Abstract

We present an effective-theory framework showing that, on a coherent domain Ωcoh where a stable rank-2 internal sector exists as a smooth bundle, the local freedom to choose an orthonormal frame induces an SU(2) gauge structure in the Wilczek–Zee sense. The analysis is explicitly conditional: we do not derive the dynamical origin of the coherence field Φ nor of the rank-2 sector, but characterize the geometric and response-theoretic consequences once a smooth rank-2 projector PΦ(x) is given on Ωcoh. The induced connection obeys the standard inhomogeneous transformation law, yields the covariant derivative and Yang–Mills curvature on Ωcoh, and admits the projector identity Fμν = −iW†[∂μPΦ, ∂ν PΦ]W (with an explicit nonzero-curvature example). We formalize “coherence gating” as a strict restriction of the domain of definition—a domain-of-validity principle rather than a claim that physical decoherence is sharp—so gauge observables are defined on Ωcoh and are not assumed to extend across ∂Ωcoh, avoiding the gauge-noncovariance that would result from multiplying a connection by a nonconstant coherence weight. Beyond this kinematic structure, we provide a response-theoretic kernel-to-normalization interface in a quadratic two-leg benchmark: Schur-complement elimination of heavy locking modes yields a dimensionless projected response fraction ωeff (benchmark ωeff ≃ 0.296), and the matching-scale inverse coupling is fixed by g−2(Λ) = ωeff /τ0, where τ0 > 0 is the coherent-domain stiffness defined via auxiliary two-form (first-order) elimination. In isotropic embeddings τ0 = τsus/(κfT) with fT = 2/3; more generally fT is a computable trace ratio determined by the microscopic (possibly anisotropic) response kernel.

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Keywords: Axis Model, non-Abelian gauge theory, SU(2) gauge structure, emergent gauge fields, coherence gating, geometric phase, Wilczek–Zee connection, Yang–Mills curvature, effective coupling matching, projector formalism, effective field theory