Abstract

We establish the quantum consistency, renormalization-group (RG) control, and domain of validity of the Axis Model as a predictive effective field theory (EFT) on the window EIR  < μ < min(ΛΦ, Λq), with ΛΦ  ∼ 105  GeV (loss of scalar coherence) and Λq ∼ 1016 GeV (morton dissolution). We implement a UV-consistent Abelian-sector extension: at μ = Λq the Abelian Axis basis terminates and matches to a pre-geometric SU (2) parent, removing the spurious Abelian Landau-pole artifact. Within this window the quartic remains positive, flows are perturbative, and dispersion/positivity bounds control leading higher-dimensional operators. We construct a BRST-invariant gauge fixing, verify anomaly cancellation via scalar-bundle triviality, and derive the one- and two-loop RGEs for the renormalizable subsector. We also exhibit the electroweak mass matrix as an emergent effect of a composite stiffness Zχ  multiplying the kinetic term of a collective SU (2) orientation coordinate χ, clarifying how veff 2  ≡ Zχv2 enters the standard mass matrix. Here, the other Standard-Model factors (SU (2)L, SU (3)C ) are treated as external inputs; consequently the low-energy electromagnetic coupling α(0) is obtained as a consistency relation between Axis U (1)Z running and the empirical weak input gL(μEW), rather than as a de novo prediction.

Keywords: Effective Field Theory, Axis Model,  Two-loop Renormalization, Threshold matching, Abelian Landau pole, Pre-geometric UV, U(1), RGEs, Vacuum Stability.