Algebraic Structure of the D4 Causal Diamond - Spectrum, Symmetry, Codes, and Gravitational Phenomenology from a Single Lorentzian Lattice

Notice: Version 1.0.4 Update (Corrigendum) This version includes a significant correction to Section 8. Monte Carlo simulation of the crossover β_c on the D_4 complex reveals that the leading-order approximation (β_c = 2.7364) used in previous versions was an artifact of the analytic expansion. The corrected value β_c ≈ 0.67 results in a deep-MOND prefactor of ≈ 3.0, which is incompatible with empirical observations. Section 8 has been reframed as a negative result. All other sections (Symmetry, CSS codes, Boundary coupling, Mass fixed point, and Rigidity) remain entirely unaffected. Abstract This paper synthesises and extends the geometric foundations of the D_4 causal diamond — the 2-complex built from the twelve lightlike nearest-neighbour vectors of the ternary Minkowski lattice \{-1, 0, +1\}^4 under η = diag(-1, +1, +1, +1) — into five mutually reinforcing algebraic results. Every quantity derived in the paper follows from a single combinatorial input: the plaquette Laplacian K = MM^T with spectrum \{0^4, 6^2, 8^3, 10^2, 28^1\} and the root count |Φ(D_4)| = 24. No free parameter is introduced at any stage. The five main results are: Symmetry group (Theorem 3.2): The stabiliser of the twelve links is a group G of order 96. V_8 carries the time-reversal-odd representation ρpvec, and V_0 carries the time-reversal-even ρvec. CSS quantum error-correcting codes (Theorem 4.3): The GF(2) rank gap generates a complete family of four CSS codes: [[12, 1, (4,3)]], [[12, 4, (4,2)]], and their Wick-rotated duals. Boundary coupling derivation (Theorem 5.3): The boundary eigenvalue λ_s = 2/13 follows uniquely from the SU(2) j = 1 Casimir distributed over Neff + 1 = 13 entities. Mass renormalisation fixed point (Theorem 6.4): The D_4 root count determines an exact fixed point γ^* = 24/13 at which the decoherence mass and geometric inertial mass coincide. MOND Functional Form (Section 8 - Corrected): The BF model predicts an interpolation function of the form νBF(x) = 1/r(β_c √x). While the functional form is a structural prediction of the D_4 complex, the exact crossover β_c ≈ 0.67 rules out this specific mechanism as the origin of the empirical MOND prefactor. The Tully-Fisher slope = 1/4 remains algebraically exact and independent of β_c. Spectral rigidity (Theorem 9.2) confirms that no perturbation simultaneously preserves all four defining constraints, establishing the D_4 causal diamond as a unique object.