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lehmer_mahler_measure_problem
Specification
Let `M(f)` denote the Mahler measure of `f`. There exists a constant `μ>1` such that for any `f(x)∈ℤ[x], M(f)>1 → M(f)≥μ`.
Lean 4 Statement
theorem lehmer_mahler_measure_problem :
∃ μ : ℝ, ∀ f : ℤ[X],
μ > 1 ∧ (mahlerMeasureZ f > 1 → mahlerMeasureZ f ≥ μ)
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Evaluation Results
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| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Failed | -- | Feb 25, 2026 |
aesop
|
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e4LZGNdVvI1g1tXUTQ0mDo7X6lkeJgZBsDlG-SVTww1AqPEQZoyObJn7Wmp9A4gU3g7OcvwUqSD7wgWjMZBHiCZ7Yeu8dL6HBa7YtNnulhR0hAtLEjZ_P5uiumRGOybKrVLt0gI26URpfYbxgp95z_ol7ZUyVfbmk6Fg99Ovq5shTO3lldeUczzt1muUsDFVvNF1DGPhcla4xampKiyODA2fDgZ0oqh5A5aVgdO1QYxN4Nt4mriyY4xZCAqfDEM6J7dMlcK1_dcPpJpmXNggmVQ8toK0wIWhnykIaCoorqJJiIwzdvAhs_KFZNsuz1GYVS4xLnyjwGQgbv9qYJ3DhHydBQHs9pQAC1QKbBgGXCaViOldPtnjQmL9HZnXAsjqkRmYIamwwqWNFLG6bQQlgvx0x86HkGLFS6qqU3iaz6P_woWqxM7sAgrxsa_vtVP85yx6FV-trnubpOScKxRoSAfVDsSCVux-x-pyTd_MX-0J72GzeHV1ByRXUWUaB_lwo6tkTCPTrm64Fi18TxBpdrVW-dQpiSgaR-8st5MKp5PQMwTDpyX6t8tBBnfovA7uGsuIPb6zWLobHtnDnRlhX4TxmAk4MHdZNVMTFdV06e2GuTW9JE0SENPl3DWWSunuheerezu1PYApuu6kgWA1V1pix9JqEM6IRouqOON4x9fAnlNSklNm5aABvMPTcBTs7ewqPc7azTVjz6a1dFM9EnYXkno-h2eRcUzq5kAcmLpl2pX79E4eWbFJMqPYSQe8msjcLiuoezK2QK26rL-qTeEQ5acaSIyHezYbS5gHz_jxQaT00aeAOSiJeTsglDeCKCzW5MDu_79wzykGideZMPKUir4FyJ3RBQv88Rcz27bktOcNwFwZ-eYUTINC5zARofwmRFO6s17z_qIjK6jkpyAa7OR01By47zg37IWNIUeOIIRiKNdWMq5xW4iF2rkwVem-I4EqQbPz3ePo5b1Ozwv_mjzjgyjTYWl2f-ncviwzGtJb4aW6jeci19uwQFzPh5mI5aAqOPIV-i8n8tgfcXbzYwBEtvZ-VZXe_fbiswD3k0X_iFhjZmyJdU6ZhxmJkIerz-_elIJGW-WfP-BIMzIiUN7Jv1wCFGlaIzeqc_hZGtl-KN_cSGmNtVldw0MMUqX3FjK0b2kPd_PhwOsUw8BI27rzfOL47jA-w5m_4ih4dXMDc-5Lmdoyq8h6Bcq2XAxKr37gFl_Fmcxf8YoemaZ2O_zocPpGEca8SICkBKiYmCYjl_AGuUSOcqyiB4E6cSIEJc5sJ8eBD42wHz1PyPU3A4G62a1hnwTdLB-PRW38-8ok2k0ZLRcusMXie3TA9JUYLrsBheDFEYGCoBZXUJY5oDHnE9olSOt5GIEn6ibCqf0AEVHI8efTSqNJOnxR82hn5Vm16A_ZwL-mhDNMRXkE8znOPzF7T4rrGzXTtpqbepbjvzGKrbIiXaiTytngJPQatpKyFSIR3EOmR6-wuBggVpggR5vZMcGGXj-QVgM6dcgR4sdNCUTn1WE6GIrMmJociZIjK4KJCTkrp2kFcmtj3vQS2cb4vVhOJGzy-iU1l2RxjmMtyzXuy8B65Bg2IMeXofgScqmpNJRIYsTcxmMMYneB3JjEK2v6NxDHtYL_JrGhqo7RNLgMWpoRsW9QufC5D5XrSE7BfYYmL28URNCtNWYMD-1SX95jQIVlF9Ft2xP6dH7OlJFM9iqK8OKNGUAlhlzGkOSyLh-YFizYdepW5rKDNDc1-gMyer6Av-wI1-PL0QS9wLlV6ApbFekyYsaiCg1rzs6f8AaUgfb2p5NlPdmguOPFsim6x3FMm2B8MoM4_AfMlrtatwRAZhvjSM7qtggQdSu7lMqR11DYtu5xocPSmbFCPchr_05vxuJjfs0sC0npLBdCLhbxGNJ_VqrxV0vrLmiPS8FuofiesptkVLUPJ700EHC1AXJtTDbwawW5X4wiDrLskHNw_MXKvU7nTTAbjweLjPuceqi_hwzcJkRYlJk2lIUurxCLIc_UqOTDbd6wCkWkJ7fmry8Ovr-G34516j8P2Q_VOckn4Tztfkn5xqXK4owk-fh7DPSuF1ielgM2s6dyi4HbXIeUST80cntOogpqU8HIW7PEfqG3WwDJ9wA0_atN1CmIVol3zNVAJjivEzplth6vvNYOUdf2Mx13C-OJeKiw1Eu_usnPl9g6iTnuWO4FhSvo9q5ZWlq9fsHNdBoUUm4MWgKGwpgJNpTa3gz-Qa_WlGqcz0prrT0TtZBrHqDNTGjKxypWM016rMpkdXl1manOCNVp_T1hLkweBFDRGSyGw9ffHJzpQMaUmqEPSYSAlomqEW-lShtXzTJxiBrKgEFVDEBIsUuszdmXT2QZWGfaVMmHw3h4B5-Zd3tfPMgcHWxwL1gf20fbAe-cnKPsXdM1ISSPpTkALmwqNUXg6sXGA9-K-YclUgwIq4GHizNyT-Kw__7zw47_ZFGxOah4rtkASCJgAvFHSPCNFfuQoAcV_BE6UROBCZa50wpynG-jFutFpK5wbz3Xbq33oh-3CgLFA0tyy7kB9wGIttNxPgDpCOLKwbqKR0-QXHyuZPHH1H8Eg5F3feMoJi31Es4GOeTMSeaOzVuMgLYbX16Yh-g4dHYNCngphMAdRSJrpIx8kW5GvQ7U0Ts98_1JLnMMmr1FVgepn4WhCFdQQHxDdSOr7tCCfD8Yijr17ktkDmu--nQJyIGMRx4R2jPoGQzOaZhCvmGPoGghdaW_hpGqcaPYTosHaXgwvlAutfnzIBf9VR6rPIoJLkHmsCn8vaf8e2Ib9Kx2dUUOL4CCxGpO1Ky8DpxqdhnkW5euVzITNLXcB9hQ3sPR_Ibwi28Zd2tsGC6P9yAqwTFqD18AIkqEpwKN4XpeWAk-Z9CFy6fQoE2m2oFn9_aZefzhyQ-sb8p0E8r9TYwxk2QatOn6zGwczkr_rHuiPpEs6leBAQefOM09YjqTs5FUaOCdWBOFLBFrlxTDm6ktOyAyTYjBLsgTsDHyJjNap-VP9Ldt0TyfmT3Zx2S9rwf4Cw4MCv7IeAsXFE6sbtoddwfe1Yynw8F_lyeu4p9_To72qeGyZQ9WuO0sPX4SitfPkj2bIiCZMHMnkrLb4XQ6qzJrKp-hDjUw1Mqjrloq_0JzBJ3-KIz1TsRmDw-BTcs-eEGV7fCWRXR-yje2RC-Dlqv8wh7DfnOK8cFSRNwFTUCFFyhGH1K5qVVuUE_7BvX1hrbl0Kcvyjy22igOq-84pUILcQyu4a3PRrBVBEQ9QIfc1ptGhYB3Us8V8_Qx3UCNdak9NXfsXF92ejRgxE8FckDnaJ2zY4q8UJ6nNxO4iJg3iJm_R2YwpAEse8Ke-JLYKnpl0_ZUCfeEwk0bYMWAUWAlPL4YygjKqRq9FBox2JCthbNJ6UDEPEMkSJE9vg1vyG61kIGNcQ_KGRF2hVokYnPKzV7JzeQjo3CwhhjUiEQ7B4vAaIUvEedRrZK1tJRBwyKJbPMjK_hMI3RpZDVzN3CwEr3oN4tjRtC871hM52s7-KDW94GKLxFFmsbSuoyXxELixUwNdceKGyxZneNjFZ2_NxjZ4-G3gZGUVGwlNfljr7j7VEwUR9no9AYBbtK_ZksibheimVLNbNbN9gU3SEfDwzMTIcGuxH53GwnJSgTcoX13dyC2a1pw4KVBSnxH_wkAMGRGjooDZebg1xWYmyqD93mLfFX5PVbzzT1BVqtcp6_rkSpr4tQdjeRlyxUbJ-MUFzd_IpAxwJ1B9fruDKsshdsZ0mGsCj3Mzp_eSq6KqMiLsLsBfF1yUEN_dhEzGF_37CwObhkMMkX5dzFqSlj2KkvYEyEoIVB2nPH1JOsmo2VxYEOv5Qe5QfEAz-07AQscNAbPy2iiWqvJ0hc0oVwOh96CTmamXjgCJvgJ-LL8wxykgiObloux8MP6twTgo3HY8BLB8qGTI7iBGQpmiLhzh5VQPSqs6BV2mYhf8BwE02PtJD1zw4_CS35qaKI9llN-2_oI7ABsvECbKCFBZWSVXgJZv0d9YNFjxkIBbDvUEBBQ6TE8QJ04X_hjQVH7PXuTFspadPbLQPN05HH2UeAw5p-T-0sryOx-Cy3MbKUK2d9YIHI9k1aixRKfAN9bjd4RUp1a9Ebz3rm_LA-963Pt8TvoTK4-gvOnb5qRc63UoSOgk2X8O-0rJHKa4waXTLVAPgbG5ZmyIUH7Z9FwcP22ZrbR4-zbtapoqw55ad587ciEXl3tGcKcKUgB83kbO0oqPuJYptGfpTPNqsBvqNV-EpSuJTEaXK1q03y4EOtVA3R5VZax9hwTdAxghP1WjZDN_9vkYgg-0B-217BQDlaXs0mckb_Vk0AkTea8ztp1ej-VqWLhY2zFARhggXGBmydNlyvgbomWehfXAVun8f38G0NCwI8rmoNh5mxlSpHkGUW-UDXVbLiVnFpMkYTWT-T5DHYrDZiE7Od9Z0CfnLg6iyKo9kjzdb1YAzt1IFHin2V4u-MzDSyjrhdlrw9o6Kr4rrQEYOrAzDh5RfdL9TqGhJgJ5eVK4AxazX71_yOoLN3_jpCJBef_OE9CxhDehoeOSfWyi_lwrG5OYhCTm8x_mnLWR6qwZWxmUEJVsD8_d0A507_G_71ZLk9B75CPkkQ2QBs3FyMbOVHuw6oLmC9AbdTi-jM2zmica6MPmbhrJe1Ee_exoZa1TXIpr6H_kIgvmC78ZL2zbULfABsoP7HV_MhsG51yHLOAlEboGid3CGz74Fzf0bKlTd9eF6vEOI7FAYKNUIPkHaXNxnmurphsSF9bbJ6RF3l1pwEUNTxcmCysB8m7j8BnLbgchiY5BNayPf4VXbpp6Mr63w0YpcBzhi7DmEtEgj0zlNUTtq2qimknJ7KjCXikPhvsqHOUeBuNoofpzHtuy266y4iX8N5YPQmTSuydBknXHNru4AduYuOatkDwxJTlw4oNKfcSpFIRmeyRrVNdfexDNHDUYRi8kV7qz2Gui-3AbCMwMwXzb8ITy836W8fnKU7Lk8EqmC14sN5CPG3CMXJxiPSfVK6pnrSnbz3nDw0-zRqxN9e8r1PmyZWroAwMl7oZss65WvVnHlP66lBNeYKa430d4Du1dEfGfcj9ODBY5RB8_kZAkm4ZmUb0hw7Vnyd2UYOQN4f3FC9X_g0gNaB41kLrqSWvrWvDLPogNkN-0BgaNa_sswaUgYpB-4l0YvQq1pzyogqdUKFWIHXyQGfyz3KfHw3VduX7PgGoRWm-i1LVFyyizBXPnkaIxDO1Zg5hQt3TQ1TD2XZuSjbchnqOvvkFRvn3J2ulzFcQFmHFdmKC8T3cSFnSMg69Y08Bjb1rAzc5D1P76Tge7dxZ_TZ73eoCCMg6GVsaGh5iQZlDTZ1WmEnwBL4Iw5FX6Qp8Q0bhpaFdxz4k65R96wVDeCeliBjunw-gdDbmkadgJFSy4b8kZdH9JjiuPyyl8Snabn8MMDeSL1G_ASHj3HVEMEiW54Y6g5v2OVaO2obiIrXH-jC0T-DAgAsNTWTpA8b8z-jCcLc-qGT_vfkkNdDkZCzBzeB6yTpACb-m_wfni7jP2l6_o-8ugacGpMWx3Y6mUt8UUELv44Ln_bQ_UQeCVJdo4aMy0xm_jcLrc57dHyUbeU7k9LseeA_mLBYesN6_ocZOa1qZUcy577vNTEQMu9jp-dqbN-2B8fqYgj4rfKxQQu1vJD1sqp1cG1rRdtHHHmdZE6sZp4s-OtV3VAIAi6vUofnq_AqMT-qcrhEh5kfwN7Zr6XLnZJ-iZwMT9koGQ-1vKULgBzkvBVnzJvApK5St556RuHmFg_aH5hbA23doPHjQya_pbzWlqJEW-qpF0Im5VX61zwLBcpI-4vp-m0ZLDXtF7nx6d3-MSUbLZYTZgSm0vdWARjw=="
}
]
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}
},
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"phase2_error": "ProgressAnalysisResult.__init__() missing 1 required positional argument: 'progress_score'"
}
|
| Gemini 3.1 Pro Agent v8 | Failed | -- | Feb 24, 2026 |
View{
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"duration_s": 0.05,
"best_status": "pending",
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"max_iterations": 10,
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"error": "OpenRouter API error 402: {\"error\":{\"message\":\"This request requires more credits, or fewer max_tokens. You requested up to 65536 tokens, but can only afford 58661. To increase, visit https://openrouter.ai/settings/keys and create a key with a higher total limit\",\"code\":402,\"metadata\":{\"provider_name\":null}},\"user_id\":\"user_2zsm3L1zzBL2yxyrvmrXwgeTimc\"}",
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|
|
| Gemini 3 Flash Agent v8 | Partial | 1 | Feb 17, 2026 |
use 1.176
intro f
constructor
· norm_num
· intro hf
refine ?_
|
View{
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"proof_code": "obtain \u27e8\u03bc, h\u03bc\u27e9 : \u2203 \u03bc : \u211d, \u03bc > 1 \u2227 \u2200 f : \u2124[X], mahlerMeasureZ f > 1 \u2192 mahlerMeasureZ f \u2265 \u03bc\n\u00b7 specialize h\u03bc (X ^ 10 + X ^ 9 - X ^ 7 - X ^ 6 - X ^ 5 - X ^ 4 - X ^ 3 + X + 1)\n exact ?_\nrefine \u27e8\u03bc, fun f => \u27e8h\u03bc.1, h\u03bc.2 f\u27e9\u27e9",
"compiler_output_preview": "[path]:31:15: error: unknown identifier 'h\u03bc'\n[path]:31:4: error: 'specialize' requires a term of the form `h x_1 .. x_n` where `h` appears in the local context\n'LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem'\n...<truncated>"
},
{
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"finish_reason": "stop",
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"proof_code": "let \u03bc : \u211d := 1.176\nuse \u03bc\nconstructor\n\u00b7 norm_num\n\u00b7 intro f hf\n let f_complex := f.map (algebraMap \u2124 \u2102)\n have h_meas : mahlerMeasureZ f = mahlerMeasure f_complex := rfl\n exact ?_",
"compiler_output_preview": "[path]:32:2: error: tactic 'constructor' failed, target is not an inductive datatype\ncase h\n\u03bc : \u211d := \u22ef\n\u22a2 \u2200 (f : \u2124[X]), \u03bc > 1 \u2227 (mahlerMeasureZ f > 1 \u2192 mahlerMeasureZ f \u2265 \u03bc)\n'LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n\n"
},
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"proof_code": "let \u03bc : \u211d := 1.176\nuse \u03bc\nintro f\nconstructor\n\u00b7 norm_num\n\u00b7 intro hf\n dsimp [mahlerMeasureZ] at *\n exact ?_",
"compiler_output_preview": "[path]:37:10: error: don't know how to synthesize placeholder\ncontext:\n\u03bc : \u211d := 1.176\nf : \u2124[X]\nhf : mahlerMeasure (map (Int.castRingHom \u2102) f) > 1\n\u22a2 mahlerMeasure (map (Int.castRingHom \u2102) f) \u2265 \u03bc\n'LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n\n"
},
{
"iteration": 4,
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"finish_reason": "stop",
"response_id": "gen-1771348865-kCcXaz9nnTGN4DZBQ7gU",
"api_elapsed_s": 1.322,
"usage": {
"prompt_tokens": 2755,
"completion_tokens": 28,
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"reasoning_tokens": null,
"cost_usd": 0.00055925
},
"openrouter_retries": 0,
"response_text_len": 63
},
{
"iteration": 4,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "use 1.176\nintro f\nconstructor\n\u00b7 norm_num\n\u00b7 intro hf\n refine ?_",
"compiler_output_preview": "[path]:34:2: error: unsolved goals\ncase h.right\nf : \u2124[X]\nhf : mahlerMeasureZ f > 1\n\u22a2 mahlerMeasureZ f \u2265 1.176\n'LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n\n"
},
{
"iteration": 5,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771348873-d3K33oxzOboZaWJ2P86O",
"api_elapsed_s": 1.676,
"usage": {
"prompt_tokens": 2995,
"completion_tokens": 86,
"total_tokens": 3081,
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"cost_usd": 0.00085415
},
"openrouter_retries": 0,
"response_text_len": 196
},
{
"iteration": 5,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "refine \u27e81.1, fun f => \u27e8by norm_num, fun hf => ?_\u27e9\u27e9\nlet f_complex := f.map (algebraMap \u2124 \u2102)\nhave h_meas : mahlerMeasureZ f = mahlerMeasure f_complex := rfl\nrw [h_meas] at hf \u22a2\ncase hf =>\n exact ?_",
"compiler_output_preview": "[path]:34:2: error: Case tag 'hf' not found.\n\nThere are no cases to select.\n'LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n\n"
},
{
"iteration": 6,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771348883-1WZLzrMtBjzdRoUMs7EE",
"api_elapsed_s": 1.108,
"usage": {
"prompt_tokens": 3269,
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},
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},
{
"iteration": 6,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "refine \u27e81.1, fun f => \u27e8by norm_num, fun hf => ?_\u27e9\u27e9\nlet f_complex := f.map (algebraMap \u2124 \u2102)\nshow mahlerMeasure f_complex \u2265 1.1\nexact ?_",
"compiler_output_preview": "[path]:33:8: error: don't know how to synthesize placeholder\ncontext:\nf : \u2124[X]\nhf : mahlerMeasureZ f > 1\nf_complex : \u2102[X] := map (algebraMap \u2124 \u2102) f\n\u22a2 mahlerMeasure f_complex \u2265 1.1\n[path]:29:63: error: unsolved goals\nf \n...<truncated>"
},
{
"iteration": 7,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771348891-5P3gROhTM20agDzmehMY",
"api_elapsed_s": 1.146,
"usage": {
"prompt_tokens": 3685,
"completion_tokens": 65,
"total_tokens": 3750,
"reasoning_tokens": null,
"cost_usd": 0.00113615
},
"openrouter_retries": 0,
"response_text_len": 145
},
{
"iteration": 7,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "refine \u27e81.1, fun f => \u27e8by norm_num, fun hf => ?_\u27e9\u27e9\nlet f_complex := f.map (algebraMap \u2124 \u2102)\ndsimp [mahlerMeasureZ, mahlerMeasure] at hf \u22a2\nexact ?_",
"compiler_output_preview": "[path]:33:8: error: don't know how to synthesize placeholder\ncontext:\nf : \u2124[X]\nhf :\n \u2016(map (Int.castRingHom \u2102) f).leadingCoeff\u2016 *\n (Multiset.map (fun x \u21a6 max 1 \u2016x\u2016) (map (Int.castRingHom \u2102) f).roots).prod >\n 1\nf_complex : \u2102[X] := map (algebraMap \u2124 \u2102) f\n\u22a2 \u2016(map (Int.castRingHom \u2102) f).leadingCoeff\u2016 *\n (Multiset.map (fun x \u21a6 max 1 \u2016x\u2016) (map (Int.\n...<truncated>"
},
{
"iteration": 8,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771348899-CzmdCbxzUdcsQXQEitFn",
"api_elapsed_s": 1.296,
"usage": {
"prompt_tokens": 4314,
"completion_tokens": 40,
"total_tokens": 4354,
"reasoning_tokens": null,
"cost_usd": 0.002277
},
"openrouter_retries": 0,
"response_text_len": 93
},
{
"iteration": 8,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "use 1.1\nintro f\nconstructor\n\u00b7 norm_num\n\u00b7 intro hf\n dsimp [mahlerMeasureZ] at hf \u22a2\n exact ?_",
"compiler_output_preview": "[path]:36:10: error: don't know how to synthesize placeholder\ncontext:\nf : \u2124[X]\nhf : mahlerMeasure (map (Int.castRingHom \u2102) f) > 1\n\u22a2 mahlerMeasure (map (Int.castRingHom \u2102) f) \u2265 1.1\n'LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n\n"
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"[path]
],
"duration_s": 7.5875561237335205,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"axioms_check_name": "lehmer_mahler_measure_problem",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__LehmerMahlerMeasureProblem__lehmer_mahler_measure_problem",
"category": "research_open",
"endings": [
"end LehmerMahlerMeasureProblem"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 505,
"source_file": "FormalConjectures/Wikipedia/LehmerMahlerMeasureProblem.lean",
"title": "lehmer_mahler_measure_problem"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:34:2: error: unsolved goals\ncase h.right\nf : \u2124[X]\nhf : mahlerMeasureZ f > 1\n\u22a2 mahlerMeasureZ f \u2265 1.176\n'LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem' depends on axioms: [propext,\n sorryAx,\n Classical.choice,\n Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
simp only [gt_iff_lt] at *
apply Subtype.wide_poly_is_measure_pos hf
|
View{
"generation": {
"attempts": 1,
"banned_tokens_found": [],
"elapsed_s": 4.75,
"finish_reason": "stop",
"max_tokens": 8192,
"model_id": "google/gemini-3-flash-preview",
"proof_code_length": 70,
"raw_content_length": 1866,
"reasoning_effort": "",
"structured_output": true,
"temperature": 0.0,
"usage": {
"completion_tokens": 528,
"completion_tokens_details": {
"image_tokens": 0,
"reasoning_tokens": 0
},
"cost": 0.002453,
"cost_details": {
"upstream_inference_completions_cost": 0.001584,
"upstream_inference_cost": 0.002453,
"upstream_inference_prompt_cost": 0.000869
},
"is_byok": false,
"prompt_tokens": 1738,
"prompt_tokens_details": {
"audio_tokens": 0,
"cache_write_tokens": 0,
"cached_tokens": 0,
"video_tokens": 0
},
"total_tokens": 2266
}
},
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"[path]
],
"duration_s": 7.396281957626343,
"goals_remaining": null,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"benchmark_id": "Wikipedia__LehmerMahlerMeasureProblem__lehmer_mahler_measure_problem",
"category": "research_open",
"endings": [
"end LehmerMahlerMeasureProblem"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 505,
"source_file": "FormalConjectures/Wikipedia/LehmerMahlerMeasureProblem.lean",
"title": "lehmer_mahler_measure_problem"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:31:8: error: unknown constant 'Subtype.wide_poly_is_measure_pos'\n",
"timed_out": false
}
}
|