Spin-parity formulas for k-differentials

For odd k≥3 and gcd(n,k)=gcd(n+1,k)=1, the spin-parity target is N_k(n)≡⌊(k+1)/4⌋ mod 2. A central reduction is the floor-sum identity N_k(n)=F_k(n+1)−F_k(n), together with a Jacobi-symbol reformulation.

Why it matters and what counts as progress

Why it matters: It supplies the arithmetic step needed to make the stated genus-zero and genus-one k-differential parity formulas unconditional.

What counts as progress: Prove the floor-sum reduction and parity formula uniformly for the coprime parameters, removing the earlier conditional step in the spin-parity formulas.

How the problem developed

2022 · Conditional result [1]
Dawei Chen and Quentin Gendron Derived genus-zero and genus-one spin-parity formulas for odd k under a number-theoretic hypothesis, stated as Conjecture A.10.
3 Feb 2026 · Theorem [2]
Dawei Chen, Evan Chen, Kenny Lau, Ken Ono and Jujian Zhang Proved the Jacobi-symbol parity identity underlying Conjecture A.10, making the genus-zero and genus-one formulas unconditional.

AI contribution

Parity of k-differentials in genus zero and one

Dawei Chen, Evan Chen, Kenny Lau, Ken Ono, Jujian Zhang

For odd k≥3 and integers n with gcd(n,k)=gcd(n+1,k)=1, AxiomProver found a Jacobi-symbol reformulation and the floor-sum identity N_k(n)=F_k(n+1)−F_k(n). The authors combine that identity with classical Jacobi facts to establish the parity formula N_k(n)≡⌊(k+1)/4⌋ mod 2 and make the earlier genus-zero and genus-one k-differential formulas unconditional.

AI involvementAI-autonomous

ImportanceNotable

Technical verificationPartial

Human understandingFull