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irrational_nine
Specification
$\zeta(9)$ is irrational.
Lean 4 Statement
theorem irrational_nine : ∃ x, Irrational x ∧ riemannZeta 9 = x
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| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Partial | 2 | Feb 25, 2026 |
refine ⟨(riemannZeta 9).re, ?_, ?_⟩
|
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"proof_code": "refine \u27e8(riemannZeta 9).re, ?_, ?_\u27e9",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772028337_4784.lean:16:67: error: unsolved goals\ncase refine_1\n\u22a2 Irrational (riemannZeta 9).re\n\ncase refine_2\n\u22a2 riemannZeta 9 = \u2191(riemannZeta 9).re\n'RiemannZetaValues.irrational_nine' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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"error_code": "theorem irrational_nine : \u2203 x, Irrational x \u2227 riemannZeta 9 = x := by"
}
}
],
"verification": {
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"stdout": "/app/asi_portal/verifier/quick_1772028387_1893.lean:16:67: error: unsolved goals\ncase refine_1\n\u22a2 Irrational (riemannZeta 9).re\n\ncase refine_2\n\u22a2 riemannZeta 9 = \u2191(riemannZeta 9).re\n'RiemannZetaValues.irrational_nine' 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{
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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 | Failed | -- | Feb 17, 2026 |
use_exists (riemannZeta 9)
constructor
· rw [Irrational]
intro h
rcases h with ⟨q, hq⟩
induction q using Rat.casesOn
let n := 9
have hn : n = 9 := rfl
have : riemannZeta 9 = (9 : ℚ) := hq
clear hq
rw [hn] at this
norm_num at this
· rfl
|
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"proof_code": "use_exists (riemannZeta 9)\nconstructor\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n induction q using Rat.casesOn\n let n := 9\n have hn : n = 9 := rfl\n have : riemannZeta 9 = (9 : \u211a) := hq\n clear hq\n rw [hn] at this\n norm_num at this\n\u00b7 rfl",
"compiler_output_preview": "[path]:17:3: error: unknown tactic\n[path]:16:67: error: unsolved goals\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 9 = \u2191x\n'RiemannZetaValues.irrational_nine' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "refine \u27e8riemannZeta 9, \u27e8?_, rfl\u27e9\u27e9\nrw [Irrational]\nintro h\ncases h with\n| mk q hq =>\n have h_val := hq\n clear hq\n ring_nf at h_val\n nomatch h_val",
"compiler_output_preview": "[path]:17:10: error: Application type mismatch: In the application\n Exists.intro (riemannZeta 9)\nthe argument\n riemannZeta 9\nhas type\n \u2102 : Type\nbut is expected to have type\n \u211d : Type\n'RiemannZetaValues.irrational_nine' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "refine \u27e8(riemannZeta 9).re, \u27e8?_, ?_\u27e9\u27e9\n\u00b7 rw [Irrational]\n intro h\n cases h with | mk q hq =>\n have : riemannZeta 9 = (q : \u211d) := by\n rw [\u2190 hq]\n have : (riemannZeta 9).im = 0 := riemannZeta_im_nat 9 (by norm_num)\n exact Complex.ext rfl (by simp [this])\n nomatch this\n\u00b7 have : (riemannZeta 9).im = 0 := riemannZeta_im_nat 9 (by norm_num)\n nth_rw 2 [\u2190 Complex.re_add_im (riemannZeta 9)]\n simp [this]",
"compiler_output_preview": "[path]:20:17: error: Invalid alternative name 'mk': Expected 'intro'\n[path]:26:37: error: unknown identifier 'riemannZeta_im_nat'\n[path]:26:2: err\n...<truncated>"
},
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"proof_code": "refine \u27e8(riemannZeta 9).re, ?_, ?_\u27e9\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n have h_im : (riemannZeta 9).im = 0 := by\n rw [\u2190 Complex.ofReal_re (riemannZeta 9)]\n have : riemannZeta (9 : \u2102) = (riemannZeta 9 : \u2102) := rfl\n simp\n have : riemannZeta 9 = (q : \u211d) := by\n ext\n \u00b7 simp [hq]\n \u00b7 simp [h_im]\n norm_num at this\n\u00b7 have h_im : (riemannZeta 9).im = 0 := by\n have : (9 : \u2102) = (9 : \u211d) := rfl\n rw [this]\n exact riemannZeta_conj 9\n nth_rw 2 [\u2190 Complex.re_add_im (riemannZeta 9)]\n simp [h_im]",
"compiler_output_preview": "[path]:22:30: error: Application type mismatch: In the application\n Complex.ofReal_re (riemannZeta 9)\nthe argument\n riemannZeta 9\nhas type\n \u2102 : Type\nbut is expected to have type\n \u211d : Type\n[path]:26:6: error: no applicable extensionality theorem found for\n \u2102\n/\n...<truncated>"
},
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"proof_code": "refine \u27e8(riemannZeta 9).re, ?_, ?_\u27e9\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n have h_im : (riemannZeta 9).im = 0 := by\n have : (riemannZeta (9 : \u2102)).conj = riemannZeta (9 : \u2102).conj := by\n apply riemannZeta_conj\n simp at this\n rw [Complex.conj_eq_iff_im] at this\n exact this\n have : riemannZeta 9 = (q : \u2102) := by\n ext\n \u00b7 simp [hq]\n \u00b7 simp [h_im]\n norm_num at this\n\u00b7 have h_im : (riemannZeta 9).im = 0 := by\n have : (riemannZeta (9 : \u2102)).conj = riemannZeta (9 : \u2102).conj := riemannZeta_conj (9 : \u2102)\n simp at this\n rw [Complex.conj_eq_iff_im] at this\n exact this\n nth_rw 2 [\u2190 Complex.re_add_im (riemannZeta 9)]\n simp [h_im]",
"compiler_output_preview": "[path]:22:13: error: Invalid field `conj`: The environment does not contain `Complex.conj`\n riemannZeta 9\nhas type\n \u2102\n[path]:22:54: error: Invalid field `conj`: The environment does not contain `Complex.conj`\n 9\nhas type\n \u2102\n[path]
},
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"proof_code": "refine \u27e8(riemannZeta 9).re, ?_, ?_\u27e9\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n have h_im : (riemannZeta 9).im = 0 := by\n suffices star (riemannZeta 9 : \u2102) = riemannZeta 9 from\n Complex.conj_eq_iff_im.mp this\n have : (9 : \u2102) = (9 : \u211d) := rfl\n rw [this, \u2190 riemannZeta_conj]\n simp\n have h_eq : riemannZeta 9 = (q : \u2102) := by\n apply Complex.ext\n \u00b7 simp [hq]\n \u00b7 simp [h_im]\n norm_num at h_eq\n\u00b7 have h_im : (riemannZeta 9).im = 0 := by\n have : (9 : \u2102) = (9 : \u211d) := rfl\n rw [this, \u2190 riemannZeta_conj]\n simp\n apply Complex.conj_eq_iff_im.mp\n simp\n nth_rw 2 [\u2190 Complex.re_add_im (riemannZeta 9)]\n simp [h_im]",
"compiler_output_preview": "[path]:25:18: error: unknown identifier 'riemannZeta_conj'\n[path]:18:2: error: unsolved goals\ncase refine_1.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 9).re\nh_im : (riemannZeta 9).im = 0\nh_eq : riemannZeta 9 = \u2191q\n\u22a2 False\n[path]
},
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"proof_code": "refine \u27e8(riemannZeta 9).re, ?_, ?_\u27e9\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n have h_im : (riemannZeta 9).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im]\n have : (9 : \u2102) = (\u2191(9 : \u211d) : \u2102) := rfl\n rw [this]\n exact riemannZeta_conj (9 : \u211d)\n have h_eq : riemannZeta 9 = (q : \u2102) := by\n apply Complex.ext\n \u00b7 simp [hq]\n \u00b7 simp [h_im]\n norm_num at h_eq\n\u00b7 have h_im : (riemannZeta 9).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im]\n have : (9 : \u2102) = (\u2191(9 : \u211d) : \u2102) := rfl\n rw [this]\n exact riemannZeta_conj (9 : \u211d)\n nth_rw 2 [\u2190 Complex.re_add_im (riemannZeta 9)]\n simp [h_im]",
"compiler_output_preview": "[path]:25:12: error: unknown identifier 'riemannZeta_conj'\n[path]:18:2: error: unsolved goals\ncase refine_1.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 9).re\nh_im : (riemannZeta 9).im = 0\nh_eq : riemannZeta 9 = \u2191q\n\u22a2 False\n[path]
},
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"proof_code": "refine \u27e8(riemannZeta 9).re, ?_, ?_\u27e9\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n have h_im : (riemannZeta 9).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im]\n exact riemannZeta_conj 9\n have h_eq : riemannZeta 9 = (q : \u2102) := by\n apply Complex.ext\n \u00b7 simp [hq]\n \u00b7 simp [h_im]\n norm_num at h_eq\n\u00b7 have h_im : (riemannZeta 9).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im]\n exact riemannZeta_conj 9\n nth_rw 2 [\u2190 Complex.re_add_im (riemannZeta 9)]\n simp [h_im]",
"compiler_output_preview": "[path]:23:12: error: unknown identifier 'riemannZeta_conj'\n[path]:18:2: error: unsolved goals\ncase refine_1.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 9).re\nh_im : (riemannZeta 9).im = 0\nh_eq : riemannZeta 9 = \u2191q\n\u22a2 False\n[path]
}
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"problem": {
"axioms_check_name": "irrational_nine",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__RiemannZetaValues__irrational_nine",
"category": "research_open",
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"source_file": "FormalConjectures/Wikipedia/RiemannZetaValues.lean",
"title": "irrational_nine"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:17:3: error: unknown tactic\n[path]:16:67: error: unsolved goals\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 9 = \u2191x\n'RiemannZetaValues.irrational_nine' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Preview | Partial | 1 | Feb 14, 2026 |
rw [riemannZeta_eq_tsum_one_div_nat_pow (by norm_num)]
apply tsum_pos
· apply summable_one_div_nat_pow
norm_num
· intro n
positivity
· use 1
positivity
have h_val : riemannZeta 9 = (q.num : ℝ) / (q.den : ℝ) := by
rw [hq]
simp
norm_num at h_val
apply absurd hq
intro h_false
rw [h_false] at h_pos
norm_num at h_pos
· rfl
|
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"stdout": "[path]:17:6: error: unknown identifier 'riemannZeta_eq_tsum_one_div_nat_pow'\n[path]:16:67: error: unsolved goals\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 9 = \u2191x\n[path]:18:6: error: unexpected identifier; expected command\n",
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|