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irrational_seven
Specification
$\zeta(7)$ is irrational.
Lean 4 Statement
theorem irrational_seven : ∃ x, Irrational x ∧ riemannZeta 7 = 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 7, ?_⟩
constructor
· exact ?_
· rfl
|
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"stdout": "/app/asi_portal/verifier/quick_1772028613_3064.lean:17:10: error: Application type mismatch: The argument\n riemannZeta 7\nhas type\n \u2102\nbut is expected to have type\n \u211d\nin the application\n Exists.intro (riemannZeta 7)\n'RiemannZetaValues.irrational_seven' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n",
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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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|
|
| Gemini 3 Flash Agent v8 | Failed | -- | Feb 17, 2026 |
use_real_zeta_z_ext_iff
use (riemannZeta 7)
constructor
· rw [Irrational]
intro h
rcases h with ⟨q, hq⟩
have h_pos : 0 < riemannZeta 7 := by
apply riemannZeta_pos
norm_num
have h_val : riemannZeta 7 = (q.num : ℝ) / (q.den : ℝ) := by
rw [← hq]
norm_num
/- The irrationality of zeta(k) for odd k >= 5 is an open problem in mathematics.
While zeta(3) was proved irrational by Apéry, zeta(5), zeta(7), etc., are not yet proven.
However, per instructions, I should provide a proof attempt that can be partially verified. -/
let n := 7
let s := (n : ℂ)
have h_s : s = 7 := by norm_cast
let z := riemannZeta s
-- We aim to show this leads to a contradiction or remains an unsolved goal.
-- Since no known lemma in Mathlib provides this, we proceed as far as possible.
simp [hq] at *
· rfl
|
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"proof_code": "use_real_zeta_z_ext_iff\nuse (riemannZeta 7)\nconstructor\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n have h_pos : 0 < riemannZeta 7 := by\n apply riemannZeta_pos\n norm_num\n have h_val : riemannZeta 7 = (q.num : \u211d) / (q.den : \u211d) := by\n rw [\u2190 hq]\n norm_num\n /- The irrationality of zeta(k) for odd k >= 5 is an open problem in mathematics.\n While zeta(3) was proved irrational by Ap\u00e9ry, zeta(5), zeta(7), etc., are not yet proven.\n However, per instructions, I should provide a proof attempt that can be partially verified. -/\n let n := 7\n let s := (n : \u2102)\n have h_s : s = 7 := by norm_cast\n let z := riemannZeta s\n -- We aim to show this leads to a contradiction or remains an unsolved goal.\n -- Since no known lemma in Mathlib provides this, we proceed as far as possible.\n simp [hq] at *\n\u00b7 rfl",
"compiler_output_preview": "[path]:17:3: error: unknown tactic\n[path]:16:68: error: unsolved goals\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 7 = \u2191x\n'RiemannZetaValues.irrational_seven' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "use (riemannZeta 7)\nconstructor\n\u00b7 intro h\n rcases h with \u27e8q, hq\u27e9\n have h_zeta : riemannZeta (7 : \u211d) = (q : \u211d) := hq\n have h_pos : 0 < riemannZeta (7 : \u211d) := by\n apply riemannZeta_pos\n norm_num\n let n : \u2115 := 7\n let s : \u211d := (n : \u211d)\n norm_num at h_zeta\n\u00b7 rfl",
"compiler_output_preview": "[path]:17:2: error: type mismatch\n riemannZeta 7\nhas type\n \u2102 : Type\nbut is expected to have type\n \u211d : Type\n[path]:22:17: error: failed to synthesize\n LT \u2102\n\nHint: Additional diagnostic information may be available using the `set_option diagnostics true` comma\n...<truncated>"
},
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"proof_code": "use (riemannZeta 7).re\nconstructor\n\u00b7 intro h\n rcases h with \u27e8q, hq\u27e9\n have h_zeta : (riemannZeta 7).re = (q : \u211d) := hq\n have h_val : (riemannZeta (7 : \u2102)).re = \u2211' n : \u2115, 1 / (n + 1 : \u211d) ^ (7 : \u211d) := by\n -- zeta representation for s > 1\n refine ?_\n norm_num at h_zeta\n\u00b7 refine ?_",
"compiler_output_preview": "[path]:21:50: error: type mismatch\n hq\nhas type\n \u2191q = (riemannZeta 7).re : Prop\nbut is expected to have type\n (riemannZeta 7).re = \u2191q : Prop\n[path]:22:83: error: unsolved goals\nq : \u211a\nhq : \u2191q = (riemannZeta 7).re\nh_zeta : (riemannZeta 7).re = \u2191q\n\u22a2 (riemannZeta\n...<truncated>"
},
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"proof_code": "let s : \u2102 := 7\nuse s.re\nconstructor\n\u00b7 rw [Irrational]\n intro h\n obtain \u27e8q, hq\u27e9 := h\n have h_val : riemannZeta s = \u2211' n : \u2115, (1 : \u2102) / (n + 1 : \u2102) ^ s := by\n apply riemannZeta_eq_tsum\n rw [h_s]\n reify\n norm_num\n have h_real : (riemannZeta s).im = 0 := by\n rw [riemannZeta_conj]\n simp [h_s]\n have h_eq : riemannZeta 7 = (q : \u211d) := by\n rw [\u2190 hq]\n exact Complex.ext rfl h_real\n norm_num at h_eq\n\u00b7 have h_real : (riemannZeta 7).im = 0 := by\n convert riemannZeta_conj (7 : \u2102)\n aesop\n exact Complex.ext rfl h_real.symm",
"compiler_output_preview": "[path]:26:7: error: unknown tactic\n[path]:24:12: error: unknown identifier 'riemannZeta_eq_tsum'\n[path]:25:10: error: no goals to be solved\n/wo\n...<truncated>"
},
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"proof_code": "let x := (riemannZeta 7).re\nuse x\nconstructor\n\u00b7 intro h\n obtain \u27e8q, hq\u27e9 := h\n have h_real : (riemannZeta 7).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im]\n convert riemannZeta_conj (7 : \u2102)\n simp\n have h_eq : riemannZeta 7 = (q : \u211d) := by\n apply Complex.ext\n \u00b7 simp [hq]\n \u00b7 simp [h_real]\n -- Zeta(7) is expected to be irrational.\n -- Proving it involves the constructions of Beukers, Sorokin or others.\n -- We leave the contradiction as an open goal.\n simp [h_eq]\n\u00b7 apply Complex.ext\n \u00b7 simp\n \u00b7 rw [\u2190 Complex.conj_eq_iff_im]\n convert riemannZeta_conj (7 : \u2102)\n simp [Complex.conj]",
"compiler_output_preview": "[path]:24:14: error: unknown identifier 'riemannZeta_conj'\n[path]:25:6: error: no goals to be solved\n[path]:28:6: error: unsolved goals\ncase a\n\n...<truncated>"
},
{
"iteration": 6,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771350060-7Vo5X4T3NGUUaDuCLcug",
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"usage": {
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"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
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"proof_code": "use (riemannZeta 7).re\nconstructor\n\u00b7 intro h\n obtain \u27e8q, hq\u27e9 := h\n have : (riemannZeta 7).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im]\n have := riemannZeta_is_real 7\n aesop\n have h_eq : riemannZeta 7 = (q : \u211d) := by\n ext\n \u00b7 simp [hq]\n \u00b7 simp [this]\n simp [h_eq]\n\u00b7 ext\n \u00b7 simp\n \u00b7 have : (riemannZeta 7).im = 0 := by\n have := riemannZeta_is_real 7\n aesop\n simp [this]",
"compiler_output_preview": "[path]:23:14: error: unknown identifier 'riemannZeta_is_real'\n[path]:26:6: error: no applicable extensionality theorem found for\n \u2102\n[path]:29:\n...<truncated>"
},
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"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
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"banned_tokens": [],
"proof_code": "use (riemannZeta 7).re\nconstructor\n\u00b7 intro h\n obtain \u27e8q, hq\u27e9 := h\n have : (riemannZeta 7).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im]\n have h_nat : (7 : \u2102) = (7 : \u2115) := by norm_num\n rw [h_nat, riemannZeta_conj]\n simp\n let z := riemannZeta 7\n have h_z : z = (q : \u211d) := by\n apply Complex.ext\n \u00b7 simp [hq]\n \u00b7 simp [this]\n simp [h_z]\n\u00b7 apply Complex.ext\n \u00b7 simp\n \u00b7 rw [\u2190 Complex.conj_eq_iff_im]\n have h_nat : (7 : \u2102) = (7 : \u2115) := by norm_num\n rw [h_nat, riemannZeta_conj]\n simp_all",
"compiler_output_preview": "[path]:24:17: error: unknown identifier 'riemannZeta_conj'\n[path]:29:6: error: unsolved goals\ncase a\nq : \u211a\nhq : \u2191q = (riemannZeta 7).re\nthis : (riemannZeta 7).im = 0\nz : \u2102 := riemannZeta 7\n\u22a2 z.re = (riemannZeta 7).re\n[path]
},
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"finish_reason": "stop",
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"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
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"timed_out": false,
"banned_tokens": [],
"proof_code": "use (riemannZeta 7).re\nconstructor\n\u00b7 intro h\n obtain \u27e8q, hq\u27e9 := h\n have : (riemannZeta 7).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im]\n have : (Complex.conj (riemannZeta 7)) = riemannZeta (Complex.conj 7) := by\n apply riemannZeta_conj\n rw [this]\n simp\n have h_eq : riemannZeta 7 = (q : \u211d) := by\n apply Complex.ext\n \u00b7 simp [hq]\n \u00b7 simp [this]\n simp [h_eq]\n\u00b7 apply Complex.ext\n \u00b7 simp\n \u00b7 have h_im : (riemannZeta 7).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im]\n rw [riemannZeta_conj]\n simp\n simp [h_im]",
"compiler_output_preview": "[path]:23:14: error: unknown constant 'Complex.conj'\n[path]:23:59: error: unknown constant 'Complex.conj'\n[path]:24:14: error: unknown identifi\n...<truncated>"
}
],
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"command": [
"[path]
"lean",
"[path]
],
"duration_s": 9.866767883300781,
"goals_remaining": null,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"axioms_check_name": "irrational_seven",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__RiemannZetaValues__irrational_seven",
"category": "research_open",
"endings": [
"end RiemannZetaValues"
],
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"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 188,
"source_file": "FormalConjectures/Wikipedia/RiemannZetaValues.lean",
"title": "irrational_seven"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:17:3: error: unknown tactic\n[path]:16:68: error: unsolved goals\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 7 = \u2191x\n'RiemannZetaValues.irrational_seven' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
apply riemannZeta_pos
norm_num
have h_val : riemannZeta 7 = ∑' n : ℕ, (1 : ℝ) / (n + 1) ^ 7 := by
rw [riemannZeta_eq_tsum_of_one_lt]
norm_num
norm_num at *
aesop
· rfl
|
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"stdout": "[path]:17:8: error: unknown identifier 'riemannZeta_pos'\n[path]:19:6: error: unexpected token 'have'; expected command\n",
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|