Article 24: Geometric Origin of the Topological Coupling alpha_topo 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 alpha_topo [B]: The bare topological coupling 1/alpha_topo = 16*pi^2*phi^2/3 = 137.808 is derived from a three-step chain: - spinorial 4*pi phase return of the Hopf double-cover [A], - golden metric dilation phi^2 from R/r = phi [A], - and effective radius identification R_eff^3 = S_flux = 4*pi*phi^2 [B]. (II) GABRIEL TRUMPET HIERARCHY [B]: The polar escape geometry is identified with a generalised Gabriel Trumpet family xy = phi^n,with volume V_n = pi*phi²ⁿ 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 R_S 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 + u^2 is exactly isochronous: period T = pi/sqrt(2) independent of amplitude, with turning points satisfying u_min * u_max = 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 m_H ≈ 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 R_eff [A]: The energy functional E[R] = alpha_c * S_flux/R + sigma * 4*pi*R^2, with gradient term from alpha_c*(grad A)^2, has its minimum at R_eff^3 = S_flux if and only if the EM sphere surface tension satisfies sigma = alpha_c/kappa^2 exactly, connecting R_eff to the two fundamental PEW couplings (alpha_c and kappa^2 = 8*pi)with no free parameter [A]. The virial ratio E_grad:E_surf = 2:1 at equilibrium follows directly [A]. Deriving the R^2 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 Delta_V = 0.7718, - and relative fraction Delta_V/V_topo = 0.56%.The perfect magnetic dipole is identified as the limit Delta_V -> 0. All results carry explicit epistemic classification [A]/[B]/[D].