A|Ω⟩ Lattice Vacuum Engine

1. Abstract

The A|Ω⟩ Research Program establishes a specialized Lattice Gauge Theory (LGT) framework designed to probe the non-perturbative structure of the SU(3) Yang-Mills vacuum.

Distinct from general-purpose lattice QCD packages (e.g., Chroma, Grid) which optimize for large-volume hadronic spectroscopy, this engine is architected specifically for Twisted Eguchi-Kawai (TEK) Reduction. It utilizes $Z_N$-twisted boundary conditions to simulate infinite-volume confinement physics on compact, topologically non-trivial manifolds ($T^4$). This approach allows for high-precision tomography of vacuum structure—specifically center vortices and topological defects—while suppressing finite-volume deconfining artifacts.

2. Methodology

The simulation kernel isolates the mechanism of color confinement via three geometric constraints:

• Twisted Boundary Conditions (TBC): Implementation of 't Hooft flux to preserve Center Symmetry ($Z_3$) in reduced volumes, preventing the breaking of confinement at small physical scales ($L \ll \Lambda_{QCD}^{-1}$).
• Toroidal Compactification: Simulation on $T^4$ hypertoroidal geometries to enforce specific topological sectors.
• Action Density Tomography: Real-time calculation of the local plaquette action $S(x)$ to map the formation of chromoelectric and chromomagnetic flux tubes.

3. Technical Specifications

Component	Specification	Description
Formalism	Pure SU(3) Yang-Mills	Wilson Gauge Action (Quenched)
Algorithm	Metropolis-Hastings	Local update algorithm with $O(N^2)$ scaling
Boundary	't Hooft Twist	Symmetric twist tensor $z_{\mu\nu} \in Z_3$
Architecture	Julia 1.10+	Just-In-Time (JIT) compiled for Apple Silicon (M-Series)
Precision	Float64 / ComplexF64	Double-precision arithmetic for unitary stability
Criticality	$\beta_c \approx 6.0$	Scaling window verification active