S12: The Strong Coupling at the Planck Scale from Wave Projection Geometry

This paper derives the strong coupling at the Planck scale from the geometry of wave projections on a two-dimensional pre-geometric surface. The derivation uses three independently motivated ingredients: a stability bound from vacuum spectral energy, projection factors from the marginalisation of the third SU(3) gauge axis onto the 2D substrate, and a dimensional reduction from 2D canvas dynamics to 4D spacetime. These combine into a self-consistency condition whose unique non-trivial solution is g_3 = 5\pi/32, giving \alpha_s(M_P) = 25\pi/4096 \approx 0.019175. The value required to reproduce the measured \alpha_s(m_Z) = 0.1179 under Standard Model renormalisation group running is \alpha_s(M_P)_{\text{required}} \approx 0.019054. The prediction agrees to 0.63\%. No parameters are fitted. The ratios of the three gauge couplings follow from the same projection geometry: g_1^2 : g_2^2 : g_3^2 = 1 : 2/3 : 2/\pi. The ratio g_2^2/g_1^2 = 2/3 is tested against high-scale extrapolations and agrees within current uncertainties. Why this matters: The Standard Model contains three gauge couplings whose values are measured, not derived. A fundamental theory should explain why they take the values they do. This paper presents such a derivation for the strong coupling at the Planck scale, with the ratios of all three couplings following from the same geometric principles. The result depends on a single declared premise: that the fundamental substrate is two-dimensional. This premise is stated explicitly. All additional assumptions are listed. The prediction is falsifiable: if improved measurements or lattice calculations shift the required \alpha_s(M_P) outside the predicted range, the derivation is ruled out. The derivation is self-contained and does not depend on any broader theoretical framework beyond the single premise of a two-dimensional substrate. It is a testable prediction of the wave projection geometry. Keywords: strong coupling, gauge coupling unification, Planck scale, wave projection, dimensional reduction, SU(3), quantum chromodynamics, running coupling, canvas model