Abstract

We present a coherence-gated mechanism showing that, on coherent domains Ωcoh, a stable rank-2 internal sector forces an SU(2) gauge structure in the precise Wilczek–Zee sense. Assume a smooth rank-2 projector PΦ(x) selects a subbundle of an underlying internal fiber on Ωcoh. The freedom to choose a local orthonormal frame of Im PΦ(x) induces a non-Abelian connection with the standard inhomogeneous transformation law, and hence a covariant derivative and Yang–Mills curvature on Ωcoh. The curvature is typically nonzero whenever the projector varies, and admits the compact projector form Fμν ∝ W †[∂μPΦ, ∂ν PΦ]W; an explicit F≠ 0 example is given. We formalize coherence gating by an idempotent domain projector ΠΦ that restricts gauge observables to Ωcoh without modifying the principal-bundle connection law. Beyond the geometric existence of the gauge structure, we provide a computable kernel-to-normalization interface in a quadratic two-leg benchmark: integrating out heavy locking modes via a Schur complement yields a dimensionless projector weight 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 by auxiliary two-form (BF-type) elimination. In the benchmark kernel, τ0 is related to an averaged heavy-mode compliance τsus by a fixed trace/transverse bookkeeping identity τ0 = τsus/(κfT), separating the geometric necessity of the gauge field from its microphysical stiffness input.

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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