{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:unrestricted-multiplicative-complexity-mul4", "title": "Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication", "canonical_statement": "Let Mul4: F_2^8 -> F_2^7 output the seven coefficients of the product of two four-term binary polynomials. The manuscript proves that its unrestricted XOR--AND multiplicative complexity is exactly 9. The upper bound is a nine-AND Karatsuba--Ofman construction. The lower bound rules out every unrestricted eight-AND circuit, including circuits that reuse nonlinear intermediate wires and exploit Boolean idempotence. Thus, for this natural vector-valued quadratic function, allowing nonlinear feedback does not improve on the optimal quadratic circuit.", "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "partial", "solution_events": [ { "id": "vibemathed:unrestricted-multiplicative-complexity-mul4:event", "problem_id": "vibemathed:unrestricted-multiplicative-complexity-mul4", "type": "proved", "status": "partial", "occurred_at": "2026-08-31T00:00:00.000Z", "title": "arXiv", "summary": "The exact Mul4 instance is resolved: its unrestricted XOR--AND multiplicative complexity is 9. The restricted bilinear/quadratic value 9 was classical; the new theorem proves that arbitrary nonlinear Boolean reuse cannot lower it. This is a natural positive special case of the Boyar--Find question, not a solution of the general (n,m) problem. The lower bound is a conceptual structural argument. A complete Lean 4 formalization checks the unrestricted circuit semantics and the exact equality, while the development-time Python/C++ programs remain independent regression checks rather than logical premises. The corresponding unrestricted questions for five or more terms remain open. Independent statement-fidelity review remains pending.", "ai_contribution": "ai_assisted", "attempt_ids": [ "vibemathed:unrestricted-multiplicative-complexity-mul4:attempt" ], "method_family_ids": [ "vibemathed:unrestricted-multiplicative-complexity-mul4:method" ], "source_assertion_ids": [ "vibemathed:unrestricted-multiplicative-complexity-mul4:assertion" ] } ], "attempts": [ { "id": "vibemathed:unrestricted-multiplicative-complexity-mul4:attempt", "problem_id": "vibemathed:unrestricted-multiplicative-complexity-mul4", "model": "OpenAI GPT-5.6 Sol (extra-high); Anthropic Opus 5 (high, referee)", "provider": "OpenAI; Anthropic", "attempted_at": "2026-08-31T00:00:00.000Z", "human_collaborators": [], "exposure": "unknown", "prompt_public": null, "outcome": "OpenAI GPT-5.6 Sol in extra-high thinking mode was used for research, proof exploration and development, computational checking, Lean 4 formalization and formal-proof engineering, literature and citation verification, manuscript drafting and revision, and release and submission preparation. Anthropic Opus 5 in high thinking mode was used as a critical referee, not as the discovery model. The public disclosure does not attribute a named essential lemma or the central informal proof solely to a model, so the conservative AI-assisted tier is retained rather than AI co-developed or AI-discovered. Gregory Morse directed and reviewed the mathematical claims, informal and formal proofs, computations, citations, code, and manuscript text and assumes full responsibility.", "failed_routes": [], "artifacts": [ { "label": "Boyar--Find finite-field question", "url": "https://arxiv.org/abs/1407.6169v1", "kind": "problem-record", "license": null }, { "label": "Complete Lean proof and immutable verification release", "url": "https://github.com/GregoryMorse/unrestricted-boolean-mul/releases/tag/n4-arxiv-v2", "kind": "lean-proof", "license": null }, { "label": "Headline theorem MC(Mul 4) = 9", "url": "https://github.com/GregoryMorse/unrestricted-boolean-mul/blob/n4-arxiv-v2/UnrestrictedBooleanMul/N4/Main.lean", "kind": "lean-statement", "license": null }, { "label": "Rebuilt here: lake build, axiom audit and leanchecker replay", "url": "https://github.com/mrconter1/vibemathed/actions/runs/33722071409", "kind": "independent", "license": null } ], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2608.30238", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication", "url": "https://vibemathed.com/problem/unrestricted-multiplicative-complexity-mul4", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:unrestricted-multiplicative-complexity-mul4:verification", "problem_id": "vibemathed:unrestricted-multiplicative-complexity-mul4", "solution_event_id": "vibemathed:unrestricted-multiplicative-complexity-mul4:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Site-confirmed: rebuilt here on 3 September 2026, not taken from the author's CI badge. This site's verify-lean workflow checked out release n4-arxiv-v2 (commit 2ebc0cf4) of GregoryMorse/unrestricted-boolean-mul, installed the toolchain the repository pins (leanprover/lean4:v4.32.1), ran $\\texttt{lake build}$ over every module, then the project's own AxiomAudit.lean, then $\\texttt{lake env leanchecker}$ replaying the whole UnrestrictedBooleanMul environment. 24m40s, every step green.\n\nAll six headline theorems - the exact values for $n = 0,1,2,3$, the eight-gate impossibility and $\\mathrm{MC}(\\mathrm{Mul}\\,4) = 9$ - report $\\texttt{propext}$, $\\texttt{Classical.choice}$ and $\\texttt{Quot.sound}$ and nothing else. No $\\texttt{sorryAx}$, no project axiom, no $\\texttt{native\\_decide}$, no $\\texttt{bv\\_decide}$.\n\nThe release tag is one commit ahead of 1533276b, which the author's own run used; that commit touches only READMEs, a citation file and checksum lists, no Lean source.\n\nNot Lean-verified. The kernel checks the proof against the statement the author wrote; whether that statement expresses the Boyar-Find question is the anchoring half, and nobody without a stake has audited it.", "sources": [ { "label": "VibeMathed: Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication", "url": "https://vibemathed.com/problem/unrestricted-multiplicative-complexity-mul4", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "arXiv", "url": "https://arxiv.org/abs/2608.30238", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2608.30238", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication", "url": "https://vibemathed.com/problem/unrestricted-multiplicative-complexity-mul4", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "expand", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:unrestricted-multiplicative-complexity-mul4:signal:recent_partial_result", "problem_id": "vibemathed:unrestricted-multiplicative-complexity-mul4", "kind": "recent_partial_result", "reason": "A source-reported partial result may support bounded expansion.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "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" }