best known347.195+
Verified here357.333806
Relative improvement+2.919909%
Verified contributor
今天#84
Solution method
Human
Verified

Review: The credited historical answer was fetched and independently checked by integer interval branch-and-bound over the whole square. Its certified intensity lower bound clears the complete three-decimal truncated source interval; this is not a finite-sampling estimate.

The external P51 benchmark and publication interval reviewed on 2026-09-05

Earlier certifications (3)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

Current verifiable construction for Lighting a unit square, n = 26This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
10 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 26

Published best known234.759+
Verified here239.67203
Relative improvement+2.092362%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (4)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

邱仲普 #72

234.759+239.538939+2.03567%

AI · MoonshotAI: Kimi K3

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
今天 #84

234.759+239.463706+2.003623%

Human

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Lighting a unit square, n = 34This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
11 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 34

Published best known337.906+
Verified here344.23623
Relative improvement+1.873069%
Verified contributor
今天#84
Solution method
Human
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (5)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

游侠 #116

337.906+344.235704+1.872913%

AI · OpenAI: GPT-5.6 Sol

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
hadsajnc #146

337.906+344.088056+1.829218%

Human

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Lighting a unit square, n = 10This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
12 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 10

Published best known62.408+
Verified here63.54872
Relative improvement+1.826211%
Verified contributor
215590053#88
Solution method
Human
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (6)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

hadsajnc #146

62.408+63.548691+1.826164%

Human

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
今天 #84

62.408+63.548718+1.826208%

Human

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Lighting a unit square, n = 15This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
13 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 15

Published best known113.191+
Verified here114.941985
Relative improvement+1.546032%
Verified contributor
hadsajnc#146
Solution method
Human
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (6)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

今天 #84

113.191+114.941974+1.546023%

Human

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
215590053 #88

113.191+114.941977+1.546025%

Human

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Lighting a unit square, n = 25This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
14 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 25

Published best known225.100+
Verified here228.046741
Relative improvement+1.308631%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (3)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

邱仲普 #72

225.100+227.942881+1.262492%

AI · MoonshotAI: Kimi K3

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Lighting a unit square, n = 24This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
15 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 24

Published best known215.573+
Verified here218.349474
Relative improvement+1.287481%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (3)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

邱仲普 #72

215.573+218.304698+1.26671%

AI · MoonshotAI: Kimi K3

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Lighting a unit square, n = 33This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
16 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 33

Published best known330.595+
Verified here334.037191
Relative improvement+1.040905%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The credited historical answer was fetched and independently checked by integer interval branch-and-bound over the whole square. Its certified intensity lower bound clears the complete three-decimal truncated source interval; this is not a finite-sampling estimate.

The external P51 benchmark and publication interval reviewed on 2026-09-05

Current verifiable construction for Unit circles in a minimum-area triangle, n = 47This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
17 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 47

Published best known183.09423+
Verified here181.356482386744450603
Relative improvement+0.9491%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 183.09423+, while the exact site certificate 181.356482386744450603 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for Lighting a unit square, n = 16This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
18 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 16

Published best known124.553+
Verified here125.436726
Relative improvement+0.708709%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The credited historical answer was fetched and independently checked by integer interval branch-and-bound over the whole square. Its certified intensity lower bound clears the complete three-decimal truncated source interval; this is not a finite-sampling estimate.

The external P51 benchmark and publication interval reviewed on 2026-09-05

Earlier certifications (2)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

Current verifiable construction for Lighting a unit square, n = 17This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
19 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 17

Published best known133.907+
Verified here134.776663
Relative improvement+0.648701%
Verified contributor
邱仲普#72
Solution method
AI · MoonshotAI: Kimi K3
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (3)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

Current verifiable construction for Unit circles in a minimum-area triangle, n = 50This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
20 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 50

Published best known194.49896+
Verified here193.3181016005745782485
Relative improvement+0.607128%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 194.49896+, while the exact site certificate 193.3181016005745782485 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for Unit circles in a minimum-area triangle, n = 49This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
21 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 49

Published best known84 + 61√3 ≈ 189.655099261702
Verified here188.5863160312695963975
Relative improvement+0.56354%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The public construction has the exact area 84 + 61√3. The site's exact integer certificate is still substantially smaller, so the improvement cannot come from decimal conversion or grid quantisation.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for Lighting a unit square, n = 29This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
22 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 29

Published best known282.856+
Verified here283.920489
Relative improvement+0.375981%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (2)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

Current verifiable construction for Lighting a unit square, n = 32This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
23 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 32

Published best known317.203+
Verified here318.204745
Relative improvement+0.315489%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (2)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

Current verifiable construction for Lighting a unit square, n = 5This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
24 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 5

Published best known22.213+
Verified here22.252208
Relative improvement+0.172%
Verified contributor
Tmrg#34
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The credited historical answer was fetched and independently checked by integer interval branch-and-bound over the whole square. Its certified intensity lower bound clears the complete three-decimal truncated source interval; this is not a finite-sampling estimate.

The external P51 benchmark and publication interval reviewed on 2026-09-05

Earlier certifications (2)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

Current verifiable construction for Lighting a unit square, n = 31This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
25 · P51VERIFIED BREAKTHROUGH · +3

Lighting a unit square · n = 31

Published best known305.298+
Verified here305.8207
Relative improvement+0.170882%
Verified contributor
游侠#116
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark

Earlier certifications (3)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

邱仲普 #72

305.298+305.683836+0.126052%

AI · MoonshotAI: Kimi K3

Review: This player's pinned historical submission was retrieved and reverified by integer interval branch-and-bound over the whole square. Its exact score clears the external publication benchmark used by the prior review. UUID and construction SHA-256 are pinned. Later improvements update the showcase without another award for the same player and instance. This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.

Friedman's table rechecked on 2026-09-06; the previously reviewed three-decimal publication benchmark
Current verifiable construction for Unit circles in a minimum-area triangle, n = 48This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
26 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 48

Published best known184.75052+
Verified here184.614248542190787528
Relative improvement+0.07376%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 184.75052+, while the exact site certificate 184.614248542190787528 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for Unit circles in a minimum-area triangle, n = 37This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
27 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 37

Published best known144.93603+
Verified here144.877235000371301433
Relative improvement+0.040566%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 144.93603+, while the exact site certificate 144.877235000371301433 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for The most separated family of subspaces, d = 5, n = 19This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
28 · P64VERIFIED BREAKTHROUGH · +3

The most separated family of subspaces · d = 5, n = 19

Published best known[1.1761278309, 1.1761278310]
Verified here1.17612899
Relative improvement+0.000099%
Verified contributor
邱仲普#72
Solution method
AI · MoonshotAI: Kimi K3
Verified

Review: The historical bases were reverified through exact rational orthogonal projectors. The minimum squared chordal distance exceeds the entire ten-decimal downward-truncated source interval. A small percentage is not an automatic rejection; the benchmark is the actual published value, not a degraded nine-place reconstruction.

The external P64 benchmark and publication interval reviewed on 2026-09-05

Current verifiable construction for Codebook packing in complex projective space, d = 4, n = 11This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
29 · P61VERIFIED BREAKTHROUGH · +3

Codebook packing in complex projective space · d = 4, n = 11

Published best known[0.425147265, 0.425147275]
Verified here0.425147173
Relative improvement+0.000022%
Verified contributor
Stonebubu#75
Solution method
AI · Z.ai: GLM 5.3 Flash
Verified

Review: The historical vectors were reverified using exact rational inner products, norms and adverse rounding. Coherence is below the eight-decimal source interval and meets the source project's explicit eighth-decimal improvement threshold, rather than comparing unpublished recomputed tails.

The external P61 benchmark and publication interval reviewed on 2026-09-05

Earlier certifications (1)

These reviews retain their original scores, sources and contributors. Being surpassed does not revoke the honour; each player still receives only one permanent award per sub-problem.

Current verifiable construction for Variable-radius circles in a fixed-perimeter rectangle, n = 26This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
30 · P67VERIFIED BREAKTHROUGH · +3

Variable-radius circles in a fixed-perimeter rectangle · n = 26

Published best known2.63931+
Verified here2.639320556
Relative improvement+0.000021%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The exact site certificate 2.639320556 clears the old public interval [2.63931, 2.63932). Friedman subsequently updated n=26 to 2.63932+ and credited Shaoheng Lai, providing independent evidence of adoption; that later adoption does not revoke the permanent certification.

The 2.63931+ best-known value in Friedman's table immediately before its 2026-08-18 update

Current verifiable construction for The most separated family of subspaces, d = 5, n = 20This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
31 · P64VERIFIED BREAKTHROUGH · +3

The most separated family of subspaces · d = 5, n = 20

Published best known[1.1619417514, 1.1619417515]
Verified here1.161941835
Relative improvement+0.000007%
Verified contributor
邱仲普#72
Solution method
AI · MoonshotAI: Kimi K3
Verified

Review: The historical bases were reverified through exact rational orthogonal projectors. The minimum squared chordal distance exceeds the entire ten-decimal downward-truncated source interval. A small percentage is not an automatic rejection; the benchmark is the actual published value, not a degraded nine-place reconstruction.

The external P64 benchmark and publication interval reviewed on 2026-09-05

Current verifiable construction for Unit circles in a minimum-area triangle, n = 39This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
32 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 39

Published best known151.83073+
Verified here151.830724848038989335
Relative improvement+0.000003%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 151.83073+, while the exact site certificate 151.830724848038989335 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for Unit circles in a minimum-area triangle, n = 41This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
33 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 41

Published best known160.11055+
Verified here160.110546188577149009
Relative improvement+0.000002%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 160.11055+, while the exact site certificate 160.110546188577149009 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for Unit circles in a minimum-area triangle, n = 26This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
34 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 26

Published best known102.70487+
Verified here102.704868171923083464
Relative improvement+0.000002%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 102.70487+, while the exact site certificate 102.704868171923083464 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for Unit circles in a minimum-area triangle, n = 43This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
35 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 43

Published best known163.71933+
Verified here163.719327388545428618
Relative improvement+0.000002%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 163.71933+, while the exact site certificate 163.719327388545428618 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for Unit circles in a minimum-area triangle, n = 33This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
36 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 33

Published best known129.20635+
Verified here129.206348152361515763
Relative improvement+0.000001%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 129.20635+, while the exact site certificate 129.206348152361515763 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for Unit circles in a minimum-area triangle, n = 46This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
37 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 46

Published best known177.88737+
Verified here177.8873691417777755165
Relative improvement+<0.000001%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 177.88737+, while the exact site certificate 177.8873691417777755165 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for Unit circles in a minimum-area triangle, n = 38This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
38 · P66VERIFIED BREAKTHROUGH · +3

Unit circles in a minimum-area triangle · n = 38

Published best known148.37141+
Verified here148.3714097309017328625
Relative improvement+<0.000001%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source publishes only 148.37141+, while the exact site certificate 148.3714097309017328625 is already below the lower edge of that truncation interval. It is strictly smaller regardless of the unpublished digits.

The then-current best-known value in Erich Friedman's equal-circle-in-a-triangle table

Current verifiable construction for The most separated family of subspaces, d = 6, n = 15This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
39 · P64VERIFIED BREAKTHROUGH · +3

The most separated family of subspaces · d = 6, n = 15

Published best known[1.4285668598, 1.4285668599]
Verified here1.428566861
Relative improvement+<0.000001%
Verified contributor
邱仲普#72
Solution method
AI · MoonshotAI: Kimi K3
Verified

Review: The historical bases were reverified through exact rational orthogonal projectors. The minimum squared chordal distance exceeds the entire ten-decimal downward-truncated source interval. A small percentage is not an automatic rejection; the benchmark is the actual published value, not a degraded nine-place reconstruction.

The external P64 benchmark and publication interval reviewed on 2026-09-05