Two determinant evaluations in Sun's conjectures involving Legendre symbols

We prove two determinant evaluations attached to Sun’s conjectures on matrices of Legendre symbols. The first one resolves the $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ part of Conjecture 4.8(i) by reducing the determinant with four indeterminates to a four-entry inverse package for the adjacent minor $[\chi (j-k+1){]}_{0\le j,k<(p-1)\mathrm{/}2}$. The core evaluation is $$\det H=\text{leg}2p(b_p^{\mathrm{\prime }}-a_p^{\mathrm{\prime }}),U^T{H}^{-1}U=\left(\begin{array}{cc}\text{leg}2p{\displaystyle \frac{p{b}_p^{\mathrm{\prime }}-a_p^{\mathrm{\prime }}}{b_p^{\mathrm{\prime }}-a_p^{\mathrm{\prime }}}}& 1\\ {\displaystyle \frac{b_p^{\mathrm{\prime }}-a_p^{\mathrm{\prime }}-1}{b_p^{\mathrm{\prime }}-a_p^{\mathrm{\prime }}}}& 1\end{array}\right),$$ where $U=(\mathbf{1},\eta )$ and ${\eta }_j=\chi (j)$. The proof uses Vsemirnov’s factorisation of Chapman’s matrix and an adjacent cofactor calculation. The second result gives a uniform exact congruence modulo $p$ for the determinant underlying Sun’s Conjecture 4.10(i), valid for any ordered half-system modulo sign and all $u,v\in {\mathbb{F}}_p$. Its standard specialization recovers the asserted square class. The square-class assertion itself also follows from Sun’s earlier evaluation of $T(d,p)$; the contribution here is an exact and half-system refinement.