Article 22: Progress on Three Priority-1 Open Problems of the PEW Series: Asymptotic Analysis of the Vortex ODE, Three-Strand Hopf Braid, and Metric Correction to the Fine-Structure Constant

The Primordial Energy Waves (PEW) framework derives fundamentalphysical constants from a single Lagrangian built on a golden-ratioHopf torus. This article reports progress on three priority-1 openproblems identified in Article 21. (PO-iota) The asymptotic mode function of the PEW vortex ODEu'' + u'/rho - kappa^2 u/rho^2 - u + 1/u^3 = 0 is analysed viaWronskian methods and a virial identity. The field tail decaysalgebraically as delta(rho) ~ kappa^2/(4 rho^2), not exponentially— a previously corrected result confirmed analytically. Theparameter iota = 1 is established numerically [B]; analytic proofremains open [D]. (PO-1b) The three-strand Hopf braid on the golden torus is shownto generate an SU(3)-type simplex: the three strand cross-sectionsform a regular simplex in charge space with sum q_i = 1,sum q_i^2 = 1, sumi<j q_i q_j = 0. The Casimir C_F = 4/3 isidentified as the ratio of the squared circumradius to the squarededge length of this simplex [A]. A full derivation from the Hopffibration structure remains open [D]. (PO-alpha-b) The metric correction to alpha_em via Riemanncurvature at rho=1 is partially addressed. The correction factor(kappa-4)/e is identified as a curvature-induced leakage term [B].The logarithmic divergence of the vortex leakage integral(coefficient kappa^2/2) is confirmed numerically: no intrinsiccutoff reproduces Delta_V = 0.7718 without an additional geometricinput, and PO-alpha-b remains open [D]. All results carry explicit epistemic classification [A]/[B]/[D].