Exact Discretisation and Boundary Observables in Lorentzian Causal Diamonds

Abstract This paper studies the ternary Minkowski lattice L = -1, 0, +1⁴ 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 R⁴. Lorentzian boundary sum (Theorem 4. 2): The discrete boundary sum of a U (1) gauge field over ∂D evaluates to n^μₑff = (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 βc ≈ 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).