lean artifact · pending
Artifact ↗Source authenticated
Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication
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.
Exact FrontierDelta
Prior state unknown→proved
Scope and record
Occurred: Aug 1, 2026
Delta type: SOURCE CLAIM
Assumptions: VibeMathed verification: lean-checked. Publication: preprint. AI contribution: ai-assisted. Imported under CC BY 4.0.
Canonical aliases: Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication · Unrestricted multiplicative complexity of Mul4
Confidence: Not scored
Registry verification: lean checked · preprint · partial
Attribution
VibeMathed
registry · event recorded by
OpenAI GPT-5.6 Sol (extra-high)
model · ai model contributor · OpenAI
Anthropic Opus 5 (high, referee)
model · ai model contributor · Anthropic
Artifacts and verifiers
formal registration · pending
Artifact ↗Compute record
No linked compute attempts recorded.
Lineage and corrections
Boyar--Find finite-field question evidence for this event
VibeMathed record: Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication evidence for this event
This event attributed to OpenAI GPT-5.6 Sol (extra-high)
Headline theorem MC(Mul 4) = 9 evidence for this event
Complete Lean proof and immutable verification release evidence for this event
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. parent of this event
This event attributed to Anthropic Opus 5 (high, referee)
Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication evidence for this event
Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication parent of this event