The PEW (Primordial Energy Waves) framework identifies the electron as a stable topological vortex on a golden Hopf torus (R/r = phi, kappa² = 8pi). This article derives the complete set of conditions under which a free photon can condense into such a vortex. (I) UNIVERSAL BIRTH CONDITION A: The law of minimum action applied to a closed orbit gives S = h per loop, which for a photon of energy E = Mc² yields the unique birth radius rbirth = hbar/ (Mc), the reduced Compton wavelength of the resulting particle. For the electron: lambdaₚhoton = lambdaC =h/ (mₑ*c) exactly. The PEW coherence length xiₚ = hbar/ (mₚ*c) (postulated in Article 1) is now derived from this same condition. (II) GOLDEN ORBIT A: Three simultaneous exact identities, all from phi² = phi+1: V (phi) = phi^-2 + phi² = 3 (exact), orbit 1/phi, phi: uₘin * uₘax = 1, DeltaE = 1, amplitude = 1. The golden ratio selects this unique orbit from all isochrones. Physical consequence: once captured in the golden orbit, the electron never dephases — it is stable because its capture is isochronous. (III) FIELD AMPLITUDE AT CAPTURE B: Flux conservation gives the global amplitude A₀ = phi at the torus surface (r=1, PEW units). The local vortex profile uᵥortex (1) = 0. 4431 (numerical) is a distinct quantity; its analytical form is unknown. New result: at rho = kappa/2 = sqrt (2*pi), the local energy balance (centrifugal = 4 x mass) gives 5*u⁴ = 1, hence uᵥortex (kappa/2) = 5^-1/4 = 0. 66874, verified to 0. 0007% from the exact BVP solution. (IV) POLARISATION AND CHIRALITY B: Circular polarisation is required to wind onto a torus. Left-circular (helicity +hbar) -> electron (CCW winding, negative charge). Right-circular (helicity -hbar) -> positron (CW winding). Falsifiable prediction: in Breit-Wheeler pair creation (gamma + gamma -> e+ + e-), same-chirality photon pairs cannot produce a pair. (V) UNIVERSAL SCALING: PROTON AND MINIMAL BLACK HOLE A: The birth condition r = hbar/ (Mc) is universal. For a black hole of mass M, the birth radius equals the Schwarzschild radius rS = 2GM/c²when M = mPlanck/sqrt (2), the minimal black-hole mass. For M above this threshold, the born object is trapped inside its own Schwarzschild radius, consistent with black-hole physics. Harmonic level n and birth wavelength are strictly dual: nborn lambdaₚhoton = lambdaC (n). (VI) NEGATIVE RESULTS A: The birth analysis was explicitly tested as a potential new ingredient for open problems PO-Reff and PO-cone. Both tests gave negative results: the birth energy DeltaE = 1 does not shift the variational minimum of ER, and the leakage integral from rho=1 to Rₑff gives 2. 733 >> 0. 7718. Both POs remain fully open D. A new open problem PO-u1 is stated: derive the analytical form of uᵥortex (1) = 0. 44310. . . The new result uᵥortex (kappa/2) = 5^-1/4 provides a second algebraically motivated point on the profile and a new target for a perturbative ODE approach.
Michel ALdon (Sun,) studied this question.