What question did this study set out to answer?

The central aim is to make progress on three key problems related to primordial energy waves and fundamental constants.

July 6, 20260 citationsOpen Access

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

Key Points

The central aim is to make progress on three key problems related to primordial energy waves and fundamental constants.
Asymptotic analysis of the vortex ODE using Wronskian methods and a virial identity.
Analysis of the three-strand Hopf braid leading to an SU(3)-type simplex structure.
Partial examination of the metric correction to alpha_em through curvature at a specific parameter.
The vortex ODE's tail decay is confirmed to be algebraic rather than exponential, characterized by a specific decay rate.
A regular simplex in charge space is generated from the three-strand Hopf braid, with specific summation properties identified.
The metric correction factor is identified, yet full conclusions about the metric correction remain open.

Abstract

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 - kappa² u/rho² - u + 1/u³ = 0 is analysed viaWronskian methods and a virial identity. The field tail decaysalgebraically as delta (rho) ~ kappa²/ (4 rho²), 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 qᵢ = 1, sum qᵢ² = 1, sum₈<₉ qᵢ qⱼ = 0. The Casimir CF = 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 alphaₑm 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 kappa²/2) is confirmed numerically: no intrinsiccutoff reproduces DeltaV = 0. 7718 without an additional geometricinput, and PO-alpha-b remains open D. All results carry explicit epistemic classification A/B/D.

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