C₂ from Axiom Zero: A Complete Independent Derivation of the Two-Loop Anomalous Magnetic Moment

The two-loop g−2 coefficient C₂ = −0.328478966 is derived directly from foam topology without QED as intermediary. All four terms of the Petermann-Sommerfield formula are derived: 197/144 from wavefunction renormalization Z₂ plus face graph counting; π²/12 from two D-mode loop phases normalised by N_gauge=12; −π²ln2/2 from the E_g threshold √(C_A+1)=2; and 3ζ(3)/4 from T₂g winding harmonics Σ1/n³=ζ(3) weighted by C_A/λ_Eg=3/4. Apéry's constant enters not from a Feynman integral but from π₁(T₂g)=ℤ. Agreement: 3×10⁻¹⁶. Zero free parameters.