Carlet's cyclic-additive conjecture for the Kasami monomials

Gábor P. Nagy, Douglas S. McNeil, Attila Vajda

Let K be a field of characteristic two with |K|=2ⁿ, gcd(k,n)=1 and dₖ=4ᵏ−2ᵏ+1. If Δ is the image of b↦(b+1)^dₖ+b^dₖ+1, then for distinct nonzero v₁,v₂∈K exactly 2^(2n−3) triples (x,y,z)∈Δ³ satisfy v₁x+v₂y+(v₁+v₂)z=0.

Sources for this record

What was already known

Carlet proposed the Kasami cyclic-additive condition. Nagy and Vajda proved selected residue classes of k and checked n≤13; their reductions and averaging argument are inputs to the new general proof.

AI involvement AI-centered

The project disclosure credits interactive GPT work with the main new argument, and Claude agents with later formal development. Nagy, McNeil and Vajda reviewed and edited the paper and retain credit for earlier partial results. The paper names a different GPT version than the code disclosure.

The cited disclosure calls GPT interactive discovery the main source of the new mathematical argument, while human authors guided the work and supplied significant antecedents. The named model version conflicts with the later paper and prevents a stronger attribution to one system.

Importance Notable

Resolves the cyclic-additive count for all admissible Kasami parameters.

Limitations

The formal development reports a complete proof of the count and a separate statement comparison. The code disclosure names GPT-5.6 Sol while the later paper names ChatGPT 6 Pro for the present proof; the model identity and phases need reconciliation. The authors’ editing does not document a complete human proof check.

Technical verification Formal (Lean 4/Mathlib kernel, Comparator and NanoDa)

The cited revision reports a complete Lean proof of the coefficient-count statement, a separate Mathlib-only literature specification and semantic bridge, standard-axiom audit, Comparator matching and NanoDa replay. Earlier exhaustive n≤13 tests are not the basis of the universal Formal rating. This review did not rerun Lean.

Independent checking

complete formal proof, semantic statement bridge, axiom audit, protected statement comparison and independent proof replay

Checker
Lean kernel, Comparator and NanoDa, with reported Aristotle statement review
Scope
KasamiCyclicAdditive.carlet_kasami_cyclic_additive for all admissible n, k and distinct nonzero coefficients
Basis
Reported by the linked checker
Evidence source

Human understanding Full

The named authors give an explicit, detailed explanation of the Fourier correction, Fermat-cubic incidence and exact-average forcing mechanism in their later paper, which they reviewed and edited. Their review is evidence of understanding, not a separately documented complete human proof check.

Correction and publication history