Coding Theory of the Priority-Axis Z-Algebra: Nilpotent Subspace Construction and Intrinsic Protocol Sections

Within the axiomatic framework of the priority-axis Z-algebra (a 16-dimensional real semisimple associative algebra, isomorphic to M_2(C) + M_2(C)), an intrinsic coding theory based on Peirce nilpotent subspaces is systematically established, with optimality, applicability boundaries and performance limits calibrated by contrast models and exhaustive enumeration. Main results: (1) for any idempotent pi, the Peirce corner Z(pi) = pi Z (1-pi) is a linear submanifold of the null surface N = {X : X^2 = 0}; the standard rank-1 construction attains the maximal real dimension 4 of linear nilpotent subspaces, and all maximal ones lie in a single automorphism orbit; (2) the standard code space lies exactly inside the active subspace of the derivation D_tau = [tau, .], realizing a nilpotent-geometry/spectral-decomposition coupling for which the non-uniform temporal coupling is necessary; (3) the construction is valid for every odd prime: the quadratic extension survives for p = 3 (mod 4), while for p = 1 (mod 4) the algebra splits into four copies of M_2(F_p) with nilpotency, weight distribution and tight-frame property preserved, yielding a [16,4,2]_p code with exact weight distribution A₂ₖ = C(4,k)(p-1)^k; (4) the kinetic-density discriminator is corrected: [tau, X]^2 = -4X^2 on active elements, hence K(X) = 0 iff tr(X^2) = 0; (5) frozen-coordinate errors are 100% detectable, single-coordinate errors are detected with average rate 1 - (2p^4)⁻¹, and exact detection rates from exhaustive enumeration at p = 3 are tabulated; (6) at p = 3 the null surface admits an explicit block characterization X = diag(A,B) with A, B in M_2(F_9) (X^2 = 0 iff both blocks are traceless with zero determinant), an exact count |N| = 3^8 and three identically vanishing coordinates, and an explicitly constructed 81-codeword nonlinear intrinsically error-correcting code (d_min = 4) is given, advancing intrinsic coding from detection to correction as enumeration-level existence evidence. All algebraic conclusions are machine-verified item by item (Appendix C: 39 basic checks plus 11 extended check groups; Appendix F: 25 null-surface nonlinear-code checks; all passed).