This is the definitive paper on the fine structure constant derivation. Version 4 presents two representations of α⁻¹: the prime spectrum form (exponents = first four primes, coefficients composed of primes) and the zeta-rung form (corrections indexed by odd Riemann zeta values ζ(3), ζ(5), ...). A bridge identity connects them. A computational stress test proves that φ-series can approximate any number to sub-ppm precision — demolishing precision as evidence — and shows that the derivation survives this challenge because the structure is forced, not searched. Coefficient rule Cₖ = 2^(k²) derived from directed coupling configurations at maximum entropy.