The Theory of Phase-Spatial Invariance - A Theorem-Proof Reconstruction of Light-Speed Invariance, the Mass Shell, and Spacetime Duality

This paper develops the Theory of Phase-Spatial Invariance, a theorem-proof reconstruction of light-speed invariance, the photon energy relation, the mass shell, Lorentz observer symmetry, spacetime duality, and free wave equations from a single phase-spatial closure norm. The theory begins with the minimal closure grammar PSOC4(3) = U(1)_phi ⊕ SO(3)_J, joining one cyclic phase degree of freedom with three spatial-orientation degrees of freedom. From phase sensitivity, spatial-orientation sensitivity, rotational invariance, and nonzero null admissibility, the closure norm C(K) = (omega/c)^2 - |k|^2 is obtained as the minimal forced invariant. The null case C(K)=0 reconstructs light-speed invariance and E=pc; the non-null residue C(K)=(mc/hbar)^2 reconstructs E^2=p^2c^2+m^2c^4 and mass as inverse closure length. Observer transformations preserving the closure norm recover Lorentz symmetry, while invariant phase pairing reconstructs spacetime as the dual measurement geometry of phase-spatial closure. The paper preserves standard equations while reversing their explanatory order: spacetime is interpreted not as the primitive container of phase, but as the observer-geometry disclosed by phase-spatial invariance.