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irrational_five
Specification
$\zeta(5)$ is irrational.
Lean 4 Statement
theorem irrational_five : ∃ x, Irrational x ∧ riemannZeta 5 = 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 5, ?_, rfl⟩
|
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"proof_code": "refine \u27e8riemannZeta 5, ?_, rfl\u27e9",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772028145_2907.lean:17:10: error: Application type mismatch: The argument\n riemannZeta 5\nhas type\n \u2102\nbut is expected to have type\n \u211d\nin the application\n Exists.intro (riemannZeta 5)\n'RiemannZetaValues.irrational_five' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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"message": "Application type mismatch: The argument\n riemannZeta 5\nhas type\n \u2102\nbut is expected to have type\n \u211d\nin the application\n Exists.intro (riemannZeta 5)",
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"goal_before_error": "```lean\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 5 = \u2191x\n```",
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"line": 16,
"code": "refine \u27e8riemannZeta 5, ?_, rfl\u27e9",
"goal": "```lean\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 5 = \u2191x\n```",
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"error_code": "refine \u27e8riemannZeta 5, ?_, rfl\u27e9"
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6pnQxzyrVef5t3_CFPENmct5w2bNRR_X19RAC1BFCMR317WN25xYxfACDv-yE7zIOZ9_6Mu8_Ky0Y9LT6vYuoVhuHj6SBjA0Y3xbnUm_jrE-RTp6GY9jnbW_yoowcXB2E0R2bF8-9Mw0j4my_JMdj1FCzVk3d3oOdS79eR6pfrG4-GtX43f_3XzMktJKW_XR0auZo5MOiXD1owv24ImRUcmesluvBdyTgEEXz6dAwFJKYNj4aUfzWaFtLomkTsMrIU4P3TJEvoRbkufVf-6tJI7GE3cYeXWCvst7SBIv65i3H8K4TucvYxrM0SP6QlhSPut3cujT72bJcnX0OWd5xZ1QxfGTwhSyVS9B9xvS-o-Ytt1nJW63vUtY1eH4-a3tIpg1-KtCfHVbcS4U61CymdAdHBmiBHHg12NJvVvM-SCMtLimDqcax4n5MhZ8CA6mNs_ob6IrcxnJpdIsNoP4nhHYXMfvU1SrF_lFwxaVDqwXsbtiNIUKSz2NkNS47EHGglPLsw7TOAS7DYgjkDqXYhPExW8PJp36rOmN_yA9BHkA9ooj5nVmR41kghwUUSAe7H-nJKxdrLf9-2dSNGSpLOrw-bdFncMo9l51nxUistTreljMTikwaeF6pS8jE_fo4T5Lr7cp70E5No6WpT-d17Sxhy9fV9NEv5On5F2-iZcwfl8QTkE7OZjG0hcioK7XgaYk8WdVO1WZyZBL1l0nyIvb5LuyngzLo4GlgtehM8GV4chd28K17hJ_D4oPoB04OJtBqM3j1Uvf_Q0-tK6EOIvuZVYD2K6fmKONWEt6kg6jvPEpok5RZR6JDidxSP1h2CsDqGxDYssgyBB6n2oIuPE7fY9pJz3aew3hBqjx7n11V8TRthMVG9lhI0lzkYN7nwKNoh4TVB64Tf351epO9xKdQYruJFc8tRjSRz5yrEh6fsdEe-aDBaS40Jrln_ZGknM1CFmzCalwICMp6zxmqaRf2rBZu4SNlusn_4A1hFoWi9NkwcmppunDtcgvOCVO-7BIQXsbYeawFphH1AVuM4uUmo5A_g1D8jZqBNkudYM95VyStLYpG7I-hCG4hu1go7n0XmwNIQ_ehaqi8d06O8k4Z6A=="
}
]
},
{
"iteration": 9,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h : Irrational (riemannZeta 5).re := irrational_riemannZeta_five\nrefine \u27e8(riemannZeta 5).re, h, ?_\u27e9\nexact by\n simp",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772028228_4251.lean:17:44: error(lean.unknownIdentifier): Unknown identifier `irrational_riemannZeta_five`\n/app/asi_portal/verifier/quick_1772028228_4251.lean:20:4: error: `simp` made no progress\n'RiemannZetaValues.irrational_five' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 38.48,
"tactic_replay": {
"tactic_count": 4,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1,
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"code": "lean.unknownIdentifier",
"fullRange": {
"end": {
"character": 71,
"line": 16
},
"start": {
"character": 44,
"line": 16
}
},
"message": "Unknown identifier `irrational_riemannZeta_five`",
"range": {
"end": {
"character": 71,
"line": 16
},
"start": {
"character": 44,
"line": 16
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 8,
"line": 19
},
"start": {
"character": 4,
"line": 19
}
},
"message": "`simp` made no progress",
"range": {
"end": {
"character": 8,
"line": 19
},
"start": {
"character": 4,
"line": 19
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 21
},
"start": {
"character": 0,
"line": 21
}
},
"message": "'RiemannZetaValues.irrational_five' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 21
},
"start": {
"character": 0,
"line": 21
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 16,
"error_char": 44,
"error_msg": "Unknown identifier `irrational_riemannZeta_five`",
"goal_before_error": "```lean\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 5 = \u2191x\n```",
"full_timeline": [
{
"line": 16,
"code": "have h : Irrational (riemannZeta 5).re := irrational_riemannZeta_five",
"goal": "```lean\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 5 = \u2191x\n```",
"goal_count": 1,
"ast_complexity": 52
},
{
"line": 17,
"code": "refine \u27e8(riemannZeta 5).re, h, ?_\u27e9",
"goal": "```lean\nh : Irrational (riemannZeta 5).re\n\u22a2 \u2203 x, Irrational x \u2227 riemannZeta 5 = \u2191x\n```",
"goal_count": 1,
"ast_complexity": 86
},
{
"line": 18,
"code": "exact by",
"goal": "```lean\nh : Irrational (riemannZeta 5).re\n\u22a2 riemannZeta 5 = \u2191(riemannZeta 5).re\n```",
"goal_count": 1,
"ast_complexity": 83
},
{
"line": 19,
"code": "simp",
"goal": "```lean\nh : Irrational (riemannZeta 5).re\n\u22a2 riemannZeta 5 = \u2191(riemannZeta 5).re\n```",
"goal_count": 1,
"ast_complexity": 83
}
],
"tactic_count": 4,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "have h : Irrational (riemannZeta 5).re := irrational_riemannZeta_five"
}
},
{
"iteration": 10,
"event": "search_mathlib",
"query": "zeta_five",
"results_count": 0
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772028284_9494.lean"
],
"comparator_passed": null,
"duration_s": 19.50013780593872,
"goals_remaining": null,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "irrational_five",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__RiemannZetaValues__irrational_five",
"category": "research_open",
"endings": [
"end RiemannZetaValues"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 188,
"source_file": "FormalConjectures/Wikipedia/RiemannZetaValues.lean",
"title": "irrational_five"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772028284_9494.lean:17:10: error: Application type mismatch: The argument\n riemannZeta 5\nhas type\n \u2102\nbut is expected to have type\n \u211d\nin the application\n Exists.intro (riemannZeta 5)\n'RiemannZetaValues.irrational_five' 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": 0.06,
"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 |
use x := riemannZeta 5
constructor
· rw [Irrational]
intro h
cases h with
| mk a b _ h_rational =>
let ζ5 := riemannZeta 5
have h_eval : ζ5 = (a : ℝ) / (b : ℝ) := by
rw [h_rational]
exact rfl
norm_num at h_eval
· rfl
|
View{
"agent_telemetry": {
"compilations": 8,
"iterations": 8,
"duration_s": 104.04,
"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": 24947,
"completion_tokens": 1248,
"total_tokens": 26195,
"reasoning_tokens": null,
"cost_usd": 0.0162175
},
"reasoning_effort": "",
"response_trace_count": 16
},
"response_trace": [
{
"iteration": 1,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349865-fXgoOePFNZevVy9acBwz",
"api_elapsed_s": 2.01,
"usage": {
"prompt_tokens": 1580,
"completion_tokens": 99,
"total_tokens": 1679,
"reasoning_tokens": null,
"cost_usd": 0.001087
},
"openrouter_retries": 0,
"response_text_len": 246
},
{
"iteration": 1,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "use x := riemannZeta 5\nconstructor\n\u00b7 rw [Irrational]\n intro h\n cases h with\n | mk a b _ h_rational =>\n let \u03b65 := riemannZeta 5\n have h_eval : \u03b65 = (a : \u211d) / (b : \u211d) := by\n rw [h_rational]\n exact rfl\n norm_num at h_eval\n\u00b7 rfl",
"compiler_output_preview": "[path]:17:6: error: unknown identifier 'x'\n[path]:16:67: error: unsolved goals\ncase h\n\u22a2 Irrational sorry \u2227 riemannZeta 5 = \u2191sorry\n[path]:17:8: err\n...<truncated>"
},
{
"iteration": 2,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349877-a7U9Iz4PvVoLlSmkmhJo",
"api_elapsed_s": 2.234,
"usage": {
"prompt_tokens": 1993,
"completion_tokens": 73,
"total_tokens": 2066,
"reasoning_tokens": null,
"cost_usd": 0.0012155
},
"openrouter_retries": 0,
"response_text_len": 161
},
{
"iteration": 2,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "refine \u27e8riemannZeta 5, ?_\u27e9\nconstructor\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, h_rat\u27e9\n have h_val : riemannZeta 5 = q := h_rat\n norm_num at h_val\n\u00b7 rfl",
"compiler_output_preview": "[path]:17:10: error: Application type mismatch: In the application\n Exists.intro (riemannZeta 5)\nthe argument\n riemannZeta 5\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\nh_rat : \u2191q = sorry\nh\n...<truncated>"
},
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"proof_code": "refine \u27e8(riemannZeta 5).re, ?_\u27e9\nconstructor\n\u00b7 intro (h : Rational (riemannZeta 5).re)\n rcases h with \u27e8q, hq\u27e9\n have h_irr : Irrational (riemannZeta 5).re := by\n simp [Irrational]\n exact h_irr \u27e8q, hq\u27e9\n\u00b7 norm_num\n congr\n ext\n \u00b7 rfl\n \u00b7 exact Complex.zero_re.symm",
"compiler_output_preview": "[path]:19:15: error: unknown identifier 'Rational'\n[path]:26:4: error: no applicable extensionality theorem found for\n \u2102\n'RiemannZetaValues.irrational_five' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 4,
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"proof_code": "refine \u27e8(riemannZeta 5).re, ?_, ?_\u27e9\n\u00b7 rw [Irrational]\n intro h\n rcases h with \u27e8q, hq\u27e9\n norm_num at hq\n\u00b7 have h_real : (riemannZeta 5).im = 0 := by\n norm_num\n exact Complex.ext rfl h_real.symm",
"compiler_output_preview": "[path]:18:2: error: unsolved goals\ncase refine_1.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 5).re\n\u22a2 False\n[path]:22:44: error: unsolved goals\n\u22a2 (riemannZeta 5).im = 0\n[path]
},
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"response_id": "gen-1771349914-J2WkX5gIixTyDK28dSID",
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"proof_code": "refine \u27e8(riemannZeta 5).re, ?_, ?_\u27e9\n\u00b7 intro h\n rcases h with \u27e8q, hq\u27e9\n have : riemannZeta 5 = (q : \u211d) := by\n apply Complex.ext\n \u00b7 exact hq\n \u00b7 have h_im : (riemannZeta 5).im = 0 := by\n rw [riemannZeta]\n simp\n rw [h_im]\n simp\n norm_num at this\n\u00b7 apply Complex.ext\n \u00b7 rfl\n \u00b7 have h_im : (riemannZeta 5).im = 0 := by\n rw [riemannZeta]\n simp\n rw [h_im]\n simp",
"compiler_output_preview": "[path]:22:8: error: type mismatch\n hq\nhas type\n \u2191q = (riemannZeta 5).re : Prop\nbut is expected to have type\n (riemannZeta 5).re = (\u2191\u2191q).re : Prop\n[path]:25:10: error: simp made no progress\n[path]
},
{
"iteration": 6,
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"finish_reason": "stop",
"response_id": "gen-1771349933-Bv1nJ5YqQSBQxZEsBBe3",
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"iteration": 6,
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"proof_code": "refine \u27e8(riemannZeta 5).re, ?_, ?_\u27e9\n\u00b7 intro h\n rcases h with \u27e8q, hq\u27e9\n have h_val : riemannZeta 5 = q := by\n apply Complex.ext\n \u00b7 exact hq.symm\n \u00b7 rw [zeta_re_five_im_zero]\n simp\n norm_num at h_val\n\u00b7 apply Complex.ext\n \u00b7 rfl\n \u00b7 rw [zeta_re_five_im_zero]\n simp\n\nwhere\n zeta_re_five_im_zero : (riemannZeta 5).im = 0 := by\n have h_real : (riemannZeta (5 : \u2102)).im = 0 := by\n apply riemannZeta_conj\n simp\n simp [riemannZeta] at *\n exact h_real",
"compiler_output_preview": "[path]:34:14: error: unknown identifier 'riemannZeta_conj'\n[path]:35:8: error: no goals to be solved\n[path]:18:2: error: unsolved goals\ncase refin\n...<truncated>"
},
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"iteration": 7,
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"proof_code": "refine \u27e8(riemannZeta 5).re, ?_, ?_\u27e9\n\u00b7 intro h\n rcases h with \u27e8q, hq\u27e9\n have h_zeta_real : (riemannZeta 5).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im_eq_zero]\n rw [riemannZeta_conj]\n simp\n have h_eq : riemannZeta 5 = (q : \u2102) := by\n apply Complex.ext\n \u00b7 exact hq.symm\n \u00b7 simp [h_zeta_real]\n norm_num at h_eq\n\u00b7 rw [\u2190 Complex.conj_eq_iff_im_eq_zero] at *\n have h_im : (riemannZeta 5).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_im_eq_zero]\n rw [riemannZeta_conj]\n simp\n apply Complex.ext\n \u00b7 simp\n \u00b7 simp [h_im]",
"compiler_output_preview": "[path]:21:12: error: unknown constant 'Complex.conj_eq_iff_im_eq_zero'\n[path]:18:2: error: unsolved goals\ncase refine_1.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 5).re\nh_zeta_real : (riemannZeta 5).im = 0\nh_eq : riemannZeta 5 = \u2191q\n\u22a2 False\n[path]
},
{
"iteration": 8,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349957-wVFbZNmsAlJRteEcqQ3t",
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"proof_code": "refine \u27e8(riemannZeta 5).re, ?_, ?_\u27e9\n\u00b7 intro h\n rcases h with \u27e8q, hq\u27e9\n have h_zeta_real : (riemannZeta 5).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_re_eq_self]\n have := Complex.conj_riemannZeta 5\n simp at this\n exact this\n have h_eq : riemannZeta 5 = (q : \u2102) := by\n apply Complex.ext\n \u00b7 exact hq.symm\n \u00b7 simp [h_zeta_real]\n norm_num at h_eq\n\u00b7 apply Complex.ext\n \u00b7 rfl\n \u00b7 have h_zeta_real : (riemannZeta 5).im = 0 := by\n rw [\u2190 Complex.conj_eq_iff_re_eq_self]\n have := Complex.conj_riemannZeta 5\n simp at this\n exact this\n simp [h_zeta_real]",
"compiler_output_preview": "[path]:21:12: error: unknown constant 'Complex.conj_eq_iff_re_eq_self'\n[path]:18:2: error: unsolved goals\ncase refine_1.intro\nq : \u211a\nhq : \u2191q = (riemannZeta 5).re\nh_zeta_real : (riemannZeta 5).im = 0\nh_eq : riemannZeta 5 = \u2191q\n\u22a2 False\n[path]
}
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"problem": {
"axioms_check_name": "irrational_five",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__RiemannZetaValues__irrational_five",
"category": "research_open",
"endings": [
"end RiemannZetaValues"
],
"imports": [
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],
"preamble_chars": 188,
"source_file": "FormalConjectures/Wikipedia/RiemannZetaValues.lean",
"title": "irrational_five"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:17:6: error: unknown identifier 'x'\n[path]:16:67: error: unsolved goals\ncase h\n\u22a2 Irrational sorry \u2227 riemannZeta 5 = \u2191sorry\n[path]:17:8: error: unexpected token ':='; expected command\n'RiemannZetaValues.irrational_five' 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 5 = ∑' n : ℕ, (1 : ℝ) / (n + 1) ^ 5 := by
rw [riemannZeta]
simp
norm_num at hq
· 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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|