Fundamental Reform of the Circular Metric: A Geometric Derivation of Mass and the Resolution of the Proton Radius Puzzle

We propose a fundamental revision of the space-time metric by replacing the ideal Euclidean constant π eucl with an operational torsion constant, KGeo ≈ 3, 1434, derived from the PR-TAU 5.2.3 Protocol. We demonstrate that the vacuum is not an inert void but a rigid lattice characterized by an intrinsic rigidity constant ϵ ≈ 5, 76 × 10-4. Under this paradigm, we derive a first-principles equation for the proton mass (mp) as a function of the topological closure defect in the Planck lattice: mp = ℏ c 2 • ℓP • ϵ • (KGeo-π eucl) • Φ TAU (1) This geometric model yields a predicted mass of 938.272 MeV/c 2 , achieving a 10-7 convergence with CODATA values without the use of free parameters. Furthermore, we resolve the Proton Radius Puzzle by showing that the muonic-electronic discrepancy is a projection error of Euclidean modeling, predicting a unified radius of 0.8412 fm. We also propose the « Moulin Shift »-a systematic-0.057% frequency drift in high-finesse optical resonators-as a definitive experimental test of vacuum torsion. This framework provides a nonparticle alternative to Dark Matter and regularizes the Navier-Stokes equations through vacuum entropy dissipation. Furthermore, we demonstrate that the Fine Structure Constant α is not an arbitrary parameter, but the coupling ratio between planar wave-fronts and volumetric torsion nodes. Our geometric derivation yields 1/α ≈ 137.036, providing the first structural explanation for the strength of electromagnetic interactions based on vacuum rigidity ϵ.