MinMax Arena
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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 breakthroughs14
Pending candidates80
Problem families7
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 two-point award only once per sub-problem, and keeps it after the record changes hands.

VERIFIED LEDGER

Verified breakthroughs

Ranked by relative improvement; percentages use the conservative edge of the published precision interval.

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.
01 · P51VERIFIED BREAKTHROUGH · +2

Lighting a unit square · n = 34

Published best known337.906+
→
Verified here341.395093
Relative improvement+1.032264%
Verified contributor
Sigmoid with Codex#71
Solution method
AI · OpenAI: GPT-5.6 Sol
Verified

Review: The source reports 337.906+. Even after treating its unpublished digits as the most conservative truncation interval, 341.395093 clears the entire interval by more than 1%; this is not an artefact of printing more digits.

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The then-current 337.906+ best-known value in Erich Friedman's table

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.
02 · P66VERIFIED BREAKTHROUGH · +2

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.

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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 = 50This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
03 · P66VERIFIED BREAKTHROUGH · +2

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.

External source ↗Inspect construction and certificate →

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.
04 · P66VERIFIED BREAKTHROUGH · +2

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.

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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 = 48This is the sub-problem's current construction; the verified score and contributor are pinned in the snapshot at right.
05 · P66VERIFIED BREAKTHROUGH · +2

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.

External source ↗Inspect construction and certificate →

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.
06 · P66VERIFIED BREAKTHROUGH · +2

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.

External source ↗Inspect construction and certificate →

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

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.
07 · P67VERIFIED BREAKTHROUGH · +2

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.

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

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.
08 · P66VERIFIED BREAKTHROUGH · +2

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.

External source ↗Inspect construction and certificate →

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.
09 · P66VERIFIED BREAKTHROUGH · +2

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.

External source ↗Inspect construction and certificate →

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.
10 · P66VERIFIED BREAKTHROUGH · +2

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.

External source ↗Inspect construction and certificate →

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.
11 · P66VERIFIED BREAKTHROUGH · +2

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.

External source ↗Inspect construction and certificate →

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.
12 · P66VERIFIED BREAKTHROUGH · +2

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.

External source ↗Inspect construction and certificate →

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.
13 · P66VERIFIED BREAKTHROUGH · +2

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.

External source ↗Inspect construction and certificate →

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.
14 · P66VERIFIED BREAKTHROUGH · +2

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.

External source ↗Inspect construction and certificate →

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