Lorentzian Four-Sheet Geometry and Maxwell Propagation in the Theta Vacuum Framework

This work presents a reproducible Lorentzian and electromagnetic completion of the Theta vacuum framework based on a four-sheet permutation sector.The four-dimensional permutation representation of S4 decomposes exactly into a radial singlet and a three-dimensional standard representation. After explicitly introducing orientation, Lorentzian signature and a tetrad as completion assumptions, the construction yields a coherent 1+3 geometry, the six electromagnetic components (E1, E2, E3, B1, B2, B3), the electromagnetic Hodge operator and Maxwell equations on flat and assigned curved backgrounds.The accompanying analytical and numerical audits verify the Maxwell principal symbol, two transverse photon polarizations, the null light cone, the geometric-optics null-geodesic limit, second-order convergence of the lattice Maxwell symbol, local Lorentz frame invariance, a torsion-free metric-compatible spin connection, agreement between spin and Riemann curvature, the first Bianchi identity, and the exact flat-tetrad limit.The package includes the manuscript in PDF and DOCX formats, machine-readable audit reports, reproducibility scripts, Zenodo metadata and SHA-256 checksums.Scope limitation: this work does not claim that the four-sheet sector, Lorentzian signature, tetrad dynamics, Einstein equations, Newton’s constant or a massless spin-2 sector are derived from the frozen Theta Mother operator. It establishes a mathematically consistent geometric and Maxwell completion, not a completed theory of general relativity.