Sun's Conjecture 3.5 on an Apéry-like supercongruence

Theorem 1.1 claims the complete stated conjecture modulo $p^2$ for every prime $p>3$, covering both residue classes modulo 3, with $p=5$ treated directly. For inert primes, a finite quadratic differential invariant lifts Sun's published modulo-$p$ vanishing. For split primes, certified modular periods, an integral Hecke polynomial with a degree bound and complex multiplication give the evaluation. The Franel sum with denominator 50 is evaluated separately for $p>5$ and credited to Sun's different 2018 Conjecture 3.5. No modulo-$p^3$ result, resolution of adjacent conjectures, or first-proof claim is made. Candidate reflects pending authoritative review of a claimed complete proof of this specific question.

Theorem 1.1 claims the complete stated conjecture modulo $p^2$ for every prime $p>3$, covering both residue classes modulo 3, with $p=5$ treated directly. For inert primes, a finite quadratic differential invariant lifts Sun's published modulo-$p$ vanishing. For split primes, certified modular periods, an integral Hecke polynomial with a degree bound and complex multiplication give the evaluation. The Franel sum with denominator 50 is evaluated separately for $p>5$ and credited to Sun's different 2018 Conjecture 3.5. No modulo-$p^3$ result, resolution of adjacent conjectures, or first-proof claim is made. Candidate reflects pending authoritative review of a claimed complete proof of this specific question.