A Shell-Cone Dirac Identity for 2x2 Hermitian Matrices over Finite Fields

The association scheme on n x n Hermitian matrices over Fₐℂ, with associate classes indexed by the rank of the difference, was constructed by Wan in 1965 and its parameters for n=2 were computed there. Within this classical setting, we introduce a shell-cone incidence operator d: l² (Oₘu) -> l² (C*) between a fixed determinantal level set Oₘu in Herm₂ (Fₐℂ) and the punctured null cone, and we prove the operator identity d* d = (A - (q-1) ) (A + 1) = A² - (q-2) A - (q-1) I, where A is the null-adjacency operator on the shell. The proof is entrywise, splitting according to the value of delta = det (X-Y), with a separate treatment of the antipodal subcase when delta = 4*mu, and uses standard quadratic Gauss sums together with the classical cardinalities of unitary geometry. The identity expresses the squared shell-cone operator as a quadratic polynomial in the null-adjacency operator A, with integer coefficients depending only on q. As a corollary, d* d admits an explicit decomposition in the algebra generated by the Hecke operators Tdelta and the antipodal operator on the shell; in particular it commutes with A. We close with a short computer-assisted example at q=7 exhibiting a three-dimensional invariant subspace on which T₅/q has spectrum -sqrt (2), 0, +sqrt (2) and T₆ = -T₅. A brute-force verification of all entrywise statements at q=3 is recorded in an appendix. --- Version 3. 0. 0 (May 2026): Major revision in preparation for journal submission. - Five-case structure (vs. four-case in v1): the degenerate value delta = 4*mu is now explicitly split into antipodal (Y = -X) and non-antipodal subcases, with separate elementary geometric and character-sum arguments. - Adjugate Lemma: proof rewritten via the 2x2 identity adj (U) = tr (U) *I - U, replacing an earlier chain that contained an incorrect intermediate step. - Counting formula (Proposition 8): scope hypothesis Y != +/- X made explicit; treatment of the R = 0 contribution clarified. - Hecke decomposition: explicit formula d* d = q (q-1) *I + sum cdelta * Tdelta - q*S, with antipodal operator S introduced as a new generator. d* d does not lie in spanTdelta alone. - Adriaensen-De Boeck (2024) cited and positioned: their projective setting on anisotropic points of polar spaces is distinguished from the present affine determinantal-level-set setting. - Brute-force verification at q = 3 added as Appendix. - Conclusion reformulated in terms of algebraic rank (quadratic-rank window) rather than matrix size, distinguishing variations within 4-dimensional quadratic models (M₂, Sym₂) from genuine higher-rank generalization (Hermₙ, n >= 3). - Octonionic Peirce-Yukawa connection externalized to a separate companion note in preparation; this paper is now self-contained as a finite-field result. - Title revised to "A Shell-Cone Dirac Identity for 2x2 Hermitian Matrices over Finite Fields" for journal submission focus. Submitted to Finite Fields and Their Applications (May 2026).