Parent-Channel Closure for Composite Complex Scalars: A Generic EFT Criterion and an Axis Model Application

This manuscript formulates a conditional parent-channel closure criterion for composite complex scalar effective fields. It separates parent-channel certification, minimal faithful representation of a compact charged scalar block, response matching, and downstream coefficient provenance. The final section applies the criterion to the Axis Model as a scalar-sector closure audit, identifying the conditions under which the suite-wide scalar field can be treated as one parent-selected complex scalar coherence coordinate.