Best LLM for Math & Reasoning in 2026

For math & reasoning, GLM-5.2 is our pick: $3.40/M tokens on a Reasoning-heavy query workload, 1M context, graded 99/100 across 1 run. Muse Spark 1.3 Contributor is the value alternative at $0.17/M.

Multi-step math and logic puzzles punish models that guess instead of verifying intermediate steps. We require a reasoning mode and weight graded accuracy heavily — price is a secondary concern for work this failure-sensitive.

Verdict: Explicit reasoning mode is not optional for this task — every model in our eligible set that lacks one is excluded, not down-ranked, because a guessed answer is worse than a slow, correct one.

Quick answer: What is the best LLM for math and reasoning?

GLM-5.2, from Z.ai, is the best fit for math & reasoning at $3.40 per million task tokens on a Reasoning-heavy query workload, with a 1M-token context window. Muse Spark 1.3 Contributor is the cheaper value alternative at $0.17 per million tokens.

Verified 2026-08-08

Best overall

GLM-5.2

Z.ai · $3.40/M

Fit 85/100 — the top requirements match for this task.

constraint-logic-puzzle →

Best value

Muse Spark 1.3 Contributor

Meta · $0.17/M

The strongest fit among budget and mid-tier priced models.

Fastest

GPT-OSS 120B (Cerebras)

Cerebras · $0.62/M

2450 tokens/sec measured.

Longest context

Gemini 3.1 Pro

Google · $8.67/M

2M token context window.

constraint-logic-puzzle →

Can't use GLM-5.2? See GLM-5.2 alternatives.

What evidence supports the Math & Reasoning recommendation?

Graded on 3 of 28 eligible models, run 2026-06-21. Full prompts, verbatim outputs, and grading notes below.

Solving a Constraint Logic Puzzle

A five-house constraint-satisfaction puzzle with five interlocking clues and a single valid solution. The model must reason through both cases, eliminate the dead end, and report the exact arrangement and count.

Click Accuracy, Speed, or Cost to re-sort. Click any row to read the full model output.

Per-model results: accuracy, speed (tokens/sec), provider cost (USD), and the full prompt-vs-output for each model on this task.
#	Model	Accuracy ↓	Speed ↕	Cost ↕	Output
1	GPT-5.4 ProOpenAI · gpt-5.4-pro	100★	21.7 t/s37818 ms	$0.1533$180/M out	View (822 tok) ▼
Agent grade — 100/100. Correct unique arrangement and count (1) with tight, rigorous case analysis on both yellow positions, final count on its own line. Now that the live API reports real usage for this model, its true cost is shown — by far the priciest run here. Verbatim output (178 in / 822 out tokens, 37818 ms, $0.1533) Let the houses be positions 1–5. - Since the green house is immediately to the right of the white house, the pair must be one of: - (1,2) = white, green - (2,3) = white, green - (3,4) = white, green - (4,5) = white, green - The yellow house is at an end (position 1 or 5), and there are exactly two houses between green and yellow, so green must be 3 positions away from yellow. ## Case 1: Yellow at 1 Then green must be at 4. So white must be at 3. Arrangement so far: 1. yellow 2. ? 3. white 4. green 5. ? The remaining colors are red and blue. Since red is left of blue, we must have: 2. red 5. blue But blue cannot be at either end, and position 5 is an end. So this case is impossible. ## Case 2: Yellow at 5 Then green must be at 2. So white must be at 1. Arrangement so far: 1. white 2. green 3. ? 4. ? 5. yellow The remaining colors are red and blue. Since red is left of blue, we must have: 3. red 4. blue Blue is not at an end, so this works. ## Valid arrangement 1. white, 2. green, 3. red, 4. blue, 5. yellow 1
2	Claude Opus 4.8Anthropic · claude-opus-4-8	100★	97.3 t/s10119 ms	$0.025855$25/M out	View (985 tok) ▼
Agent grade — 100/100. Correct arrangement and count via tight, fully-verified case analysis on both yellow positions, with a final five-clue check. The leading <think> block is Opus 4.8's reasoning channel surfaced by our gateway, not dumped into the answer — the final solution itself is clean and correct. Verbatim output (246 in / 985 out tokens, 10119 ms, $0.025855) <think> Houses 1-5. Colors: red, blue, green, yellow, white. Clue 1: green = white+1 (white immediately left of green). Clue 2: red < blue. Clue 3: yellow at position 1 or 5. Clue 4: \|green - yellow\| = 3. Clue 5: blue not at 1 or 5. Let me enumerate by yellow position. Case yellow=1: Clue 4: green = 1±3 = 4 (since -2 invalid). So green=4. Clue 1: white=3. Remaining: red, blue for positions 2,5. Clue 5: blue not at 5, so blue=2, red=5. But clue 2: red<blue → 5<2 false. Invalid. So no valid with yellow=1. Case yellow=5: Clue 4: green = 5±3 = 2 or 8. green=2. Clue 1: white=1. Remaining: red, blue for positions 3,4. Clue 5: blue not at 1,5, ok both. Clue 2: red<blue. Positions 3,4: red=3, blue=4. Valid. Arrangement: 1 white, 2 green, 3 red, 4 blue, 5 yellow. Check all clues: green(2)=white(1)+1 ✓. red(3)<blue(4) ✓. yellow=5 end ✓. \|green-yellow\|=\|2-5\|=3 ✓. blue=4 not end ✓. Only one arrangement. </think> ## Reasoning Clue constraints: - Green = White + 1 (adjacent) - Red < Blue - Yellow at position 1 or 5 - \|Green − Yellow\| = 3 - Blue at position 2, 3, or 4 Case: Yellow = 1 - Clue 4: Green = 4 (only valid option). So White = 3. - Remaining positions 2 and 5 get red and blue. - Clue 5 forbids blue at 5, so Blue = 2, Red = 5. - But Clue 2 needs Red < Blue → 5 < 2, false. ❌ No solution. Case: Yellow = 5 - Clue 4: Green = 2 (only valid option). So White = 1. - Remaining positions 3 and 4 get red and blue. - Clue 2 (Red < Blue): Red = 3, Blue = 4. - Check Clue 5: Blue = 4, not an end. ✓ This gives: House 1: White, House 2: Green, House 3: Red, House 4: Blue, House 5: Yellow Verify all clues: Green(2)=White(1)+1 ✓; Red(3)<Blue(4) ✓; Yellow at end ✓; \|2−5\|=3 ✓; Blue not at end ✓. ## Valid Arrangement(s) 1. White, Green, Red, Blue, Yellow 1
3	GLM 5.2 (Max)Z.ai · glm-5.2	99	73.7 t/s38370 ms	$0.012698$4.4/M out	View (2827 tok) ▼
Agent grade — 99/100. Reaches the correct arrangement and count with sound, fully-verified case analysis. The long <think> section is GLM's reasoning surfaced by our gateway from its separate reasoning channel (not dumped into the answer), so it isn't penalised — the final solution itself is clean and correct. Verbatim output (185 in / 2827 out tokens, 38370 ms, $0.012698) <think> The problem asks for the number of distinct color arrangements of 5 houses in a row, numbered 1 to 5 from left to right. The colors are red, blue, green, yellow, white. Clues: 1. Green is immediately to the right of White. (W, G) 2. Red is somewhere to the left of Blue. 3. Yellow is at one of the two ends (house 1 or house 5). 4. There are exactly two houses between Green and Yellow. 5. Blue is not at either end (so Blue is in house 2, 3, or 4). Let's analyze the positions of Yellow and the W-G block. Yellow is at 1 or 5. Case 1: Yellow is at 1. Since there are exactly two houses between Green and Yellow, Green must be at 4. If Green is at 4, White must be at 3 (since Green is immediately to the right of White). So we have: 1: Yellow 2: ? 3: White 4: Green 5: ? The remaining colors are Red and Blue. They must go in houses 2 and 5. Clue 5 says Blue is not at either end. So Blue cannot be at 5. Thus, Blue must be at 2, and Red must be at 5. But Clue 2 says Red is somewhere to the left of Blue. If Blue is at 2 and Red is at 5, Red is to the right of Blue. This violates Clue 2. So Case 1 yields no valid arrangements. Case 2: Yellow is at 5. Since there are exactly two houses between Green and Yellow, Green must be at 2. If Green is at 2, White must be at 1 (since Green is immediately to the right of White). So we have: 1: White 2: Green 3: ? 4: ? 5: Yellow The remaining colors are Red and Blue. They must go in houses 3 and 4. Clue 5 says Blue is not at either end. Houses 3 and 4 are not ends, so this is satisfied. Clue 2 says Red is to the left of Blue. So Red must be at 3, and Blue must be at 4. Let's check all clues for this arrangement: 1: White 2: Green 3: Red 4: Blue 5: Yellow Clue 1: Green is immediately to the right of White. (W at 1, G at 2) - Yes. Clue 2: Red is to the left of Blue. (R at 3, B at 4) - Yes. Clue 3: Yellow is at one of the two ends. (Y at 5) - Yes. Clue 4: Exactly two houses between Green and Yellow. (G at 2, Y at 5, houses 3 and 4 are between) - Yes. Clue 5: Blue is not at either end. (B at 4) - Yes. Are there any other possibilities? Let's double check. Could Yellow be at 1 and Green at 4? We checked this, it leads to W at 3, Y at 1. Remaining houses 2, 5 for R, B. B cannot be at 5, so B at 2, R at 5. But R must be left of B. Contradiction. Could Yellow be at 5 and Green at 2? We checked this, it leads to W at 1, Y at 5. Remaining houses 3, 4 for R, B. B cannot be at 5 (already satisfied). R must be left of B, so R at 3, B at 4. This works. Wait, are there any other interpretations of "exactly two houses between"? If Yellow is at 1, Green is at 4. Houses 2 and 3 are between. That's exactly two houses. If Yellow is at 5, Green is at 2. Houses 3 and 4 are between. That's exactly two houses. Are there any other positions for Yellow? No, clue 3 says Yellow is at one of the two ends. Are there any other positions for Green? No, because the distance between 1 and 4 is 2 houses, and between 2 and 5 is 2 houses. So Green must be at 2 or 4. If Green is at 4, White is at 3. If Green is at 2, White is at 1. Let's quickly review the steps. Colors: R, B, G, Y, W. Positions: 1, 2, 3, 4, 5. Y is at 1 or 5. If Y=1, G=4. W=3. Remaining: 2, 5 for R, B. B cannot be at 5 (end), so B=2, R=5. But R must be left of B. 5 is not left of 2. Invalid. If Y=5, G=2. W=1. Remaining: 3, 4 for R, B. B cannot be at ends (3 and 4 are not ends, so both valid). R must be left of B, so R=3, B=4. Valid. Is there any ambiguity in "The green house is immediately to the right of the white house"? No, it means W is at x, G is at x+1. Is there any ambiguity in "exactly two houses between"? No, \|pos(G) - pos(Y)\| = 3. Is there any ambiguity in "The red house is somewhere to the left of the blue house"? No, pos(R) < pos(B). Let's check if there's any other way. What if Yellow is at 1, Green is at 4, White is at 3. Houses: 1: Y, 2: ?, 3: W, 4: G, 5: ? Colors left: R, B. If B=2, R=5. R is not left of B. If B=5, R=2. B is at an end, which violates clue 5. So no solution for Y=1. What if Yellow is at 5, Green is at 2, White is at 1. Houses: 1: W, 2: G, 3: ?, 4: ?, 5: Y. Colors left: R, B. If R=3, B=4. R is left of B. B is not at an end. Valid. If R=4, B=3. R is not left of B. Invalid. So there is exactly 1 valid arrangement. The arrangement is: House 1: White House 2: Green House 3: Red House 4: Blue House 5: Yellow Let me re-read the prompt to make sure I didn't miss anything. "Five houses stand in a row, numbered 1 to 5 from left to right." "Each house is painted a different color, one of: red, blue, green, yellow, white." "1. The green house is immediately to the right of the white house." -> W, G "2. The red house is somewhere to the left of the blue house (not necessarily adjacent)." -> R < B "3. The yellow house is at one of the two ends of the row." -> Y=1 or Y=5 "4. There are exactly two houses between the green house and the yellow house." -> \|G - Y\| = 3 "5. The blue house is not at either end." -> B != 1, B != 5 Everything seems perfectly straightforward. Let's write down the solution clearly. - State the possible positions for Yellow based on Clue 3 (1 or 5). - Use Clue 4 to find the corresponding positions for Green (4 or 2). - Use Clue 1 to find the corresponding positions for White (3 or 1). - Evaluate the two cases: - Case 1: Y=1, G=4, W=3. Remaining positions for R and B are 2 and 5. Clue 5 says B cannot be 5, so B=2, R=5. But Clue 2 says R must be left of B, which is violated. So this case is invalid. - Case 2: Y=5, G=2, W=1. Remaining positions for R and B are 3 and 4. Clue 2 says R must be left of B, so R=3, B=4. Clue 5 is satisfied since B is not at an end. This case is valid. - Conclude there is exactly 1 valid arrangement. - List the arrangement as house 1 through house 5. - State the final count on its own line. The arrangement: House 1: White House 2: Green House 3: Red House 4: Blue House 5: Yellow Final count: 1 Let me double check the "exactly two houses between" part. If Y is at 1, and G is at 4. Houses between 1 and 4 are 2 and 3. That's exactly two houses. If Y is at 5, and G is at 2. Houses between 2 and 5 are 3 and 4. That's exactly two houses. Yes, this is correct. What if "exactly two houses between" means the distance is 2? No, "two houses between" means there are two houses separating them. For example, between house 1 and house 4, there are houses 2 and 3. That's two houses. If it meant distance is 2, it would be "one house between". So \|G - Y\| = 3 is correct. Let's do a quick sanity check. Colors: R, B, G, Y, W. Arrangement: W, G, R, B, Y. 1. Green immediately right of White: W at 1, G at 2. Yes. 2. Red left of Blue: R at 3, B at 4. Yes. 3. Yellow at an end: Y at 5. Yes. 4. Exactly two houses between Green and Yellow: G at 2, Y at 5. Houses 3 and 4 are between. Yes. 5. Blue not at either end: B at 4. Yes. Everything is consistent. The logic is solid. No other interpretations seem likely. I will structure the output to show the reasoning, list the valid arrangement, and then state the final count. </think> To solve this logic puzzle, we can deduce the positions of the houses step-by-step using the given clues: Step 1: Determine the possible positions for the Yellow and Green houses. * Clue 3 states that the yellow house is at one of the two ends (house 1 or house 5). * Clue 4 states that there are exactly two houses between the green house and the yellow house. This gives us two possible scenarios: * Scenario A: Yellow is in house 1, which means Green must be in house 4. * Scenario B: Yellow is in house 5, which means Green must be in house 2. Step 2: Place the White house. * Clue 1 states that the green house is immediately to the right of the white house (meaning they are a pair: White, Green). * In Scenario A (Green in 4), White must be in house 3. * In Scenario B (Green in 2), White must be in house 1. Step 3: Place the Red and Blue houses. * Clue 5 states that the blue house is not at either end, meaning it cannot be in house 1 or house 5. Let's test Scenario A: * Houses: 1: Yellow, 2: ?, 3: White, 4: Green, 5: ? * The remaining colors for houses 2 and 5 are Red and Blue. * Since Blue cannot be at an end (house 5), Blue must be in house 2, leaving Red in house 5. * However, Clue 2 says the red house is to the left of the blue house. Here, Red (5) is to the right of Blue (2), which violates the clue. * Therefore, Scenario A is invalid. Let's test Scenario B: * Houses: 1: White, 2: Green, 3: ?, 4: ?, 5: Yellow * The remaining colors for houses 3 and 4 are Red and Blue. * Clue 2 says Red is to the left of Blue. Therefore, Red must be in house 3 and Blue must be in house 4. * Let's check Clue 5: Blue is in house 4, which is not an end. This fits perfectly. The only valid arrangement is: House 1: White House 2: Green House 3: Red House 4: Blue House 5: Yellow 1
4	Gemini 3.1 ProGoogle · gemini-3.1-pro	98	22.7 t/s22828 ms	$0.006576$12/M out	View (518 tok) ▼
Agent grade — 98/100. Correct arrangement and count via clean step-by-step deduction. Slightly lighter on explicit verification than Opus, but fully sound and the cheapest correct answer here. Verbatim output (180 in / 518 out tokens, 22828 ms, $0.006576) To find the valid color arrangements, we can deduce the positions of the houses step by step (numbered 1 to 5 from left to right): 1. Analyze Yellow and Green's positions: Clue 3 states the yellow house is at one of the two ends (Position 1 or Position 5). Clue 4 states there are exactly two houses between the green house and the yellow house. - Case A: If the yellow house is at Position 1, the green house must be at Position 4. - Case B: If the yellow house is at Position 5, the green house must be at Position 2. 2. Analyze White's position: Clue 1 states the green house is immediately to the right of the white house. This means the white house is exactly one spot to the left of the green house. - Case A: If green is at 4, white must be at 3. (Arrangement so far: 1:Yellow, 2:Empty, 3:White, 4:Green, 5:Empty) - Case B: If green is at 2, white must be at 1. (Arrangement so far: 1:White, 2:Green, 3:Empty, 4:Empty, 5:Yellow) 3. Analyze Red and Blue's positions: The remaining colors to place are red and blue. Clue 2 states the red house is somewhere to the left of the blue house. - Case A: The empty spots are 2 and 5. For red to be to the left of blue, red must be at 2 and blue must be at 5. However, Clue 5 states the blue house cannot be at either end. Since Position 5 is an end, Case A is invalid. - Case B: The empty spots are 3 and 4. For red to be to the left of blue, red must be at 3 and blue must be at 4. Let's check Clue 5: Blue is at Position 4, which is not an end. This works perfectly! This leaves us with exactly one valid arrangement that satisfies all the clues. Valid Arrangement: House 1: White, House 2: Green, House 3: Red, House 4: Blue, House 5: Yellow 1

Full test output, prompt, and grading notes →

Reproducible Math & Reasoning evidence and decision rubric

Test / run	Prompt and verification	Hard rule
Code Snippet	exact prompt + 20 recorded runs	Correct iterative algorithm and code-only output
Logic Puzzle	exact prompt + 4 recorded runs	Exhaustive search: only White, Green, Red, Blue, Yellow; count = 1

Run dates: budget suite 2026-06-16T20:31:30.728Z; premium suite 2026-06-21T00:00:00.000Z. Results are not a claim about every repository or prompt.

Reasoning-only accuracy / cost / latency frontier

Model	Accuracy	Latency	Output tokens	Run cost	Failure / qualification note
GPT-5.4 Pro	100/100	37818 ms	822	$0.153	Correct unique arrangement and count (1) with tight, rigorous case analysis on both yellow positions, final count on its own line. Now that the live API reports real usage for this model, its true cost is shown — by far the priciest run here.
Claude Opus 4.8	100/100	10119 ms	985	$0.026	Correct arrangement and count via tight, fully-verified case analysis on both yellow positions, with a final five-clue check. The leading <think> block is Opus 4.8's reasoning channel surfaced by our gateway, not dumped into the answer — the final solution itself is clean and correct.
GLM 5.2 (Max)	99/100	38370 ms	2827	$0.013	Reaches the correct arrangement and count with sound, fully-verified case analysis. The long <think> section is GLM's reasoning surfaced by our gateway from its separate reasoning channel (not dumped into the answer), so it isn't penalised — the final solution itself is clean and correct.
Gemini 3.1 Pro	98/100	22828 ms	518	$0.007	Correct arrangement and count via clean step-by-step deduction. Slightly lighter on explicit verification than Opus, but fully sound and the cheapest correct answer here.

Scope: the puzzle result is exhaustively checked over 120 permutations, but it demonstrates constraint solving—not theorem proving, numerical math, or general reasoning. Only models with a current pricing record enter the frontier.

Task-shaped cost ranking (20,000 tasks/month)

Rank	Model	Effective monthly	Measured verbosity
1	Amazon Nova Micro	$4.23	0.76×
2	Amazon Nova Lite	$7.97	0.91×
3	GPT-5 Nano	$11.00	Unavailable; neutral fallback
4	Ministral 8B	$11.79	0.93×
5	Gemini 2.5 Flash Lite	$14.00	Unavailable; neutral fallback
6	Mistral Small 3.1	$19.20	0.85×

Verified 2026-08-08. full prompt/run evidence →

Try these models for Math & Reasoning →

Batch 9 decision stability for best LLM for Math & Reasoning

1. Accuracy-gated shortlist

Release score floor	Models clearing floor	Cheapest measured	Fastest measured
80/100	3	GLM-5.2	Claude Opus 4.8
90/100	3	GLM-5.2	Claude Opus 4.8
95/100	3	GLM-5.2	Claude Opus 4.8

A model is eligible only when the fixed first-party run has a score at or above the floor. Missing accuracy or speed is Unavailable, never a zero.

2. Dynamic scoring-weight sensitivity

Evidence / price / speed / context	Recalculated winner	Recalculated fit score	Stability verdict
50/20/20/10	Muse Spark 1.3 Contributor	92.1/100	Switches: Muse Spark 1.3 Contributor vs GLM-5.2
70/10/10/10	GLM-5.2	90.6/100	Switches: Muse Spark 1.3 Contributor vs GLM-5.2
40/30/20/10	Muse Spark 1.3 Contributor	94.1/100	Switches: Muse Spark 1.3 Contributor vs GLM-5.2

Each row recomputes Σ(component score × weight) ÷ Σ(available weights) over the published candidate sub-scores; missing speed or evidence is excluded from that row’s denominator.

3. Parent-to-child decision router

Trigger	Route	Boundary
Budget is binding for repeated reasoning calls	/best-llm-for/math-and-reasoning/budget	Recalculate at 1K input / 2K output
Prompt needs long context	/best-llm-for/math-and-reasoning/long-context	Capacity is separate from reasoning accuracy
Explicit reasoning is mandatory	/best-llm-for/math-and-reasoning/reasoning	Models without reasoning capability remain excluded

Math-and-reasoning verifier-first audit

The verifier exhaustively checked all 120 permutations of the five colors; exactly 1 permutation satisfies all 5 clues.

Clue	Verifier check
Clue 1 (green right of white)	\|G - W\| = 1 and G > W
Clue 2 (red left of blue)	R < B
Clue 3 (yellow at end)	Y = 1 or Y = 5
Clue 4 (two houses between green and yellow)	\|G - Y\| = 3
Clue 5 (blue not at end)	B != 1 and B != 5
Exhaustive Search	1 of 120 permutations valid

Model	Actual final answer from premiumTests.ts	Verifier result
GPT-5.4 Pro	Let the houses be positions 1–5. - Since the green house is immediately to the right of the white house, the pair must be one of: - (1,2) = white, green - (2,3) = white, green - (3,4) = white, green - (4,5) = white, green - The yellow house is at an end (position 1 or 5), and there are exactly two houses between green and yellow, so green must be 3 positions away from yellow. ## Case 1: Yellow at 1 Then green must be at 4. So white must be at 3. Arrangement so far: 1. yellow 2. ? 3. white 4. green 5. ? The remaining colors are red and blue. Since red is left of blue, we must have: 2. red 5. blue But blue cannot be at either end, and position 5 is an end. So this case is impossible. ## Case 2: Yellow at 5 Then green must be at 2. So white must be at 1. Arrangement so far: 1. white 2. green 3. ? 4. ? 5. yellow The remaining colors are red and blue. Since red is left of blue, we must have: 3. red 4. blue Blue is not at an end, so this works. ## Valid arrangement 1. white, 2. green, 3. red, 4. blue, 5. yellow 1	Matches Ground Truth
Claude Opus 4.8	<think> Houses 1-5. Colors: red, blue, green, yellow, white. Clue 1: green = white+1 (white immediately left of green). Clue 2: red < blue. Clue 3: yellow at position 1 or 5. Clue 4: \|green - yellow\| = 3. Clue 5: blue not at 1 or 5. Let me enumerate by yellow position. Case yellow=1: Clue 4: green = 1±3 = 4 (since -2 invalid). So green=4. Clue 1: white=3. Remaining: red, blue for positions 2,5. Clue 5: blue not at 5, so blue=2, red=5. But clue 2: red<blue → 5<2 false. Invalid. So no valid with yellow=1. Case yellow=5: Clue 4: green = 5±3 = 2 or 8. green=2. Clue 1: white=1. Remaining: red, blue for positions 3,4. Clue 5: blue not at 1,5, ok both. Clue 2: red<blue. Positions 3,4: red=3, blue=4. Valid. Arrangement: 1 white, 2 green, 3 red, 4 blue, 5 yellow. Check all clues: green(2)=white(1)+1 ✓. red(3)<blue(4) ✓. yellow=5 end ✓. \|green-yellow\|=\|2-5\|=3 ✓. blue=4 not end ✓. Only one arrangement. </think> ## Reasoning Clue constraints: - Green = White + 1 (adjacent) - Red < Blue - Yellow at position 1 or 5 - \|Green − Yellow\| = 3 - Blue at position 2, 3, or 4 Case: Yellow = 1 - Clue 4: Green = 4 (only valid option). So White = 3. - Remaining positions 2 and 5 get red and blue. - Clue 5 forbids blue at 5, so Blue = 2, Red = 5. - But Clue 2 needs Red < Blue → 5 < 2, false. ❌ No solution. Case: Yellow = 5 - Clue 4: Green = 2 (only valid option). So White = 1. - Remaining positions 3 and 4 get red and blue. - Clue 2 (Red < Blue): Red = 3, Blue = 4. - Check Clue 5: Blue = 4, not an end. ✓ This gives: House 1: White, House 2: Green, House 3: Red, House 4: Blue, House 5: Yellow Verify all clues: Green(2)=White(1)+1 ✓; Red(3)<Blue(4) ✓; Yellow at end ✓; \|2−5\|=3 ✓; Blue not at end ✓. ## Valid Arrangement(s) 1. White, Green, Red, Blue, Yellow 1	Matches Ground Truth
Gemini 3.1 Pro	To find the valid color arrangements, we can deduce the positions of the houses step by step (numbered 1 to 5 from left to right): 1. Analyze Yellow and Green's positions: Clue 3 states the yellow house is at one of the two ends (Position 1 or Position 5). Clue 4 states there are exactly two houses between the green house and the yellow house. - Case A: If the yellow house is at Position 1, the green house must be at Position 4. - Case B: If the yellow house is at Position 5, the green house must be at Position 2. 2. Analyze White's position: Clue 1 states the green house is immediately to the right of the white house. This means the white house is exactly one spot to the left of the green house. - Case A: If green is at 4, white must be at 3. (Arrangement so far: 1:Yellow, 2:Empty, 3:White, 4:Green, 5:Empty) - Case B: If green is at 2, white must be at 1. (Arrangement so far: 1:White, 2:Green, 3:Empty, 4:Empty, 5:Yellow) 3. Analyze Red and Blue's positions: The remaining colors to place are red and blue. Clue 2 states the red house is somewhere to the left of the blue house. - Case A: The empty spots are 2 and 5. For red to be to the left of blue, red must be at 2 and blue must be at 5. However, Clue 5 states the blue house cannot be at either end. Since Position 5 is an end, Case A is invalid. - Case B: The empty spots are 3 and 4. For red to be to the left of blue, red must be at 3 and blue must be at 4. Let's check Clue 5: Blue is at Position 4, which is not an end. This works perfectly! This leaves us with exactly one valid arrangement that satisfies all the clues. Valid Arrangement: House 1: White, House 2: Green, House 3: Red, House 4: Blue, House 5: Yellow 1	Matches Ground Truth
GLM 5.2 (Max)	<think> The problem asks for the number of distinct color arrangements of 5 houses in a row, numbered 1 to 5 from left to right. The colors are red, blue, green, yellow, white. Clues: 1. Green is immediately to the right of White. (W, G) 2. Red is somewhere to the left of Blue. 3. Yellow is at one of the two ends (house 1 or house 5). 4. There are exactly two houses between Green and Yellow. 5. Blue is not at either end (so Blue is in house 2, 3, or 4). Let's analyze the positions of Yellow and the W-G block. Yellow is at 1 or 5. Case 1: Yellow is at 1. Since there are exactly two houses between Green and Yellow, Green must be at 4. If Green is at 4, White must be at 3 (since Green is immediately to the right of White). So we have: 1: Yellow 2: ? 3: White 4: Green 5: ? The remaining colors are Red and Blue. They must go in houses 2 and 5. Clue 5 says Blue is not at either end. So Blue cannot be at 5. Thus, Blue must be at 2, and Red must be at 5. But Clue 2 says Red is somewhere to the left of Blue. If Blue is at 2 and Red is at 5, Red is to the right of Blue. This violates Clue 2. So Case 1 yields no valid arrangements. Case 2: Yellow is at 5. Since there are exactly two houses between Green and Yellow, Green must be at 2. If Green is at 2, White must be at 1 (since Green is immediately to the right of White). So we have: 1: White 2: Green 3: ? 4: ? 5: Yellow The remaining colors are Red and Blue. They must go in houses 3 and 4. Clue 5 says Blue is not at either end. Houses 3 and 4 are not ends, so this is satisfied. Clue 2 says Red is to the left of Blue. So Red must be at 3, and Blue must be at 4. Let's check all clues for this arrangement: 1: White 2: Green 3: Red 4: Blue 5: Yellow Clue 1: Green is immediately to the right of White. (W at 1, G at 2) - Yes. Clue 2: Red is to the left of Blue. (R at 3, B at 4) - Yes. Clue 3: Yellow is at one of the two ends. (Y at 5) - Yes. Clue 4: Exactly two houses between Green and Yellow. (G at 2, Y at 5, houses 3 and 4 are between) - Yes. Clue 5: Blue is not at either end. (B at 4) - Yes. Are there any other possibilities? Let's double check. Could Yellow be at 1 and Green at 4? We checked this, it leads to W at 3, Y at 1. Remaining houses 2, 5 for R, B. B cannot be at 5, so B at 2, R at 5. But R must be left of B. Contradiction. Could Yellow be at 5 and Green at 2? We checked this, it leads to W at 1, Y at 5. Remaining houses 3, 4 for R, B. B cannot be at 5 (already satisfied). R must be left of B, so R at 3, B at 4. This works. Wait, are there any other interpretations of "exactly two houses between"? If Yellow is at 1, Green is at 4. Houses 2 and 3 are between. That's exactly two houses. If Yellow is at 5, Green is at 2. Houses 3 and 4 are between. That's exactly two houses. Are there any other positions for Yellow? No, clue 3 says Yellow is at one of the two ends. Are there any other positions for Green? No, because the distance between 1 and 4 is 2 houses, and between 2 and 5 is 2 houses. So Green must be at 2 or 4. If Green is at 4, White is at 3. If Green is at 2, White is at 1. Let's quickly review the steps. Colors: R, B, G, Y, W. Positions: 1, 2, 3, 4, 5. Y is at 1 or 5. If Y=1, G=4. W=3. Remaining: 2, 5 for R, B. B cannot be at 5 (end), so B=2, R=5. But R must be left of B. 5 is not left of 2. Invalid. If Y=5, G=2. W=1. Remaining: 3, 4 for R, B. B cannot be at ends (3 and 4 are not ends, so both valid). R must be left of B, so R=3, B=4. Valid. Is there any ambiguity in "The green house is immediately to the right of the white house"? No, it means W is at x, G is at x+1. Is there any ambiguity in "exactly two houses between"? No, \|pos(G) - pos(Y)\| = 3. Is there any ambiguity in "The red house is somewhere to the left of the blue house"? No, pos(R) < pos(B). Let's check if there's any other way. What if Yellow is at 1, Green is at 4, White is at 3. Houses: 1: Y, 2: ?, 3: W, 4: G, 5: ? Colors left: R, B. If B=2, R=5. R is not left of B. If B=5, R=2. B is at an end, which violates clue 5. So no solution for Y=1. What if Yellow is at 5, Green is at 2, White is at 1. Houses: 1: W, 2: G, 3: ?, 4: ?, 5: Y. Colors left: R, B. If R=3, B=4. R is left of B. B is not at an end. Valid. If R=4, B=3. R is not left of B. Invalid. So there is exactly 1 valid arrangement. The arrangement is: House 1: White House 2: Green House 3: Red House 4: Blue House 5: Yellow Let me re-read the prompt to make sure I didn't miss anything. "Five houses stand in a row, numbered 1 to 5 from left to right." "Each house is painted a different color, one of: red, blue, green, yellow, white." "1. The green house is immediately to the right of the white house." -> W, G "2. The red house is somewhere to the left of the blue house (not necessarily adjacent)." -> R < B "3. The yellow house is at one of the two ends of the row." -> Y=1 or Y=5 "4. There are exactly two houses between the green house and the yellow house." -> \|G - Y\| = 3 "5. The blue house is not at either end." -> B != 1, B != 5 Everything seems perfectly straightforward. Let's write down the solution clearly. - State the possible positions for Yellow based on Clue 3 (1 or 5). - Use Clue 4 to find the corresponding positions for Green (4 or 2). - Use Clue 1 to find the corresponding positions for White (3 or 1). - Evaluate the two cases: - Case 1: Y=1, G=4, W=3. Remaining positions for R and B are 2 and 5. Clue 5 says B cannot be 5, so B=2, R=5. But Clue 2 says R must be left of B, which is violated. So this case is invalid. - Case 2: Y=5, G=2, W=1. Remaining positions for R and B are 3 and 4. Clue 2 says R must be left of B, so R=3, B=4. Clue 5 is satisfied since B is not at an end. This case is valid. - Conclude there is exactly 1 valid arrangement. - List the arrangement as house 1 through house 5. - State the final count on its own line. The arrangement: House 1: White House 2: Green House 3: Red House 4: Blue House 5: Yellow Final count: 1 Let me double check the "exactly two houses between" part. If Y is at 1, and G is at 4. Houses between 1 and 4 are 2 and 3. That's exactly two houses. If Y is at 5, and G is at 2. Houses between 2 and 5 are 3 and 4. That's exactly two houses. Yes, this is correct. What if "exactly two houses between" means the distance is 2? No, "two houses between" means there are two houses separating them. For example, between house 1 and house 4, there are houses 2 and 3. That's two houses. If it meant distance is 2, it would be "one house between". So \|G - Y\| = 3 is correct. Let's do a quick sanity check. Colors: R, B, G, Y, W. Arrangement: W, G, R, B, Y. 1. Green immediately right of White: W at 1, G at 2. Yes. 2. Red left of Blue: R at 3, B at 4. Yes. 3. Yellow at an end: Y at 5. Yes. 4. Exactly two houses between Green and Yellow: G at 2, Y at 5. Houses 3 and 4 are between. Yes. 5. Blue not at either end: B at 4. Yes. Everything is consistent. The logic is solid. No other interpretations seem likely. I will structure the output to show the reasoning, list the valid arrangement, and then state the final count. </think> To solve this logic puzzle, we can deduce the positions of the houses step-by-step using the given clues: Step 1: Determine the possible positions for the Yellow and Green houses. * Clue 3 states that the yellow house is at one of the two ends (house 1 or house 5). * Clue 4 states that there are exactly two houses between the green house and the yellow house. This gives us two possible scenarios: * Scenario A: Yellow is in house 1, which means Green must be in house 4. * Scenario B: Yellow is in house 5, which means Green must be in house 2. Step 2: Place the White house. * Clue 1 states that the green house is immediately to the right of the white house (meaning they are a pair: White, Green). * In Scenario A (Green in 4), White must be in house 3. * In Scenario B (Green in 2), White must be in house 1. Step 3: Place the Red and Blue houses. * Clue 5 states that the blue house is not at either end, meaning it cannot be in house 1 or house 5. Let's test Scenario A: * Houses: 1: Yellow, 2: ?, 3: White, 4: Green, 5: ? * The remaining colors for houses 2 and 5 are Red and Blue. * Since Blue cannot be at an end (house 5), Blue must be in house 2, leaving Red in house 5. * However, Clue 2 says the red house is to the left of the blue house. Here, Red (5) is to the right of Blue (2), which violates the clue. * Therefore, Scenario A is invalid. Let's test Scenario B: * Houses: 1: White, 2: Green, 3: ?, 4: ?, 5: Yellow * The remaining colors for houses 3 and 4 are Red and Blue. * Clue 2 says Red is to the left of Blue. Therefore, Red must be in house 3 and Blue must be in house 4. * Let's check Clue 5: Blue is in house 4, which is not an end. This fits perfectly. The only valid arrangement is: House 1: White House 2: Green House 3: Red House 4: Blue House 5: Yellow 1	Matches Ground Truth

Model	Output tokens	Amplification vs Gemini 3.1 Pro (518 tok)	Latency	Run cost
GPT-5.4 Pro	822	1.59x	37.8s	$0.153300
Claude Opus 4.8	985	1.90x	10.1s	$0.025855
Gemini 3.1 Pro	518	1.00x	22.8s	$0.006576
GLM 5.2 (Max)	2,827	5.46x	38.4s	$0.012698

Verifier scope	Result	Boundary
Exhaustive assignment search	120 permutations checked; 1 satisfies all 5 clues	This verifies the puzzle answer, not general reasoning ability
Arithmetic consistency	Unavailable	No independent arithmetic proof

Evidence class	Supported by verifier transfer	Unsupported boundary	Exact promotion test
Constraint solving	Yes — same 120-permutation verifier and prompt	Different verifier/model/version is not transferable	Re-run exact prompt and require the same unique permutation
Arithmetic	No	No independent arithmetic proof	Add a deterministic arithmetic verifier and pass every assertion
Theorem proving	No	No formal proof kernel evidence	Replay in a named proof assistant with accepted proof artifact
Tool-assisted math	No	No tool trace or sandbox evidence	Replay with tool trace, tool outputs, and independently checked final answer

Evidence transfers only at the same prompt, verifier, model version, and dated run boundary. Output-token amplification is recorded output tokens divided by Gemini 3.1 Pro’s 518-token baseline.

Verified 2026-08-08. Luna is the data owner. “Unavailable” means no compatible dated evidence was found; it is never treated as zero or an inferred successor. Dated task evidence · Run this evidence in All AI Ask.

Batch 13 · math verification topology and transfer boundary

1. Verification-topology planner

Topology	Samples	Model token cost	Latency	Checker / human success	Decision
Single sample	1	$0.01	Unavailable	User-supplied	Select only with measured checker/human result
3-sample vote	3	$0.03	Unavailable	User-supplied	Select only with measured checker/human result
5-sample vote	5	$0.05	Unavailable	User-supplied	Select only with measured checker/human result
Model + symbolic checker	1	$0.01	Unavailable	User-supplied	Select only with measured checker/human result
Human review	1	$0.01	Unavailable	User-supplied	Select only with measured checker/human result

The exhaustive constraint-puzzle observation remains an observed result. Checker and human success rates are not inferred from it.

2. Reasoning-output budget and truncation surface

Output expansion	Output tokens	Calculated token bill	Context / max output	Measured speed	Stop rule
1×	2000	$0.01	1,000,000 / 64,000	Unavailable	Stop at cap/truncation; no quality uplift assigned
2×	4000	$0.02	1,000,000 / 64,000	Unavailable	Stop at cap/truncation; no quality uplift assigned
4×	8000	$0.04	1,000,000 / 64,000	Unavailable	Stop at cap/truncation; no quality uplift assigned
8×	16000	$0.07	1,000,000 / 64,000	Unavailable	Stop at cap/truncation; no quality uplift assigned

3. Domain-transfer evidence ledger

Domain	Current evidence	Candidate verdict	Matched unlock test / rubric
arithmetic	Unavailable	Unavailable	Freeze arithmetic prompt, grader, tool policy, and dated multi-model run
algebra	Unavailable	Unavailable	Freeze algebra prompt, grader, tool policy, and dated multi-model run
geometry	Unavailable	Unavailable	Freeze geometry prompt, grader, tool policy, and dated multi-model run
theorem proof	Unavailable	Unavailable	Freeze theorem proof prompt, grader, tool policy, and dated multi-model run
constraint solving	premium:constraint-logic-puzzle · 2026-06-16T20:31:30.728Z	GLM-5.2	Repeat puzzle with independent samples
tool-assisted math	Unavailable	Unavailable	Freeze tool-assisted math prompt, grader, tool policy, and dated multi-model run

Verified 2026-08-08. Data owner: Luna. “Unavailable” means no compatible dated evidence was found; it is not zero or an estimate. Re-verify dated rates, specs, and policy before production use. First-party source · Run this scenario →

Batch 14 · math solver-verifier pairing, evidence strata, and policy loss

1. Solver-verifier pairing matrix

Solver	Verifier	Generation cost	Verification cost	Retry cost	Human review	Success input / decision
cheap solver	model verifier	$0.01	$0.0050	Unavailable	N/A	Unavailable
cheap solver	symbolic verifier	$0.01	Unavailable	Unavailable	N/A	Unavailable
cheap solver	human verifier	$0.01	Unavailable	Unavailable	User-supplied	Unavailable
premium solver	model verifier	$0.01	$0.0050	Unavailable	N/A	Unavailable
premium solver	symbolic verifier	$0.01	Unavailable	Unavailable	N/A	Unavailable
premium solver	human verifier	$0.01	Unavailable	Unavailable	User-supplied	Unavailable

Total pairing cost = generation + verification + retry + human review. Success rates are observed, user-supplied, or Unavailable; no broad winner is emitted from a cost-only row.

2. Difficulty-stratified evidence plan

Difficulty	Prompt count	Arithmetic	Algebra	Geometry	Proof	Constraint/tool	Duplicate runs	Duplicate-run spend	Explicit rubric	Minimum gate
easy	20	4	4	4	4	4	2	$0.80	Correctness 0/1; proof validity; constraint satisfaction; no unsupported claim; abstain when unsolved	Shared dated runs across every declared stratum
medium	20	4	4	4	4	4	2	$0.80	Correctness 0/1; proof validity; constraint satisfaction; no unsupported claim; abstain when unsolved	Shared dated runs across every declared stratum
hard	20	4	4	4	4	4	2	$0.80	Correctness 0/1; proof validity; constraint satisfaction; no unsupported claim; abstain when unsolved	Shared dated runs across every declared stratum

The plan freezes 20 prompts per difficulty (four per declared stratum), two duplicate runs, the displayed duplicate-run spend, and the rubric before testing. A winner requires the shared-coverage gate; puzzle evidence cannot transfer to other domains.

3. Answer-policy loss surface

Action	Wrong-answer loss	Delay loss	Review cost	Confidence calibration	Expected loss / decision
answer	User-supplied	User-supplied	N/A	Unavailable	Unavailable
abstain	User-supplied	User-supplied	N/A	Unavailable	Unavailable
retry	User-supplied	User-supplied	N/A	Unavailable	Unavailable
escalate	User-supplied	User-supplied	User-supplied	Unavailable	Unavailable

Expected loss = wrong-answer cost + delay cost + retry/review cost, weighted by user-supplied outcome frequencies. Calibration is Unavailable until a dated calibration run exists.

Verified 2026-08-08. Data owner: Luna. “Unavailable” means no compatible dated evidence was found; it is not zero or an estimate. Source / registry · Run this scenario →

Batch 15 · math equivalence, tool-assisted evidence, and confidence calibration

1. Answer-equivalence verifier matrix

Answer form	False accept	False reject	Review cost	Rule
exact integer	Unavailable	Unavailable	Unavailable	Frozen cases required
fraction	Unavailable	Unavailable	Unavailable	Frozen cases required
decimal/tolerance	Unavailable	Unavailable	Unavailable	Frozen cases required
units	Unavailable	Unavailable	Unavailable	Frozen cases required
symbolic form	Unavailable	Unavailable	Unavailable	Frozen cases required
proof obligation	Unavailable	Unavailable	Unavailable	Frozen cases required

Formula / rule: report rates over the frozen cases; string mismatch is not model failure when the verifier proves equivalence.

2. Model-only versus calculator/Python/symbolic-tool matched runs

Task family	Model-only	Tool invocation	Latency/errors/retries	Winner
arithmetic	Unavailable	Unavailable	Unavailable	Withheld
algebra	Unavailable	Unavailable	Unavailable	Withheld
geometry	Unavailable	Unavailable	Unavailable	Withheld
constraint	Unavailable	Unavailable	Unavailable	Withheld

Formula / rule: tool-assisted winner requires matched arithmetic/algebra/geometry/constraint runs and separately attributed tool, token, latency, error, retry, and review evidence.

3. Confidence-calibration protocol

Score bin	Frozen n	Predictions/outcomes	Brier/ECE	Duplicate-run cost	Routing
0–.2	20	Unavailable	Unavailable	Unavailable	Prohibited
.2–.4	20	Unavailable	Unavailable	Unavailable	Prohibited
.4–.6	20	Unavailable	Unavailable	Unavailable	Prohibited
.6–.8	20	Unavailable	Unavailable	Unavailable	Prohibited
.8–1.0	20	Unavailable	Unavailable	Unavailable	Prohibited

Formula / rule: Brier = mean((p − outcome)^2); ECE = sum(bin count / n × |mean p − accuracy|). No confidence routing until every bin has dated predictions and outcomes.

Verified 2026-08-08. Data owner: Luna. Source / registry: dated repository pricing and provider records. “Unavailable” means compatible dated evidence is missing; it is not zero, an estimate, or an inferred capability. Run this evidence scenario →

Batch 16 · reasoning diversity, proof localization, and numeric precision

1. Self-consistency diversity audit

Samples	Distinct paths	Correlated wrong answers	Vote outcome	Verifier	Token/time cost	Decision
1	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable	No quality verdict
3	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable	No quality verdict
5	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable	No quality verdict
10	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable	No quality verdict

Formula / rule: agreement = winning vote share, not correctness; correctness requires an independent verifier on the same frozen problem.

2. Proof-step localization suite

Derivation	First invalid step	Justification	False alarms	Repair success	Reviewer effort	Coverage/verdict
valid derivation	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable	Shared coverage required
injected invalid step	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable	Shared coverage required
repair after localization	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable	Shared coverage required

Formula / rule: localization requires identifying the first invalid step on valid and injected-invalid derivations; final-answer match alone is insufficient.

3. Numeric precision and dimensional-analysis ladder

Case	Equivalence rule	Tool/retries	Latency	Spend	Verdict
exact	Unavailable	Unavailable	Unavailable	Unavailable	No quantitative verdict
floating point	Unavailable	Unavailable	Unavailable	Unavailable	No quantitative verdict
tolerance	Unavailable	Unavailable	Unavailable	Unavailable	No quantitative verdict
significant figures	Unavailable	Unavailable	Unavailable	Unavailable	No quantitative verdict
unit conversion	Unavailable	Unavailable	Unavailable	Unavailable	No quantitative verdict
overflow/cancellation	Unavailable	Unavailable	Unavailable	Unavailable	No quantitative verdict

Formula / rule: numeric result is accepted only under the declared exact/tolerance/unit rule; precision handling is not inferred from a final numeric string.

Verified 2026-08-08. Data owner: Luna. Source / registry: dated repository pricing and provider records. “Unavailable” means no compatible dated evidence or observed run; it is not zero or an inferred capability. Run this evidence scenario →

Batch 17 · representation invariance, triage, and error localization

1. Representation-invariance suite

Transformation	Answer equivalence	Proof-path stability	Regression	Retries/latency	Spend
variable rename	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
equation reorder	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
unit change	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
diagram to text	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
irrelevant format	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable

Formula / rule: invariance = equivalent answers across matched transformations, with proof-path stability reported separately; formatting agreement is not correctness.

2. Underspecified-and-inconsistent problem triage

Case	Solve/clarify/abstain	Hallucinated assumptions	False refusals	Corrections	Cost
solvable	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
missing data	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
contradictory	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
multiple answer	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable

Formula / rule: triage correctness = expected solve, clarify, or abstain behavior under the frozen case label; unsupported assumptions are failures, not successful solves.

3. Extraction-to-verification error attribution

Stage	Correctness	Tool calls	Repair scope	Reviewer effort	Total cost
fact extraction	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
equation setup	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
plan	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
calculation	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
unit handling	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
final answer	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable
checker	Unavailable	Unavailable	Unavailable	Unavailable	Unavailable

Formula / rule: error localization assigns the first failed stage among extraction, setup, plan, calculation, units, final answer, and checker; final-answer match alone is insufficient.

Batch 18 · counterexamples, independently checked certificates, and probability reasoning

1. False-conjecture counterexample suite

Domain	Valid witness / minimality	Premises satisfied	Verifier rejection	Retries / reviewer / cost
arithmetic	Unavailable	Unavailable	Unavailable	Unavailable
algebra	Unavailable	Unavailable	Unavailable	Unavailable
geometry	Unavailable	Unavailable	Unavailable	Unavailable
combinatorics	Unavailable	Unavailable	Unavailable	Unavailable
program invariant	Unavailable	Unavailable	Unavailable	Unavailable

Formula / rule: counterexample accepted only when a valid minimal witness satisfies all premises and an independent verifier accepts it.

2. Answer-plus-certificate gate

Task	Answer/objective	Certificate completeness	Independent checker	Gap/invalid witness / spend
constraint	Unavailable	Unavailable	Unavailable	Unavailable
optimization	Unavailable	Unavailable	Unavailable	Unavailable
satisfiability	Unavailable	Unavailable	Unavailable	Unavailable

Formula / rule: gate = answer correct ∧ certificate complete ∧ independent checker accepts; unverifiable final answers do not pass.

3. Probabilistic and base-rate reasoning suite

Case	Equation / numeric answer	Assumptions	Simulation/tool / verifier	Failures/retries / cost
conditional probability	Unavailable	Unavailable	Unavailable	Unavailable
expected value	Unavailable	Unavailable	Unavailable	Unavailable
dependence	Unavailable	Unavailable	Unavailable	Unavailable
calibration	Unavailable	Unavailable	Unavailable	Unavailable
rare event	Unavailable	Unavailable	Unavailable	Unavailable

Formula / rule: correctness requires the stated equation, compatible assumptions, numeric result, and verifier agreement; confidence alone is not correctness.

Verified 2026-08-08. Data owner: Luna. Source / registry: dated repository records and matched-run evidence. “Unavailable” means no compatible dated source or observed run; it is not zero or an inferred capability. Run this Batch 18 evidence scenario →

Batch 19 · causal inference, combinatorial counting, and optimization sensitivity

Observed benchmark window: 2026-08-26 UTC. Every row is a page-specific frozen fixture with controls, field observations, reviewer decision, token measurement, and exact registry cost.

1. Causal-inference suite

Dated matched run / case	Frozen controls	Field-level observation	Reviewer decision	Token measurement	Exact cost
run-20260826-b19-math-01-01 · observational	DAG confounder; backdoor estimand	graph valid; set={Z}; estimate=1.84; checker agree	ACCEPT	6,200 in + 1,420 out	$0.053200
run-20260826-b19-math-01-02 · collider	DAG X→C←Y; forbid conditioning C	conditioned first; checker flagged; repair removed C	ACCEPT repaired; claim false	7,100 in + 1,660 out	$0.061600
run-20260826-b19-math-01-03 · mediation	direct/indirect; assumptions disclosed	simulator delta=0.02; estimand valid	ACCEPT	8,300 in + 1,940 out	$0.072000

Formula / rule: pass=graph∧adjustment∧estimand∧numeric/checker∧assumptions Source: pricing registry verified 2026-08-26. Rate: GPT-5.6 Sol, $4.0000 input/M + $20.0000 output/M.

2. Combinatorial-counting gate

Dated matched run / case	Frozen controls	Field-level observation	Reviewer decision	Token measurement	Exact cost
run-20260826-b19-math-02-01 · permutations	7 books; 2 adjacent	5!×2×6=1,440; brute force=1,440	ACCEPT certificate	4,200 in + 920 out	$0.035200
run-20260826-b19-math-02-02 · occupancy	8 balls / 3 nonempty	inclusion-exclusion=5,796; verifier=5,796	ACCEPT	5,600 in + 1,280 out	$0.048000
run-20260826-b19-math-02-03 · recurrence	F(n)=F(n−1)+F(n−2); F10	closed form=89; verifier=89	ACCEPT	3,800 in + 810 out	$0.031400

Formula / rule: accepted=decomposition∧closed form equals verifier Source: pricing registry verified 2026-08-26. Rate: GPT-5.6 Sol, $4.0000 input/M + $20.0000 output/M.

3. Optimization-sensitivity suite

Dated matched run / case	Frozen controls	Field-level observation	Reviewer decision	Token measurement	Exact cost
run-20260826-b19-math-03-01 · linear	3 coefficients +10%; LP certificate	objective 124→131.2; active set unchanged	ACCEPT local	5,100 in + 1,100 out	$0.042400
run-20260826-b19-math-03-02 · integer	capacity bound −1; integer feasibility	active set changed; objective +7; certificate valid	ACCEPT change	6,400 in + 1,380 out	$0.053200
run-20260826-b19-math-03-03 · scheduling	2 jobs unavailable; recognize infeasible	status=infeasible; certificate matched	ACCEPT no stability claim	5,800 in + 1,260 out	$0.048400

Formula / rule: perturbation joins solution+certificate+active-set/objective delta Source: pricing registry verified 2026-08-26. Rate: GPT-5.6 Sol, $4.0000 input/M + $20.0000 output/M.

Verified 2026-08-08. Data owner: Luna. Run IDs are match keys; missing vendor fields are scoped to their named run. Run the math evidence scenario →

Batch 20 · numerical-precision propagation, graph-algorithmic verification, and hypothesis-test correctness

Observed benchmark window: 2026-08-26 UTC. Every row is a page-specific frozen fixture with visible controls, a distinct field-level source/run identifier, a registry-computed cost or a scoped Unavailable reason — never a blanket matrix.

1. Numerical-precision and rounding-error propagation suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch20-math-m1-r1 · 3-step arithmetic chain	3-step chain with 2-decimal intermediate rounding specified; 400 input tokens; 350 output tokens	Unavailable — no matched precision-propagation run recorded for the 3-step chain as of 2026-08-26	HOLD — deviation/attribution unverified; fixture cost is reproducible from the registry rate	$0.008600
batch20-math-m1-r2 · 6-step arithmetic chain	6-step chain with 2-decimal intermediate rounding specified; 650 input tokens; 600 output tokens	Unavailable — no matched precision-propagation run recorded for the 6-step chain as of 2026-08-26	HOLD — deviation/attribution unverified; fixture cost is reproducible from the registry rate	$0.014600
batch20-math-m1-r3 · 10-step arithmetic chain	10-step chain with 2-decimal intermediate rounding specified; 950 input tokens; 900 output tokens	Unavailable — no matched precision-propagation run recorded for the 10-step chain as of 2026-08-26	HOLD — deviation/attribution unverified; fixture cost is reproducible from the registry rate	$0.021800

Formula / rule: Matched-run cost = frozen arithmetic-chain token bill at the gpt-5.6 Sol registry rate. Final-answer deviation from an exact-arithmetic reference, error-source attribution to a specific step, and unsupported precision claims require a matched run, which is not present in the registry, so only the fixture cost below is reproducible. Source: pricing registry verified 2026-08-26.

2. Graph-theory algorithmic-reasoning suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch20-math-m2-r1 · Shortest-path instance — 12-node graph	12-node weighted graph; requires distance + path certificate; 700 input tokens; 500 output tokens	Unavailable — no matched shortest-path verification run recorded as of 2026-08-26	HOLD — correctness/certificate validity unverified; instance cost is reproducible from the registry rate	$0.012800
batch20-math-m2-r2 · Bipartite-matching instance — 10×10 graph	10×10 bipartite graph; requires matching + augmenting-path certificate; 900 input tokens; 650 output tokens	Unavailable — no matched bipartite-matching verification run recorded as of 2026-08-26	HOLD — correctness/certificate validity unverified; instance cost is reproducible from the registry rate	$0.016600
batch20-math-m2-r3 · Max-flow instance — 15-node network	15-node flow network; requires max-flow value + min-cut certificate; 1,100 input tokens; 800 output tokens	Unavailable — no matched max-flow verification run recorded as of 2026-08-26	HOLD — correctness/certificate validity unverified; instance cost is reproducible from the registry rate	$0.020400

Formula / rule: Matched-run cost = frozen graph-instance token bill at the gpt-5.6 Sol registry rate. Solution correctness against an independent solver, certificate validity, and false-optimality claims require a matched run, which is not present in the registry, so only the instance cost below is reproducible. Source: pricing registry verified 2026-08-26.

3. Statistical-hypothesis-test suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch20-math-m3-r1 · Two-sample scenario — α = 0.05	two-sample dataset, n=30/32; declared significance level 0.05; 550 input tokens; 480 output tokens	Unavailable — no matched hypothesis-test run recorded for the two-sample scenario as of 2026-08-26	HOLD — test-selection correctness unverified; scenario cost is reproducible from the registry rate	$0.011800
batch20-math-m3-r2 · Paired scenario — α = 0.01	paired dataset, n=24; declared significance level 0.01; 500 input tokens; 460 output tokens	Unavailable — no matched hypothesis-test run recorded for the paired scenario as of 2026-08-26	HOLD — test-selection correctness unverified; scenario cost is reproducible from the registry rate	$0.011200
batch20-math-m3-r3 · Multiple-comparison scenario — 5 groups, α = 0.05	5-group dataset, n=20 each; declared family-wise significance level 0.05; 700 input tokens; 620 output tokens	Unavailable — no matched multiple-comparison run recorded as of 2026-08-26	HOLD — correction-application correctness unverified; scenario cost is reproducible from the registry rate	$0.015200

Formula / rule: Matched-run cost = frozen scenario-data token bill at the gpt-5.6 Sol registry rate. Correct test selection, assumption-checking disclosure, p-value interpretation accuracy, multiple-comparison correction, and misinterpretation rate require a matched run, which is not present in the registry, so only the scenario cost below is reproducible. Source: pricing registry verified 2026-08-26.

Verified 2026-08-08. Data owner: Luna. Run identifiers are per-row match keys; an Unavailable field names the exact missing dated record or matched run and is never inferred as zero. Run the math-and-reasoning evidence scenario →

Batch 21 · symbolic-algebra correctness, unit-conversion/dimensional-analysis correctness, and seeded-error detection-and-recovery

1. Symbolic-algebra manipulation correctness suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch21-math-m1-r1 · Equation-simplification task	frozen rational-expression simplification task; 350 input tokens; 300 output tokens	Unavailable — no matched CAS-reference-scoring run recorded for the simplification task as of 2026-08-26	HOLD — correctness/false-simplification rate unverified; fixture cost is reproducible from the registry rate	$0.007400
batch21-math-m1-r2 · Factoring task	frozen polynomial-factoring task; 380 input tokens; 320 output tokens	Unavailable — no matched CAS-reference-scoring run recorded for the factoring task as of 2026-08-26	HOLD — correctness/false-simplification rate unverified; fixture cost is reproducible from the registry rate	$0.007920
batch21-math-m1-r3 · Expansion task	frozen multi-term expansion task; 400 input tokens; 340 output tokens	Unavailable — no matched CAS-reference-scoring run recorded for the expansion task as of 2026-08-26	HOLD — correctness/false-simplification rate unverified; fixture cost is reproducible from the registry rate	$0.008400

Formula / rule: Matched-run cost = frozen equation-task token bill at the gpt-5.6 Sol registry rate. Final-expression correctness and intermediate-step validity against an independent computer-algebra-system reference, plus false-simplification rate, require a matched scoring run, which is not present in the registry, so only the fixture cost below is reproducible. Source: pricing registry verified 2026-08-26.

2. Unit-conversion and dimensional-analysis correctness suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch21-math-m2-r1 · Physics word problem — mixed SI/imperial units	frozen kinematics problem requiring mixed SI/imperial conversion; 450 input tokens; 380 output tokens	Unavailable — no matched dimensional-analysis-scoring run recorded for the physics problem as of 2026-08-26	HOLD — conversion-factor correctness unverified; problem cost is reproducible from the registry rate	$0.009400
batch21-math-m2-r2 · Engineering word problem — multi-step unit chain	frozen fluid-flow problem requiring a 3-step unit-conversion chain; 520 input tokens; 420 output tokens	Unavailable — no matched dimensional-analysis-scoring run recorded for the engineering problem as of 2026-08-26	HOLD — conversion-factor correctness unverified; problem cost is reproducible from the registry rate	$0.010480
batch21-math-m2-r3 · Compound-unit word problem — derived-unit consistency	frozen energy/power problem requiring derived-unit consistency; 560 input tokens; 460 output tokens	Unavailable — no matched dimensional-analysis-scoring run recorded for the compound-unit problem as of 2026-08-26	HOLD — conversion-factor correctness unverified; problem cost is reproducible from the registry rate	$0.011440

Formula / rule: Matched-run cost = frozen word-problem token bill at the gpt-5.6 Sol registry rate. Dimensional consistency, correct conversion-factor application, and final-answer accuracy against a worked reference solution require a matched scoring run, which is not present in the registry, so only the problem cost below is reproducible. Source: pricing registry verified 2026-08-26.

3. Seeded-error-detection-and-recovery suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch21-math-m3-r1 · 3-step solution with a seeded sign error	frozen 3-step solution, error seeded at step 2; 500 input tokens; 300 output tokens	Unavailable — no matched error-detection-scoring run recorded for the 3-step solution as of 2026-08-26	HOLD — detection/propagation outcome unverified; presented-solution cost is reproducible from the registry rate	$0.008000
batch21-math-m3-r2 · 6-step solution with a seeded unit-conversion error	frozen 6-step solution, error seeded at step 4; 750 input tokens; 450 output tokens	Unavailable — no matched error-detection-scoring run recorded for the 6-step solution as of 2026-08-26	HOLD — detection/propagation outcome unverified; presented-solution cost is reproducible from the registry rate	$0.012000
batch21-math-m3-r3 · 10-step solution with a seeded algebraic-substitution error	frozen 10-step solution, error seeded at step 7; 1,050 input tokens; 620 output tokens	Unavailable — no matched error-detection-scoring run recorded for the 10-step solution as of 2026-08-26	HOLD — detection/propagation outcome unverified; presented-solution cost is reproducible from the registry rate	$0.016600

Formula / rule: Matched-run cost = frozen multi-step-solution token bill at the gpt-5.6 Sol registry rate, presenting a deliberately incorrect intermediate step. Whether the model detects, names, and corrects the error versus silently propagating it to a wrong final answer requires a matched scoring run, which is not present in the registry, so only the presented-solution cost below is reproducible. Source: pricing registry verified 2026-08-26.

Batch 22 · answer-format compliance, matrix/linear-algebra operation correctness, and distribution parameter estimation

1. Answer-format compliance suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch22-math-m1-r1 · Exact-fraction format required	frozen arithmetic problem; declared exact-fraction output format; 300 input tokens; 200 output tokens	Unavailable — no matched format-compliance-scoring run recorded for the exact-fraction fixture as of 2026-08-26	HOLD — format-compliance rate unverified; fixture cost is reproducible from the registry rate	$0.005200
batch22-math-m1-r2 · Decimal-to-3-places format required	frozen arithmetic problem; declared 3-decimal-place output format; 320 input tokens; 210 output tokens	Unavailable — no matched format-compliance-scoring run recorded for the decimal-places fixture as of 2026-08-26	HOLD — format-compliance rate unverified; fixture cost is reproducible from the registry rate	$0.005480
batch22-math-m1-r3 · Boxed-final-answer format required	frozen multi-step problem; declared boxed-final-answer output format; 450 input tokens; 380 output tokens	Unavailable — no matched format-compliance-scoring run recorded for the boxed-answer fixture as of 2026-08-26	HOLD — format-compliance rate unverified; fixture cost is reproducible from the registry rate	$0.009400

Formula / rule: Matched-run cost = frozen problem-plus-format-instruction token bill at the gpt-5.6 Sol registry rate against a declared output format (exact fraction, decimal-to-N-places, boxed final answer). Adherence to the declared format versus a numerically correct but non-compliant answer requires a matched format-scoring run, which is not present in the registry, so only the fixture cost below is reproducible. Source: pricing registry verified 2026-08-26.

2. Matrix/linear-algebra operation correctness suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch22-math-m2-r1 · 2×2 matrix — determinant and inverse	frozen 2×2 matrix task requesting determinant and inverse; 300 input tokens; 250 output tokens	Unavailable — no matched linear-algebra-reference-scoring run recorded for the 2×2 fixture as of 2026-08-26	HOLD — correctness unverified; fixture cost is reproducible from the registry rate	$0.006200
batch22-math-m2-r2 · 4×4 matrix — eigenvalues	frozen 4×4 matrix task requesting the eigenvalue set; 450 input tokens; 400 output tokens	Unavailable — no matched linear-algebra-reference-scoring run recorded for the 4×4 fixture as of 2026-08-26	HOLD — correctness unverified; fixture cost is reproducible from the registry rate	$0.009800
batch22-math-m2-r3 · 6×6 matrix — determinant, inverse, and eigenvalues	frozen 6×6 matrix task requesting determinant, inverse, and eigenvalue set; 650 input tokens; 600 output tokens	Unavailable — no matched linear-algebra-reference-scoring run recorded for the 6×6 fixture as of 2026-08-26	HOLD — correctness and numerical-stability drift unverified; fixture cost is reproducible from the registry rate	$0.014600

Formula / rule: Matched-run cost = frozen matrix-task token bill at the gpt-5.6 Sol registry rate. Correctness of the computed determinant, inverse, or eigenvalue set against an independent linear-algebra reference, and numerical-stability drift at larger matrix sizes, require a matched scoring run, which is not present in the registry, so only the fixture cost below is reproducible. Source: pricing registry verified 2026-08-26.

3. Distribution parameter-estimation correctness suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch22-math-m3-r1 · 10-point dataset — normal-distribution mean/variance	frozen 10-point dataset; requested mean and variance estimate; 350 input tokens; 220 output tokens	Unavailable — no matched parameter-estimation-scoring run recorded for the 10-point dataset as of 2026-08-26	HOLD — estimate correctness unverified; fixture cost is reproducible from the registry rate	$0.005800
batch22-math-m3-r2 · 30-point dataset — linear-regression coefficients	frozen 30-point dataset; requested slope/intercept estimate; 600 input tokens; 380 output tokens	Unavailable — no matched parameter-estimation-scoring run recorded for the 30-point dataset as of 2026-08-26	HOLD — estimate correctness unverified; fixture cost is reproducible from the registry rate	$0.010000
batch22-math-m3-r3 · 80-point dataset — Poisson-rate parameter	frozen 80-point count dataset; requested Poisson rate-parameter estimate; 900 input tokens; 500 output tokens	Unavailable — no matched parameter-estimation-scoring run recorded for the 80-point dataset as of 2026-08-26	HOLD — estimate correctness unverified; fixture cost is reproducible from the registry rate	$0.013600

Formula / rule: Matched-run cost = frozen dataset-plus-task token bill at the gpt-5.6 Sol registry rate. Correctness of estimated distribution parameters (mean, variance, or fitted-model parameters) against a computed ground-truth reference requires a matched scoring run, which is not present in the registry, so only the fixture cost below is reproducible. Source: pricing registry verified 2026-08-26.

Batch 23 · answer-format compliance, matrix/linear-algebra operation correctness, and distribution parameter estimation

1. Answer-format compliance suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch23-math-m1-r1 · Exact-fraction format required	frozen arithmetic problem; declared exact-fraction output format; 300 input tokens; 200 output tokens	Unavailable — no matched format-compliance-scoring run recorded for the exact-fraction fixture as of 2026-08-26	HOLD — format-compliance rate unverified; fixture cost is reproducible from the registry rate	$0.005200
batch23-math-m1-r2 · Decimal-to-3-places format required	frozen arithmetic problem; declared 3-decimal-place output format; 320 input tokens; 210 output tokens	Unavailable — no matched format-compliance-scoring run recorded for the decimal-places fixture as of 2026-08-26	HOLD — format-compliance rate unverified; fixture cost is reproducible from the registry rate	$0.005480
batch23-math-m1-r3 · Boxed-final-answer format required	frozen multi-step problem; declared boxed-final-answer output format; 450 input tokens; 380 output tokens	Unavailable — no matched format-compliance-scoring run recorded for the boxed-answer fixture as of 2026-08-26	HOLD — format-compliance rate unverified; fixture cost is reproducible from the registry rate	$0.009400

2. Matrix/linear-algebra operation correctness suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch23-math-m2-r1 · 2×2 matrix — determinant and inverse	frozen 2×2 matrix task requesting determinant and inverse; 300 input tokens; 250 output tokens	Unavailable — no matched linear-algebra-reference-scoring run recorded for the 2×2 fixture as of 2026-08-26	HOLD — correctness unverified; fixture cost is reproducible from the registry rate	$0.006200
batch23-math-m2-r2 · 4×4 matrix — eigenvalues	frozen 4×4 matrix task requesting the eigenvalue set; 450 input tokens; 400 output tokens	Unavailable — no matched linear-algebra-reference-scoring run recorded for the 4×4 fixture as of 2026-08-26	HOLD — correctness unverified; fixture cost is reproducible from the registry rate	$0.009800
batch23-math-m2-r3 · 6×6 matrix — determinant, inverse, and eigenvalues	frozen 6×6 matrix task requesting determinant, inverse, and eigenvalue set; 650 input tokens; 600 output tokens	Unavailable — no matched linear-algebra-reference-scoring run recorded for the 6×6 fixture as of 2026-08-26	HOLD — correctness and numerical-stability drift unverified; fixture cost is reproducible from the registry rate	$0.014600

3. Distribution parameter-estimation correctness suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch23-math-m3-r1 · 10-point dataset — normal-distribution mean/variance	frozen 10-point dataset; requested mean and variance estimate; 350 input tokens; 220 output tokens	Unavailable — no matched parameter-estimation-scoring run recorded for the 10-point dataset as of 2026-08-26	HOLD — estimate correctness unverified; fixture cost is reproducible from the registry rate	$0.005800
batch23-math-m3-r2 · 30-point dataset — linear-regression coefficients	frozen 30-point dataset; requested slope/intercept estimate; 600 input tokens; 380 output tokens	Unavailable — no matched parameter-estimation-scoring run recorded for the 30-point dataset as of 2026-08-26	HOLD — estimate correctness unverified; fixture cost is reproducible from the registry rate	$0.010000
batch23-math-m3-r3 · 80-point dataset — Poisson-rate parameter	frozen 80-point count dataset; requested Poisson rate-parameter estimate; 900 input tokens; 500 output tokens	Unavailable — no matched parameter-estimation-scoring run recorded for the 80-point dataset as of 2026-08-26	HOLD — estimate correctness unverified; fixture cost is reproducible from the registry rate	$0.013600

Batch 24 · ordinary differential equations, numerical integration/error bounds, and finite-game equilibria

Observed benchmark window: 2026-08-27 UTC. Every row is a frozen fixture with visible controls, a distinct field-level source/run ID, a registry-computed baseline or scoped Unavailable state, and a named decision boundary.

1. ordinary-differential-equation suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch24-math-m1-r1 · Small fixture	Fixed separable/smooth/dominance problem; 500 input; 350 output tokens	Unavailable — no matched ordinary-differential-equation suite small run or dated rate recorded as of 2026-08-27	HOLD — reference correctness unverified; fixture cost is reproducible	$0.009000
batch24-math-m1-r2 · Medium fixture	Fixed IVP/oscillatory/mixed-game problem; 1,200 input; 800 output tokens	Unavailable — no matched ordinary-differential-equation suite medium run or dated rate recorded as of 2026-08-27	HOLD — residual/convergence or equilibrium validity unverified	$0.020800
batch24-math-m1-r3 · Hard fixture	Fixed stiff/improper/sequential-game problem; 2,500 input; 1,500 output tokens	Unavailable — no matched ordinary-differential-equation suite hard run or dated rate recorded as of 2026-08-27	HOLD — invalid claims, solver exploitability, and spend unverified	$0.040000

Formula / scoring rule: Matched-run cost = frozen math fixture token bill at the gpt-5.6 Sol registry rate. Solver residuals, convergence, equilibrium validity, and correction counts require the matched reference run and are not inferred from a correct-looking answer. Source: pricing registry verified 2026-08-27.

2. numerical-integration and error-bound gate

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch24-math-m2-r1 · Small fixture	Fixed separable/smooth/dominance problem; 500 input; 350 output tokens	Unavailable — no matched numerical-integration and error-bound gate small run or dated rate recorded as of 2026-08-27	HOLD — reference correctness unverified; fixture cost is reproducible	$0.009000
batch24-math-m2-r2 · Medium fixture	Fixed IVP/oscillatory/mixed-game problem; 1,200 input; 800 output tokens	Unavailable — no matched numerical-integration and error-bound gate medium run or dated rate recorded as of 2026-08-27	HOLD — residual/convergence or equilibrium validity unverified	$0.020800
batch24-math-m2-r3 · Hard fixture	Fixed stiff/improper/sequential-game problem; 2,500 input; 1,500 output tokens	Unavailable — no matched numerical-integration and error-bound gate hard run or dated rate recorded as of 2026-08-27	HOLD — invalid claims, solver exploitability, and spend unverified	$0.040000

3. finite-game equilibrium benchmark

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost (registry-computed or Unavailable)
batch24-math-m3-r1 · Small fixture	Fixed separable/smooth/dominance problem; 500 input; 350 output tokens	Unavailable — no matched finite-game equilibrium benchmark small run or dated rate recorded as of 2026-08-27	HOLD — reference correctness unverified; fixture cost is reproducible	$0.009000
batch24-math-m3-r2 · Medium fixture	Fixed IVP/oscillatory/mixed-game problem; 1,200 input; 800 output tokens	Unavailable — no matched finite-game equilibrium benchmark medium run or dated rate recorded as of 2026-08-27	HOLD — residual/convergence or equilibrium validity unverified	$0.020800
batch24-math-m3-r3 · Hard fixture	Fixed stiff/improper/sequential-game problem; 2,500 input; 1,500 output tokens	Unavailable — no matched finite-game equilibrium benchmark hard run or dated rate recorded as of 2026-08-27	HOLD — invalid claims, solver exploitability, and spend unverified	$0.040000

Verified 2026-08-08. Data owner: Luna. Unavailable fields name their exact missing dated record or matched run and are never inferred as zero. Run the math evidence scenario →

Batch 25 · Partial-differential-equation conditions, complex-analysis contours/residues, and linear-control-system design

Observed benchmark window: 2026-08-27 UTC. Frozen inputs, field-level run IDs, reproducible formulas, provenance, and fail-closed evidence decisions are rendered in the initial server response.

1. Partial-differential-equation boundary/initial-condition suite

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost breakdown
batch25-math-m1-r1 · Heat equation · observed 2026-08-27	Frozen heat problem; classification, separation method, boundary satisfaction, residual, and domain	heat equation u_t = 0.5u_xx on [0,1]: Dirichlet residual max 2.1e-10; initial residual 4.8e-9; boundary residuals 3.2e-10/2.7e-10; 41/41 grid checks · run batch25-math-m1-r1 · observed 2026-08-27	PASS — heat PDE, boundary conditions, and initial condition satisfy the numeric residual gate	model 1320×$4.00/M + 612×$20.00/M = $0.017520; specialized units = $0.000000; total = $0.017520
batch25-math-m1-r2 · Wave equation · observed 2026-08-27	Frozen wave problem; initial conditions, characteristic method, residual, and repairs	wave equation u_tt = 4u_xx: initial displacement residual 7.4e-9; velocity residual 8.1e-9; boundary max 1.6e-8; 36/36 characteristic checks · run batch25-math-m1-r2 · observed 2026-08-27	PASS — wave initial/boundary conditions and characteristic solution agree within tolerance	model 1510×$4.00/M + 704×$20.00/M = $0.020120; specialized units = $0.000000; total = $0.020120
batch25-math-m1-r3 · Laplace/transport · observed 2026-08-27	Frozen Laplace/transport problem; condition satisfaction, reference residual, and accepted cost	Laplace equation on unit square: interior residual max 3.6e-7; four-edge boundary residual 2.2e-6; corner compatibility fail 1/4; 2 invalid domain claims · run batch25-math-m1-r3 · observed 2026-08-27	BOUNDARY — residuals exceed the 1e-7 gate and corner data are incompatible; reject solution	model 1690×$4.00/M + 786×$20.00/M = $0.022480; specialized units = $0.000000; total = $0.022480

Formula / scoring rule: Score = correct PDE classification/method + boundary/initial-condition satisfaction − normalized residual against symbolic/numeric reference − invalid domain claims; cost is matched run bill. Source: pricing registry and dated evidence index verified 2026-08-27.

2. Complex-analysis contour and residue benchmark

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost breakdown
batch25-math-m2-r1 · Pole-order fixture · observed 2026-08-27	Frozen pole-order contour; singularities, residues, orientation, and numeric check	contour \|z\|=2, poles z=0 and z=1: orders 2/1 identified; CCW orientation; residue sum numeric 3.14159265i; quadrature 3.14159264i; relative error 3.2e-9 · run batch25-math-m2-r1 · observed 2026-08-27	PASS — residue theorem result matches independent contour quadrature	model 1280×$4.00/M + 598×$20.00/M = $0.017080; specialized units = $0.000000; total = $0.017080
batch25-math-m2-r2 · Branch-cut fixture · observed 2026-08-27	Frozen branch-cut contour; branch convention, validity, and correction record	branch-cut contour for Log(z) on principal branch: cut and indentation stated; orientation valid; residue contribution 0.78539816; mpmath check 0.78539817; relative error 1.3e-8 · run batch25-math-m2-r2 · observed 2026-08-27	PASS — branch convention, contour validity, and numeric check are mutually consistent	model 1460×$4.00/M + 684×$20.00/M = $0.019520; specialized units = $0.000000; total = $0.019520
batch25-math-m2-r3 · Real-integral/mapping fixture · observed 2026-08-27	Frozen real integral/mapping; contour result, independent numeric check, and spend	real-integral mapping with four finite poles: pole orders 1/2/1/1; contour orientation correct; analytic result 2.46740110; 80-digit numeric check 2.46739891; relative error 8.9e-7 · run batch25-math-m2-r3 · observed 2026-08-27	BOUNDARY — numeric discrepancy exceeds the 1e-8 check threshold; require a corrected residue calculation	model 1720×$4.00/M + 808×$20.00/M = $0.023040; specialized units = $0.000000; total = $0.023040

Formula / scoring rule: Score = singularity/pole-order identification + valid contour + residue/orientation + branch convention + independent numeric check − reviewer corrections; spend uses returned tokens. Source: pricing registry and dated evidence index verified 2026-08-27.

3. Linear-control-system stability and design gate

Frozen fixture / matched run	Controls (visible inputs)	Field observation	Decision / boundary	Cost breakdown
batch25-math-m3-r1 · Transfer function · observed 2026-08-27	Frozen transfer function; poles, Routh/Nyquist reasoning, and stability verdict	transfer function poles {-1,-3}: Routh first column positive; Nyquist encirclements 0; step overshoot 0%; settling time 3.96 s; stable verdict matched · run batch25-math-m3-r1 · observed 2026-08-27	PASS — pole, frequency-domain, and time-domain checks all confirm stability	model 1260×$4.00/M + 586×$20.00/M = $0.016760; specialized units = $0.000000; total = $0.016760
batch25-math-m3-r2 · State space · observed 2026-08-27	Frozen state-space system; controllability/observability and eigenvalue extraction	state-space A eigenvalues {-0.4,-2.1}; controllability rank 2/2; observability rank 2/2; numerical eigenvalue residual 2.6e-12; stable · run batch25-math-m3-r2 · observed 2026-08-27	PASS — both state channels are controllable/observable and eigenvalues are reproduced numerically	model 1480×$4.00/M + 692×$20.00/M = $0.019760; specialized units = $0.000000; total = $0.019760
batch25-math-m3-r3 · Closed-loop design · observed 2026-08-27	Frozen design request; gain/pole placement, time-domain validation, corrections, and cost	pole-placement target {-2,-4}: Ackermann gain [6, 4]; closed-loop poles {-2.0003,-3.9997}; max pole residual 3e-4; step settling 2.11 s; simulation diverged under sign-flipped gain control · run batch25-math-m3-r3 · observed 2026-08-27	BOUNDARY — nominal placement passes, but sign-flip robustness check diverges; publish with implementation-sign restriction	model 1760×$4.00/M + 824×$20.00/M = $0.023520; specialized units = $0.000000; total = $0.023520

Formula / scoring rule: Gate score = poles/eigenvalues + controllability/observability + Routh/Nyquist reasoning + gain/pole placement + time-domain validation − unsafe claims; cost is matched accepted-run spend. Source: pricing registry and dated evidence index verified 2026-08-27.

Verified 2026-08-08. Data owner: Luna. Specialized rates and unmatched observations are never inferred from a base modality. Run the math evidence scenario →

Batch 26 · Real analysis, abstract algebra, and information-theory reasoning

Frozen verification window: 2026-08-27 UTC. Every row is an initial-response fixture with a field-level run ID, visible controls, method, result or narrowly scoped unavailable state, and dated provenance.

1. Real-analysis limit and convergence suite

Frozen fixture / run	Visible controls	Field-level observation	Decision boundary	Reproducible cost / state
epsilon-delta limit `batch26-math-m1-r1` observed 2026-08-27	epsilon; witness delta; domain	quantifiers 5/5; delta bound valid; theorem preconditions 4/4; verifier PASS	PASS — constructive proof accepted	tokens: (1440×$2.50 + 620×$10.00)/1M = $0.009800
uniform convergence `batch26-math-m1-r2` observed 2026-08-27	sequence/series; interchange condition; counterexample	bound 7/7; interchange preconditions 6/6; numeric check agrees	PASS — conditions explicitly stated	tokens: (1820×$2.50 + 770×$10.00)/1M = $0.012250
limit interchange edge case `batch26-math-m1-r3` observed 2026-08-27	counterexample; theorem preconditions; reviewer	counterexample valid; theorem preconditions 2/4; proof gap remains	BOUNDARY — reject numerical-looking conclusion without proof	tokens: (2140×$2.50 + 918×$10.00)/1M = $0.014530

Formula / scoring rule: Score = quantifier order + witness/bound + theorem preconditions + counterexample handling + independent check − proof gaps; numerical approximation never substitutes for proof. Source: pricing registry and dated evidence index verified 2026-08-27.

2. Abstract-algebra structure benchmark

Frozen fixture / run	Visible controls	Field-level observation	Decision boundary	Reproducible cost / state
group/homomorphism `batch26-math-m2-r1` observed 2026-08-27	kernel/image; axioms; verifier	axioms 8/8; kernel/image 6/6; homomorphism proof 4/4	PASS — verifier and reviewer agree	tokens: (1560×$2.50 + 668×$10.00)/1M = $0.010580
ring/ideal quotient `batch26-math-m2-r2` observed 2026-08-27	ideal test; quotient operation; cosets	ideal closure 9/9; coset arithmetic 12/12; operation well-defined	PASS — quotient construction valid	tokens: (1980×$2.50 + 812×$10.00)/1M = $0.013070
field/isomorphism counterexample `batch26-math-m2-r3` observed 2026-08-27	isomorphism claim; counterexample; repairs	counterexample valid; proof gaps 2; reviewer correction 1; repair incomplete	BOUNDARY — do not accept unclosed isomorphism claim	tokens: (2260×$2.50 + 946×$10.00)/1M = $0.015110

Formula / scoring rule: Score = axiom use + subgroup/ideal tests + kernel/image + isomorphism validity + coset arithmetic + counterexample quality − proof gaps. Source: pricing registry and dated evidence index verified 2026-08-27.

3. Information-theory calculation and reasoning gate

Frozen fixture / run	Visible controls	Field-level observation	Decision boundary	Reproducible cost / state
entropy/mutual information `batch26-math-m3-r1` observed 2026-08-27	distribution; log base; units	normalization 8/8; bits/nats explicit; reference delta 2.1e-10	PASS — calculation and interpretation agree	tokens: (1380×$2.50 + 584×$10.00)/1M = $0.009290
KL/source coding `batch26-math-m3-r2` observed 2026-08-27	support; inequality; code bound	support 12/12; KL nonnegative; code bound 4/4; verifier PASS	PASS — assumptions and bound visible	tokens: (1760×$2.50 + 742×$10.00)/1M = $0.011820
channel capacity `batch26-math-m3-r3` observed 2026-08-27	optimizer; units; unsafe extrapolation review	distribution normalized; optimizer 3/4; one unsupported extrapolation; repair pending	BOUNDARY — hold until optimizer and extrapolation are corrected	tokens: (2180×$2.50 + 906×$10.00)/1M = $0.014510

Formula / scoring rule: Gate = normalized distribution + log base/units + independence assumptions + inequality bounds + optimizer validity + reference calculation − unsafe extrapolation. Source: pricing registry and dated evidence index verified 2026-08-27.

Verified 2026-08-08. Data owner: Luna. Missing specialized units, rates, and matched runs are never inferred from a neighboring modality or provider. Run the math Batch 26 evidence scenario →

Batch 27 · Number theory, point-set topology, and category-theory reasoning gates

Frozen verification window: 2026-08-27 UTC. Every row is an initial-response fixture with visible inputs, a field-level run ID, a reproducible method/result or narrowly scoped unavailable state, dated provenance, and a decision boundary.

1. Elementary-number-theory reasoning suite

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
congruence and divisibility `batch27-math-m1-r1` observed 2026-08-27	mod 17; Euclidean witness; exact verifier	6/6 transformations valid; witness verifies; no gap	PASS — constructive modular proof accepted	$0.014850 = (2420×$2.50 + 880×$10.00)/1M
Diophantine and multiplicative function `batch27-math-m1-r2` observed 2026-08-27	integer domain; factorization; counterexample control	domain stated; 4/4 cases exact; one unsupported generalization repaired	PASS WITH REPAIR — final claim is scoped	$0.017850 = (3060×$2.50 + 1020×$10.00)/1M
quadratic residue edge case `batch27-math-m1-r3` observed 2026-08-27	Legendre symbol; prime precondition; independent arithmetic	precondition omitted; numeric answer happens to match; proof not accepted	BOUNDARY — numerical agreement cannot replace proof	Unavailable — accepted proof after prime-precondition correction

Formula / scoring rule: Score = theorem preconditions + modular transformations + witness/counterexample validity + exact verification − proof gaps. Source: pricing registry and dated evidence index verified 2026-08-27.

2. Point-set-topology proof benchmark

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
basis and continuity `batch27-math-m2-r1` observed 2026-08-27	basis definition; epsilon-free open sets; 12 fixtures	12/12 quantifiers correct; preimages open; reviewer PASS	PASS — continuity proof is topological, not numerical	$0.016350 = (2780×$2.50 + 940×$10.00)/1M
compactness/product/subspace `batch27-math-m2-r2` observed 2026-08-27	finite subcover; product topology; subspace counterexample	finite subcover valid; product precondition stated; 8/8 accepted	PASS — product and subspace reasoning agrees	$0.020450 = (3460×$2.50 + 1180×$10.00)/1M
separation axiom edge case `batch27-math-m2-r3` observed 2026-08-27	T1/T2 definitions; counterexample; reviewer	counterexample valid; implication direction reversed in first proof; repair incomplete	BOUNDARY — hold until implication is corrected	Unavailable — accepted topology proof after direction repair

Formula / scoring rule: Score = definition use + quantifier order + open/closed reasoning + theorem preconditions + counterexample quality − proof gaps. Source: pricing registry and dated evidence index verified 2026-08-27.

3. Category-theory diagram and universal-property gate

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
functor/natural transformation `batch27-math-m3-r1` observed 2026-08-27	objects/morphisms; two commuting squares; variance	types 16/16; squares commute; variance covariant correct	PASS — diagram is well typed	$0.018200 = (3120×$2.50 + 1040×$10.00)/1M
product/pullback `batch27-math-m3-r2` observed 2026-08-27	universal cone; existence/uniqueness; pullback diagram	cone commutes; uniqueness proof 3/3; reviewer accepted	PASS — universal property is explicit	$0.022900 = (3880×$2.50 + 1320×$10.00)/1M
adjunction/Yoneda edge case `batch27-math-m3-r3` observed 2026-08-27	unit/counit; hom-set direction; representability counterexample	hom-set variance reversed; counterexample repair does not close	BOUNDARY — do not accept a typed-looking but reversed diagram	Unavailable — accepted universal-property proof after variance repair

Formula / scoring rule: Gate = object/morphism typing + commutativity + existence/uniqueness + variance direction + counterexample validity − repairs. Source: pricing registry and dated evidence index verified 2026-08-27.

Verified 2026-08-08. Data owner: Luna. Missing specialized units, rates, and matched runs are never inferred from a neighboring modality or provider. Run the math Batch 27 evidence scenario →

Batch 28 · Measure theory, differential geometry, and mathematical logic gates

Frozen verification window: 2026-08-27 UTC. These are server-rendered matched fixtures, not live estimates. Each row exposes inputs, a reproducible formula/result or a narrowly scoped missing record, dated provenance, and a decision boundary.

1. Measure-theory reasoning suite

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
measurability and convergence `batch28-math-m1-r1` observed 2026-08-27	sigma-algebra; dominated convergence; 12 fixtures	12/12 definitions; domination precondition stated; reviewer PASS	PASS — theorem preconditions are explicit	$0.016950 = (2860×$2.50 + 980×$10.00)/1M
product measure/Radon–Nikodym `batch28-math-m1-r2` observed 2026-08-27	Fubini/Tonelli; density; product space	density computed; one almost-everywhere qualifier repaired	PASS WITH REPAIR — retain a.e. scope	$0.020900 = (3640×$2.50 + 1180×$10.00)/1M
counterexample edge `batch28-math-m1-r3` observed 2026-08-27	claimed convergence; adversarial sequence	counterexample is numerically plausible but proof misses measurability	BOUNDARY — numerical intuition cannot close proof	Unavailable — accepted proof with measurability check

Formula / scoring rule: Score = definitions/preconditions + quantifier order + a.e. distinctions + bounds + independent proof check − proof gaps. Source: pricing registry and dated evidence index verified 2026-08-27.

2. Differential-geometry benchmark

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
manifold charts and tangent maps `batch28-math-m2-r1` observed 2026-08-27	two charts; Jacobian; tangent/cotangent types	transition maps compose; 16/16 tensor types valid	PASS — coordinate change is explicit	$0.018200 = (3120×$2.50 + 1040×$10.00)/1M
metric, connection, geodesic `batch28-math-m2-r2` observed 2026-08-27	Christoffel symbols; sign convention; geodesic check	geodesic residual 0; curvature sign corrected in review	PASS WITH REPAIR — state sign convention	$0.022350 = (3980×$2.50 + 1240×$10.00)/1M
differential-form identity `batch28-math-m2-r3` observed 2026-08-27	chart boundary; wedge/d; theorem precondition	identity holds on chart but global precondition absent	BOUNDARY — local calculation is not global proof	Unavailable — global manifold precondition and accepted proof

Formula / scoring rule: Score = coordinate invariance + tensor typing + sign convention + computation checks + theorem preconditions − proof gaps. Source: pricing registry and dated evidence index verified 2026-08-27.

3. Mathematical-logic and model-theory gate

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
propositional/first-order derivations `batch28-math-m3-r1` observed 2026-08-27	12 derivations; variable binding; proof checker	12/12 syntax valid; 11/12 semantic entailments verified	PASS WITH REPAIR — correct one binding error	$0.016050 = (2740×$2.50 + 920×$10.00)/1M
compactness/completeness structures `batch28-math-m3-r2` observed 2026-08-27	finite models; theory; witness construction	finite satisfiability and witness agree; theorem conditions listed	PASS — model and proof are separated	$0.020750 = (3660×$2.50 + 1160×$10.00)/1M
Löwenheim–Skolem edge `batch28-math-m3-r3` observed 2026-08-27	countability claim; structure/countermodel	countermodel repair does not establish required cardinality	BOUNDARY — do not accept an untyped model claim	Unavailable — proof-checker acceptance for cardinality argument

Formula / scoring rule: Gate = syntax/semantics separation + binding + model/countermodel validity + theorem applicability − proof repairs. Source: pricing registry and dated evidence index verified 2026-08-27.

Verified 2026-08-08. Data owner: Luna. Missing specialized units, rates, and matched runs are never inferred from a neighboring modality or provider. Run the math Batch 28 evidence scenario →

Batch 29 · Functional analysis, stochastic calculus, and algebraic geometry

Frozen verification window: 2026-08-27 UTC. These are server-rendered matched fixtures, not live estimates. Each row exposes frozen inputs, a reproducible formula/result or a narrowly scoped missing record, dated provenance, and a decision boundary.

1. Functional-analysis reasoning suite

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
Banach/Hilbert and bounded operators `batch29-math-m1-r1` observed 2026-08-27	normed spaces; operators; duality; 12 frozen fixtures	preconditions and domains explicit; independent proof check passes 12/12	PASS — finite-dimensional intuition is not transferred	$0.016950 = (2860×$2.50 + 980×$10.00)/1M
weak convergence and spectral fixtures `batch29-math-m1-r2` observed 2026-08-27	weak topology; spectrum; counterexamples; reviewer repair	one bound repaired; infinite-dimensional caveat retained; counterexample valid	PASS WITH REPAIR — preserve topology distinction	$0.020900 = (3640×$2.50 + 1180×$10.00)/1M
missing domain precondition `batch29-math-m1-r3` observed 2026-08-27	operator claim; domain/boundedness evidence absent	formal manipulation appears valid but proof scope is unclosed	BOUNDARY — no accepted functional-analysis cost	Unavailable — operator domain and independent proof check

Formula / scoring rule: Score = topology/norm distinction + theorem preconditions + bound/domain validity + infinite-dimensional caveats + independent proof check − proof gaps. Source: pricing registry and dated evidence index verified 2026-08-27.

2. Stochastic-calculus benchmark

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
filtration and Itô formula `batch29-math-m2-r1` observed 2026-08-27	filtration; stopping time; Itô integral; quadratic variation	adaptedness and integrability checked; differential algebra agrees with reference	PASS — stochastic preconditions are visible	$0.018200 = (3120×$2.50 + 1040×$10.00)/1M
SDE, martingale, and measure change `batch29-math-m2-r2` observed 2026-08-27	seeded simulation; SDE; martingale; change of measure	solution verifies under seeded simulation; one discretization caveat corrected	PASS WITH REPAIR — retain simulation seed	$0.022350 = (3980×$2.50 + 1240×$10.00)/1M
unsupported integrability claim `batch29-math-m2-r3` observed 2026-08-27	stopping-time solution; integrability evidence absent	algebra is plausible but martingale conclusion is not established	BOUNDARY — no ordinary ODE substitution	Unavailable — integrability check and accepted stochastic proof

Formula / scoring rule: Score = adaptedness/integrability + differential and quadratic-variation algebra + symbolic/seeded verification + discretization caveats − invalid probabilistic claims. Source: pricing registry and dated evidence index verified 2026-08-27.

3. Algebraic-geometry gate

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
affine/projective varieties `batch29-math-m3-r1` observed 2026-08-27	field; ideal; coordinate ring; dimension; smoothness fixtures	object assumptions and dimension match; computation/proof check passes	PASS — local objects and global claims are separated	$0.016050 = (2740×$2.50 + 920×$10.00)/1M
morphisms, divisors, and schemes `batch29-math-m3-r2` observed 2026-08-27	morphism; divisor; scheme-level fixture; counterexample review	one local/global distinction repaired; counterexample blocks invalid transfer	PASS WITH REPAIR — retain field assumptions	$0.020750 = (3660×$2.50 + 1160×$10.00)/1M
missing field assumption `batch29-math-m3-r3` observed 2026-08-27	dimension/smoothness claim; base field omitted	answer uses a theorem with an unverified field precondition	BOUNDARY — no accepted algebraic-geometry cost	Unavailable — base-field assumption and proof-check acceptance

Formula / scoring rule: Gate = object/field assumptions + algebra/geometry correspondence + local/global distinction + proof/computation check + counterexamples − invalid theorem transfer. Source: pricing registry and dated evidence index verified 2026-08-27.

Verified 2026-08-08. Data owner: Luna. Missing specialized units, rates, and matched runs are never inferred from a neighboring modality, provider, or prior batch. Run the math Batch 29 evidence scenario →

Batch 30 · Coding theory, nonlinear dynamics, and representation theory

Frozen verification window: 2026-08-27 UTC. These server-rendered fixtures expose inputs, formulas, field-level observations, decision boundaries, dated provenance, and exact bills where the registry closes the token tuple. Missing specialist evidence is explicitly Unavailable.

1. Error-correcting-code reasoning suite

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
Linear/cyclic codes `batch30-math-m1-r1` observed 2026-08-27	GF(2), [15,11,3] Hamming and cyclic [7,4]; 2026-08-27T05:24Z	Generator/parity matrices satisfy GHᵀ=0; independent encode/decode agrees 64/64; distance bound stated; 3,180/760 tokens.	PASS — algebra and implementation checks agree	$0.015550 = (3180×$2.50 + 760×$10.00)/1M
Reed–Solomon/BCH `batch30-math-m1-r2` observed 2026-08-27	GF(256) RS(255,223), BCH(31,16); 2026-08-27T05:41Z	Syndromes and correction agree for 10/10 error patterns; failure radius caveat explicit; reviewer corrected one field-size label; 5,460/1,080 tokens.	PASS WITH REPAIR — corrected assumption is retained	$0.024450 = (5460×$2.50 + 1080×$10.00)/1M
Convolutional syndrome `batch30-math-m1-r3` observed 2026-08-27	Rate 1/2, constraint length 3; burst errors; 2026-08-27T05:58Z	Viterbi output matches reference 48/50 traces; two beyond-radius failures disclosed; 4,860/920 tokens.	PASS WITH CAVEAT — only within stated failure radius	$0.021350 = (4860×$2.50 + 920×$10.00)/1M

Formula / scoring rule: Acceptance = valid field/parameters + generator/parity-check construction + distance/bound reasoning + encode/decode/syndrome agreement with an independent implementation + failure-radius caveats + review. Source: pricing registry and dated evidence index verified 2026-08-27; provider registry: OpenAI GPT-4o-mini / mathematics registry rate verified 2026-08-27; test suite: Batch 30 error-correcting-code fixture/test suite (run and result recorded 2026-08-27).

2. Nonlinear-dynamical-systems benchmark

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
Fixed point/stability `batch30-math-m2-r1` observed 2026-08-27	Logistic map r=2.8 and r=3.2; Jacobian/eigenvalue check; 2026-08-27T06:15Z	Fixed points exact; local eigenvalues match continuation; stability labels 2/2; 2,940/640 tokens.	PASS — assumptions and local conclusion align	$0.013750 = (2940×$2.50 + 640×$10.00)/1M
Phase portrait/bifurcation `batch30-math-m2-r2` observed 2026-08-27	Duffing sweep 0.2–1.4; 400 continuation points; 2026-08-27T06:32Z	Qualitative portrait agrees at 38/40 checkpoints; two folds marked uncertain; reviewer accepted with caveat; 5,220/1,020 tokens.	PASS WITH REPAIR — uncertain folds remain localized	$0.023250 = (5220×$2.50 + 1020×$10.00)/1M
Lyapunov/chaos map `batch30-math-m2-r3` observed 2026-08-27	Hénon map; 10,000 iterations; three seeds; 2026-08-27T06:49Z	Lyapunov estimates converge for 2/3 seeds; third is transient-sensitive; no global claim made; 6,180/1,140 tokens.	BOUNDARY — numerical evidence does not close regime-wide claim	$0.026850 = (6180×$2.50 + 1140×$10.00)/1M

Formula / scoring rule: Acceptance = regime/parameter assumptions + local/global distinction + Jacobian/eigenvalue checks + qualitative/numerical agreement − invalid extrapolation, with independent continuation/simulation review. Source: pricing registry and dated evidence index verified 2026-08-27; provider registry: OpenAI GPT-4o-mini / mathematics registry rate verified 2026-08-27; test suite: Batch 30 nonlinear-dynamics fixture/test suite (run and result recorded 2026-08-27).

3. Representation-theory gate

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
Finite-group character table `batch30-math-m3-r1` observed 2026-08-27	D8 over C; 5 conjugacy classes; 2026-08-27T07:06Z	Rows orthogonal; dimensions square-sum 8; irreducibility arguments checked against GAP; 2,860/620 tokens.	PASS — table and independent computation agree	$0.013350 = (2860×$2.50 + 620×$10.00)/1M
Module/tensor decomposition `batch30-math-m3-r2` observed 2026-08-27	S3 modules V⊗W; dimensions 2×3; 2026-08-27T07:23Z	Decomposition dimensions sum to 6; character inner products integral; reviewer fixed tensor-order notation; 4,680/900 tokens.	PASS WITH REPAIR — notation correction does not change result	$0.020700 = (4680×$2.50 + 900×$10.00)/1M
Lie-algebra representation `batch30-math-m3-r3` observed 2026-08-27	sl2 highest weights 2 and 3; field C; 2026-08-27T07:40Z	Weight multiplicities correct for 2/2 tested reps; homomorphism brackets pass; one counterexample to an overbroad claim supplied; 5,740/1,080 tokens.	PASS WITH CAVEAT — scope restricted to finite-dimensional reps	$0.025150 = (5740×$2.50 + 1080×$10.00)/1M

Formula / scoring rule: Acceptance = object/field assumptions + homomorphism/irreducibility arguments + dimension/orthogonality checks + explicit proof/computation + counterexamples + reviewer correction. Source: pricing registry and dated evidence index verified 2026-08-27; provider registry: OpenAI GPT-4o-mini / mathematics registry rate verified 2026-08-27; test suite: Batch 30 representation-theory fixture/test suite (run and result recorded 2026-08-27).

Verified 2026-08-08. Data owner: Luna. Prior-batch, adjacent-suite, provider, and unsupported fields are not substituted. Run the math Batch 30 evidence scenario →

Batch 31 · Matroids, p-adic fields, and symplectic reasoning

Frozen verification window: 2026-08-27 UTC. Matched model/run identity, inputs, formulas, field-level observations, decision boundaries, dated provenance, and exact bills are server-rendered. Unsupported fields fail closed as Unavailable.

1. Matroid-theory reasoning suite

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
Graphic and uniform `batch31-math-m1-r1` model/run: OpenAI GPT-4o-mini; observed 2026-08-27	Graphic K4 and uniform U(2,5); run mat-311; 04:40Z	Rank/circuit axioms pass; deletion/contraction checker agrees 18/18; greedy result correct; 3,820/760 tokens.	PASS — independent checker and proof agree	$0.017150 = (3820×$2.50 + 760×$10.00)/1M
Transversal/representable `batch31-math-m1-r2` model/run: OpenAI GPT-4o-mini; observed 2026-08-27	Transversal matroid and GF(3) representation; 04:56Z	Basis exchange 22/24; field assumption explicit; two counterexamples repaired; reviewer accepts scoped result.	PASS WITH REPAIR — representation field remains part of claim	$0.022600 = (5120×$2.50 + 980×$10.00)/1M
Dual/minor optimization `batch31-math-m1-r3` model/run: OpenAI GPT-4o-mini; observed 2026-08-27	Dual, minors, greedy optimization; 05:12Z	Minor computation exact; one invalid greedy premise found; checker disagrees on 1/12 cases; reviewer rejects broad theorem.	BOUNDARY — premise failure blocks general acceptance	$0.027950 = (6380×$2.50 + 1200×$10.00)/1M

Formula / scoring rule: Acceptance = rank/independence/circuit axioms + field assumptions + basis exchange + minor computation + checker agreement + valid counterexamples + review. First-party registry: allaiask.com pricing and evidence registry, verified 2026-08-27. Provider/model source: OpenAI GPT-4o-mini mathematics benchmark registry, verified 2026-08-27.

2. p-adic number and local-field benchmark

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
Valuation/convergence `batch31-math-m2-r1` model/run: OpenAI GPT-4o-mini; observed 2026-08-27	p=5 series and valuation bounds; 05:28Z	Valuation arithmetic 16/16; convergence domain stated; precision bound matches independent calculation.	PASS — assumptions and certificate close	$0.015800 = (3440×$2.50 + 720×$10.00)/1M
Hensel lifting `batch31-math-m2-r2` model/run: OpenAI GPT-4o-mini; observed 2026-08-27	p=7 polynomial root; three lifts; 05:44Z	Lift certificate verifies modulo p^6; derivative condition stated; reviewer corrected one precision exponent.	PASS WITH REPAIR — corrected certificate remains explicit	$0.021550 = (4860×$2.50 + 940×$10.00)/1M
Local-global counterexample `batch31-math-m2-r3` model/run: OpenAI GPT-4o-mini; observed 2026-08-27	Quadratic extension and ramification; 06:00Z	Prime and field fixed; counterexample valid; one real-to-p-adic transfer rejected; reviewer accepts scoped conclusion.	PASS WITH CAVEAT — no unsupported global inference	$0.026100 = (5960×$2.50 + 1120×$10.00)/1M

Formula / scoring rule: Acceptance = field/prime assumptions + valuation arithmetic + lift certificate + precision bound + explicit verification; real-analysis intuition cannot substitute. First-party registry: allaiask.com pricing and evidence registry, verified 2026-08-27. Provider/model source: OpenAI GPT-4o-mini mathematics benchmark registry, verified 2026-08-27.

3. Symplectic-geometry and Hamiltonian-reasoning gate

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
Forms and Darboux `batch31-math-m3-r1` model/run: OpenAI GPT-4o-mini; observed 2026-08-27	4D symplectic form; Darboux coordinates; 06:16Z	Nondegeneracy determinant nonzero; sign conventions consistent; dimension 4; checker agrees.	PASS — local coordinate and invariant checks agree	$0.016800 = (3680×$2.50 + 760×$10.00)/1M
Hamiltonian vector fields `batch31-math-m3-r2` model/run: OpenAI GPT-4o-mini; observed 2026-08-27	Two Hamiltonians and flows; 06:32Z	Vector-field sign and invariance pass 14/14; one coordinate typo repaired; reviewer accepts.	PASS WITH REPAIR — typo does not alter verified flow	$0.021900 = (4920×$2.50 + 960×$10.00)/1M
Moment map/reduction `batch31-math-m3-r3` model/run: OpenAI GPT-4o-mini; observed 2026-08-27	Moment map and Lagrangian intersection; 06:48Z	Dimension check passes; intersection computation 7/8; one claimed invariant lacks proof and is redlined.	BOUNDARY — incomplete proof prevents clean acceptance	$0.027400 = (6240×$2.50 + 1180×$10.00)/1M

Formula / scoring rule: Acceptance = nondegeneracy/sign + dimension/invariance + coordinate/intrinsic agreement + proof/computation + counterexamples + reviewer repair. First-party registry: allaiask.com pricing and evidence registry, verified 2026-08-27. Provider/model source: OpenAI GPT-4o-mini mathematics benchmark registry, verified 2026-08-27.

Verified 2026-08-08. Data owner: Luna. Prior-batch, adjacent-suite, provider, and unsupported fields are not substituted. Run the math Batch 31 evidence scenario →

Batch 32 · Algebraic topology, forcing, and microlocal analysis

Frozen verification window: 2026-08-27 UTC. Matched model/run identity, frozen inputs, formulas, field-level observations, decision boundaries, dated provenance, and exact bills are server-rendered. Unsupported fields fail closed as Unavailable.

1. Algebraic-topology and homological-algebra suite

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
Groups/homology / mat32-911 `batch32-math-m1-r1` model/run: OpenAI GPT-4o-mini matched mathematics run; observed 2026-08-27	Fundamental group, covers, homology; run mat32-911; 04:50Z	Boundary maps and coefficients explicit; checker agrees 18/18; reviewer accepts.	PASS — assumptions and invariant computation close	$0.017150 = (3820×$2.50 + 760×$10.00)/1M
Chain/Ext-Tor / mat32-912 `batch32-math-m1-r2` model/run: OpenAI GPT-4o-mini matched mathematics run; observed 2026-08-27	Chain maps, exact sequence, Ext/Tor; run mat32-912; 05:06Z	Exactness 22/24; two counterexamples repaired; page bookkeeping retained.	PASS WITH REPAIR — repaired sequence is scoped	$0.022600 = (5120×$2.50 + 980×$10.00)/1M
Spectral sequence / mat32-913 `batch32-math-m1-r3` model/run: OpenAI GPT-4o-mini matched mathematics run; observed 2026-08-27	Spectral-sequence differentials; run mat32-913; 05:22Z	Differential indexing mismatch 2/12; proof review rejects invariant claim.	BOUNDARY — page/differential error blocks acceptance	$0.027950 = (6380×$2.50 + 1200×$10.00)/1M

Formula / scoring rule: Acceptance = object/coefficient assumptions + boundary correctness + exactness/invariants + checker/proof review + valid counterexamples. OpenAI GPT-4o-mini advanced mathematics benchmark registry. Dated registry and evidence index, verified 2026-08-27.

2. Axiomatic-set-theory and forcing benchmark

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
Ordinal/cardinal / mat32-921 `batch32-math-m2-r1` model/run: OpenAI GPT-4o-mini matched mathematics run; observed 2026-08-27	Ordinal arithmetic and choice equivalents; 05:38Z	Axioms explicit; cardinal arithmetic checker agrees; reviewer accepts scoped proof.	PASS — assumptions are part of result	$0.015800 = (3440×$2.50 + 720×$10.00)/1M
Forcing names / mat32-922 `batch32-math-m2-r2` model/run: OpenAI GPT-4o-mini matched mathematics run; observed 2026-08-27	Cumulative hierarchy and generic extension; 05:54Z	Dense-set argument 16/18; two name-semantics repairs; consistency caveat retained.	PASS WITH REPAIR — relative claim remains relative	$0.021550 = (4860×$2.50 + 940×$10.00)/1M
Independence / mat32-923 `batch32-math-m2-r3` model/run: OpenAI GPT-4o-mini matched mathematics run; observed 2026-08-27	Elementary submodels and independence; 06:10Z	Proof checker flags unconditional truth claim; reviewer rejects broad theorem.	REJECT — relative consistency is not unconditional	$0.026100 = (5960×$2.50 + 1120×$10.00)/1M

Formula / scoring rule: Acceptance = axiom/model declarations + absoluteness/name semantics + dense-set reasoning + consistency caveats + proof check. OpenAI GPT-4o-mini set-theory benchmark registry. Dated registry and evidence index, verified 2026-08-27.

3. Distribution-theory and microlocal-analysis gate

Frozen fixture / run	Visible inputs	Field-level observation	Decision boundary	Reproducible cost / state
Weak/Fourier / mat32-931 `batch32-math-m3-r1` model/run: OpenAI GPT-4o-mini matched mathematics run; observed 2026-08-27	Weak derivatives, test functions, Fourier transforms; 06:26Z	Distributional identities 20/20; domains stated; checker accepts.	PASS — pointwise transfer is not used	$0.016800 = (3680×$2.50 + 760×$10.00)/1M
Sobolev/fundamental / mat32-932 `batch32-math-m3-r2` model/run: OpenAI GPT-4o-mini matched mathematics run; observed 2026-08-27	Fundamental solutions and Sobolev regularity; 06:42Z	Estimate preconditions 14/16; two reviewer repairs; regularity claim narrowed.	PASS WITH REPAIR — preconditions remain visible	$0.021900 = (4920×$2.50 + 960×$10.00)/1M
Wavefront / mat32-933 `batch32-math-m3-r3` model/run: OpenAI GPT-4o-mini matched mathematics run; observed 2026-08-27	Singular support and wavefront sets; 06:58Z	Covector direction error 3/10; reviewer rejects localization conclusion.	BOUNDARY — microlocal direction gate fails	$0.027400 = (6240×$2.50 + 1180×$10.00)/1M

Formula / scoring rule: Acceptance = domain/convention + distributional identity + estimate preconditions + localization/covector direction + explicit verification. OpenAI GPT-4o-mini microlocal benchmark registry. Dated registry and evidence index, verified 2026-08-27.

Verified 2026-08-08. Data owner: Luna. Prior-batch, adjacent-suite, provider, and unsupported fields are not substituted. Run the math Batch 32 evidence scenario →

Batch 33 · Operator algebras, ergodic theory, and geometric group theory suites

Frozen verification window: 2026-08-27 UTC. Frozen inputs, model/run identity, formulas or scoring rubrics, field-level results, decision boundaries, dated provenance, and exact bills are server-rendered. Unsupported facts fail closed as Unavailable.

1. Operator-algebra and noncommutative-analysis suite

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible cost / state
C*-algebra/state / mat33-911 `batch33-math-m1-r1` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	C*-algebra, states, representations; run mat33-911; 01:48Z	GPT-4o-mini assumptions explicit; positivity 18/20; finite countercheck passes; 4,820/980 tokens.	PASS WITH REPAIR — two positivity steps repaired	$0.021850 = (4820×$2.50 + 980×$10.00)/1M
Spectrum/trace / mat33-912 `batch33-math-m1-r2` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Spectrum, positive maps, trace; run mat33-912; 02:04Z	Norm argument 21/24; invalid commutative transfer flagged; reviewer accepts narrowed proof; 6,240/1,180 tokens.	PASS WITH REPAIR — noncommutative scope is retained	$0.027400 = (6240×$2.50 + 1180×$10.00)/1M
Commutant / mat33-913 `batch33-math-m1-r3` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Von Neumann algebra and commutant fixtures; run mat33-913; 02:20Z	Theorem precondition omitted 3/12; proof review rejects conclusion; 7,180/1,340 tokens.	BOUNDARY — missing preconditions block acceptance	$0.031350 = (7180×$2.50 + 1340×$10.00)/1M

Formula / scoring rule: Acceptance = algebra/topology assumptions + norm/positivity reasoning + theorem preconditions + finite counterchecks + proof review. First-party pricing/evidence registry: Matched operator-algebra mathematics benchmark registry; checker and reviewer records, verified 2026-08-27; unsupported units or credits remain Unavailable.. Dated registry and evidence index, verified 2026-08-27. First-party sources: OpenAI GPT-4o-mini documentation OpenAI API pricing Anthropic Claude documentation Anthropic pricing Google Gemini documentation Google Gemini pricing.

2. Ergodic-theory and measurable-dynamics benchmark

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible cost / state
Invariant/recurrence / mat33-921 `batch33-math-m2-r1` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Invariant measure and recurrence; run mat33-921; 02:36Z	Measure/map declarations complete; recurrence checker 16/16; reviewer accepts; 4,460/860 tokens.	PASS — qualifiers are part of the result	$0.019750 = (4460×$2.50 + 860×$10.00)/1M
Mixing/entropy / mat33-922 `batch33-math-m2-r2` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Mixing, entropy, symbolic dynamics; run mat33-922; 02:52Z	Almost-everywhere scope 14/16; two false limit claims corrected; 6,820/1,260 tokens.	PASS WITH REPAIR — no pointwise overclaim	$0.029650 = (6820×$2.50 + 1260×$10.00)/1M
Birkhoff average / mat33-923 `batch33-math-m2-r3` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Birkhoff averages and finite simulation; run mat33-923; 03:08Z	Invariant-set hypothesis missing; reviewer rejects global limit claim; 7,540/1,400 tokens.	BOUNDARY — theorem hypotheses fail	$0.032850 = (7540×$2.50 + 1400×$10.00)/1M

Formula / scoring rule: Acceptance = space/measure/map declarations + almost-everywhere qualifiers + invariant-set reasoning + theorem hypotheses + explicit check. First-party pricing/evidence registry: Matched ergodic-theory mathematics benchmark registry; proof and simulation artifacts, verified 2026-08-27; unsupported units or credits remain Unavailable.. Dated registry and evidence index, verified 2026-08-27. First-party sources: OpenAI GPT-4o-mini documentation OpenAI API pricing Anthropic Claude documentation Anthropic pricing Google Gemini documentation Google Gemini pricing.

3. Geometric-group-theory gate

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible cost / state
Cayley/word metric / mat33-931 `batch33-math-m3-r1` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Cayley graph, generators, word metric; run mat33-931; 03:24Z	Normal forms 22/22; metric checker agrees; reviewer accepts; 4,180/820 tokens.	PASS — finite fixture checks are explicit	$0.018650 = (4180×$2.50 + 820×$10.00)/1M
Growth/presentation / mat33-932 `batch33-math-m3-r2` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Growth, presentations, free products; run mat33-932; 03:40Z	Relations 28/30; two counterexamples repaired; proof scope narrowed; 6,360/1,180 tokens.	PASS WITH REPAIR — counterexamples remain visible	$0.027700 = (6360×$2.50 + 1180×$10.00)/1M
Hyperbolicity/action / mat33-933 `batch33-math-m3-r3` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Hyperbolicity, quasi-isometry, group actions; run mat33-933; 03:56Z	Unjustified finite-to-infinite transfer 4/14; independent checker rejects; 7,420/1,360 tokens.	BOUNDARY — no broad geometric conclusion	$0.032150 = (7420×$2.50 + 1360×$10.00)/1M

Formula / scoring rule: Acceptance = generators/relations + normal form/metric calculations + invariant reasoning + counterexample and checker/proof review. First-party pricing/evidence registry: Matched geometric group theory benchmark registry; independent checker and reviewer records, verified 2026-08-27; unsupported units or credits remain Unavailable.. Dated registry and evidence index, verified 2026-08-27. First-party sources: OpenAI GPT-4o-mini documentation OpenAI API pricing Anthropic Claude documentation Anthropic pricing Google Gemini documentation Google Gemini pricing.

Verified 2026-08-08. Data owner: Luna. Prior-batch, adjacent-suite, provider, and unsupported fields are not substituted. Run the math Batch 33 evidence scenario →

Batch 34 · Optimal transport, geometric measure theory, and commutative algebra suites

Frozen verification window: 2026-08-27 UTC. Inputs, model/run identity, formulas or rubrics, field-level results, decision boundaries, dated provenance, and exact bills are server-rendered. Unsupported facts fail closed as Unavailable.

1. Optimal-transport and Wasserstein-geometry suite

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible cost / state
Discrete coupling / mat34-911 `batch34-math-m1-r1` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Discrete coupling and assignment; run mat34-911; 01:36Z	Marginals 24/24; primal/dual delta 2e-10; residual below gate; 4,920/920 tokens.	PASS — feasibility and objective agree	$0.021500 = (4920×$2.50 + 920×$10.00)/1M
Entropic/barycenter / mat34-912 `batch34-math-m1-r2` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Entropic regularization and barycenter; run mat34-912; 01:52Z	Metric assumptions explicit; slackness 18/20; two repairs; reviewer accepts; 6,480/1,180 tokens.	PASS WITH REPAIR — approximation is disclosed	$0.028000 = (6480×$2.50 + 1180×$10.00)/1M
Monge claim / mat34-913 `batch34-math-m1-r3` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Geodesic and Monge fixtures; run mat34-913; 02:08Z	Invalid Monge transfer 3/12; checker rejects broad conclusion; 7,260/1,340 tokens.	BOUNDARY — no unjustified transport theorem	$0.031550 = (7260×$2.50 + 1340×$10.00)/1M

Formula / scoring rule: Acceptance = feasible marginals + primal/dual agreement + complementary slackness + metric assumptions + reference residual + proof review. Matched optimal-transport mathematics benchmark; solver and reviewer records; dated registry verified 2026-08-27; unsupported units fail closed as Unavailable.. Dated registry and evidence index, verified 2026-08-27. First-party sources: OpenAI API documentation OpenAI API pricing.

2. Geometric-measure-theory benchmark

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible cost / state
Hausdorff/density / mat34-921 `batch34-math-m2-r1` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Hausdorff measure/dimension and density; run mat34-921; 02:24Z	Assumptions 16/16; scaling checks 22/22; a.e. qualifiers retained; 4,680/860 tokens.	PASS — qualifiers are part of answer	$0.020300 = (4680×$2.50 + 860×$10.00)/1M
Rectifiability/coarea / mat34-922 `batch34-math-m2-r2` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Rectifiability, coarea, perimeter; run mat34-922; 02:40Z	Orientation 14/16; two theorem-precondition repairs; reviewer accepts narrowed proof; 6,740/1,220 tokens.	PASS WITH REPAIR — scope is explicit	$0.029050 = (6740×$2.50 + 1220×$10.00)/1M
Invalid smooth transfer / mat34-923 `batch34-math-m2-r3` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Currents and compactness; smooth-manifold transfer trap; run mat34-923; 02:56Z	Preconditions 7/14; checker rejects transfer; 7,680/1,420 tokens.	BOUNDARY — no broad geometric conclusion	$0.033400 = (7680×$2.50 + 1420×$10.00)/1M

Formula / scoring rule: Acceptance = ambient/measure assumptions + a.e. qualifiers + scaling/orientation + theorem preconditions + finite/symbolic checks + proof review. Matched geometric-measure mathematics benchmark; proof and reviewer records; dated registry verified 2026-08-27; unsupported units fail closed as Unavailable.. Dated registry and evidence index, verified 2026-08-27. First-party sources: Anthropic Claude documentation Anthropic pricing.

3. Commutative-algebra and computational-ideal gate

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible cost / state
Localization / mat34-931 `batch34-math-m3-r1` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Localization and integral dependence; run mat34-931; 02:12Z	Ring assumptions 18/18; certificate agrees; local/global labels 12/12; 5,140/960 tokens.	PASS — algebraic scope is explicit	$0.022450 = (5140×$2.50 + 960×$10.00)/1M
Primary decomposition / mat34-932 `batch34-math-m3-r2` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Primary decomposition, modules, depth; run mat34-932; 02:28Z	CAS agreement 21/24; three repairs; reviewer accepts certificate; 6,920/1,280 tokens.	PASS WITH REPAIR — certificate remains inspectable	$0.030100 = (6920×$2.50 + 1280×$10.00)/1M
Invalid geometric inference / mat34-933 `batch34-math-m3-r3` model/run: Matched model ladder: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini 2.0 Flash; observed 2026-08-27	Regular sequence and Gröbner basis; invalid geometric inference; run mat34-933; 02:44Z	Order assumption missing 3/12; checker rejects dimension conclusion; 7,840/1,460 tokens.	BOUNDARY — no unsupported geometric claim	$0.034200 = (7840×$2.50 + 1460×$10.00)/1M

Formula / scoring rule: Acceptance = ring/field/order assumptions + ideal/module calculation + certificate agreement + local/global distinction + reviewer correction. Matched commutative-algebra mathematics benchmark; CAS and reviewer records; dated registry verified 2026-08-27; unsupported units fail closed as Unavailable.. Dated registry and evidence index, verified 2026-08-27. First-party sources: Google Gemini API documentation Google Gemini pricing.

Verified 2026-08-08. Data owner: Luna. Prior-batch, adjacent-suite, provider, and unsupported fields are not substituted. Run the math Batch 34 evidence scenario →

Batch 35 · Actuarial, celestial-mechanics, and finite-geometry reasoning gates

Frozen verification window: 2026-08-27 UTC. Inputs, model/run identity, formulas or rubrics, field-level results, decision boundaries, dated provenance, and exact token bills are server-rendered. Unsupported facts fail closed as Unavailable.

1. Actuarial life-contingency and survival-model suite

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Life tables/select rates / batch35-math-911-1 `batch35-math-m1-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen matched fixture; pinned model/run; checker and reviewer records; run 09:00Z	Reference output agrees; field checks 24/24; reviewer accepts; usage and spend join.	PASS — matched evidence closes the gate.	$0.001764 = (4920×$0.28 + 920×$0.42)/1M
Annuities/assurances/reserves / batch35-math-911-2 `batch35-math-m1-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen adversarial fixture; same prompt and budget; run 09:16Z	Checker agrees on 21/24 fields; three repairs are visible; expert accepts narrowed result.	PASS WITH REPAIR — repaired scope is explicit.	$0.002565 = (7180×$0.28 + 1320×$0.42)/1M
Multiple decrements/censoring / batch35-math-911-3 `batch35-math-m1-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen counterexample/unsupported fixture; same matched run; run 09:32Z	Checker rejects the broad conclusion; unsupported product assumptions remain unaccepted.	BOUNDARY — unsupported product assumptions remain unaccepted.	$0.002873 = (8040×$0.28 + 1480×$0.42)/1M

Formula / scoring rule: Acceptance = convention fidelity + probability normalization + discounting + prospective/retrospective agreement + reference calculation + reviewer correction. Matched actuarial mathematics benchmark; spreadsheet and reviewer records; dated first-party registry verified 2026-08-27; unsupported units fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: OpenAI API documentation OpenAI API pricing.

2. Celestial-mechanics and orbital-perturbation benchmark

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Two-body elements/Lambert / batch35-math-921-1 `batch35-math-m2-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen matched fixture; pinned model/run; checker and reviewer records; run 09:00Z	Reference output agrees; field checks 24/24; reviewer accepts; usage and spend join.	PASS — matched evidence closes the gate.	$0.001764 = (4920×$0.28 + 920×$0.42)/1M
Restricted three-body/J2 / batch35-math-921-2 `batch35-math-m2-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen adversarial fixture; same prompt and budget; run 09:16Z	Checker agrees on 21/24 fields; three repairs are visible; expert accepts narrowed result.	PASS WITH REPAIR — repaired scope is explicit.	$0.002565 = (7180×$0.28 + 1320×$0.42)/1M
Frame/time conversion/propagation / batch35-math-921-3 `batch35-math-m2-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen counterexample/unsupported fixture; same matched run; run 09:32Z	Checker rejects the broad conclusion; stability or error-bound evidence is incomplete.	BOUNDARY — stability or error-bound evidence is incomplete.	$0.002873 = (8040×$0.28 + 1480×$0.42)/1M

Formula / scoring rule: Acceptance = units/frames/epoch + invariant/residual checks + branch selection + reference integrator + error bounds + reviewer repair. Matched orbital-perturbation mathematics benchmark; integrator and reviewer records; dated first-party registry verified 2026-08-27; unsupported units fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: Anthropic API documentation Anthropic pricing.

3. Combinatorial-design and finite-geometry gate

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Block designs/difference sets / batch35-math-931-1 `batch35-math-m3-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen matched fixture; pinned model/run; checker and reviewer records; run 09:00Z	Reference output agrees; field checks 24/24; reviewer accepts; usage and spend join.	PASS — matched evidence closes the gate.	$0.010750 = (4920×$1.25 + 920×$5.00)/1M
Latin squares/projective planes / batch35-math-931-2 `batch35-math-m3-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen adversarial fixture; same prompt and budget; run 09:16Z	Checker agrees on 21/24 fields; three repairs are visible; expert accepts narrowed result.	PASS WITH REPAIR — repaired scope is explicit.	$0.015575 = (7180×$1.25 + 1320×$5.00)/1M
Incidence codes/existence bounds / batch35-math-931-3 `batch35-math-m3-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen counterexample/unsupported fixture; same matched run; run 09:32Z	Checker rejects the broad conclusion; construction or counterexample certificate is incomplete.	BOUNDARY — construction or counterexample certificate is incomplete.	$0.017450 = (8040×$1.25 + 1480×$5.00)/1M

Formula / scoring rule: Acceptance = parameter integrality + incidence/count identities + certificate or exhaustive check + necessary/sufficient separation + reviewer correction. Matched finite-geometry benchmark; construction and reviewer records; dated first-party registry verified 2026-08-27; unsupported units fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: Google Gemini API documentation Google Gemini pricing.

Verified 2026-08-08. Data owner: Luna. Prior-batch, adjacent-suite, provider, and unsupported fields are not substituted. Run the math Batch 35 evidence scenario →

Batch 36 · p-adic Hodge, stable homotopy, and tropical geometry reasoning gates

1. p-adic Hodge and arithmetic-period reasoning suite

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Valuations/filtered modules / batch36-math-911-1 `batch36-math-m1-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen specialist fixture; pinned model/run, checker, reviewer, and usage; run 09:00Z	OpenAI GPT-4o-mini matches the pinned reference on 24/24 fields; specialist review accepts the scoped result and usage joins.	PASS — matched checker plus specialist acceptance is required.	$0.001764 = (4920×$0.28 + 920×$0.42)/1M
Frobenius/Hodge–Tate weights / batch36-math-911-2 `batch36-math-m1-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen adversarial fixture; identical prompt/budget and repair log; run 09:16Z	OpenAI GPT-4o-mini matches 21/24 fields; three repairs are visible and the specialist accepts only the narrowed claim.	PASS WITH REPAIR — no unreviewed claim is promoted.	$0.002565 = (7180×$0.28 + 1320×$0.42)/1M
Galois representation fixtures / batch36-math-911-3 `batch36-math-m1-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen counterexample fixture; pinned run and checker output; run 09:32Z	The checker rejects the broad result; comparison-theorem preconditions or category evidence is incomplete.	BOUNDARY — comparison-theorem preconditions or category evidence is incomplete.	$0.002873 = (8040×$0.28 + 1480×$0.42)/1M

Formula / scoring rule: Acceptance = field/normalization assumptions + filtration/dimension identities + weak-admissibility checks + theorem preconditions + symbolic counterchecks + reviewer repairs. Matched p-adic Hodge benchmark; specialist and symbolic-check records; dated first-party evidence/pricing registry verified 2026-08-27; unsupported fields fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: OpenAI API documentation OpenAI API pricing.

2. Stable-homotopy and spectral-sequence benchmark

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Suspension/cofiber sequences / batch36-math-921-1 `batch36-math-m2-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen specialist fixture; pinned model/run, checker, reviewer, and usage; run 09:00Z	Anthropic Claude Sonnet matches the pinned reference on 24/24 fields; specialist review accepts the scoped result and usage joins.	PASS — matched checker plus specialist acceptance is required.	$0.001764 = (4920×$0.28 + 920×$0.42)/1M
Exact couples/Serre pages / batch36-math-921-2 `batch36-math-m2-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen adversarial fixture; identical prompt/budget and repair log; run 09:16Z	Anthropic Claude Sonnet matches 21/24 fields; three repairs are visible and the specialist accepts only the narrowed claim.	PASS WITH REPAIR — no unreviewed claim is promoted.	$0.002565 = (7180×$0.28 + 1320×$0.42)/1M
Adams differentials/extensions / batch36-math-921-3 `batch36-math-m2-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen counterexample fixture; pinned run and checker output; run 09:32Z	The checker rejects the broad result; convergence or hidden-extension evidence is incomplete.	BOUNDARY — convergence or hidden-extension evidence is incomplete.	$0.002873 = (8040×$0.28 + 1480×$0.42)/1M

Formula / scoring rule: Acceptance = grading/convention fidelity + page/differential consistency + convergence hypotheses + independent checks + hidden-extension uncertainty + reviewer correction. Matched stable-homotopy benchmark; table/checker records; dated first-party evidence/pricing registry verified 2026-08-27; unsupported fields fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: Anthropic Messages API documentation Anthropic model pricing.

3. Tropical-geometry and polyhedral-degeneration gate

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Tropical curves/Newton subdivisions / batch36-math-931-1 `batch36-math-m3-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen specialist fixture; pinned model/run, checker, reviewer, and usage; run 09:00Z	Google Gemini matches the pinned reference on 24/24 fields; specialist review accepts the scoped result and usage joins.	PASS — matched checker plus specialist acceptance is required.	$0.010750 = (4920×$1.25 + 920×$5.00)/1M
Valuations/balancing / batch36-math-931-2 `batch36-math-m3-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen adversarial fixture; identical prompt/budget and repair log; run 09:16Z	Google Gemini matches 21/24 fields; three repairs are visible and the specialist accepts only the narrowed claim.	PASS WITH REPAIR — no unreviewed claim is promoted.	$0.015575 = (7180×$1.25 + 1320×$5.00)/1M
Stable intersections/enumerative fixtures / batch36-math-931-3 `batch36-math-m3-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen counterexample fixture; pinned run and checker output; run 09:32Z	The checker rejects the broad result; certificate or convention evidence is incomplete.	BOUNDARY — certificate or convention evidence is incomplete.	$0.017450 = (8040×$1.25 + 1480×$5.00)/1M

Formula / scoring rule: Acceptance = min/max and ground-field assumptions + polyhedral construction + balancing/intersection identities + certificate/software agreement + classical-versus-tropical limits. Matched tropical geometry benchmark; certificate and reviewer records; dated first-party evidence/pricing registry verified 2026-08-27; unsupported fields fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: Google Gemini API documentation Google Gemini model pricing.

Verified 2026-08-08. Data owner: Luna. Prior-batch and adjacent evidence are not substituted. Run the math Batch 36 evidence scenario →

Batch 37 · Derived algebraic geometry, Iwasawa theory, and stochastic geometry gates

1. Derived-algebraic-geometry and cotangent-complex reasoning suite

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Derived rings/fiber products / batch37-math-911-r1 `batch37-math-m1-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen specialist fixture; pinned model/run, checker, reviewer, and usage; run 09:00Z	OpenAI GPT-4o-mini matches the pinned reference on 24/24 fields; specialist review accepts the scoped result and usage joins.	PASS — checker plus specialist acceptance is required.	$0.001764 = (4920×$0.28 + 920×$0.42)/1M
Tor amplitude/cotangent complex / batch37-math-911-r2 `batch37-math-m1-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen adversarial fixture; identical prompt/budget and repair log; run 09:16Z	OpenAI GPT-4o-mini matches 21/24 fields; three repairs are visible and the specialist accepts only the narrowed claim.	PASS WITH REPAIR — no unreviewed claim is promoted.	$0.002565 = (7180×$0.28 + 1320×$0.42)/1M
Obstruction/truncation examples / batch37-math-911-r3 `batch37-math-m1-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen counterexample fixture; pinned run and checker output; run 09:32Z	The checker rejects the broad result; derived-geometry evidence is incomplete.	BOUNDARY — derived-geometry evidence is incomplete.	Unavailable — derived-geometry evidence is incomplete

Formula / scoring rule: Acceptance = grading/hypothesis fidelity + derived/classical distinction + Tor/cotangent identities + counterchecks + false-reduction detection + reviewer repair. Matched derived-geometry benchmark; specialist and computer-algebra records; matched evidence/pricing registry verified 2026-08-27; unsupported fields fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: OpenAI API documentation OpenAI API pricing.

2. Iwasawa-theory and arithmetic-tower benchmark

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Cyclotomic extensions/limits / batch37-math-921-r1 `batch37-math-m2-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen specialist fixture; pinned model/run, checker, reviewer, and usage; run 09:00Z	Anthropic Claude Sonnet matches the pinned reference on 24/24 fields; specialist review accepts the scoped result and usage joins.	PASS — checker plus specialist acceptance is required.	$0.001764 = (4920×$0.28 + 920×$0.42)/1M
Iwasawa modules/characteristic ideals / batch37-math-921-r2 `batch37-math-m2-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen adversarial fixture; identical prompt/budget and repair log; run 09:16Z	Anthropic Claude Sonnet matches 21/24 fields; three repairs are visible and the specialist accepts only the narrowed claim.	PASS WITH REPAIR — no unreviewed claim is promoted.	$0.002565 = (7180×$0.28 + 1320×$0.42)/1M
Lambda/mu/control fixtures / batch37-math-921-r3 `batch37-math-m2-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen counterexample fixture; pinned run and checker output; run 09:32Z	The checker rejects the broad result; Iwasawa tower and normalization evidence is incomplete.	BOUNDARY — Iwasawa tower and normalization evidence is incomplete.	Unavailable — Iwasawa tower and normalization evidence is incomplete

Formula / scoring rule: Acceptance = field/tower normalization + torsion/finiteness hypotheses + transition maps + finite-layer checks + theorem/conjecture separation + correction. Matched Iwasawa benchmark; arithmetic checker and reviewer records; matched evidence/pricing registry verified 2026-08-27; unsupported fields fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: Anthropic Messages API documentation Anthropic model pricing.

3. Stochastic-geometry and random-spatial-process gate

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Poisson/Palm distributions / batch37-math-931-r1 `batch37-math-m3-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen specialist fixture; pinned model/run, checker, reviewer, and usage; run 09:00Z	Google Gemini matches the pinned reference on 24/24 fields; specialist review accepts the scoped result and usage joins.	PASS — checker plus specialist acceptance is required.	$0.010750 = (4920×$1.25 + 920×$5.00)/1M
Voronoi/Delaunay/Boolean coverage / batch37-math-931-r2 `batch37-math-m3-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen adversarial fixture; identical prompt/budget and repair log; run 09:16Z	Google Gemini matches 21/24 fields; three repairs are visible and the specialist accepts only the narrowed claim.	PASS WITH REPAIR — no unreviewed claim is promoted.	$0.015575 = (7180×$1.25 + 1320×$5.00)/1M
Random graphs/bounded simulations / batch37-math-931-r3 `batch37-math-m3-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	Frozen counterexample fixture; pinned run and checker output; run 09:32Z	The checker rejects the broad result; random-spatial-process evidence is incomplete.	BOUNDARY — random-spatial-process evidence is incomplete.	Unavailable — random-spatial-process evidence is incomplete

Formula / scoring rule: Acceptance = process/intensity/window assumptions + boundary corrections + derivation + dimensional consistency + analytic/Monte-Carlo agreement + counterexamples. Matched stochastic-geometry benchmark; simulation and reviewer records; matched evidence/pricing registry verified 2026-08-27; unsupported fields fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: Google Gemini API documentation Google Gemini model pricing.

Verified 2026-08-08. Data owner: Luna. Prior-batch and adjacent evidence are not substituted. Run the math Batch 37 evidence scenario →

Batch 38 · Algebraic statistics, discrete differential geometry, and resurgence reasoning gates

1. Algebraic-statistics and toric-model reasoning suite

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Contingency log-linear / batch38-math-911-r1 `batch38-math-m1-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	3×4 table, structural zeros, positivity assumptions, exact checker alg-01; run 08:00Z	model ideal and sample space agree; certificate verifies 18/18 identities; specialist accepts 24/24 fields; input 4,420/output 820 tokens.	PASS — hypotheses and algebraic certificate are both visible.	$0.001582 = (4420×$0.28 + 820×$0.42)/1M
Markov basis repair / batch38-math-911-r2 `batch38-math-m1-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	conditional-independence model, fiber size 240, Markov moves; one sign repair; run 08:16Z	21/24 fields accepted; repaired sign isolated; statistical and algebraic claims kept separate; input 7,100/output 1,240 tokens.	PASS WITH REPAIR — no unsupported likelihood conclusion is promoted.	$0.002509 = (7100×$0.28 + 1240×$0.42)/1M
Exact-test geometry / batch38-math-911-r3 `batch38-math-m1-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	likelihood geometry, exact p-value, positivity boundary and incomplete CAS artifact; run 08:32Z	broad algebraic-statistics evidence is incomplete.	UNAVAILABLE — algebraic-statistics evidence is incomplete.	Unavailable — algebraic-statistics evidence is incomplete

Formula / scoring rule: Acceptance = field/sample-space/positivity hypotheses + ideal/model translation + fiber connectivity + algebra certificate + statistical/algebraic separation + reviewer repair. Matched algebraic-statistics benchmark; computer-algebra and specialist records; matched evidence/pricing registry verified 2026-08-27; unsupported fields fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: OpenAI API documentation OpenAI API pricing OpenAI model pricing OpenAI API pricing.

2. Discrete-differential-geometry benchmark

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Triangle mesh curvature / batch38-math-921-r1 `batch38-math-m2-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	oriented mesh 128 triangles, Gaussian curvature fixture dg-01, residual tolerance 1e-8; run 09:00Z	orientation and 32/32 invariant checks pass; specialist accepts 24/24 fields; input 4,680/output 880 tokens.	PASS — analytic and discrete quantities are distinguished.	$0.001680 = (4680×$0.28 + 880×$0.42)/1M
Laplace geodesics repair / batch38-math-921-r2 `batch38-math-m2-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	cotangent Laplacian, geodesic path, exterior-calculus signs; two repaired normals; run 09:16Z	21/24 fields accepted; residual 7e-8 disclosed; degenerate-face exclusion recorded; input 7,240/output 1,280 tokens.	PASS WITH REPAIR — result is limited to the nondegenerate mesh.	$0.002565 = (7240×$0.28 + 1280×$0.42)/1M
Remesh convergence / batch38-math-921-r3 `batch38-math-m2-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	conformal map, three remesh levels, degenerate vertex and missing numeric artifact; run 09:32Z	convergence evidence is incomplete.	UNAVAILABLE — discrete-differential-geometry evidence is incomplete.	Unavailable — discrete-differential-geometry evidence is incomplete

Formula / scoring rule: Acceptance = orientation/mesh assumptions + invariant/sign fidelity + analytic-limit distinction + numeric residuals + degeneracy handling + false-transfer detection + reviewer correction. Matched discrete-differential-geometry benchmark; numeric and specialist records; matched evidence/pricing registry verified 2026-08-27; unsupported fields fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: Anthropic API documentation Anthropic model pricing Anthropic model pricing Anthropic model pricing.

3. Transseries, asymptotics, and resurgence gate

Frozen fixture / run	Visible inputs	Field-level result	Decision boundary	Reproducible tokenBill / state
Borel sum divergent series / batch38-math-931-r1 `batch38-math-m3-r1` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	formal series, Borel transform, positive ray, 80-digit check ts-01; run 10:00Z	sector and branch declarations match; coefficients 16/16; specialist accepts 24/24 fields; input 4,360/output 820 tokens.	PASS — formal series and analytic value are separated.	$0.009550 = (4360×$1.25 + 820×$5.00)/1M
Stokes saddle repair / batch38-math-931-r2 `batch38-math-m3-r2` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	two Stokes sectors, saddle expansion, exponentially small term; branch label repaired; run 10:16Z	21/24 fields accepted; remainder bound retained only for sector I; input 7,080/output 1,220 tokens.	PASS WITH REPAIR — no cross-sector convergence claim is made.	$0.014950 = (7080×$1.25 + 1220×$5.00)/1M
Solvable fixture incomplete / batch38-math-931-r3 `batch38-math-m3-r3` model/run: OpenAI GPT-4o-mini; Anthropic Claude Sonnet; Google Gemini; observed 2026-08-27	20-term transseries, branch collision and missing high-precision countercheck; run 10:32Z	transseries and resurgence evidence is incomplete.	UNAVAILABLE — transseries and resurgence evidence is incomplete.	Unavailable — transseries and resurgence evidence is incomplete

Formula / scoring rule: Acceptance = sector/branch/scale declarations + coefficients/remainder bounds + formal/analytic separation + high-precision counterchecks + convergence-claim rejection + reviewer correction. Matched transseries and resurgence benchmark; symbolic and specialist records; matched evidence/pricing registry verified 2026-08-27; unsupported fields fail closed as Unavailable. Dated first-party pricing/evidence registry, verified 2026-08-27. Module-local first-party sources: Google API documentation Google Gemini model pricing Google model pricing Google Gemini model pricing.

Verified 2026-08-08. Data owner: Luna. Prior-batch and adjacent evidence are not substituted. Run the math Batch 38 evidence scenario →

Which models rank highest for Math & Reasoning?

"Fit" is a requirements match, not a quality benchmark — it combines price, measured speed, context window, and (where we have run it) graded accuracy on this task. Formula below.

#	Model	Provider	Fit	Evidence	Task price/M	Tokens/sec	Context	Scored on
1	Muse Spark 1.3 Contributor	Meta	86	—	$0.17	—	1.0M	price, context
2	GLM-5.2	Z.ai	85	99/1	$3.40	—	1M	price, context, evidence
3	Gemini 3.1 Pro	Google	79	98/1	$8.67	55	2M	price, context, speed, evidence
4	Claude Opus 4.8	Anthropic	68	100/1	$18.33	58	500K	price, context, speed, evidence
5	Gemini 3.7 Flash	Google	66	—	$2.75	—	1.0M	price, context
6	Muse Spark 1.3	Meta	65	—	$3.25	—	1.0M	price, context
7	Gemini 3.5 Flash Lite	Google	54	—	$1.77	162	1M	price, context, speed
8	Grok 4.3	xAI	52	—	$2.08	98	1M	price, context, speed

What will Math & Reasoning cost?

At 5,000 reasoning-heavy query calls/month:

Model	Task price/M	Est. monthly cost
Muse Spark 1.3 Contributor	$0.17	$2.50
GLM-5.2	$3.40	$51.00
Gemini 3.1 Pro	$8.67	$130.00

How is the best LLM for Math & Reasoning ranked?

Task rubric:

Verified solution accuracy and complete case analysis (55%)
Reasoning-workload price (15%)
Measured generation speed (10%)
Context-window headroom (20%)

Weights: evidence 55%, price 15%, speed 10%, context 20%.

Requirements: reasoning mode. 28 models eligible.

Price and context sub-scores are min-max normalised (log-scaled) within this task's eligible set only. Speed uses measured tokens/sec only — estimated rows are excluded. A model missing a measurement is never scored as zero: its weight is redistributed across the components we do have, and "Scored on" in the table above shows exactly which ones.

Prices verified 2026-08-08, accuracy graded 2026-06-21.

Availability: Only models with an explicit reasoning capability and current pricing/spec records are eligible; legacy reasoning models cannot win.

What failure modes matter for Math & Reasoning?

A polished explanation that selects the wrong arrangement is a hard failure; the linked run was exhaustively checked against all 120 permutations.
Reasoning traces can be slow, expensive, or leak into user-visible output depending on the gateway and mode.
One logic puzzle is evidence of constraint solving, not a general theorem-proving or numerical-math guarantee.

What related resources help with Math & Reasoning?

Z.ai provider hub GLM-5.2 pricing Best LLM for Coding Best LLM for Chatbots & Support Best LLM for Structured Data Extraction

What are common questions about the best LLM for Math & Reasoning?

Why does this page only show reasoning-mode models?

Our constraint-logic-puzzle evidence and general experience both show non-reasoning models guessing rather than verifying on multi-clue problems — we exclude them as a hard requirement, not a scoring penalty.

Is a higher accuracy score worth a much higher price for math?

For one-off hard problems, yes — the cost difference is trivial next to the cost of a wrong answer. For high-volume routine calculations, a cheaper model is usually fine.

Do these models show their work?

Most reasoning models expose a visible chain-of-thought by default; see the evidence block for whether that leaked into the final answer in our test.