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littlewood_conjecture
Specification
For any two real numbers $\alpha$ and $\beta$, $$ \liminf_{n\to\infty} n\||n\alpha\||\||n\beta\|| = 0 $$ where $\||x\|| := \min(|x - \lfloor x \rfloor|, |x - \lceil x \rceil|)$ is the distance to the nearest integer.
Lean 4 Statement
theorem littlewood_conjecture (α β : ℝ) :
atTop.liminf (fun (n : ℕ) ↦ n * distToNearestInt (n * α) * distToNearestInt (n * β)) = 0
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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 |
simp
|
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"
}
]
},
{
"iteration": 2,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "simp",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772023406_5252.lean:29:2: error: `simp` made no progress\n'LittlewoodConjecture.littlewood_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 31.396,
"tactic_replay": {
"tactic_count": 1,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 6,
"line": 28
},
"start": {
"character": 2,
"line": 28
}
},
"message": "`simp` made no progress",
"range": {
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StUHkBu-0C-Nkj4o9nWtJODWa-Q7zzNidIkVozZqKmYCCtIi7hVXy22yPVfKf9FzmDtlYO_ElZrzxdNT76vf8YniOwRvv4jnOn5MZsgigBsoZpH6Fiip1Vg_ce8tCYxvpQjhyNAvTBqyjqwbib9DI3ETmDYYITIpSSPz6_eZCvdRNe6e7Gninel0LlSl755zN-u2rxejHTfeoGtuIZFB_ZO4LYPA-5o8ZTKlHDoOyoOkf4O1DmmrkJ0KFUGequLloLB4a8fzEHT9HoUDIZJRq6eh5jECxmkKnj4mSwI-E_CFR6boXjKWNkdJYaHMWqZKA9uU8i_BqmKeE-j9r8Q23p43GxE0oXMBKwXNMNQrEIs8j_mUBhXo1cTTsO9bGtrdmcYvxSy6OkNLaBiG0ryrFP1cwGgzydD4BrfZ2AFHCtYcctr4hfAWaexe6kBjtccWUYt_dzksdfRdEHRoNv2wUcZcl9ZwN-sAdHFKH688YwAkzW3VE0fJIIvt1tvpneMbVkIU9_MosW-ZWhE0l6qivjX2iFnSqqEUsT0HzFWZU05tQUo4xM9_fprPHmkhY2lXAh2fAsqNlqtrCCXGg7S8mpwHEND3EVz27T-pc_gL9RAehW6oCCBkwEPfwgWLJs09Vu-wunsMh4CsgO53udekasWjdWS7pVA26NzKidigNarl18tL94T-kgixPUczvB-Gt2y2G1C346jrRw3am6hrL24uaPkRkMUJ1mE-CCym68W77L85vIMkuvJSLNgh_d3BC4jHeoFSZrBn8q4CYDwLw6CSswM-R7M00wt9klNnoQRMKr_YiTNwekEncY0xlyQsBAqSBXSgOuONGBdcTLBTWSMj325piUfieUR4Qy9apGsYWrqtaLsTxZcePvnwiFdbeD8olpAKfIVDvWskXFjA3SM_0mKW3rh0o41-79HF0-kIMYtAUV0iiQaXQeVirsmVDMt9uE61jeUPo0v313_C15VIQjSzP1aJdJO5OcJQO3H6UYFKFWCiu3D0_7r_D_LjVHb86Iw6zaqPlAZiidI3lb6yKE4QB4QiPZMo6qzbYlVdKKcUK55cb8WISbCmd"
}
]
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lEJVopfp2Z1Gh3W2bFt_SBY9EHioEDwReb1LgaBTpx6eAWdaH0yydHwopvd3Jf3b0j5jdhB7NrIf1_RO0svY_iGEXeiVF2EnclYvl4kBdnKKYeWFuT7jxEwSPk426QY5Ygc2BAQnWwr87a93nrZanXH5MuupMiXelEYME5GwwAs1LkIb87NrFPVgyaQz7mqJU3Qg6drx31L9gSGMRTC42sDmU03EBPEVi8S_NRIYmSGxkw-d1Ttv5hN2TfxNhfHdlBWWg7Lonr5alF4eoxqQzlJRpkmpYhN3eiheaPc0zlFeQrOPU2LY1o33sJPY7ztRhYA_V_7OHL-jxf4yUJo8EOHtm7P_wnQeX6_t6VS_XqoN38Q7vMS0mXG4G5Xw929V0vxO0-JjUxd7aDDtxfV4Mh2ffd78qb9GHGKrTDFHTgDJ1g8a1Ye5LhAGLWJrrEX3H4KX9Cn1HMoZ1xElwORZb_JdMof9oz1PtpZrJeqIjnfekB97XK6T_Z1kEa8p_w_SufSGoC--4w88rEYuMsUj6wdcU9msMgsI0d4Wf1VfHuKLdr40Qouuwzq17yZeL-qy3hGo17wxwr4SLFgy1rSJhgNIEHIYJOR0U30PKq0PGOt49hp6edj89XwMV-bicRmHM2px3HliSp_eMWKBVBrJew-3Igm2Un-mz7yCp7pkU7u9wnq0xeZymda-56P-xb7zhZe0nKOMJblgnkpGtJNichztRuiONm9Q1OYKsnuSjLg6wzJhnCwgkj16L4gR0C3GIRxzD5LzGJs92x-mpOaSeY_yIh46cE9gaEsHPiUhMMkkakmuHfJqlDKL_ef7PyfPEjxCxFqOaT7dKQZcSfaydgXSifhG4C-JV0dfkXSQyOizuYHFNA81hIDO9I4sGJvL15VORNmtD3lySJPf7HZ7Tm81G9m3or2UdE0fUXUVd9dAbVWvxuxlCUoMQnXim9lJRHCsjJ_iZRabV5Ibq9RTxFA4fPFKou2pexKxmx4hNLh2gYp3dAWovtrKDtmpl3iacZ-Iv4WJ6OgmmZR2wo_p3ndOdkAHSWV21Gqir9WycxAXcq1eS6cjGynyGwXuTdSvm2XeNs7k7XB2hwvJdk5aw6RKG6xPxCQqRM3o455f9h01ztTv8XdimEtvI8IwqfODMWJEws2pVzoffjmm7O6_fvVTUF4NQGRKlxVbhqKWP8xO9HOxC0tS6qcLxhLm7joIfMQS7xd-pIyDr1zTiaTHUHPQMreCZpFVJg8cwBoP6IiBnWywUql6pBmq5XV6nIjpN54FyCYLVDABTVRtPm31mltso5DSu1__-Kr_zqEIkyVlHJrW5bx5MBF5Qnbr-dUS37HO8Wn0rBMFAdVfCOuTgw14F6VH_AZC2wCDe8xRdvP5dm1Dk07F5a_xzVvduTqNLPxIc_LrC6atrMDxWZoNAJy_gUeY2J6m-LS-U2AZKQYJusAfd8Fbodal7lb-19QcceRFewiARYv57ix_XMsjquPKu9o64jtbcokMmCkfSFe7xouZjemcYNzm6nzj84wWhP9imTBV1ycZ1uK3a6QPi1_BCK8SxeeaxtnrqpRmhkvezn-u5UbxuJMLMUPgn1L3poAOAhwBl-S-KB9fXCYdHjo363GuTLEWCGXA3uEGt3om8FeAI2Nn1tKuxxT2Mvh7HmqOyIs-z6FZF3BRv9pUqY7zyWowxFIc6fAl_rqj7ggOvBH76zC5PzSqA005szoLtySbEp2YPHKMW7wfiOHiIksQN4lcpwdiGBPZSthn8uUP4SS7MA7ERJow4Kt2Blr_iZmxC4Cse-GtoFbmd2_CGP7sv3yTb2JDpubyF4Ao31zWuMVUCYjoclW-CinzqSsMCsq4QbyZjRzA4RNHcuv3MLd6WRUu4qsRHEwDnURJtq5hOINmF7bhLJNc01ZeqKgjmFr3qSbw2xcMCw8AIdIMwwqLK49Wdee16S5YpyWXkxRTBKdlmho_JIVcGIzpt_o12SSmo-A5nSeK3S04KoH20hM179vwkVImMhfzf8uKfEVYS5TuJ58e6VBmYorV6kzBr910w02bFKailBd_fmXwtHWjs_v6U-ZbCT1qGsGNg-S3xLATdIvOFfuwvZct19ofjyuJnOWCdqn8RQOdIAxyda_c6bAhGUeVSww_J82-MNy2fZLngO-wO_Wabrs7fHyuHMa46rbUndS2WnfW9D7jwgOK4WnRHjZa3PUZF0krK9hdqyGhCecMbJedW4yQVtuDfOlVI1C2UjpTzrLI352X3dY0kI94h-mlSe494zTgcZDsZgeZD1KHeZ96_QQLGgyimjnial6xUBfdYpQ3AEhXdARwP6MCEwJYv2wWk9OW6xeeBdcoaBdTaKYcySw65oLROiaOzOphb4WD5WBemGP4s1FMWMFs9WuxG__tfcW2HdXf4Jz8pzRqXFjV4B8Qr3sltUALMhDO2k3AQm1x5GbRs5-KwoDTvg4MCl0_Rn6eUaU0-jb4nd_t7UcASZKhLP6Z0h0W6zzeMsG8H-pKaWR7mjZZquWao-AXbWA19mj7cikCW7Pw6PM0SuCgjNa9dojx31gUTXkqxcey-a8eS_ZU7D3ObQRcqNhIMu3ajl-BXbHoHnIMvTGR4Hh2AajnQJj4Lvi7eEJMTBOwENJjBfKXEare8JCCfYZwA1W8MsZUwRjWabijNe5KiWz3MS4leSmArcwtI5paeZL54bUvByXrT-WD0Qq6pKLqUJbZhEXhoifnVQKN4QkAFtCYqCIxd6wCeZbvuAoGVml0tjGp4qfzTootraZYlv2ouWlI5AF7UBP88KkUvsie6-mkwctUuC1wIlwIL08FtsN8NViGPkNdZZejgjZRh3UTtdMQWGGmuYupKW8UpeW6rbNnbfgWtqufpenSiPgvNEaqijoNMMurosQ0IhOW_ov6v7NvRxWsvhkrPF5151gx16YjNiZLSq65gIi1ImDcfgDPHsXSKR5ssPIui5OBXFH7Rs0R5hfOFfCZ8wLUdRiC8ZpNFFN1cODOAZWZgDVjNnA3sLOCyj36iwvStaJskFhEeOgR1AIEN-5Nu-AmoKJmW7a0fICQSfmnkeqtAQ53SE6LZN30DQPHlVtInD6gvrDxcf0mKIC1BndJXlCnr9PC0Cs1zS6gRxfUBGN_DUwbOR4UCrINpY1EKqXDt6a6zf8eBiya1Bvigce9Ru_ZURx1B-mWrkRVBZxB5ZnxNnAoLCExjQsOx9BpgdcQD5GHVZEISKPza8edWUKEiadFPL1eyRebfcT93HmeADjfA9mpMZzoZbh9Z8X2obIONe5Cge7yrgxJx7iX59AdFBlRiLQnR2xU7BuunoXVaZyoWZA0_OcDEv_S6TuZSIoWhAFI3yl2Dv1BmiAKoiLtztE0P_ZIgJLHs1jyjyQUZ6I48zjhvWTPQDNqjW77HBp13-3Rf2N8nr0GQOHWUcLb_LLgc8qNHjqaj4Tniw6PVVj6UEkvLVHNJ9VRz65WPbrWRoGhYtz52qEx2tDNoAeNWmDlvJ2zbAH8xhsZ5u6Se9_dF2yRi9wUbnCg6nWKH7SnYRNNOEkjmNrOszZ8YefrGDF70e_9KT_L2c0ijxVAKS7GwHew01NO7W4LSKSf4fS1qnLkp7AwDS-kHv5Kwps6y8kVCfwzgbCdDx_EGc-fh93u5zdlLceiWIaw_T1b_ttgOCXYp9WPWYhH2CVZw1AfS_3Hf8HYYYPGxolwOLbr_nPTr0CNhKXLN4MyXr8wPNwg6uKHFWj2YS9-4fTowAh7aC7pZS9xb_2rhEEmCgzEKlV2-Wp5YiG_DeJYf279JcPYvS0tabmtOpG3nMhGYdJyvg6qjCNoWt3r0GX5GxSwO31JB3aBMguLyj-GgjSfYnCOEBv_M_U28KsB16gfxNsiKg_N6KbpJllFk4NCUiQy9NzYdJ-79jz8M70zDrHbfv7ptoewdYT92Y4GTbgZW52vF4HJBMM6cvHTH7W7jiJQv9GEaUp3nfL74SOFXICbGvsu0gbszm3uxR_3HxTXkZjOh-UDnvAs-UOj9gf_uBHygu1qDaNEAqZhTvxmKLDvSHHajPplrLpjHo5rCD2STXWFqmDqkrCLm1qAvY94JcwGaE7IliKQXG-ljku-A36IJWZEeiSq7YODbDw6uD7vhosEz_h7uAWZTFRN-TbfRxwaqclgwKVQ_1lMneEFKRTy1ihKqpdELP2Gn1TTj6sJk3b1RPRXUuHsKBvF1y5myko37cDdC52ZGnDya9t3_QNeQZGzltV8uev2MtshPtAomKcZH-NQmud0hHdEKmLFD2PqN8U6CZ9Hn2YzQiMS6sxh9HTDHUj-ThHa1G2a-3p3YSYLmJxBozrZMpy38a47IBNh2OOwEnY1MGG4otEU4AB0mWSExMXqEBdFqO9iQB84PC0iGuaMLv1CXVgWFHFgkJRH_Dmm90HQ-SJ_JAa_DUBwNP3FKuclgD3ALiNOsSIQJnMDRHXmCu6FK5Pq3bBH97KZPOD3qgQGkefbc-b5ZKq-Ho4-0Ocw26aiYHNMa9zWR9x6Gyb91XIw5AQBivBYVF2Bxor4vXA_wwQPH5vy36ZOjUs6PKQbR8GnR59dcnfdrm_pfT_95ANL9biOxUEKOkZErMfEZn37ysBIXffee9-PP2sNGeIJ1ZXA51mgEeyFEJOtuVHLqKIbfX3GASnplHhzOR7RT2oyTGH-IDIu6EwYPbnF0kSUijdFVgbbqVaP9xmbglxjE1Gel94r6IiWp2go7r-s6DtWeUb12VvgxTTlKjIbiAaDr4nqRHUCgKbogTYUkiBEBzpZKzP165sH0zv5_YilOmHx4J8yszHZTqkozyjKUcFnx7CgjSu61-mIsadsa4XCvu5rGfMRNNSwTz-Mn2LjbG7FrOFKh27LONpZytG7nRPwziXKtOPydP4IrCRvqQeeqjqXXEPR9dFGn1P6GuUvwhTgF5tBIpvULvU1tbxeRFRKsD7wt5SuYgsEEB6_rXOZCQtPvn6jUuDm8dGiNh-CnmZ8dNiZqz65jAclCPmSwYYFVUxeiMx5I312aU-YCLKctHpK7Lwp8CFPBFGcudDI7VZCW3oqmsdXiNrNAFTec8unIUURbrPXdChU87amr_RoVGRZB1Cb2qhrV-CdRGoAesDMQwKk5t2GRyJ7VgdcrM1OqA_rYeDVgf2M4NZE3s1V_RqLMLyb6j3ZdX-t69Z2j6cK9ifRA-GSvibu23_F5vIboGu2K-9tP845R6AFYlE3aAfzll0Nr3A95Ih2rrw7vGg0aW0CXGS6z48s4Xrd1tSlV6JcSDUhAucvBp35ntLNI_2KZtzpa3EsHgzPBpu_ABbjjKX46jiB1WLD00DxJ1gXtEWPf6sG8E4oG76LLBcysqRSxhocg5EEBoK4T3FDf0BS0VAmto1vofADuGe_8Wz7QPm10UIQEF__o-meFcYqbyZ6aj5kLAmuEC0KntforbszjVYcu9zYwWGe99OUs4YMyLJu70Ytxf6yGkSTOe5TwRxQLlJcUc7DICtRKgpnZm39iwQM-kOyigmkXGpK_xRF9rh7Egu51CBoiPylUHNqTyw0FWRd7kfcA5E6K_DQiGmUpo_LDhwIAj_sW-fQccyAQNSKwt3QBuWK0IWXUiHf3P0ews7BWqamB353sR8kOJ3ULerR0ij2chxxNvUamxyPuq6ZhGHBtX42TBJHmMt3mchPbDr6DCr05iScpGOTiFzMKPEG5C3Zjhhlds7HWNv3jKXLjG6LdNXlhU2GXT070DDMklLeVCakPtr3UmBilQIva2hdvjReI20_RP9VzZ-fmPD2H_sWU0o_ZncSaHDYAM-g--i6FobvlsvS2ss8MTYQ0nTgrNBSJFY3BAn7gx-BZm8xw0m6qDtN_iQO5i09BTcd-eOQeRh13Hf8g88tAmV77ytgyUfPLgH0feDYUT1O0VCFy77QeuhTFv6hXtV9-RJLeluz7Pm0dgXUuh1az0erTumR9slyxLr553gorTjGbCqRZE7itlDv3ZSnXw8llFxP-biR1uVwqPwilxkz6YN_BFipQKiUKkn8UKXFzWLz7ECdEHXE_qF8q8nZp6OpU6_KKKbGb_Ab84yPNhUG36DyxKpyMONZf8647N8UJ-FaY5ZojL5aKMwtdfYFEMG1HQ_hcpQzQdWq7GEZQKA0HCTEKnIO5IfS8ms7pFb-tjIvEXEJkXZdlc625iEeB4LuLXmdIkeIadm8cXH3DWAy50DluGgDvDqsxnhpnJ9WYl5AzAwTLdIQg24TM08wwbzH_709vOvJ5MEXa9VTVX3WX1AIqNpzlcbxfjYVs7zmABWqHv9Me1XG1EmxHTCjRY4YJKtpIpXNR2xox1YX6f6QGvL6LJj5WlLbw4HeM2vMYwV2ag_4aVDtRh0eY4jn6SK50hYd3Ryp52aFbJcSaQod9knhEp3HUk4ErVBMm7PLe4aRh9EaqTJUlSVOx4mtldy7Wu8ZRi2XGRSP0tk_HqMUoLgkIL74VuXAEOZLJ0I-4J5TrJdtsDcbYbiHW3cRbsbSyFNWTymke0dzoODS5-7byFZ0pbFCgsFGn3ZuxobxMy2O9sWcJDzqhPfBKbumt0DHmE-QluWc9L3S4EW6dc_NCLJBu2fHTRJSTfxZOT3JbOKG1pvJnKkwkjnnWZinXKPkqRBlWQl3JluM4bjHXy0lVVRn9CXo46ZrCgPsyxt-OH4MDF_PxJCV1eoD6MFlOpT4H_WJUCG8gKJ_aQtKK8us1IgO37uASO8V6DpxuM_k_v9mJ1NokolVslyaKsbBkI58Z7JbeMRB48sEIHScP73F3Tzq5IzSkXMPSTcJ97oKcc3dixysLHRPj8zoqWd7eWZJ2KjXTgmFr_wJK4AHSpu2bae3wLema9U2bbIKvSGINHm6ryWLDRPQjtgkx_Sw8I2mgWFjKSZQCOLF7lba5mBAHejtf5aW9K5agTy3I2l6q5TALNpriYT-c6QR5gveEkqa3ttpNv0VBQQEXzI0eREg2TKg-WWV6GybW2fvW697u6m3aVd9TVRvCruzohxQ416DrqGu-SC9J7iSq_1KVCP2Aw0fRR9zVJzorlEqMMlTx0BhDm1MGO71DlBEZZ8GyNl5Q4zYKL8fqO14_Y_o0OO_FfZa_WwgWGidxcWJc8XDyaE-4InVJB8ErrKgLFt-atTEnabiMMt3B9GeRa52xG8wvTSTvMLOjJTEXuX-wuheQkP_R3yy0kZTJqrITytQ4uXyOkAxvFLTqzS5KtccSta8Cv2mU_ciMijRLoUDJI6xxDZqdbXTfE2_cMBVbxiBzid6pWle8vBSQuLrkHjeOCm_vdUgzOAvSw5XTiokOnNi3S8jyP-m-jk5Y1K7KjtUyZifo-h511GJtG6lDiWzERNVEmDptqHMmJrPo1_U7_1yF3MdaO8W3UgN5tsxjsdbRKBU5ijVKGtG9BdFGhmo8QjAm0easL6ABeD4gBJDQ3-dUZ8CSe9Eap0qEF3VhIN3LMlOFA_TVI5AeAtoCnGJSO_SSRRLNa_7kmGllLk87iVE4RIPTEayg_2O9GiYRKnc6auTE6K8FrFWlt_G0XNdJbkovEHYT7_Va1Vufwoa98WAiAeFDZsajeWLDeN4eUkwyaWgdINJhwtMD1RmmCtVOTSAVNa2tp9mPIunZc6f9wppVg7zHhAZU8safTg-ZiUxt0HisvpnKDivcXkHd3KwsLUvcjgXAKkUF_h6IyC-Wy5gd1z7MCIHD4J7FPazLd7E9Ol-FreySJjnrPv2LayCZlrItUdVpDzIStgYK3qf4pASANQLPOBkuIrZ7mg94sDDCtfT3AHRO7Eq6cdM9Gk1H-EOio-ySMQa3uvH3UQ=="
}
]
},
{
"iteration": 10,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "exact littlewood",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772023714_1094.lean:29:8: error(lean.unknownIdentifier): Unknown identifier `littlewood`\n'LittlewoodConjecture.littlewood_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 28.069,
"tactic_replay": {
"tactic_count": 1,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"code": "lean.unknownIdentifier",
"fullRange": {
"end": {
"character": 18,
"line": 28
},
"start": {
"character": 8,
"line": 28
}
},
"message": "Unknown identifier `littlewood`",
"range": {
"end": {
"character": 18,
"line": 28
},
"start": {
"character": 8,
"line": 28
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 30
},
"start": {
"character": 0,
"line": 30
}
},
"message": "'LittlewoodConjecture.littlewood_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 30
},
"start": {
"character": 0,
"line": 30
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 28,
"error_char": 8,
"error_msg": "Unknown identifier `littlewood`",
"goal_before_error": "```lean\n\u03b1 \u03b2 : \u211d\n\u22a2 liminf (fun n \u21a6 \u2191n * distToNearestInt (\u2191n * \u03b1) * distToNearestInt (\u2191n * \u03b2)) atTop = 0\n```",
"full_timeline": [
{
"line": 28,
"code": "exact littlewood",
"goal": "```lean\n\u03b1 \u03b2 : \u211d\n\u22a2 liminf (fun n \u21a6 \u2191n * distToNearestInt (\u2191n * \u03b1) * distToNearestInt (\u2191n * \u03b2)) atTop = 0\n```",
"goal_count": 1,
"ast_complexity": 107
}
],
"tactic_count": 1,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "exact littlewood"
}
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772023742_2287.lean"
],
"comparator_passed": null,
"duration_s": 12.965830087661743,
"goals_remaining": null,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "littlewood_conjecture",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__LittlewoodConjecture__littlewood_conjecture",
"category": "research_open",
"endings": [
"end LittlewoodConjecture"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 503,
"source_file": "FormalConjectures/Wikipedia/LittlewoodConjecture.lean",
"title": "littlewood_conjecture"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772023742_2287.lean:29:2: error: `simp` made no progress\n'LittlewoodConjecture.littlewood_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n",
"timed_out": false
},
"phase2_error": "ProgressAnalysisResult.__init__() missing 1 required positional argument: 'progress_score'"
}
|
| Gemini 3.1 Pro Agent v8 | Failed | -- | Feb 24, 2026 |
View{
"agent_telemetry": {
"compilations": 0,
"iterations": 0,
"duration_s": 0.12,
"best_status": "pending",
"best_goals": null,
"max_iterations": 10,
"model": "google/gemini-3.1-pro-preview",
"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\"}",
"provider": "openrouter_primary",
"finish_reason": "",
"usage": {
"prompt_tokens": 0,
"completion_tokens": 0,
"total_tokens": 0,
"reasoning_tokens": null,
"cost_usd": 0.0
},
"reasoning_effort": "high",
"response_trace_count": 0
},
"response_trace": []
}
|
|
| Gemini 3 Flash Agent v8 | Failed | -- | Feb 17, 2026 |
unfold distToNearestInt
apply Filter.liminf_eq_zero_of_nonnegative
· intro n
positivity
· intro ε hε
obtain ⟨n, hn_pos, hn_val⟩ : ∃ n : ℕ, 0 < n ∧ (n : ℝ) * |(n : ℝ) * α - round ((n : ℝ) * α)| * |(n : ℝ) * β - round ((n : ℝ) * β)| < ε :=
by
-- This conjecture is a famous open problem in Diophantine approximation.
-- Currently, it is unknown if it holds for all α, β.
-- We represent the goal state for the solver.
refine ?_
use n
exact hn_val
|
View{
"agent_telemetry": {
"compilations": 8,
"iterations": 8,
"duration_s": 76.5,
"best_status": "failed",
"best_goals": null,
"max_iterations": 8,
"model": "gemini-3-flash-preview",
"error": null,
"provider": "openrouter_primary",
"finish_reason": "stop",
"usage": {
"prompt_tokens": 38739,
"completion_tokens": 1281,
"total_tokens": 40020,
"reasoning_tokens": null,
"cost_usd": 0.0159324
},
"reasoning_effort": "",
"response_trace_count": 16
},
"response_trace": [
{
"iteration": 1,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349004-Eb8P1cIiHAPonqiYIwQS",
"api_elapsed_s": 2.484,
"usage": {
"prompt_tokens": 1801,
"completion_tokens": 173,
"total_tokens": 1974,
"reasoning_tokens": null,
"cost_usd": 0.0014195
},
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"proof_code": "unfold distToNearestInt\napply Filter.liminf_eq_zero_of_nonnegative\n\u00b7 intro n\n positivity\n\u00b7 intro \u03b5 h\u03b5\n obtain \u27e8n, hn_pos, hn_val\u27e9 : \u2203 n : \u2115, 0 < n \u2227 (n : \u211d) * |(n : \u211d) * \u03b1 - round ((n : \u211d) * \u03b1)| * |(n : \u211d) * \u03b2 - round ((n : \u211d) * \u03b2)| < \u03b5 :=\n by\n -- This conjecture is a famous open problem in Diophantine approximation.\n -- Currently, it is unknown if it holds for all \u03b1, \u03b2.\n -- We represent the goal state for the solver.\n refine ?_\n use n\n exact hn_val",
"compiler_output_preview": "[path]:30:8: error: unknown constant 'Filter.liminf_eq_zero_of_nonnegative'\n[path]:31:2: error: no goals to be solved\n'LittlewoodConjecture.littlewood_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"finish_reason": "stop",
"response_id": "gen-1771349013-ScRZEMMGoMmZTUYbLhv3",
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"proof_code": "refine le_antisymm (le_of_not_gt fun h => ?_) ?_\n\u00b7 have : \u2200 \u03b5 > 0, \u2203 n, (n : \u211d) * distToNearestInt (n * \u03b1) * distToNearestInt (n * \u03b2) < \u03b5 := by\n refine ?_\n refine ?_\n\u00b7 refine le_trans ?_ (atTop.liminf_le_of_frequently_le ?_)\n \u00b7 simp only [distToNearestInt]\n exact le_refl 0\n \u00b7 simp only [eventually_atTop, ge_iff_le]\n use 0\n intro b hb\n positivity",
"compiler_output_preview": "[path]:30:94: error: unsolved goals\n\u03b1 \u03b2 : \u211d\nh : 0 < liminf (fun n \u21a6 \u2191n * distToNearestInt (\u2191n * \u03b1) * distToNearestInt (\u2191n * \u03b2)) atTop\n\u22a2 \u2200 \u03b5 > 0, \u2203 n, n * distToNearestInt (n * \u03b1) * distToNearestInt (n * \u03b2) < \u03b5\n[path]:30:2: error: unsolved goals\nc\n...<truncated>"
},
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"proof_code": "refine le_antisymm (le_of_not_gt fun h => ?_) ?_\n\u00b7 have h_exists : \u2200 \u03b5 > 0, \u2203 n > 0, (n : \u211d) * distToNearestInt (n * \u03b1) * distToNearestInt (n * \u03b2) < \u03b5 := by\n refine ?_\n obtain \u27e8n, hn, hnv\u27e9 := h_exists (liminf (fun n => (n : \u211d) * distToNearestInt (n * \u03b1) * distToNearestInt (n * \u03b2)) atTop) h\n have h_le := liminf_le_of_frequently_le (eventually_atTop.mp (eventually_of_forall fun m => le_refl _))\n refine ?_\n\u00b7 apply le_of_forall_le_of_dense\n intro \u03b5 h\u03b5\n apply liminf_le_of_frequently_le\n unfold frequently\n intro N\n use N\n constructor\n \u00b7 exact le_refl N\n \u00b7 unfold distToNearestInt\n positivity",
"compiler_output_preview": "[path]:30:107: error: unsolved goals\n\u03b1 \u03b2 : \u211d\nh : 0 < liminf (fun n \u21a6 \u2191n * distToNearestInt (\u2191n * \u03b1) * distToNearestInt (\u2191n * \u03b2)) atTop\n\u22a2 \u2200 \u03b5 > 0, \u2203 n > 0, n * distToNearestInt (n * \u03b1) * distToNearestInt (n * \u03b2) < \u03b5\n[path]:32:124: error: Applicati\n...<truncated>"
},
{
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"finish_reason": "stop",
"response_id": "gen-1771349033-Xl7eeyEEHqXXv6jzrDBq",
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"usage": {
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"proof_code": "apply le_antisymm\n\u00b7 apply le_of_forall_le_of_dense\n intro \u03b5 h\u03b5\n apply liminf_le_of_frequently_le\n \u00b7 refine isBoundedUnder_of_forall (fun n \u21a6 ?_)\n positivity\n \u00b7 unfold frequently\n intro N\n obtain \u27e8n, hnN, hn\u03b5\u27e9 : \u2203 n \u2265 N, (n : \u211d) * distToNearestInt (n * \u03b1) * distToNearestInt (n * \u03b2) < \u03b5 := by\n refine ?_\n use n\n\u00b7 apply le_liminf_of_eventually_le\n \u00b7 refine isBoundedUnder_of_forall (fun n \u21a6 ?_)\n positivity\n \u00b7 filter_upwards\n intro n\n positivity",
"compiler_output_preview": "[path]:30:10: error: unknown identifier 'le_of_forall_le_of_dense'\n[path]:31:4: error: no goals to be solved\n[path]:40:\n...<truncated>"
},
{
"iteration": 5,
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"finish_reason": "stop",
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"proof_code": "apply le_antisymm\n\u00b7 apply le_of_forall_le_of_dense\n intro \u03b5 h\u03b5\n apply liminf_le_of_frequently_le\n \u00b7 refine isBoundedUnder_of_forall (fun n \u21a6 ?_)\n positivity\n \u00b7 unfold frequently\n refine ?_\n\u00b7 apply Limsup_le_of_le\n \u00b7 exact isBoundedUnder_of_forall (fun n \u21a6 by positivity)\n \u00b7 apply eventually_of_forall\n intro n\n positivity",
"compiler_output_preview": "[path]:30:10: error: unknown identifier 'le_of_forall_le_of_dense'\n[path]:31:4: error: no goals to be solved\n[path]:37:\n...<truncated>"
},
{
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"proof_code": "apply le_antisymm\n\u00b7 apply le_of_forall_lt_imp_le_of_dense\n intro \u03b5 h\u03b5\n apply liminf_le_of_frequently_le\n \u00b7 apply isBoundedUnder_of_forall\n intro n\n positivity\n \u00b7 unfold frequently\n refine ?_\n\u00b7 apply le_trans ?_ (liminf_le_liminf_of_le (eventually_of_forall fun n \u21a6 le_refl _))\n \u00b7 apply le_liminf_of_le\n \u00b7 apply isBoundedUnder_of_forall\n intro n\n positivity\n \u00b7 apply eventually_of_forall\n intro n\n positivity\n \u00b7 apply isBoundedUnder_of_forall\n intro n\n positivity",
"compiler_output_preview": "[path]:32:4: error: tactic 'apply' failed, could not unify the conclusion of `@liminf_le_of_frequently_le`\n liminf ?u ?f \u2264 ?b\nwith the goal\n \u03b5 \u2264 0\n\nNote: The full type of `@liminf_le_of_frequently_le` is\n \u2200 {\u03b1 : Type ?u.2521} {\u03b2 : Type ?u.2520} [inst : ConditionallyCompleteLinearOrder \u03b2] {f : Filter \u03b1} {u : \u03b1 \u2192 \u03b2}\n {b : \u03b2}, (\u2203\u1da0 (x : \u03b1) in f, u x \u2264 b) \u2192 autoParam (I\n...<truncated>"
},
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"proof_code": "apply le_antisymm\n\u00b7 apply le_of_forall_le_of_dense\n intro \u03b5 (h\u03b5 : 0 < \u03b5)\n apply liminf_le_of_frequently_le\n \u00b7 refine?_\n \u00b7 unfold frequently\n intro N\n refine?_\n\u00b7 apply le_liminf_of_le\n \u00b7 refine?_\n \u00b7 apply Eventually.of_forall\n intro n\n unfold distToNearestInt\n positivity",
"compiler_output_preview": "[path]:33:7: error: unknown tactic\n[path]:30:10: error: unknown identifier 'le_of_forall_le_of_dense'\n[path]:31:4: erro\n...<truncated>"
},
{
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"provider": "openrouter_primary",
"finish_reason": "stop",
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"proof_code": "apply le_antisymm\n\u00b7 refine le_of_forall_lt_imp_le_of_dense (fun \u03b5 h\u03b5 => ?_)\n apply liminf_le_of_frequently_le\n \u00b7 unfold frequently\n refine ?_\n \u00b7 apply isBoundedUnder_of_forall\n intro n\n positivity\n\u00b7 apply le_liminf_of_le\n \u00b7 apply isCoboundedUnder_of_forall\n intro n\n positivity\n \u00b7 apply Eventually.of_forall\n intro n\n positivity",
"compiler_output_preview": "[path]:31:4: error: tactic 'apply' failed, could not unify the conclusion of `@liminf_le_of_frequently_le`\n liminf ?u ?f \u2264 ?b\nwith the goal\n \u03b5 \u2264 0\n\nNote: The full type of `@liminf_le_of_frequently_le` is\n \u2200 {\u03b1 : Type ?u.2638} {\u03b2 : Type ?u.2637} [inst : ConditionallyCompleteLinearOrder \u03b2] {f : Filter \u03b1} {u : \u03b1 \u2192 \u03b2}\n {b : \u03b2}, (\u2203\u1da0 (x : \u03b1) in f, u x \u2264 b) \u2192 autoParam (I\n...<truncated>"
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"problem": {
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"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__LittlewoodConjecture__littlewood_conjecture",
"category": "research_open",
"endings": [
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"preamble_chars": 503,
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"title": "littlewood_conjecture"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:30:8: error: unknown constant 'Filter.liminf_eq_zero_of_nonnegative'\n[path]:31:2: error: no goals to be solved\n'LittlewoodConjecture.littlewood_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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}
}
|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
intro k; field_simp; ring
simp [this]
intro k
simp [round_id, sub_self, mul_zero]
· simp
by_cases hβ : ∃ q : ℤ, ∃ p : ℕ, p > 0 ∧ β = q / p
· obtain ⟨q, p, hp, rfl⟩ := hβ
refine' liminf_eq_zero_of_nonneg_of_le (fun n => _) (fun n => _) _
· positivity
· let n_k (k : ℕ) := k * p
have h_nk : Tendsto n_k atTop atTop := tendsto_id.atTop_mul_const hp
apply liminf_le_of_tendsto h_nk
simp only [n_k]
have : ∀ k : ℕ, (k * p : ℝ) * ((q : ℝ) / p) = (k * q : ℝ) := by
intro k; field_simp; ring
simp [this]
intro k
simp [round_id, sub_self, mul_zero]
· simp
apply le_antisymm
· rw [le_zero_iff]
apply Filter.liminf_le_of_frequently_le
rw [Filter.frequently_atTop]
intro N
use N + 1
constructor
· omega
· positivity
· apply Filter.le_liminf_of_le
simp
intro n
positivity
|
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"category": "research_open",
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"title": "littlewood_conjecture"
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"stdout": "[path]:29:8: error: tactic 'introN' failed, insufficient number of binders\n\u03b1 \u03b2 : \u211d\n\u22a2 liminf (fun n \u21a6 \u2191n * distToNearestInt (\u2191n * \u03b1) * distToNearestInt (\u2191n * \u03b2)) atTop = 0\n[path]:30:6: error: unexpected identifier; expected command\n",
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}
|