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 breakthroughs38
Pending candidates15
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.

Reviews preserve the historical achievement. The current site #1 earns 1 point; a reviewed world #1 earns an additional 3. An internal takeover removes both; an external defeat removes the world bonus. Historical honours remain, without accumulating points.

VERIFIED ACHIEVEMENTS

Verified breakthroughs

One strongest historical review per sub-problem, ordered by its original conservative gain. Current records determine eligibility for world #1 points.

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 · P51World #1 · currently +3

Lighting a unit square · n = 21

Published best known[172.658, 172.659)
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 This is neither a global optimality proof nor a claim to know unpublished digits of the external configuration.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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

NUE_13 #31

[172.658, 172.659)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, 172.659)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 · P51Historical honour · no current points

Lighting a unit square · n = 28

Published best known[252.447, 252.448)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

邱仲普 #72

[252.447, 252.448)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 · P51Historical honour · no current points

Lighting a unit square · n = 20

Published best known[165.272, 165.273)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

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 · P51Historical honour · no current points

Lighting a unit square · n = 27

Published best known[242.055, 242.056)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

邱仲普 #72

[242.055, 242.056)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 · P51Historical honour · no current points

Lighting a unit square · n = 18

Published best known[141.556, 141.557)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

iqy #106

[141.556, 141.557)147.338946+4.084536%

AI · MoonshotAI: Kimi K3

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 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 · P51Historical honour · no current points

Lighting a unit square · n = 19

Published best known[152.854, 152.855)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

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.
07 · P51Historical honour · no current points

Lighting a unit square · n = 30

Published best known[285.345, 285.346)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

邱仲普 #72

[285.345, 285.346)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.
08 · P51Historical honour · no current points

Lighting a unit square · n = 35

Published best known[347.195, 347.196)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

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.
09 · P51Historical honour · no current points

Lighting a unit square · n = 26

Published best known[234.759, 234.760)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

邱仲普 #72

[234.759, 234.760)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, 234.760)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.
10 · P51Historical honour · no current points

Lighting a unit square · n = 34

Published best known[337.906, 337.907)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

游侠 #116

[337.906, 337.907)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, 337.907)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.
11 · P51Historical honour · no current points

Lighting a unit square · n = 10

Published best known[62.408, 62.409)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

hadsajnc #146

[62.408, 62.409)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, 62.409)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.
12 · P51Historical honour · no current points

Lighting a unit square · n = 15

Published best known[113.191, 113.192)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

今天 #84

[113.191, 113.192)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, 113.192)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.
13 · P51Historical honour · no current points

Lighting a unit square · n = 25

Published best known[225.100, 225.101)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

邱仲普 #72

[225.100, 225.101)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.
14 · P51Historical honour · no current points

Lighting a unit square · n = 24

Published best known[215.573, 215.574)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

邱仲普 #72

[215.573, 215.574)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.
15 · P51World #1 · currently +3

Lighting a unit square · n = 33

Published best known[330.595, 330.596)
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.
16 · P66World #1 · currently +3

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

Published best known[183.09423, 183.09424)
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.
17 · P51Historical honour · no current points

Lighting a unit square · n = 16

Published best known[124.553, 124.554)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

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.
18 · P51Historical honour · no current points

Lighting a unit square · n = 17

Published best known[133.907, 133.908)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

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.
19 · P66World #1 · currently +3

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

Published best known[194.49896, 194.49897)
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.
20 · P66World #1 · currently +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.
21 · P51Historical honour · no current points

Lighting a unit square · n = 29

Published best known[282.856, 282.857)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

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.
22 · P51Historical honour · no current points

Lighting a unit square · n = 32

Published best known[317.203, 317.204)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

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.
23 · P51Historical honour · no current points

Lighting a unit square · n = 5

Published best known[22.213, 22.214)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

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.
24 · P51Historical honour · no current points

Lighting a unit square · n = 31

Published best known[305.298, 305.299)
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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

邱仲普 #72

[305.298, 305.299)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.
25 · P66World #1 · currently +3

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

Published best known[184.75052, 184.75053)
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.
26 · P66World #1 · currently +3

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

Published best known[144.93603, 144.93604)
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.
27 · P64World #1 · currently +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.
28 · P61World #1 · currently +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 results, sources and contributors. Historical honours remain; current points belong only to an eligible current holder and do not accumulate per review.

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.
29 · P67World #1 · currently +3

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

Published best known[2.63931, 2.63932)
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.
30 · P64World #1 · currently +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.
31 · P66World #1 · currently +3

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

Published best known[151.83073, 151.83074)
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.
32 · P66World #1 · currently +3

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

Published best known[160.11055, 160.11056)
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.
33 · P66World #1 · currently +3

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

Published best known[102.70487, 102.70488)
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.
34 · P66World #1 · currently +3

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

Published best known[163.71933, 163.71934)
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.
35 · P66World #1 · currently +3

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

Published best known[129.20635, 129.20636)
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.
36 · P66World #1 · currently +3

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

Published best known[177.88737, 177.88738)
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.
37 · P66World #1 · currently +3

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

Published best known[148.37141, 148.37142)
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.
38 · P64World #1 · currently +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