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

This archive contains the manuscript and verification scripts associated with: "The Discrete Dirac Identity on Hermitian Shells and Spectral Consequences for Hecke Operators" For every odd prime power q and every non-zero determinant shell Oₘu in Herm₂ (Fₐℂ), the paper proves the exact shell-cone identity d^† d = (A - (q - 1) ) (A + 1), where d is the shell-cone incidence operator from the massive shell to the punctured null cone, and A is the null-difference adjacency operator on the shell. The proof is entrywise and splits according to the geometric value of delta = det (X-Y), with separate treatment of the diagonal, null-adjacent, non-degenerate, antipodal, and degenerate non-antipodal cases. It uses additive characters, quadratic Gauss sums, and elementary counting in Herm₂ (Fₐℂ). The construction is placed in the context of classical association schemes of Hermitian forms over finite fields, following the work of Wan, Stanton, Bannai-Ito, and Schmidt. The identity proved here is not a restatement of the standard rank-metric spectrum: it is a shell-level incidence identity involving a fixed determinant shell and the punctured null cone. This archive also contains: verifyq3. py: finite exact verification of all entrywise statements at q=3, including an exact character-sum check in Zomega. q7ₕermitiandiagnostic. sage: finite exact Sage verification of the q=7 shell-side spectral sector E₆ inside the 11-dimensional H-spherical Hecke algebra. The script verifies the construction of E₆ as a joint spectral sector and the exact restricted Hecke identities: chi₁₅/₇ (x) = x³ - 2x, Pₛing = -B2/14, B4 = 7 (2Pₛing - I), B6 = -B5. The shell-side q=7 statements are finite exact computations. The comparison with the Peirce-Yukawa framework is empirical and is not a consequence of the general identity.