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 c on the D₄ complex reveals that the leading-order approximation (c = 2. 7364) used in previous versions was an artifact of the analytic expansion. The corrected value 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 D₄ causal diamond — the 2-complex built from the twelve lightlike nearest-neighbour vectors of the ternary Minkowski lattice \-1, 0, +1\⁴ under = 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 = MMT with spectrum \0⁴, 6², 8³, 10², 28¹\ and the root count | (D₄) | = 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. V₈ carries the time-reversal-odd representation ₏ₕ₄₂, and V₀ carries the time-reversal-even ₕ₄₂. 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 ₛ = 2/13 follows uniquely from the SU (2) j = 1 Casimir distributed over N₄₅₅ + 1 = 13 entities. Mass renormalisation fixed point (Theorem 6. 4): The D₄ root count determines an exact fixed point ^* = 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 ₁₅ (x) = 1/r (c x). While the functional form is a structural prediction of the D₄ complex, the exact crossover 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 c. Spectral rigidity (Theorem 9. 2) confirms that no perturbation simultaneously preserves all four defining constraints, establishing the D₄ causal diamond as a unique object.