The density of hyperbolicity conjecture, stating that the set of all parameters `c` for which
`fun z ↦ z ^ 2 + c` has an attracting cycle is dense in the Mandelbrot set.
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intro c hc
have hmain : c ∈ closure {c | ∃ m z, IsAttractingCycle (fun z ↦ z ^ 2 + c) m z} := ?_
exact hmain
View
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"goal": "```lean\n\u22a2 mandelbrotSet \u2286 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n```",
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"goal": "```lean\nc : \u2102\nhc : c \u2208 mandelbrotSet\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n```",
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"compiler_output_preview": "/app/asi_portal/verifier/quick_1772024248_9770.lean:39:30: error(lean.unknownIdentifier): Unknown identifier `density_of_hyperbolicity_multibrot`\n'Mandelbrot.density_of_hyperbolicity' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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"line": 38
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{
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}
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}
},
{
"iteration": 6,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
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},
{
"iteration": 6,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
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"banned_tokens": [],
"proof_code": "intro c hc\nsimp [mandelbrotSet, multibrotSet] at hc\nhave hmain : c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z} := ?_\nexact hmain",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772024297_4114.lean:38:86: error: unsolved goals\ncase refine_1\nc : \u2102\nhc : \u00acTendsto (fun k \u21a6 (fun z \u21a6 z ^ 2 + c)^[k] 0) atTop (cobounded \u2102)\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n'Mandelbrot.density_of_hyperbolicity' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 31.852,
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1,
1,
1,
2
],
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{
"fullRange": {
"end": {
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"line": 41
},
"start": {
"character": 86,
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},
"leanTags": [
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"message": "unsolved goals\ncase refine_1\nc : \u2102\nhc : \u00acTendsto (fun k \u21a6 (fun z \u21a6 z ^ 2 + c)^[k] 0) atTop (cobounded \u2102)\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}",
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},
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{
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},
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"range": {
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},
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"character": 0,
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}
},
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],
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"error_char": 86,
"error_msg": "unsolved goals\ncase refine_1\nc : \u2102\nhc : \u00acTendsto (fun k \u21a6 (fun z \u21a6 z ^ 2 + c)^[k] 0) atTop (cobounded \u2102)\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}",
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{
"line": 38,
"code": "intro c hc",
"goal": "```lean\n\u22a2 mandelbrotSet \u2286 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n```",
"goal_count": 1,
"ast_complexity": 92
},
{
"line": 39,
"code": "simp [mandelbrotSet, multibrotSet] at hc",
"goal": "```lean\nc : \u2102\nhc : c \u2208 mandelbrotSet\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n```",
"goal_count": 1,
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},
{
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"code": "have hmain : c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z} := ?_",
"goal": "```lean\nc : \u2102\nhc : \u00acTendsto (fun k \u21a6 (fun z \u21a6 z ^ 2 + c)^[k] 0) atTop (cobounded \u2102)\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n```",
"goal_count": 1,
"ast_complexity": 156
},
{
"line": 41,
"code": "exact hmain",
"goal": "```lean\ncase refine_2\nc : \u2102\nhc : \u00acTendsto (fun k \u21a6 (fun z \u21a6 z ^ 2 + c)^[k] 0) atTop (cobounded \u2102)\nhmain : c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n```\n---\n```lean\ncase refine_1\nc : \u2102\nhc : \u00acTendsto (fun k \u21a6 (fun z \u21a6 z ^ 2 + c)^[k] 0) atTop (cobounded \u2102)\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n```",
"goal_count": 2,
"ast_complexity": 420
}
],
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"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "mandelbrotSet \u2286 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z} := by"
}
},
{
"iteration": 7,
"event": "search_mathlib",
"query": "Function.IsPeriodicPt",
"results_count": 1
},
{
"iteration": 8,
"event": "search_mathlib",
"query": "IsPeriodicPt 1",
"results_count": 0
},
{
"iteration": 9,
"event": "search_mathlib",
"query": "isPeriodicPt",
"results_count": 8
},
{
"iteration": 10,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772024345-6K7z6lXEcKgD3yHuruHK",
"api_elapsed_s": 21.094,
"usage": {
"prompt_tokens": 13343,
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"total_tokens": 14354,
"reasoning_tokens": 988,
"cost_usd": 0.01714265
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Testing automated theorem search**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Testing automated theorem search**"
},
{
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"format": "openai-responses-v1",
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"data": 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}
]
},
{
"iteration": 10,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "aesop",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772024366_7056.lean:39:2: error: tactic 'aesop' failed, made no progress\nInitial goal:\n \u22a2 mandelbrotSet \u2286 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n'Mandelbrot.density_of_hyperbolicity' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 30.478,
"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": 7,
"line": 38
},
"start": {
"character": 2,
"line": 38
}
},
"message": "tactic 'aesop' failed, made no progress\nInitial goal:\n \u22a2 mandelbrotSet \u2286 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}",
"range": {
"end": {
"character": 7,
"line": 38
},
"start": {
"character": 2,
"line": 38
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 40
},
"start": {
"character": 0,
"line": 40
}
},
"message": "'Mandelbrot.density_of_hyperbolicity' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 40
},
"start": {
"character": 0,
"line": 40
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 38,
"error_char": 2,
"error_msg": "tactic 'aesop' failed, made no progress\nInitial goal:\n \u22a2 mandelbrotSet \u2286 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}",
"goal_before_error": "```lean\n\u22a2 mandelbrotSet \u2286 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n```",
"full_timeline": [
{
"line": 38,
"code": "aesop",
"goal": "```lean\n\u22a2 mandelbrotSet \u2286 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n```",
"goal_count": 1,
"ast_complexity": 92
}
],
"tactic_count": 1,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "aesop"
}
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772024396_8133.lean"
],
"comparator_passed": null,
"duration_s": 13.33488917350769,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "density_of_hyperbolicity",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Mandelbrot__density_of_hyperbolicity",
"category": "research_open",
"endings": [
"end Mandelbrot"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 1477,
"source_file": "FormalConjectures/Wikipedia/Mandelbrot.lean",
"title": "density_of_hyperbolicity"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772024396_8133.lean:38:86: error: unsolved goals\ncase refine_1\nc : \u2102\nhc : c \u2208 mandelbrotSet\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n'Mandelbrot.density_of_hyperbolicity' 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": {
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"duration_s": 0.04,
"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": {
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"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
Partial
1
Feb 17, 2026
intro c hc
let Hyp := {c | ∃ m z, IsAttractingCycle (fun z ↦ z ^ 2 + c) m z}
show c ∈ closure Hyp
by_cases h : c ∈ closure Hyp
· exact h
· have h_not_in : c ∉ closure Hyp := h
rw [Metric.mem_closure_iff] at h
push_neg at h
obtain ⟨ε, ε_pos, h_ball⟩ := h
-- This contradicts the density of hyperbolicity conjecture.
-- Since the conjecture is open, we leave the resulting state.
refine ?_
simp [Hyp] at *
refine ?_
View
{
"agent_telemetry": {
"compilations": 8,
"iterations": 8,
"duration_s": 95.77,
"best_status": "partial",
"best_goals": 1,
"max_iterations": 8,
"model": "gemini-3-flash-preview",
"error": null,
"provider": "openrouter_primary",
"finish_reason": "stop",
"usage": {
"prompt_tokens": 29006,
"completion_tokens": 1487,
"total_tokens": 30493,
"reasoning_tokens": null,
"cost_usd": 0.013528
},
"reasoning_effort": "",
"response_trace_count": 16
},
"response_trace": [
{
"iteration": 1,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349129-laSS7XlMRrfJgRzrZWQl",
"api_elapsed_s": 1.722,
"usage": {
"prompt_tokens": 2069,
"completion_tokens": 107,
"total_tokens": 2176,
"reasoning_tokens": null,
"cost_usd": 0.0013555
},
"openrouter_retries": 0,
"response_text_len": 273
},
{
"iteration": 1,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro c hc\nrefine closure_minimal ?_ ?_ hc\n\u00b7 intros c' hc'\n rw [mem_setOf_eq]\n refine ?_\n\u00b7 exact isClosed_closure\n\u00b7 intro c' hc'\n by_cases h_hyper : \u2203 m z, IsAttractingCycle (fun z => z ^ 2 + c') m z\n \u00b7 apply subset_closure\n exact h_hyper\n \u00b7 refine ?_\n exact hc'",
"compiler_output_preview": "[path]:40:31: error: Application type mismatch: In the application\n closure_minimal ?m.3097 ?m.3098 hc\nthe argument\n hc\nhas type\n c \u2208 mandelbrotSet : Prop\nbut is expected to have type\n c \u2208 closure ?m.3095 : Prop\n[path]:38:86: error: unsolved goals\nc : \u2102\nhc\n...<truncated>"
},
{
"iteration": 2,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349138-d5W1bqByn30th5B3IR93",
"api_elapsed_s": 2.368,
"usage": {
"prompt_tokens": 2487,
"completion_tokens": 213,
"total_tokens": 2700,
"reasoning_tokens": null,
"cost_usd": 0.00097665
},
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},
{
"iteration": 2,
"event": "compile_proof",
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"status": "failed",
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"timed_out": false,
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"proof_code": "intro c hc\nlet Hyp := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nsuffices c \u2208 closure Hyp from this\nby_cases h : c \u2208 closure Hyp\n\u00b7 exact h\n\u00b7 exfalso\n have h_not_hyp : c \u2209 closure Hyp := h\n let h_int := (isOpen_complement (isClosed_closure (s := Hyp))).interior_mem (c := c)\n simp at h_not_hyp\n refine ?_\n -- This is a deep result in complex dynamics known as the Density of Hyperbolicity.\n -- It is equivalent to the fact that every non-hyperbolic component is a ghost component\n -- or that there are no invariant line fields on the Julia set.\n -- This remains an open problem in the general case (the Fatou conjecture).\n -- We leave the goal open as it is a major open conjecture in mathematics.",
"compiler_output_preview": "[path]:46:18: error: unknown identifier 'isOpen_complement'\n'Mandelbrot.density_of_hyperbolicity' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 3,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349148-5B3cDyI5rS7mN7ssDIw5",
"api_elapsed_s": 2.096,
"usage": {
"prompt_tokens": 2878,
"completion_tokens": 172,
"total_tokens": 3050,
"reasoning_tokens": null,
"cost_usd": 0.00104915
},
"openrouter_retries": 0,
"response_text_len": 618
},
{
"iteration": 3,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro c hc\nlet Hyp := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nshow c \u2208 closure Hyp\nby_cases h : c \u2208 closure Hyp\n\u00b7 exact h\n\u00b7 -- This is the conjecture. We proceed by contradiction assuming the complement.\n -- The interior of the Mandelbrot set consists of hyperbolic components and ghost components.\n -- The density of hyperbolicity is equivalent to the absence of ghost components.\n have : c \u2208 mandelbrotSet \\ closure Hyp := \u27e8hc, h\u27e9\n refine ?_\n -- Subgoals remain as this is an open conjecture.\n -- The proof would require the result that hyperbolic parameters are dense.\n simp [Hyp] at *\n aesop",
"compiler_output_preview": "[path]:52:4: error: (deterministic) timeout at `aesop`, maximum number of heartbeats (200000) has been reached\n\nNote: Use `set_option maxHeartbeats <num>` to set the limit.\n\nHint: Additional diagnostic information may be available using the `set_option diagnostics true` command.\n'Mandelbrot.density_of_hyperbolicity' depends on axioms: [propext, sorryAx, Classical.choice, Quot.s\n...<truncated>"
},
{
"iteration": 4,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349174-95IxpDJfB5exVoleu9ws",
"api_elapsed_s": 4.85,
"usage": {
"prompt_tokens": 3241,
"completion_tokens": 155,
"total_tokens": 3396,
"reasoning_tokens": null,
"cost_usd": 0.0020855
},
"openrouter_retries": 0,
"response_text_len": 429
},
{
"iteration": 4,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro c hc\nlet Hyp := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nshow c \u2208 closure Hyp\nby_cases h : c \u2208 closure Hyp\n\u00b7 exact h\n\u00b7 have h_not_in : c \u2209 closure Hyp := h\n rw [Metric.mem_closure_iff] at h\n push_neg at h\n obtain \u27e8\u03b5, \u03b5_pos, h_ball\u27e9 := h\n -- This contradicts the density of hyperbolicity conjecture.\n -- Since the conjecture is open, we leave the resulting state.\n refine ?_\n simp [Hyp] at *\n refine ?_",
"compiler_output_preview": "[path]:44:2: error: unsolved goals\ncase neg.intro.intro\nc : \u2102\nhc : c \u2208 mandelbrotSet\nHyp : Set \u2102 := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nh_not_in : c \u2209 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nh_ball : \u2200 (b : \u2102) (x : \u2115) (x_1 : \u2102), IsAttractingCycle (fun z \u21a6 z ^ 2 + b) x x_1 \u2192 \u03b5 \u2264 dist c b\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAt\n...<truncated>"
},
{
"iteration": 5,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349186-BvILxUMP3ZtCOdiKybcn",
"api_elapsed_s": 2.499,
"usage": {
"prompt_tokens": 3755,
"completion_tokens": 258,
"total_tokens": 4013,
"reasoning_tokens": null,
"cost_usd": 0.0026515
},
"openrouter_retries": 0,
"response_text_len": 694
},
{
"iteration": 5,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro c hc\nlet Hyp := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nshow c \u2208 closure Hyp\nby_cases h : c \u2208 closure Hyp\n\u00b7 exact h\n\u00b7 exfalso\n rw [Metric.mem_closure_iff] at h\n push_neg at h\n obtain \u27e8\u03b5, \u03b5_pos, h_ball\u27e9 := h\n -- The density of hyperbolicity states that Hyp is dense in mandelbrotSet.\n -- This case (c \u2208 mandelbrotSet \\ closure Hyp) is conjectured to be impossible.\n -- We provide the specific contradiction goal for the density of hyperbolicity.\n have : \u2200 c' \u2208 mandelbrotSet, \u2200 \u03b5 > 0, \u2203 c'' \u2208 Hyp, dist c' c'' < \u03b5 := ?_\n specialize this c hc \u03b5 \u03b5_pos\n obtain \u27e8c'', h_c'', h_dist\u27e9 := this\n have : \u03b5 \u2264 dist c c'' := h_ball c'' h_c''.1 h_c''.2.1 h_c''.2.2.1\n linarith",
"compiler_output_preview": "[path]:54:40: error: Invalid projection: Cannot project a value of non-propositional type\n \u2115\nfrom the expression\n h_c''\nwhich has propositional type\n c'' \u2208 Hyp\n[path]:44:2: error: unsolved goals\ncase neg.intro.intro.refine_1\nc : \u2102\nhc : c \u2208 mandelbrotSet\nHyp\n...<truncated>"
},
{
"iteration": 6,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349196-9x39ZEd5Sbg2tF30KlXL",
"api_elapsed_s": 2.085,
"usage": {
"prompt_tokens": 4390,
"completion_tokens": 189,
"total_tokens": 4579,
"reasoning_tokens": null,
"cost_usd": 0.002762
},
"openrouter_retries": 0,
"response_text_len": 569
},
{
"iteration": 6,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro c hc\nlet Hyp := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nshow c \u2208 closure Hyp\nby_cases h : c \u2208 closure Hyp\n\u00b7 exact h\n\u00b7 exfalso\n rw [Metric.mem_closure_iff] at h\n push_neg at h\n obtain \u27e8\u03b5, \u03b5_pos, h_ball\u27e9 := h\n -- The density of hyperbolicity conjecture is that Hyp is dense in mandelbrotSet.\n -- This contradicts the existence of a parameter c in the Mandelbrot set\n -- that is not a limit point of hyperbolic parameters.\n have h_dense : mandelbrotSet \u2286 closure Hyp := ?_\n have h_inc : c \u2208 closure Hyp := h_dense hc\n contradiction\n exact ?_",
"compiler_output_preview": "[path]:53:4: error: tactic 'contradiction' failed\ncase neg.intro.intro.refine_2\nc : \u2102\nhc : c \u2208 mandelbrotSet\nHyp : Set \u2102 := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh_ball : \u2200 b \u2208 Hyp, \u03b5 \u2264 dist c b\nh_dense : mandelbrotSet \u2286 closure Hyp\nh_inc : c \u2208 closure Hyp\n\u22a2 False\n'Mandelbrot.density_of_hyperbolicity' depends on axioms: [propext, sorryAx, Cl\n...<truncated>"
},
{
"iteration": 7,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349205-ewSTNXlJYPrgmZ6VG5qc",
"api_elapsed_s": 1.952,
"usage": {
"prompt_tokens": 4857,
"completion_tokens": 203,
"total_tokens": 5060,
"reasoning_tokens": null,
"cost_usd": 0.00122535
},
"openrouter_retries": 0,
"response_text_len": 576
},
{
"iteration": 7,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro c hc\nlet Hyp := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nshow c \u2208 closure Hyp\nby_cases h : c \u2208 closure Hyp\n\u00b7 exact h\n\u00b7 exfalso\n have h_not_in : c \u2209 closure Hyp := h\n rw [Metric.mem_closure_iff] at h\n push_neg at h\n obtain \u27e8\u03b5, \u03b5_pos, h_ball\u27e9 := h\n -- The conjecture is that mandelbrotSet is contained in the closure of Hyp.\n -- This proof step represents the core of the Density of Hyperbolicity conjecture.\n have h_dense : mandelbrotSet \u2286 closure Hyp := ?_\n have h_in_closure : c \u2208 closure Hyp := h_dense hc\n exact h_not_in h_in_closure\n exact ?_",
"compiler_output_preview": "[path]:54:10: error: don't know how to synthesize placeholder\ncontext:\nc : \u2102\nhc : c \u2208 mandelbrotSet\nHyp : Set \u2102 := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nh_not_in : c \u2209 closure Hyp\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh_ball : \u2200 b \u2208 Hyp, \u03b5 \u2264 dist c b\n\u22a2 mandelbrotSet \u2286 closure Hyp\n'Mandelbrot.density_of_hyperbolicity' depends on axioms: [propext, sorryAx, Classical.choice, Quot.s\n...<truncated>"
},
{
"iteration": 8,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771349215-7WsasWdP6cOZd0680vEo",
"api_elapsed_s": 2.218,
"usage": {
"prompt_tokens": 5329,
"completion_tokens": 190,
"total_tokens": 5519,
"reasoning_tokens": null,
"cost_usd": 0.00142235
},
"openrouter_retries": 0,
"response_text_len": 635
},
{
"iteration": 8,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro c hc\nlet Hyp := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nshow c \u2208 closure Hyp\nby_cases h : c \u2208 closure Hyp\n\u00b7 exact h\n\u00b7 -- The \"Density of Hyperbolicity\" is a major open conjecture in complex dynamics.\n -- It states that hyperbolic parameters are dense in the Mandelbrot set.\n -- For parameters c in the Mandelbrot set, every neighborhood contains a parameter\n -- with an attracting cycle.\n suffices mandelbrotSet \u2286 closure Hyp from this hc\n refine ?_\n intro c' hc'\n rw [Metric.mem_closure_iff]\n intro \u03b5 \u03b5_pos\n -- This step requires the density proof (e.g. from any potential local connectivity)\n refine ?_",
"compiler_output_preview": "[path]:44:2: error: unsolved goals\ncase neg\nc : \u2102\nhc : c \u2208 mandelbrotSet\nHyp : Set \u2102 := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nh : c \u2209 closure Hyp\nc' : \u2102\nhc' : c' \u2208 mandelbrotSet\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\n\u22a2 \u2203 b \u2208 Hyp, dist c' b < \u03b5\n'Mandelbrot.density_of_hyperbolicity' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"[path]
],
"duration_s": 7.732987403869629,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"axioms_check_name": "density_of_hyperbolicity",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Mandelbrot__density_of_hyperbolicity",
"category": "research_open",
"endings": [
"end Mandelbrot"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 1477,
"source_file": "FormalConjectures/Wikipedia/Mandelbrot.lean",
"title": "density_of_hyperbolicity"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:44:2: error: unsolved goals\ncase neg.intro.intro\nc : \u2102\nhc : c \u2208 mandelbrotSet\nHyp : Set \u2102 := {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\nh_not_in : c \u2209 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nh_ball : \u2200 (b : \u2102) (x : \u2115) (x_1 : \u2102), IsAttractingCycle (fun z \u21a6 z ^ 2 + b) x x_1 \u2192 \u03b5 \u2264 dist c b\n\u22a2 c \u2208 closure {c | \u2203 m z, IsAttractingCycle (fun z \u21a6 z ^ 2 + c) m z}\n'Mandelbrot.density_of_hyperbolicity' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
Gemini 3 Flash Preview
Failed
--
Feb 14, 2026
intro c' hc' h_exists
exact h_empty c' ⟨hc', h_exists⟩
have h_bounded : ∀ k, ‖(fun z ↦ z ^ 2 + c)^[k] 0‖ ≤ 2 := by
intro k
induction k with
| zero => simp
| succ k ih =>
simp [f]
sorry_placeholder_replacement_tactic
exact ih
exact h_empty c ⟨hmem, sorry_placeholder_replacement_tactic⟩
