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irrational_eleven
Specification
$\zeta(11)$ is irrational.
Lean 4 Statement
theorem irrational_eleven : ∃ x, Irrational x ∧ riemannZeta 11 = x
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| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Failed | -- | Feb 25, 2026 |
refine ⟨riemannZeta 11, ?_, rfl⟩
|
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"proof_code": "refine \u27e8riemannZeta 11, ?_, rfl\u27e9",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772028043_4803.lean:17:10: error: Application type mismatch: The argument\n riemannZeta 11\nhas type\n \u2102\nbut is expected to have type\n \u211d\nin the application\n Exists.intro (riemannZeta 11)\n'RiemannZetaValues.irrational_eleven' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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"message": "Application type mismatch: The argument\n riemannZeta 11\nhas type\n \u2102\nbut is expected to have type\n \u211d\nin the application\n Exists.intro (riemannZeta 11)",
"range": {
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"goal_before_error": "```lean\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 11 = \u2191x\n```",
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"code": "refine \u27e8riemannZeta 11, ?_, rfl\u27e9",
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"error_code": "refine \u27e8riemannZeta 11, ?_, rfl\u27e9"
}
},
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}
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"title": "irrational_eleven"
},
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"stdout": "/app/asi_portal/verifier/quick_1772028100_1103.lean:17:10: error: Application type mismatch: The argument\n riemannZeta 11\nhas type\n \u2102\nbut is expected to have type\n \u211d\nin the application\n Exists.intro (riemannZeta 11)\n'RiemannZetaValues.irrational_eleven' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n",
"timed_out": false
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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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"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 riemannZeta 11
constructor
· rw [Irrational]
intro h
rcases h with ⟨q, hq⟩
have h_val : riemannZeta 11 = q := hq
clear hq
suffices : False
· exact this
norm_num
· rfl
|
View{
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"proof_code": "use RiemannZetaValues in\nexists (riemannZeta 11)\nconstructor\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n induction q using Quotient.inductionOn\n exact hq.symm.not_lt (by simp; norm_num; admit)\n\u00b7 rfl",
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"proof_code": "use riemannZeta 11\nconstructor\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n have h_val : riemannZeta 11 = q := hq\n clear hq\n suffices : False\n \u00b7 exact this\n norm_num\n\u00b7 rfl",
"compiler_output_preview": "[path]:17:2: error: type mismatch\n riemannZeta 11\nhas type\n \u2102 : Type\nbut is expected to have type\n \u211d : Type\n[path]:19:2: error: unsolved goals\ncase this\nq : \u211a\nh_val : riemannZeta 11 = \u2191q\n\u22a2 False\n'RiemannZetaValues.irrational_eleven' depends on axioms: [prop\n...<truncated>"
},
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"proof_code": "exists (riemannZeta (11 : \u211d))\nconstructor\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n have h_val : riemannZeta 11 = q := hq\n norm_num at h_val\n\u00b7 rfl",
"compiler_output_preview": "[path]:17:9: error: Application type mismatch: In the application\n Exists.intro (riemannZeta \u219111)\nthe argument\n riemannZeta \u219111\nhas type\n \u2102 : Type\nbut is expected to have type\n \u211d : Type\n[path]:19:2: error: unsolved goals\ncase left.intro\nq : \u211a\nhq : \u2191q = sor\n...<truncated>"
},
{
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"proof_code": "use (riemannZeta 11).re\nconstructor\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n have h_val : (riemannZeta 11).re = \u2191q := hq\n norm_num at h_val\n\u00b7 simp\n rw [riemannZeta]\n norm_num\n rfl",
"compiler_output_preview": "[path]:22:45: error: type mismatch\n hq\nhas type\n \u2191q = (riemannZeta 11).re : Prop\nbut is expected to have type\n (riemannZeta 11).re = \u2191q : Prop\n[path]:19:2: error: unsolved goals\ncase h.left.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 11).re\nh_val : (riemannZeta 11).\n...<truncated>"
},
{
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"iteration": 5,
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"proof_code": "exists (riemannZeta 11).re\nconstructor\n\u00b7 intro h\n rcases h with \u27e8q, hq\u27e9\n have h_val : (riemannZeta 11).re = \u2191q := hq.symm\n norm_num at h_val\n\u00b7 congr\n apply Complex.ext\n \u00b7 rfl\n \u00b7 simp\n rw [riemannZeta]\n norm_num\n exact Complex.zero_re.symm",
"compiler_output_preview": "[path]:19:2: error: unsolved goals\ncase left.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 11).re\nh_val : (riemannZeta 11).re = \u2191q\n\u22a2 False\n[path]:29:6: error: type mismatch\n Eq.symm Complex.zero_re\nhas type\n 0 = Complex.re 0 : Prop\nbut is expected to have type\n (Hurwi\n...<truncated>"
},
{
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"proof_code": "exists (riemannZeta 11).re\nconstructor\n\u00b7 intro h\n rcases h with \u27e8q, hq\u27e9\n have h_val : (riemannZeta 11).re = \u2191q := hq.symm\n norm_num at h_val\n\u00b7 apply Complex.ext\n \u00b7 rfl\n \u00b7 rw [riemannZeta]\n norm_num\n rfl",
"compiler_output_preview": "[path]:19:2: error: unsolved goals\ncase left.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 11).re\nh_val : (riemannZeta 11).re = \u2191q\n\u22a2 False\n[path]:27:6: error: tactic 'rfl' failed, the left-hand side\n (HurwitzZeta.hurwitzZetaEven 0 11).im\nis not definitionally equal to t\n...<truncated>"
},
{
"iteration": 7,
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"proof_code": "exists (riemannZeta 11).re\nconstructor\n\u00b7 intro h\n rcases h with \u27e8q, hq\u27e9\n have h_val : (riemannZeta 11).re = \u2191q := hq.symm\n norm_num at h_val\n\u00b7 apply Complex.ext\n \u00b7 rfl\n \u00b7 rw [riemannZeta]\n simp\n apply HurwitzZeta.hurwitzZetaEven_im\n \u00b7 simp\n \u00b7 norm_num",
"compiler_output_preview": "[path]:19:2: error: unsolved goals\ncase left.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 11).re\nh_val : (riemannZeta 11).re = \u2191q\n\u22a2 False\n[path]:27:12: error: unknown identifier 'HurwitzZeta.hurwitzZetaEven_im'\n[path]
},
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"proof_code": "exists (riemannZeta 11).re\nconstructor\n\u00b7 intro h\n rcases h with \u27e8q, hq\u27e9\n have h_val : (riemannZeta 11).re = \u2191q := hq.symm\n norm_num at h_val\n\u00b7 apply Complex.ext\n \u00b7 rfl\n \u00b7 rw [riemannZeta]\n simp [HurwitzZeta.hurwitzZetaEven_zero]\n rfl",
"compiler_output_preview": "[path]:19:2: error: unsolved goals\ncase left.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 11).re\nh_val : (riemannZeta 11).re = \u2191q\n\u22a2 False\n[path]:27:6: error: tactic 'rfl' failed, the left-hand side\n (riemannZeta 11).im\nis not definitionally equal to the right-hand side\n...<truncated>"
}
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"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__RiemannZetaValues__irrational_eleven",
"category": "research_open",
"endings": [
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],
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],
"preamble_chars": 188,
"source_file": "FormalConjectures/Wikipedia/RiemannZetaValues.lean",
"title": "irrational_eleven"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:17:2: error: type mismatch\n riemannZeta 11\nhas type\n \u2102 : Type\nbut is expected to have type\n \u211d : Type\n[path]:19:2: error: unsolved goals\ncase this\nq : \u211a\nh_val : riemannZeta 11 = \u2191q\n\u22a2 False\n'RiemannZetaValues.irrational_eleven' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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}
}
|
| Gemini 3 Flash Preview | Partial | 1 | Feb 14, 2026 |
rw [riemannZeta_eq_tsum_of_one_lt h_pos]
simp only [h_eq] at h_zeta
norm_cast at h_zeta
have : (a : ℝ) / (b : ℝ) = ∑' n : ℕ, 1 / (n : ℝ) ^ 11 := h_zeta
aesop
· rfl
|
View{
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"raw_content_length": 1384,
"reasoning_effort": "",
"structured_output": true,
"temperature": 0.0,
"usage": {
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"completion_tokens_details": {
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},
"cost": 0.002062,
"cost_details": {
"upstream_inference_completions_cost": 0.00129,
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"problem": {
"benchmark_id": "Wikipedia__RiemannZetaValues__irrational_eleven",
"category": "research_open",
"endings": [
"end RiemannZetaValues"
],
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"FormalConjectures.Util.ProblemImports"
],
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"source_file": "FormalConjectures/Wikipedia/RiemannZetaValues.lean",
"title": "irrational_eleven"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:17:6: error: unknown identifier 'riemannZeta_eq_tsum_of_one_lt'\n[path]:16:70: error: unsolved goals\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 11 = \u2191x\n[path]:18:6: error: unexpected identifier; expected command\n",
"timed_out": false
}
}
|