Article 21: Resolution of PO-α: Complete Derivation of the Fine-Structure Constant from PEW First Principles — Three-Step Chain, Dynamic Tympan, and Fibonacci Backbone

Article 3 derived the fine-structure constant to 0.013% (residual +0.018 on 1/α_em). Article 19 identified a metric correction reducing the residual to 0.000039% but left its origin as open problem PO-α. This article closes PO-α at level [B] by deriving the complete three-step chain from the PEW Lagrangian. Step 1 [A]: Ideal closed Hopf torus with R/r = φ gives α_topo = 3/(16π²φ²), yielding 1/α = 137.808.Step 2 [A]: Gate correction f_gate = sin(1/φ)/(32π) (open spiral, geodesic curvature 1/φ), yielding 1/α = 137.018.Step 3 [B]: Metric correction ε = (κ−5)²(κ−4)/e from the curvature of the PEW conformal metric at the gate location (ρ = 1). The D₅ pentagonal symmetry kills the first-order term; the surviving second-order correction, renormalized by the Z₄→Z₅ symmetry-transition factor (κ−4) and damped by the Mother Wave growth rate e, yields 1/α = 137.036053 (precision 0.000039%, improvement 334× over Article 3). Two exact algebraic identities anchor the derivation: κ²/π = 8 = F₆ (exact) — the PEW coupling encodes the sixth Fibonacci number. 13 − 8φ = φ⁻⁶ (Binet, exact) — after F₆ = 8 spiral revolutions, the phase deficit is φ⁻⁶. The gate is interpreted as a dynamic tympan: a spiralled elastic membrane at equilibrium radius r_tympan = 1/e. The electromagnetic surface deficit ΔS = (4πφ)² − 3/α_em = 2.315 divided by 2π gives the tympan width w = 0.3684 ≈ 1/e ≈ ξ_p (three-way identification to within 0.4%). The hierarchy charge ∝ φ⁻³, g−2 ∝ φ⁻⁶ is identified as a first-order / second-order vibration structure of the tympan. A compact exact formula is also established [A]: 1/α_em = (16π²φ²e − 2π)/(3e), which gives 137.03730 (first-order approximation, 0.0009%). PO-α is resolved at level [B]. The remaining open problem PO-α-b [D] is the formal Lagrangian derivation of the factor (κ−4)/e from the Riemann curvature of the PEW metric at ρ = 1.