Strong Coupling Constant α_s(M_Z)

τ-Formula

α_s(M_Z) = W₃(4) NLO readout = 0.1183

Derivation

The three electroweak observables at the $Z$-pole are derived in the $τ$ framework as readouts of the A-sector and B-sector couplings:

Observable & $τ$-value & PDG 2024 & ppm $^2θ_W$ & $0.231\,19$ & $0.231\,21(4)$ & $-0.65$ $M_W$ & $80.3696$ GeV & $80.3692(13)$ GeV & $-0.42$ $α_s(M_Z)$ & $0.11794$ & $0.11789(10)$ & $+43$

The Waring number $W_3(4) = 5$ plays the role in the $τ$ framework that the loop order plays in orthodox perturbation theory. Specifically:

• The tree-level Weinberg angle is $^2θ_W^(0) = 1/4$ (the cross-coupling ratio $κ(A;1)/κ(B;2)$ at zeroth order).
• The NLO correction involves $ι_τ^W_3(4) = ι_τ^5$, which shifts $^2θ_W$ from $1/4 = 0.2500$ to $0.2312$.
• The same $ι_τ^5$ correction enters $M_W$ and $α_s$, producing the correlated triple agreement.

Source

This prediction is derived in the Numerical Physics Ledger (Chapter 60 — mass-spectrum), Books IV–V of Panta Rhei.