Let k≥ 3 be an odd integer. For any integer n, let N_k(n) be the number of pairs of positive integers (b_1,b_2) such that b_1,b_2≤ (k-1)/2, b_1+b_2≥ (k+1)/2, and b_2 equiv nb_1 pmodk. If n satisfies gcd(n,k)=gcd(n+1,k)=1, then N_k(n)equiv lfloor frack+14rfloor pmod2.

Spin parity for k-differentials

The problem asks whether the stated parity formula always holds for odd $k$ under the given coprimality conditions. No source in the scan identifies its original proposer or date.

February 2026 proof claim

“Parity of $k$ -differentials in genus zero and one” states that Conjecture 1.1 is true. Its proof uses $N_k(n)=F_k(n+1)-F_k(n)$ and Jacobi-symbol arguments, and claims to make earlier conditional spin-parity results unconditional. The source attributes the key observation and formal proof of a main lemma to AxiomProver, with Lean verification artifacts discussed in an appendix.

Current status (as of February 2026): The conjecture is claimed proved in a preprint, but independent mathematical verification is still absent.

Spin parity for $k$ -differentials

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