The Massless Photon and the Nonlinear Einstein Field Equation from Order-Two Taub–NUT Geometry

The order-two Taub–NUT geometry selected by the real–complex scalar system identifies the massless photon through the symmetry of the Higgs vacuum and determines the nonlinear Einstein field equation through its long-wavelength response to matter. Exactly one electroweak gauge direction leaves the full Higgs vacuum vector fixed, including its phase. Along this direction the Higgs mass form vanishes, while a simple transverse pole with positive residue identifies a propagating electromagnetic field. Its two helicities share one spacetime null cone. The Hopf geometry of their polarization amplitudes gives a geometric phase equal to minus half the oriented solid angle on the Poincaré sphere; an octant cycle therefore shifts a single-photon interference fringe by one eighth of a period. The same electromagnetic normalization fixes the electron’s Coulomb length and time scales: in one Hartree atomic time, light traverses 137.0359991916… Bohr radii, the value of the geometric inverse fine-structure constant. The photon cone determines the metric up to scale, and a specified electron phase gradient fixes its local scale. In the Lorentzian spin-two sector, the curvature response along all null directions and covariant stress conservation recover the Einstein tensor through two spacetime derivatives. Static exchange gives a Newtonian coupling of 6.674552 × 10⁻¹¹ m³ kg⁻¹ s⁻², 1.68 standard uncertainties above the 2022 CODATA value. The cosmological term drops out of every null contraction, but the reference-subtracted fiber volume fixes its coefficient at 1.14784 × 10⁻⁵² m⁻².