A Proof of a Conjecture of Zhi-Wei Sun on a Truncated Legendre-Symbol Determinant

We prove Conjecture~3.4 of Zhi-Wei Sun by evaluating, for every prime $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, a truncated Legendre-symbol determinant as $\lfloor (p-2)\mathrm{/}3{\rfloor }^{2}x$. The argument reduces the determinant to inverse data for Chapman’s full Legendre-symbol matrix and evaluates those data using Vsemirnov’s factorization and a Schur-Pfaffian resolvent identity. OpenAI’s ChatGPT produced the proof, which the authors independently checked and confirmed.