The Photon as a Harmonic 1-Form on T^3 : A Geometric Derivation of Masslessness and the Constancy of the Speed of Light

            We present a geometric model of the photon emerging from the harmonic 1-forms on the three-dimensional torus        T^3 = S^1 × S^1× S^1. The space of such forms has dimension b1(T^3) = 3, corresponding to three formal polarizations. Gauge invariance reduces these to the two observed transverse modes, with the longitudinal mode corresponding to an exact form removable by a gauge transformation. The photon’s strict zero mass follows directly from the Laplace equation ∆ω = 0 on T^3, placing it in the kernel of the Laplacian.
The constancy of the speed of light c is derived as a synchronization invariant between the compact internal torus T^3 and the spectral gap ∆E of the Dirac operator on a compact hyperbolic 3-manifold, which is fixed by Mostow rigidity [1]. The model requires no free parameters and provides a natural geometric origin for the photon’s properties.