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/αₑm). 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 αₜopo = 3/ (16π²φ²), yielding 1/α = 137. 808. Step 2 A: Gate correction fgate = 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 rₜympan = 1/e. The electromagnetic surface deficit ΔS = (4πφ) ² − 3/αₑm = 2. 315 divided by 2π gives the tympan width w = 0. 3684 ≈ 1/e ≈ ξₚ (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/αₑm = (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.