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 Fₐℂ, 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 Tdelta restricted to the invariant eigenspaces of A satisfy explicit polynomial relations, and the operator d†d commutes with all Tdelta. At q = 7, the restriction to the three-dimensional SL (2, F₇) -singlet space yields a mass operator Hₘass = T₅/q with spectrum -√2, 0, +√2, a rank-1 projector Pₛing = -T₂/ (2q) onto the null mode, and the spectral morphism Y†Y = Hₘass (3Hₘass + 4I) /16 connecting the shell algebra to the Peirce-Yukawa framework of octonionic flavor structure.