Zhi-Hong Sun's Conjecture 3.7: two Apéry-like sums at discriminant -20

Theorem 1.1 claims the complete conjecture: both signed sums, every prime except 2 and 5, all representation classes, and modulus p squared. Split primes use two modular periods, an integral Hecke polynomial and separate finite specializations, including the degree-p endpoint and nonprincipal CM sign. For inert primes, a degree-five endomorphism with square [-5] forces Hasse vanishing, then a differential invariant lifts each original sum. The prime 3 is checked directly. The paper credits Sun's mod-p and Legendre antecedents, Straub's coefficient congruence and Beukers's framework. The stronger Sun-Ye quartic congruence concerns an auxiliary sum; the available finite transports alone have only mod-p precision. No uniform mod-p-cubed equality is claimed: at p=7 the unsigned target sums are 192 and 290 modulo 343.

Theorem 1.1 claims the complete conjecture: both signed sums, every prime except 2 and 5, all representation classes, and modulus p squared. Split primes use two modular periods, an integral Hecke polynomial and separate finite specializations, including the degree-p endpoint and nonprincipal CM sign. For inert primes, a degree-five endomorphism with square [-5] forces Hasse vanishing, then a differential invariant lifts each original sum. The prime 3 is checked directly. The paper credits Sun's mod-p and Legendre antecedents, Straub's coefficient congruence and Beukers's framework. The stronger Sun-Ye quartic congruence concerns an auxiliary sum; the available finite transports alone have only mod-p precision. No uniform mod-p-cubed equality is claimed: at p=7 the unsigned target sums are 192 and 290 modulo 343.