The Scaling Cascade

The fine-structure constant is not produced by a single geometric level of the funneledspring recursion. It is produced by a scaling cascade: under the Cohesion UFT pressureaxiom, every recursion level generates sub-recursions at one-third the curvature scale,which generate sub-sub-recursions, and so on indefinitely. The top-level couplingC0 =3/2 (from the Hexpolar n = 6 geometry) decreases geometrically as the cascadedescends:Ck = C03k = 3/23k ,k =0,1,2,...The effective coupling of the full cascade is the weighted sum over all levels. For ageometric cascade with per-level depth probability 1 − p:Ceff = 3pC02 +p .This cascade formula reproduces the CODATA value 1/α = 137.036 when p = 0.981(i.e., 1 − p = 0.019 per-level attenuation). The cascade ratio r = 3 is purely geometric(r = κ6/κ2 = 3). The per-level attenuation 1 − p ≈ 0.019 is the one quantity thatrequires the physical scale of the recursion. The cascade is the geometric equivalentof the QED vacuum polarisation sum — each level corresponds to one loop order —described without virtual particles, Feynman diagrams, or α as an input to the structure.