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 $p\equiv1\pmod4$ part of Conjecture 4.8(i) by reducing the determinant with four indeterminates to a four-entry inverse package for the adjacent minor $[\chi(j-k+1)]_{0\le j,k<(p-1)/2}$. The core evaluation is $$\det H=\leg{2}{p}(b'_p-a'_p),\qquad U^TH^{-1}U= \begin{pmatrix} \leg{2}{p}\dfrac{pb'_p-a'_p}{b'_p-a'_p}&1\\[2mm] \dfrac{b'_p-a'_p-1}{b'_p-a'_p}&1 \end{pmatrix},$$ where $U=(\mathbf1,\eta)$ 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.