Overview

The Exact Discretisation and Boundary Observables in Lorentzian Causal Diamonds repository presents a comprehensive examination of the ternary Minkowski lattice represented as L = {-1, 0, +1}^4, utilizing the metric η = diag(-1, +1, +1, +1). This work establishes a sequence of precise geometric and algebraic findings related to its lightlike structure and the discrete causal diamond two-complex derived from it.

Key Findings

The study articulates five primary results:

1. Lightlike Enumeration (Lemma 2.3): The lattice comprises exactly 12 lightlike nearest-neighbour vectors, which partition into two null sheets, each consisting of 6 mutually spacelike channels.
2. D4 Root Identification (Proposition 2.5): These 12 vectors correspond to the mixed roots of the D4 root system, leading to a Minkowski partition of 24 = 12 + 12.
3. Causal Diamond Geometry (Theorem 3.2): The boundary ∂D adheres to five independent geometric conditions and spans R^4, showcasing the geometric richness of the structure.
4. Lorentzian Boundary Sum (Theorem 4.2): The discrete boundary sum of a U(1) gauge field over ∂D results in an effective vector n^μ_eff = (12, 0, 0, 0), indicating a purely temporal configuration.
5. Plaquette Laplacian Spectrum (Propositions 5.4–5.5): The 21 order-4 plaquettes of the diamond yield a Laplacian with an exact integer spectrum of {0^(4), 6^(2), 8^(3), 10^(2), 28^(1)} along with a 4-dimensional flat-connection null space.

A leading-order U(1) lattice BF partition function is constructed, demonstrating a finite-system crossover at approximately β_c ≈ 2.7364.

Repository Structure

The repository is systematically organized as follows:

• /paper: Contains LaTeX source files and a PDF preprint of the manuscript.
• /script: Dedicated to the verification script, which confirms all numerical claims made in the paper.

Verification Script

The numerical claims are validated through exhaustive enumeration in the script verification_ED_BO_CD_paper.py, which derives every structure from the ground up without hardcoded matrices or enumerations. The output confirms each verification with a [PASS] or [FAIL] status.

Verification Checks

The script includes seven parts aligned with the sections of the paper:

Part	Checks
1 — Lattice Classification	81 lattice points; 12 lightlike, 66 spacelike, 2 timelike
2 — D4 Root System	`
3 — Causal Diamond Conditions	Verification of all five conditions from Theorem 3.2
4 — Boundary Sum	Lorentzian sum resulting in (12,0,0,0); the Riemannian sum equates to zero
5 — Plaquettes & Laplacian	21 plaquettes, yielding an exact integer spectrum and flat-connection space
5b–5d — CW-complex & Extended Complex	Examination of obstruction in Remark 5.7 and extended K_{6,6} Betti numbers
6 — BF Theory	Spectra of K_bdy and K_total; boundary coupling 2/13
7 — Character Expansion	60 compatible plaquette pairs; crossover at β_c ≈ 2.7364, C(β_c) ≈ 12.69

Key Numerical Results

The work presents pivotal numerical results detailed in the following table:

Quantity	Value
Lightlike Vectors	12
Order-4 Plaquettes	21
Plaquette Laplacian Spectrum	{0^4, 6^2, 8^3, 10^2, 28^1}
Flat-Connection Dimension	4
Effective Boundary Vector	(12, 0, 0, 0)
Boundary Coupling	β/13
BF Crossover	β_c ≈ 2.7364

Related Research

This repository includes references to related papers that further explore the topics introduced:

• Algebraic Structure of the D4 Causal Diamond — Examines the geometric framework alongside symmetry group analytics, discussing mass renormalisation and entanglement entropy. (Read more)
• A Lorentzian CSS Duality in Causal Diamond Quantum Error-Correcting Codes — A comprehensive analysis of the four-code CSS family, including No-Go theorems and thresholds at the circuit level. (Read more)

This project endeavors to contribute significantly to the understanding of causal diamonds and their algebraic structures within the realm of theoretical physics.