A structural trace identity and certified spectra for the Richelot-Brandt graph

Abstract: Let $B_2(2)$ be the degree-2 Brandt operator on the principal genus of binary quaternion Hermitian lattices of discriminant $p$. Geometrically, $B_2(2)$ is the weighted adjacency operator of the Richelot $(2,2)$-isogeny graph on superspecial principally polarized abelian surfaces, and it commutes with an Atkin-Lehner involution $R(π)$. We prove a structural formula for the trace of $R(π)$ on the principal genus. For every prime $p \ge 7$, this trace is the sum of an explicit lift contribution, determined by the Atkin-Lehner eigenspaces of elliptic newforms of weights 2 and 4, and a signed defect of the weight-3 paramodular non-lift space. The proof compares term by term the closed principal- and non-principal-genus trace evaluations of Ibukiyama. The resulting closed formula for the signed defect yields the Fricke-sign bias $d(p) \ge 0$ for every prime. We next formulate an eigenvalue-sign refinement of Ibukiyama's principal-genus multiplicity conjectures. It predicts a factorization of $\mathrm{charpoly}\, B_2(2)$ into Eisenstein, Saito-Kurokawa, opposite-sign Yoshida, type-Va, and general-type blocks, and specifies the $R(π)$-sign on each block. In particular, it predicts that the two members of every type-Va pair are separated by opposite $R(π)$-eigenvalues. Finally, exact-arithmetic certificates verify this prediction for every prime $11 \le p \le 149$. The verification constructs the Richelot matrices, checks their weighted graph structure, matches the lift blocks with separately computed elliptic data, and certifies the signed isotypic multiplicities by projector traces. At $p = 19$ the first verified type-Va pair is separated by $R(π)$; at $p = 61$ the graph realizes the general-type factor $x + 7$.