The Eight-Tick Projection Theorem A Parameter-Free Derivation of the Recognition Gap Weight

The inverse fine-structure constant in Recognition Science is predicted by α⁻¹ = 44π exp(−w₈ log φ/(44π)), where φ = (1 + √5)/2 and w₈ is the gap weight of the fundamental eight-tick recognition cycle. This paper derives w₈ from first principles. The input is the canonical pattern a_t = φᵗ on the cyclic group ℤ/8ℤ. We take its discrete Fourier transform, weight the nonzero modes by the spectrum of the one-step difference operator and by the reciprocal φ-ladder envelope, normalize by Parseval energy, and multiply by the cardinality 8 · 8 = 64 of the tick-vertex interface cell. The resulting projection weight is proved to be w₈ = (348 + 210√2 − (204 + 130√2)φ)/(246 + 145√2 − 102√5 − 65√10) = 7. Thus the integers in the gap weight are not fitted. They arise from the DFT-8 denominators, the sin²(kπ/8) difference spectrum, Parseval normalization, the 64-cell tick-vertex measure, and Fibonacci identities for powers of φ. Substitution gives 137.030 < α⁻¹ < 137.039, with the CODATA value 137.035999084(21) inside the interval.