A Conjecture on Triple Counts for the Kasami APN Function

Precise statement

For the Kasami APN function $F(x) = x^{4^k - 2^k + 1}$ on $\mathrm{GF}(2^n)$ with $\gcd(k, n) = 1$, the conjecture asserts that for $\Delta = \{F(b) + F(b+1) + 1\}$ and all distinct nonzero $v_1, v_2$, the number of triples in $\Delta^3$ with $v_1 x + v_2 y + (v_1 + v_2) z = 0$ is exactly $2^{2n-3}$. Proved for $k \bmod n \in \{1, 2, n-2, n-1\}$ and verified exhaustively for $n \le 13$; the general case remains open.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

Claude Fable 5, Aristotle

The paper's statement: "Every proof in this paper was obtained by the AI assistant Claude Fable 5 and has subsequently been formally verified in the Lean theorem prover by Aristotle (Harmonic)." The model was prompted with the conjecture statement together with background hints and a pointer to the companion repository.

Provider: Anthropic, Harmonic · Prompt public: unknown · Independence: unknown

Verification boundary

lean checked statement unaudited

Checked by this site on 21 August 2026 against the paper (arXiv:2608.18584v1): the AI and Lean statements are verbatim as quoted. Recorded lean-checked rather than lean-verified deliberately - the companion repository is cited but its URL is not exposed in the HTML, so no artifact has been audited here for sorry or for declared axioms, and statement fidelity is unaudited. Entered as Partial on the paper's own words, "The general case remains open".

Correctness: supported · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-08-19

   On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for k mod n in {1,2,n-2,n-1}, and exhaustive verification for n<=13

   VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• 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.