Abstract

Complex scalar fields can enter effective field theories as elementary fields, order parameters, auxiliary variables, condensate coordinates, or composite response coordinates. This paper formulates a parent-channel closure criterion for the case in which a one-complex-scalar effective field is claimed to arise from a fixed microscopic scalar coherence channel. The criterion separates four layers: a parent-channel certificate, the minimal faithful representation of a compact charged scalar block, the parent response data fixing the effective coefficients, and the downstream matching maps using those coefficients. The representation-theoretic step is standard: a connected compact U(1) has no nontrivial one-real-dimensional charged representation; its smallest faithful real charged block is two-dimensional, equivalently one complex scalar. The substantive condition is provenance. The parent certificate must select a single compact scalar channel with positive-norm local response, stabilizing higher response data, an effective cutoff, and no additional light charged scalar block below that cutoff. Downstream scalar-sector coefficients must then factor through the same parent response and matching prescription. A spectral-compression theorem gives a stronger sufficient condition: if a self-adjoint parent response contains exactly one unstable compact charged Lorentz-scalar two-plane with positive quartic stabilization, while all other charged scalar blocks are heavy or stable below the cutoff, the one-complex-scalar effective coordinate is generated by that single spectral block. The final section applies the criterion to the Axis Model as a conditional EFT closure audit. Using the cited Axis provenance ledger, the retained parent branch satisfies the one-field scalar audit within the stated parent-response class.

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Keywords: complex scalar field; effective field theory; order parameter; compact U(1); composite scalar; scalar coherence; Axis Model; parent-channel closure; provenance matching