Article 25: Birth of the Electron in the PEW Framework: Resonance Condition, Golden Orbit, and Universal Scaling

The PEW (Primordial Energy Waves) framework identifies the electron as a stable topological vortex on a golden Hopf torus (R/r = phi, kappa^2 = 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 = Mc^2 yields the unique birth radius r_birth = hbar/(Mc), the reduced Compton wavelength of the resulting particle. For the electron: lambda_photon = lambda_C =h/(m_e*c) exactly. The PEW coherence length xi_p = hbar/(m_p*c) (postulated in Article 1) is now derived from this same condition. (II) GOLDEN ORBIT [A]:Three simultaneous exact identities, all from phi^2 = phi+1: V(phi) = phi⁻² + phi^2 = 3 (exact), orbit [1/phi, phi]: u_min * u_max = 1, Delta_E = 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 A_0 = phi at the torus surface (r=1, PEW units). The local vortex profile u_vortex(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*u^4 = 1, hence u_vortex(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 r_S = 2GM/c^2when M = m_Planck/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: n_born <=> lambda_photon = lambda_C(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 Delta_E = 1 does not shift the variational minimum of E[R], and the leakage integral from rho=1 to R_eff gives 2.733 >> 0.7718.Both POs remain fully open [D]. A new open problem PO-u1 is stated: derive the analytical form of u_vortex(1) = 0.44310... The new result u_vortex(kappa/2) = 5-1/4 provides a second algebraically motivated point on the profile and a new target for a perturbative ODE approach.