Theorem 171: The Koide Scale Problem — Why RG Running Cannot Bridge the Gap to Singh's Electroweak Value, and What This Reveals (Photonic Universe Hypothesis)

Tejinder P. Singh’s electroweak-scale Koide parameter K_th = 0.66916 departs from the PUH low-energy value Q_PDG = 0.66666051 by Δ +0.002499. We test whether renormalization group running can bridge this gap. Three mechanisms are tested numerically: dielectric scaling of the Z3 breaking parameter ε; standard QED logarithmic mass running; and geometric differential running with anomalous dimensions fixed by E8 lattice weights. All three fail — producing |ΔQ| less than 10⁻⁴, three orders of magnitude below the required shift — and standard QED running moves Q in the wrong direction. The conclusion is that Singh’s K_th is not a renormalization scale effect on pole masses. It is an algebraic quantity from Jordan algebra eigenvalues before mass identification. Theorem 171 establishes: (1) PUH’s Z3 breaking ε = -9.62×10⁻⁶ and Singh’s triality breaking are different geometric objects at different algebraic levels; (2) the correct bridge is a geometric map from Jordan eigenvalues to E8 lattice mode frequencies, not RG running; (3) the derivation of ΔQ = 0.002499 from E8 geometry in closed form is identified as the key open calculation connecting PUH to Singh’s framework.