Topological Constraint on the Electromagnetic Coupling Scale from a Discrete ℓ¹ Obstruction

This paper derives the inverse electromagnetic fine-structure constant (alphaₑm inverse, approximately 137) from a single topological seed — a sign contradiction on a minimal directed graph — with zero free parameters between the seed and the UV coupling scale. The derivation proceeds in six stages: (1) A discrete l1 coboundary obstruction is identified on a two-edge directedgraph, producing a non-zero first cohomology class. (2) Barycentric subdivision heals the obstruction, yielding a three-node lattice whose geometry is fixed by the coboundary condition. (3) A discrete Fourier transform on this lattice produces the unique transport amplitude a = sinc (pi/6) = 3/pi, proven to be the only value consistent with both the l1 norm constraint and a representation-theoretic uniqueness theorem. (4) The Born rule converts this amplitude to a transition probability P = 9/pi². (5) An SU (3) Casimir bridge maps the unitarity deficit (1 - P) to a GUT-scale inverse coupling alphaGUT inverse of approximately 25. 54. (6) Standard one-loop renormalization-group flow with SM particle content runs the coupling from the GUT scale down to the Z-boson mass, yielding alphaₑm inverse of approximately 137. 03, within 0. 02% of the experimental value. The key structural claim is that the sinc kernel amplitude is not fitted but forced: it is the unique function satisfying three independent constraints (coboundary consistency, Fourier-analytic smoothness, and l1 minimality) simultaneously. If any step in the chain fails, the entire derivation collapses — there are no tunable parameters to absorb discrepancies. The paper includes a falsification protocol: seven independent tests, any one of which would invalidate the result, and a computational verification suite.