{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture", "title": "The Ibragimov–Iosifescu conjecture for φ-mixing sequences", "canonical_statement": "Ibragimov conjectured that if $(X_n)$ is a strictly stationary, $\\varphi$-mixing sequence with\n$$\n\\mathbb E X_0=0,\\qquad \\mathbb E X_0^2<\\infty,\n$$\nand\n$$\n\\sigma_n^2=\\operatorname{Var}(S_n)\\to\\infty,\\qquad S_n=\\sum_{j=0}^{n-1}X_j,\n$$\nthen\n$$\n\\frac{S_n}{\\sigma_n}\\Rightarrow N(0,1).\n$$\nGPT-6 Astra constructs a counterexample satisfying all these hypotheses for which, along a subsequence $n_j\\to\\infty$,\n$$\n\\frac{S_{n_j}}{\\sigma_{n_j}}\\to 0\n$$\nin probability. Hence the conjectured central limit theorem fails.", "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "candidate", "solution_events": [ { "id": "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture:event", "problem_id": "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture", "type": "disproved", "status": "candidate", "occurred_at": "2026-09-05T00:00:00.000Z", "title": "Github", "summary": "Astra constructs a strictly stationary real process\n$$\nX_t=\\xi_t+K(\\xi_{t-1},\\xi_{t-2},\\ldots),\n$$\nwhere the innovations $\\xi_t$ are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and $K$ is bounded, continuous, and odd. The process is $\\varphi$-mixing, centered, square-integrable, and satisfies\n$$\n\\operatorname{Var}(S_n)\\to\\infty.\n$$\nNevertheless there are times $n_j\\to\\infty$ such that\n$$\n\\frac{S_{n_j}}{\\sqrt{\\operatorname{Var}(S_{n_j})}}\\to0\n$$\nin probability. Therefore the normalized sums cannot converge in distribution to $N(0,1)$.\n\nThe same example also rules out Iosifescu's stronger weak invariance-principle conjecture, since Brownian convergence would imply the CLT at time $1$.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture:attempt" ], "method_family_ids": [ "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture:method" ], "source_assertion_ids": [ "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture:assertion" ] } ], "attempts": [ { "id": "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture:attempt", "problem_id": "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture", "model": "GPT-6 Astra (pre-release)", "provider": "OpenAI", "attempted_at": "2026-09-05T00:00:00.000Z", "human_collaborators": [ "Tom Adamczewski" ], "exposure": "unknown", "prompt_public": null, "outcome": "GPT-6 Astra autonomously constructed the counterexample and wrote its Lean proof. The run lasted about 12 hours and 1054 turns. The core idea is to build a stationary causal process from Gaussian-smoothed rare spikes plus a bounded nonlinear feedback correction. The feedback suppresses ordinary fluctuations along selected times, while extremely rare large spikes keep $\\operatorname{Var}(S_n)\\to\\infty$. Astra also proves that the resulting process remains genuinely $\\varphi$-mixing by constructing a bounded causal inverse with square-summable tail variation.", "failed_routes": [], "artifacts": [ { "label": "Challenge.lean: the compared statement", "url": "https://github.com/tadamcz/phi-mixing-clt/blob/main/Challenge.lean", "kind": "lean-statement", "license": null }, { "label": "Peligrad (1990), On Ibragimov–Iosifescu conjecture for φ-mixing sequences", "url": "https://doi.org/10.1016/0304-4149(90)90008-6", "kind": "problem-record", "license": null } ], "sources": [ { "label": "Github", "url": "https://github.com/tadamcz/phi-mixing-clt", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The Ibragimov–Iosifescu conjecture for φ-mixing sequences", "url": "https://vibemathed.com/problem/ibragimov-iosifescu-varphi-mixing-clt-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": 261, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture:verification", "problem_id": "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture", "solution_event_id": "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture:event", "level": "lean_checked_statement_unaudited", "mathematical_correctness": "supported", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Lean-checked, statement unaudited. Checked here on 6 September 2026 from a clone of tadamcz/phi-mixing-clt at 8e08498: 13,047 lines, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; the repository's recorded verifier accepted the disproof with the default kernel and only the three standard axioms. The statement, however, was produced by Epoch's AI-autoformalized 'wikipedia' run, not by a human-curated repository, and this site has not audited the formal definitions of strict stationarity and the φ-mixing coefficient against the sources beyond reading the docstring, which is careful about conventions. The repository's own audit finds no mismatch, but that audit is machine-written. No probabilist outside the run has read the 12,900-line construction. Hence Candidate.", "sources": [ { "label": "VibeMathed: The Ibragimov–Iosifescu conjecture for φ-mixing sequences", "url": "https://vibemathed.com/problem/ibragimov-iosifescu-varphi-mixing-clt-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "Github", "url": "https://github.com/tadamcz/phi-mixing-clt", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "Github", "url": "https://github.com/tadamcz/phi-mixing-clt", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: The Ibragimov–Iosifescu conjecture for φ-mixing sequences", "url": "https://vibemathed.com/problem/ibragimov-iosifescu-varphi-mixing-clt-conjecture", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture:signal:verify_now", "problem_id": "vibemathed:ibragimov-iosifescu-varphi-mixing-clt-conjecture", "kind": "verify_now", "reason": "The source labels this result as a candidate; verification should precede reuse.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "The source status is candidate and must not be represented as solved.", "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }