1/α_EM

Bare product (BA5)

At μ* = 15, branch q_+ = 1 − 2/μ* = 13/15, compute sin²(θ_p) on the three active primes {3, 5, 7}:

$$ \sin^2\theta_3 = 0.21916, \quad \sin^2\theta_5 = 0.19397, \quad \sin^2\theta_7 = 0.17261. $$

Their product gives the bare value:

$$ \alpha_{\rm bare} = \sin^2\theta_3 \cdot \sin^2\theta_5 \cdot \sin^2\theta_7 = \frac{1}{136.278}. $$

This is bridge axiom BA5 (derived theorem: T6 + Pontryagin on the arithmetic torus T³).

Binary-channel dressing (F(2))

The difference between 1/136.28 and observed 1/137.036 (~0.55%) comes from the binary channel p = 2. This prime carries the spin/parity dynamics (info / anti-info boundary); its echo polarization dresses the coupling at very short distance.

The leading correction is:

$$ F(2) = \frac{(\mu^* - 1)(\mu^* - 2)\,\Phi_6(\mu^*)}{\mu^{*4}} \cdot \cos^2\!\left(\frac{\arccos(((\mu^*-1)^2+1)/\mu^{*2})}{(p_1+1)^{p_1+1} - 1}\right) = 0.7582727826. $$

Closed form, 99.96% rational, derived in chapter 10 (R51, Binary Leakage Dressing).

Feedback spiral

The dressing F(2) re-perturbs the primary cascade, creating a feedback spiral that converges to a fixed point:

$$ \Delta^* = \frac{F(2)}{1 + \gamma_3 \cdot r \cdot \Pi}, \quad r = \alpha_1 \sum_{p \in \{3,5,7\}} \gamma_p^2. $$

Contraction $|\rho| \approx 8 \times 10^{-4}$ ensures convergence; relative truncation error is $O(\rho^2) \approx 6 \times 10^{-7}$.

Echo and 2-loop terms

Echo primes {11, 13} traverse the binary boundary: $\delta_{\rm echo} = \sin^2\theta_2 \cdot \beta_{\rm echo} \cdot \alpha^2$. 2-loop vacuum polarisation: $\delta_{2\ell} = (\alpha/\pi)^2 / N_c$.

Total

$$ \frac{1}{\alpha_{\rm EM}} = \frac{1}{\alpha_{\rm bare}} + F(2) + \Delta_{\rm spiral} + \delta_{\rm echo} + \delta_{2\ell} = 137.035\,999\,083. $$

Gap to CODATA: 0.004 ppb. Zero fitted parameters.