The Discrete Dirac Identity on Hermitian Shells and Spectral Consequences for Hecke Operators

For every odd prime q, every non-zero mass shell in the space of 2x2 Hermitian matrices over the finite field Fq², and the shell-cone incidence operator d, we prove the identity d†d = (A - (q-1))(A + 1), where A is the null-adjacency operator on the shell. The proof is entrywise, splitting into four cases according to delta = det(X - Y). We then derive the spectral consequences: the Hecke operators T_delta restricted to the invariant eigenspaces of A satisfy explicit polynomial relations, and the operator d†d commutes with all T_delta. At q = 7, the restriction to the three-dimensional SL(2, F_7)-singlet space yields a mass operator H_mass = T_5/q with spectrum {-√2, 0, +√2}, a rank-1 projector P_sing = -T_2/(2q) onto the null mode, and the spectral morphism Y†Y = H_mass(3H_mass + 4I)/16 connecting the shell algebra to the Peirce-Yukawa framework of octonionic flavor structure.