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{"family": "matmul", "model": "fake", "target": "(3, 3, 3, 23)", "proposal_raw": "I believe Strassen solved this in 1969.", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 0.0}
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{"family": "matmul", "model": "fake", "target": "(2, 2, 3, 11)", "proposal_raw": "I believe Strassen solved this in 1969.", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 0.0}
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Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-23 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "316b3249d2527580", "latency_s": 0.002}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "663245c9b1a7efd6", "latency_s": 2.373}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-6 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "9c5daf0650a4773e", "latency_s": 1.852}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "fc30dc31e71a7253", "latency_s": 2.629}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 1, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 1],\n    [1, 1, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 1, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 1, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 1, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 1, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 1, 0, 0, 1],\n    [0, 1, 0, 0, 1, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [1, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [1, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 1, 0, 0, 1, 0],\n    [1, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 1, 0, 0, 0],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 1, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 1, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 1, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-24 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "fd7c74eb1b31e16e", "latency_s": 8.55}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "134bf0e3573e6b97", "latency_s": 2.392}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "1ebdb18624530981", "latency_s": 2.122}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "fc30dc31e71a7253", "latency_s": 2.515}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-23 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "d5fe884ed100f94f", "latency_s": 7.681}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "a95c621b44e548c3", "latency_s": 2.257}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-6 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "e2e7a6e51b3f389e", "latency_s": 1.826}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "b65bba9c68f243e1", "latency_s": 2.469}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"]\n  ],\n  \"v\": [\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"]\n  ],\n  \"w\": [\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-23 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "c2dfa1435b16572e", "latency_s": 8.203}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0]\n  ]\n}\n```\n\nWait, let me reconsider. The standard rank of matrix multiplication tensor for 2x2 is known to be 7 (Strassen's algorithm). Let me provide the correct decomposition:\n\n```json\n{\n  \"u\": [\n    [1, 1, 0, 0],\n    [0, 1, 1, 0],\n    [1, 0, 0, 1],\n    [0, 0, 1, 1],\n    [1, 0, 1, 0],\n    [0, 1, 0, 1],\n    [0, 1, 1, 1]\n  ],\n  \"v\": [\n    [1, 0, 1, 0],\n    [0, 1, 0, 1],\n    [1, 0, 0, 1],\n    [0, 1, 1, 0],\n    [1, 1, 0, 0],\n    [0, 0, 1, 1],\n    [1, 1, 1, 1]\n  ],\n  \"w\": [\n    [1, 0, 0, 1],\n    [0, 0, 1, 1],\n    [0, 1, 0, 1],\n    [0, 1, 1, 0],\n    [1, 1, 0, 0],\n    [1, 0, 0, 1],\n    [1, 1, 1, 0]\n  ]\n}\n```", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 4.081}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-6 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "0dd403a4f58eee18", "latency_s": 1.779}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "fc30dc31e71a7253", "latency_s": 2.468}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 1, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 1],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [1, 0, 0, 0, 0, 0, 0, 0, 1],\n    [0, 1, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 1, 0],\n    [1, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 1, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 1, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 1]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 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Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-23 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "9b807d36ee538743", "latency_s": 8.004}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "183ca76c2513ddd4", "latency_s": 3.11}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "e202d0431d8cb414", "latency_s": 2.157}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "d468bddae0c59e89", "latency_s": 2.418}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"]\n  ],\n  \"v\": [\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"]\n  ],\n  \"w\": [\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-23 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "ce10893084f14005", "latency_s": 8.409}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "134bf0e3573e6b97", "latency_s": 2.248}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "a95c621b44e548c3", "latency_s": 2.014}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "fc30dc31e71a7253", "latency_s": 2.591}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\"]\n  ],\n  \"v\": [\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\"]\n  ],\n  \"w\": [\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-23 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "b9089b9cf534c263", "latency_s": 7.855}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "663245c9b1a7efd6", "latency_s": 2.277}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0]\n  ]\n}\n```\n\nWait, let me reconsider. The standard rank for 2x2 matrix multiplication is 7. Let me provide the correct decomposition:\n\n```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 1, 0, 0],\n    [0, 1, 1, 0],\n    [1, 0, 1, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0]\n  ]\n}\n```", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 3.678}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "9919bb000184623f", "latency_s": 2.773}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\"]\n  ],\n  \"v\": [\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\"]\n  ],\n  \"w\": [\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"1\", \"0\", \"0\", \"0\"]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-23 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "35c7d7cb14137bcd", "latency_s": 8.404}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 1, 0, 0],\n    [0, 1, 0, 0],\n    [0, 1, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "5956b89b1599fc82", "latency_s": 2.248}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "4b3a91fb7b67d2c5", "latency_s": 2.294}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "fc30dc31e71a7253", "latency_s": 2.427}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 1, 0, 0, 0, 1],\n    [0, 1, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 1, 0, 0],\n    [1, 0, 0, 0, 1, 0, 0, 0, 1],\n    [0, 1, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 1, 0, 0],\n    [1, 0, 0, 0, 1, 0, 0, 0, 1],\n    [0, 1, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 1, 0, 0],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), 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Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-24 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "0bfad6fb497ff3f6", "latency_s": 8.392}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "a95c621b44e548c3", "latency_s": 2.204}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "a95c621b44e548c3", "latency_s": 2.045}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "fc30dc31e71a7253", "latency_s": 2.528}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [1, 1, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 1, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 1, 1],\n    [\"1/2\", \"1/2\", 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, \"1/2\", \"1/2\", 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, \"1/2\", \"1/2\", 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [1, 1, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 1, 1, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 1, 1],\n    [\"1/2\", 0, 0, \"1/2\", 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, \"1/2\", 0, 0, \"1/2\", 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, \"1/2\"]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 0],\n    [1, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 1, 0],\n    [0, 0, 1, 0, 0, 0, 1, 0, 0],\n    [0, 1, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 1, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1],\n    [\"1/3\", \"1/3\", \"1/3\", 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, \"1/3\", \"1/3\", \"1/3\", 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, \"1/3\", \"1/3\", \"1/3\"],\n    [1, 0, 0, 0, 0, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 0, 0, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 2), Fraction(1, 2), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 2), Fraction(1, 2), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 2), Fraction(1, 2), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 2), Fraction(0, 1), Fraction(0, 1), Fraction(1, 2), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 2), Fraction(0, 1), Fraction(0, 1), Fraction(1, 2), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 2))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 3), Fraction(1, 3), Fraction(1, 3), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 3), Fraction(1, 3), Fraction(1, 3), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 3), Fraction(1, 3), Fraction(1, 3)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-23 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "52070a39dca31f6c", "latency_s": 8.656}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "a66e0f095044dcd0", "latency_s": 2.276}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "a95c621b44e548c3", "latency_s": 2.096}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "fc30dc31e71a7253", "latency_s": 2.532}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [\"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"1\"],\n    [\"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"1\", \"1\", \"0\", \"0\", \"0\", \"0\", \"0\"],\n    [\"0\", \"0\", \"0\", \"0\", \"1\", \"1\", \"0\", \"0\", \"0\"],\n    [\"1\", \"0\", \"0\", \"0\", 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{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 8)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1]],\n\"v\": [[1,0,0,1],[1,0,0,0],[0,1,0,-1],[-1,0,1,0],[0,0,0,1],[1,1,0,0],[0,0,1,1]],\n\"w\": [[1,0,0,1],[0,0,1,-1],[0,1,0,1],[1,0,1,0],[-1,1,0,0],[0,0,0,1],[1,0,0,0]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 8)", "witness": "exact tensor identity holds; rank 7 <= 8", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 8, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 7.844}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 7)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1]],\n\"v\": [[1,0,0,1],[1,0,0,0],[0,1,0,-1],[-1,0,1,0],[0,0,0,1],[1,1,0,0],[0,0,1,1]],\n\"w\": [[1,0,0,1],[0,0,1,-1],[0,1,0,1],[1,0,1,0],[-1,1,0,0],[0,0,0,1],[1,0,0,0]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": "exact tensor identity holds; rank 7 <= 7", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 7, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 9.658}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 3, 11)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],\n\"v\": [[1,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,-1,0],[-1,0,0,1,0,0],[0,0,0,0,1,0],[1,1,0,0,0,0],[0,0,0,1,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,0,1]],\n\"w\": [[1,0,0,0,1,0],[0,0,0,1,-1,0],[0,1,0,0,1,0],[1,0,0,1,0,0],[-1,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,1,0,0,0],[0,0,0", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 43.463}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(3, 3, 3, 23)", "proposal_raw": "", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 42.768}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 8)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1]],\n\"v\": [[1,0,0,1],[1,0,0,0],[0,1,0,-1],[-1,0,1,0],[0,0,0,1],[1,1,0,0],[0,0,1,1]],\n\"w\": [[1,0,0,1],[0,0,1,-1],[0,1,0,1],[1,0,1,0],[-1,1,0,0],[0,0,0,1],[1,0,0,0]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 8)", "witness": "exact tensor identity holds; rank 7 <= 8", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 8, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 13.826}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 7)", "proposal_raw": "{\"u\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"]],\n\"v\": [[\"1\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"1\",\"1\"]],\n\"w\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"-1\"], [\"0\",\"1\",\"0\",\"1\"], [\"1\",\"0\",\"1\",\"0\"], [\"-1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": "exact tensor identity holds; rank 7 <= 7", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 7, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 9.214}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1,0,0,1],\n    [0,0,1,1],\n    [1,0,0,0],\n    [0,0,0,1],\n    [1,1,0,0],\n    [-1,0,1,0],\n    [0,1,0,-1],\n    [1,0,0,0],\n    [0,1,0,0],\n    [0,0,1,0],\n    [0,0,0,1]\n  ],\n  \"v\": [\n    [1,0,0,0,1,0],\n    [1,0,0,0,0,0],\n    [0,1,0,0,-1,0],\n    [-1,0,0,1,0,0],\n    [0,0,0,0,1,0],\n    [1,1,0,0,0,0],\n    [0,0,0,1,1,0],\n    [0,0,1,0,0,0],\n    [0,0,0,0,0,1],\n    [0,0,1,0,0,0],\n    [0,0,0,0,0,1]\n  ],\n  \"w\": [\n    [1,0,0,0,1,0],\n    [0,0,0,1,-1,0],\n    [0,1,0,0,1,0],\n    [1,0,0,1,0,0],\n    [-1,1,0,0,0,0],\n    [0,0,0,0,1,0],\n    [1,0,0,0,0,0],\n    [0,0,1,0,0,0],\n    [0,0,1,0,0,0],\n    [0,0,0,0,0,1],\n    [0,0,0,0,0,1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(-1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "certified", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": "exact tensor identity holds; rank 11 <= 11", "certificate": {"dims": [2, 2, 3], "rank": 11, "target_rank": 11, "first_violation": null}, "key": "cb6d70a944fe9b22", "latency_s": 40.296}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(3, 3, 3, 23)", "proposal_raw": "", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 46.77}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 8)", "proposal_raw": "{\"u\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"]], \"v\": [[\"1\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"1\",\"1\"]], \"w\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"-1\"], [\"0\",\"1\",\"0\",\"1\"], [\"1\",\"0\",\"1\",\"0\"], [\"-1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 8)", "witness": "exact tensor identity holds; rank 7 <= 8", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 8, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 10.903}
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{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 8)", "proposal_raw": "{\"u\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"]], \"v\": [[\"1\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"1\",\"1\"]], \"w\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"-1\"], [\"0\",\"1\",\"0\",\"1\"], [\"1\",\"0\",\"1\",\"0\"], [\"-1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 8)", "witness": "exact tensor identity holds; rank 7 <= 8", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 8, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 8.607}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 7)", "proposal_raw": "{\"u\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"]], \"v\": [[\"1\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"1\",\"1\"]], \"w\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"-1\"], [\"0\",\"1\",\"0\",\"1\"], [\"1\",\"0\",\"1\",\"0\"], [\"-1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": "exact tensor identity holds; rank 7 <= 7", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 7, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 10.061}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 3, 11)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],\n\"v\": [[1,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,-1,0],[-1,0,0,1,0,0],[0,0,0,0,1,0],[1,1,0,0,0,0],[0,0,0,1,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,0,1]],\n\"w\": [[1,0,0,0,1,0],[0,0,0,1,-1,0],[0,1,0,0,1,0],[1,0,0,1,0,0],[-1,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,0,1]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(-1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "certified", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": "exact tensor identity holds; rank 11 <= 11", "certificate": {"dims": [2, 2, 3], "rank": 11, "target_rank": 11, "first_violation": null}, "key": "cb6d70a944fe9b22", "latency_s": 36.964}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(3, 3, 3, 23)", "proposal_raw": "", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 44.167}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 8)", "proposal_raw": "{\"u\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"]], \"v\": [[\"1\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"1\",\"1\"]], \"w\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"-1\"], [\"0\",\"1\",\"0\",\"1\"], [\"1\",\"0\",\"1\",\"0\"], [\"-1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 8)", "witness": "exact tensor identity holds; rank 7 <= 8", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 8, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 8.787}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 7)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1]],\n\"v\": [[1,0,0,1],[1,0,0,0],[0,1,0,-1],[-1,0,1,0],[0,0,0,1],[1,1,0,0],[0,0,1,1]],\n\"w\": [[1,0,0,1],[0,0,1,-1],[0,1,0,1],[1,0,1,0],[-1,1,0,0],[0,0,0,1],[1,0,0,0]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": "exact tensor identity holds; rank 7 <= 7", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 7, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 9.781}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 1],\n    [0, 0, 1, 1],\n    [1, 0, 0, 0],\n    [0, 0, 0, 1],\n    [1, 1, 0, 0],\n    [-1, 0, 1, 0],\n    [0, 1, 0, -1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 1, 0],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, -1, 0],\n    [-1, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [1, 1, 0, 0, 0, 0],\n    [0, 0, 0, 1, 1, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 1, 0],\n    [0, 0, 0, 1, -1, 0],\n    [0, 1, 0, 0, 1, 0],\n    [1, 0, 0, 1, 0, 0],\n    [-1, 1, 0, 0, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [1, 0, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(-1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "certified", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": "exact tensor identity holds; rank 11 <= 11", "certificate": {"dims": [2, 2, 3], "rank": 11, "target_rank": 11, "first_violation": null}, "key": "cb6d70a944fe9b22", "latency_s": 30.625}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(3, 3, 3, 23)", "proposal_raw": "", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 53.38}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 8)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1]],\n\"v\": [[1,0,0,1],[1,0,0,0],[0,1,0,-1],[-1,0,1,0],[0,0,0,1],[1,1,0,0],[0,0,1,1]],\n\"w\": [[1,0,0,1],[0,0,1,-1],[0,1,0,1],[1,0,1,0],[-1,1,0,0],[0,0,0,1],[1,0,0,0]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 8)", "witness": "exact tensor identity holds; rank 7 <= 8", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 8, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 11.534}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 7)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1]],\n\"v\": [[1,0,0,1],[1,0,0,0],[0,1,0,-1],[-1,0,1,0],[0,0,0,1],[1,1,0,0],[0,0,1,1]],\n\"w\": [[1,0,0,1],[0,0,1,-1],[0,1,0,1],[1,0,1,0],[-1,1,0,0],[0,0,0,1],[1,0,0,0]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": "exact tensor identity holds; rank 7 <= 7", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 7, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 11.305}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 3, 11)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],\n\"v\": [[1,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,-1,0],[-1,0,0,1,0,0],[0,0,0,0,1,0],[1,1,0,0,0,0],[0,0,0,1,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,0,1]],\n\"w\": [[1,0,0,0,1,0],[0,0,0,1,-1,0],[0,1,0,0,1,0],[1,0,0,1,0,0],[-1,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,0,1]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(-1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "certified", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": "exact tensor identity holds; rank 11 <= 11", "certificate": {"dims": [2, 2, 3], "rank": 11, "target_rank": 11, "first_violation": null}, "key": "cb6d70a944fe9b22", "latency_s": 45.272}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(3, 3, 3, 23)", "proposal_raw": "", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 42.683}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 8)", "proposal_raw": "{\"u\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"]], \"v\": [[\"1\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"], [\"0\",\"1\",\"0\",\"-1\"], [\"-1\",\"0\",\"1\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"1\",\"1\"]], \"w\": [[\"1\",\"0\",\"0\",\"1\"], [\"0\",\"0\",\"1\",\"-1\"], [\"0\",\"1\",\"0\",\"1\"], [\"1\",\"0\",\"1\",\"0\"], [\"-1\",\"1\",\"0\",\"0\"], [\"0\",\"0\",\"0\",\"1\"], [\"1\",\"0\",\"0\",\"0\"]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 8)", "witness": "exact tensor identity holds; rank 7 <= 8", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 8, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 9.76}
{"family": "matmul", "model": "claude-sonnet-5", "target": "(2, 2, 2, 7)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1]],\n\"v\": [[1,0,0,1],[1,0,0,0],[0,1,0,-1],[-1,0,1,0],[0,0,0,1],[1,1,0,0],[0,0,1,1]],\n\"w\": [[1,0,0,1],[0,0,1,-1],[0,1,0,1],[1,0,1,0],[-1,1,0,0],[0,0,0,1],[1,0,0,0]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": "exact tensor identity holds; rank 7 <= 7", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 7, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 10.709}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]], \"w\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "364bff7828b8ec13", "latency_s": 1.611}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "7d80172821465195", "latency_s": 1.854}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "405604d2f941e4c9", "latency_s": 3.037}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]], \"v\": [[1,0,0,0,1,0,0,0,1],[0,1,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0],[1,0,0,0,1,0,0,0,1],[0,1,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0],[1,0,0,0,1,0,0,0,1],[0,1,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0]], \"w\": [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]}\n}\n```", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 4.755}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]], \"w\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[0,0,1,0],[0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "ddfa65dc9db36e75", "latency_s": 1.493}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]], \"w\": [[1,0,0,0],[0,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0],[0,0,0,0],[0,0,0,0],[0,0,1,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "90dacb39980c699a", "latency_s": 1.723}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]], \"v\": [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]], \"w\": [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "be893451aab8a31d", "latency_s": 2.161}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]], \"v\": [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]], \"w\": [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-9 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "04dee67a4f1a6665", "latency_s": 2.658}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]], \"w\": [[1,0,0,0],[0,0,0,0],[0,1,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0],[0,0,1,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "5dae3e853592a083", "latency_s": 1.645}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0]], \"w\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "6d84939b469ff8da", "latency_s": 1.478}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[1,0,0,0],[0,1,0,0],[1,0,0,0],[0,1,0,0],[0,0,1,0]], \"v\": [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]], \"w\": [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,0,0,0],[1,0,0,0,0,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "5a7242128976d328", "latency_s": 2.134}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]], \"v\": [[1,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[1,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[1,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0]], \"w\": [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-9 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "26040ac881435f42", "latency_s": 2.747}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]], \"w\": [[1,0,0,0],[0,0,0,0],[0,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,0],[0,0,0,0],[0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "134bf0e3573e6b97", "latency_s": 1.754}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0]], \"w\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "cca85df7682c9092", "latency_s": 1.754}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[1,0,0,0],[0,1,0,0],[1,0,0,0],[0,1,0,0],[0,0,1,0]], \"v\": [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]], \"w\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "44182eed2c01e19d", "latency_s": 2.062}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1,0,0,0,0,0,0,0,0],\n    [0,1,0,0,0,0,0,0,0],\n    [0,0,1,0,0,0,0,0,0],\n    [0,0,0,1,0,0,0,0,0],\n    [0,0,0,0,1,0,0,0,0],\n    [0,0,0,0,0,1,0,0,0],\n    [0,0,0,0,0,0,1,0,0],\n    [0,0,0,0,0,0,0,1,0],\n    [0,0,0,0,0,0,0,0,1],\n    [1,0,0,0,1,0,0,0,0],\n    [0,0,0,0,0,0,1,0,0],\n    [1,0,0,0,0,0,0,0,0],\n    [0,1,0,0,0,0,0,0,0],\n    [0,0,0,1,0,0,0,0,0],\n    [0,0,0,0,0,1,0,0,0],\n    [0,0,0,0,0,0,0,1,0],\n    [0,0,0,0,0,0,0,0,1],\n    [0,1,0,0,0,0,0,0,0],\n    [0,0,1,0,0,0,0,0,0],\n    [0,0,0,0,1,0,0,0,0],\n    [0,0,0,0,0,1,0,0,0],\n    [0,0,0,0,0,0,1,0,0],\n    [0,0,0,0,0,0,0,1,0]\n  ],\n  \"v\": [\n    [1,0,0,0,0,0,0,0,0],\n    [0,0,0,1,0,0,0,0,0],\n    [0,0,0,0,0,0,1,0,0],\n    [0,1,0,0,0,0,0,0,0],\n    [0,0,0,0,1,0,0,0,0],\n    [0,0,0,0,0,0,0,1,0],\n    [0,0,1,0,0,0,0,0,0],\n    [0,0,0,0,0,1,0,0,0],\n    [0,0,0,0,0,0,0,0,1],\n    [1,0,0,1,0,0,0,0,0],\n    [0,0,1,0,0,0,0,0,0],\n    [0,1,0,0,0,0,0,0,0],\n    [0,0,0,0,1,0,0,0,0],\n    [0,0,0,1,0,0,0,0,0],\n    [0,0,0,0,0,1,0,0,0],\n    [0,0,0,0,0,0,1,0,0],\n    [0,0,0,0,0,0,0,1,0],\n    [1,0,0,0,0,0,0,0,0],\n    [0,1,0,0,0,0,0,0,0],\n    [0,0,0,1,0,0,0,0,0],\n    [0,0,0,0,1,0,0,0,0],\n    [0,0,0,0,0,1,0,0,0],\n    [0,0,0,0,0,0,1,0,0]\n  ],\n  \"w\": [\n    [1,0,0,0,0,0,0,0,0],\n    [0,0,0,0,1,0,0,0,0],\n    [0,0,0,0,0,0,0,0,1],\n    [0,1,0,0,0,0,0,0,0],\n    [0,0,0,0,1,0,0,0,0],\n    [0,0,0,0,0,0,0,0,1],\n    [0,0,1,0,0,0,0,0,0],\n    [0,0,0,0,1,0,0,0,0],\n    [0,0,0,0,0,0,0,0,1],\n    [1,0,0,0,1,0,0,0,0],\n    [0,0,1,0,0,0,0,0,0],\n    [1,0,0,0,0,0,0,0,0],\n    [0,1,0,0,0,0,0,0,0],\n    [0,0,0,1,0,0,0,0,0],\n    [0,0,0,0,0,1,0,0,0],\n    [0,0,0,0,0,0,1,0,0],\n    [0,0,0,0,0,0,0,1,0],\n    [0,1,0,0,0,0,0,0,0],\n    [0,0,1,0,0,0,0,0,0],\n    [0,0,0,0,1,0,0,0,0],\n    [0,0,0,0,0,1,0,0,0],\n    [0,0,0,0,0,0,1,0,0],\n    [0,0,0,0,0,0,0,1,0]\n  ]\n}\n```", "obj": 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1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-23 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "73da4a6225cc85b6", "latency_s": 6.392}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[1,0,0,0],[0,0,1,0]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]], \"w\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[0,0,1,0],[0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "cbca896ae14a9857", "latency_s": 1.499}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 0, 1, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "b458daaf4f7590c8", "latency_s": 2.221}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]], \"v\": [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]], \"w\": [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "697ce26232857a19", "latency_s": 2.122}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]], \"v\": [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]], \"w\": [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-9 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "2bbbe63569c8fbbb", "latency_s": 2.801}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]], \"w\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[0,0,1,0],[0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "ddfa65dc9db36e75", "latency_s": 1.489}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0]], \"w\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "6d84939b469ff8da", "latency_s": 1.461}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0],\n    [0, 0, 1, 0, 0, 0],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [0, 0, 0, 1, 0, 0],\n    [0, 0, 0, 0, 1, 0],\n    [0, 0, 0, 0, 0, 1],\n    [1, 0, 0, 0, 0, 0],\n    [0, 1, 0, 0, 0, 0]\n  ]\n}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "a16244fe3cb9e91d", "latency_s": 2.808}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]], \"v\": [[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]], \"w\": [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-9 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "ef84ba5681f730dd", "latency_s": 2.72}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]], \"w\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[0,0,1,0],[0,0,0,1]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "b0a2121707cdfe1d", "latency_s": 1.551}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0]], \"w\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[0,0,1,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "32cbefaa8d525193", "latency_s": 1.84}
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1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-24 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "0716a56e2b734dfa", "latency_s": 5.463}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]], \"w\": [[1,0,0,0],[0,0,0,0],[0,0,0,0],[0,1,0,0],[0,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "1ef3bae6f8305b5b", "latency_s": 1.659}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0]], \"w\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[0,0,1,0],[0,1,0,0],[0,1,0,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": null, "certificate": null, "key": "101984f3bfc53cc8", "latency_s": 1.533}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[1,0,0,0],[0,1,0,0],[1,0,0,0],[0,1,0,0],[0,0,1,0]], \"v\": [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]], \"w\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "295b2e977f07e7d8", "latency_s": 2.099}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]], \"v\": [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]], \"w\": [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), 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{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], \"v\": [[1,0,0,0],[1,0,0,0],[0,1,0,0],[0,1,0,0],[0,0,1,0],[0,0,1,0],[0,0,0,1],[0,0,0,1]], \"w\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "915cf297d8648024", "latency_s": 1.601}
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1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-24 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "54789287820de856", "latency_s": 5.95}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], \"v\": [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1],[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]], \"w\": [[1,0,0,0],[0,0,0,0],[0,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,0],[0,0,0,0],[0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-8 decomposition of <2,2,2> (target <= 8)", "witness": null, "certificate": null, "key": "134bf0e3573e6b97", "latency_s": 1.66}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 2, 7)", "proposal_raw": "```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0]\n  ]\n}\n```\n\nWait, let me reconsider this more carefully for the correct rank-7 decomposition:\n\n```json\n{\n  \"u\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0]\n  ],\n  \"v\": [\n    [1, 0, 0, 0],\n    [0, 0, 1, 0],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 1, 0, 0],\n    [0, 0, 0, 1],\n    [0, 0, 1, 0]\n  ],\n  \"w\": [\n    [1, 0, 0, 0],\n    [0, 1, 0, 0],\n    [0, 0, 1, 0],\n    [0, 0, 0, 1],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0],\n    [0, 0, 0, 0]\n  ]\n}\n```", "obj": null, "outcome": "malformed", "statement": null, "witness": null, "certificate": null, "key": null, "latency_s": 3.242}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(2, 2, 3, 11)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]], \"v\": [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]], \"w\": [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "rejected", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": null, "certificate": null, "key": "ba9c2b8a2afbf89e", "latency_s": 2.144}
{"family": "matmul", "model": "claude-haiku-4-5-20251001", "tag": "v2", "target": "(3, 3, 3, 23)", "proposal_raw": "```json\n{\"u\": [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1]], \"v\": [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]], \"w\": [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]}\n```", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "rejected", "statement": "a rank-9 decomposition of <3,3,3> (target <= 23)", "witness": null, "certificate": null, "key": "50edbad14e7e6b83", "latency_s": 2.629}
{"family": "matmul", "model": "claude-sonnet-5", "tag": "v2", "target": "(2, 2, 2, 8)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1]],\n \"v\": [[1,0,0,1],[1,0,0,0],[0,1,0,-1],[-1,0,1,0],[0,0,0,1],[1,1,0,0],[0,0,1,1]],\n \"w\": [[1,0,0,1],[0,0,1,-1],[0,1,0,1],[1,0,1,0],[-1,1,0,0],[0,0,0,1],[1,0,0,0]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 8)", "witness": "exact tensor identity holds; rank 7 <= 8", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 8, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 15.81}
{"family": "matmul", "model": "claude-sonnet-5", "tag": "v2", "target": "(2, 2, 2, 7)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1]],\n\"v\": [[1,0,0,1],[1,0,0,0],[0,1,0,-1],[-1,0,1,0],[0,0,0,1],[1,1,0,0],[0,0,1,1]],\n\"w\": [[1,0,0,1],[0,0,1,-1],[0,1,0,1],[1,0,1,0],[-1,1,0,0],[0,0,0,1],[1,0,0,0]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1))))", "outcome": "certified", "statement": "a rank-7 decomposition of <2,2,2> (target <= 7)", "witness": "exact tensor identity holds; rank 7 <= 7", "certificate": {"dims": [2, 2, 2], "rank": 7, "target_rank": 7, "first_violation": null}, "key": "e6efec940f23c256", "latency_s": 8.526}
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{"family": "matmul", "model": "claude-sonnet-5", "tag": "v2", "target": "(2, 2, 3, 11)", "proposal_raw": "{\"u\": [[1,0,0,1],[0,0,1,1],[1,0,0,0],[0,0,0,1],[1,1,0,0],[-1,0,1,0],[0,1,0,-1],[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]], \"v\": [[1,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,-1,0],[-1,0,0,1,0,0],[0,0,0,0,1,0],[1,1,0,0,0,0],[0,0,0,1,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,0,1]], \"w\": [[1,0,0,0,1,0],[0,0,0,1,-1,0],[0,1,0,0,1,0],[1,0,0,1,0,0],[-1,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,0,1]]}", "obj": "(((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(-1, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(-1, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))), ((Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(-1, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(-1, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)), (Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(1, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)), (Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(1, 1))))", "outcome": "certified", "statement": "a rank-11 decomposition of <2,2,3> (target <= 11)", "witness": "exact tensor identity holds; rank 11 <= 11", "certificate": {"dims": [2, 2, 3], "rank": 11, "target_rank": 11, "first_violation": null}, "key": "cb6d70a944fe9b22", "latency_s": 53.14}
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