Exact Discretisation and Boundary Observables in Lorentzian Causal Diamonds

This paper studies the ternary Minkowski lattice L = {-1, 0, +1}^4 under the metric η = diag(-1, +1, +1, +1) and establishes a sequence of exact geometric and algebraic results about its lightlike structure and the discrete causal diamond two-complex built from it. Five main results are established: Lightlike enumeration (Lemma 2.3): The lattice contains exactly 12 lightlike nearest-neighbour vectors, partitioning into two null sheets of 6 mutually spacelike channels each. D4 root identification (Proposition 2.5): These 12 vectors are precisely the mixed roots of the D4 root system, giving the exact Minkowski partition 24 = 12 + 12. Causal diamond geometry (Theorem 3.2): The boundary ∂D satisfies five independent geometric conditions and spans R^4. Lorentzian boundary sum (Theorem 4.2): The discrete boundary sum of a U(1) gauge field over ∂D evaluates to n^μ_eff = (12, 0, 0, 0) — purely temporal. Plaquette Laplacian spectrum (Propositions 5.4–5.5): The 21 order-4 plaquettes of D yield a Laplacian with exact integer spectrum {0^(4), 6^(2), 8^(3), 10^(2), 28^(1)} and a 4-dimensional flat-connection null space. A leading-order U(1) lattice BF partition function is also constructed, exhibiting a finite-system crossover at β_c ≈ 2.7364. Repository Contents Code: Numerical verification script Paper: Full PDF pre-print and original LaTeX source files. License Information Please note the dual-licensing structure of this repository: Software/Source Code: Licensed under the Apache License 2.0. PDF Document & LaTeX Source: Licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0).