Article 24: Geometric Origin of the Topological Coupling alphaₜopo and the Gabriel Trumpet Hierarchy of Polar Cones in PEW

The Primordial Energy Waves (PEW) framework builds on a golden-ratio Hopf torus (R/r = phi) to derive fundamental physical constants. This article addresses three open problems and reports four results. (I) GEOMETRIC DERIVATION OF alphaₜopo B: The bare topological coupling 1/alphaₜopo = 16*pi²*phi²/3 = 137. 808 is derived from a three-step chain: - spinorial 4*pi phase return of the Hopf double-cover A, - golden metric dilation phi² from R/r = phi A, - and effective radius identification Rₑff³ = Sflux = 4*pi*phi² B. (II) GABRIEL TRUMPET HIERARCHY B: The polar escape geometry is identified with a generalised Gabriel Trumpet family xy = phiⁿ, with volume Vₙ = pi*phi^2n and infinite surface area — the exact topological signature of an electromagnetic coupling. The classical Gabriel solid (n=0, volume pi) is recovered exactly. Sgr A*'s Schwarzschild radius lands at harmonic level n = 21. 004 (stable integer, Delta n = 0. 004) B; the Sun's RS at n = 17. 906, inside the topological hole n = 15-17 of Article 10, confirming the Sun cannot be a black hole B. (III) ISOCHRONOUS BREATHING OSCILLATOR A: The PEW vacuum potential V (u) = u^-2 + u² is exactly isochronous: period T = pi/sqrt (2) independent of amplitude, with turning points satisfying uₘin * uₘax = 1 exactly. This is a known classical result (isotropic 2D harmonic oscillator reduction) applied here for the first time to PEW. A falsifiable consequence: the radial boson resonance at mH ≈ 2656 MeV should show no amplitude-dependent broadening B. The 0D oscillator does not validate a torus-to-sphere dynamical trajectory; PO-electron-dynamics remains open D. (IV) VARIATIONAL DERIVATION OF Rₑff A: The energy functional ER = alphac * Sflux/R + sigma * 4*pi*R², with gradient term from alphac* (grad A) ², has its minimum at Rₑff³ = Sflux if and only if the EM sphere surface tension satisfies sigma = alphac/kappa² exactly, connecting Rₑff to the two fundamental PEW couplings (alphac and kappa² = 8*pi) with no free parameter A. The virial ratio Egrad: Eₛurf = 2: 1 at equilibrium follows directly A. Deriving the R² surface term from the phase sector of the Lagrangian remains open D. Three distinct descriptions of the polar cone deficit are clarified: - angular deficit epsilon = 0. 0704, - absolute volume DeltaV = 0. 7718, - and relative fraction DeltaV/Vₜopo = 0. 56%. The perfect magnetic dipole is identified as the limit DeltaV -> 0. All results carry explicit epistemic classification A/B/D.