On Symmetry, Projection, and Noetherian Closure of Physical Invariants

Noether’s first theorem relates conserved currents and charges to continuous global symmetries of an action, while Noether’s second theorem yields identities and constraints associated with local gauge symmetries. This paper addresses a distinct closure problem that remains once equivalence has been fixed: which global normalizations, completions, and scale selections are admissible so that dimensionless quantities and global consistency constraints arise as invariants forced by structure rather than as independently specifiable inputs. We formulate this as an extension of Noether’s symmetry–invariant correspondence from dynamical laws to admissible normalization. The framework assumes a universal symmetry substrate with a unique symmetry-native spectral generator, an additive gluing law with controlled associativity defect, a functorial projection to observables preserving positivity and composition, and an upstream completion lock that fixes boundary normalization prior to comparison. Under these hypotheses, the completed generating functional is unique up to gauge equivalence and an overall unit choice. Canonical projections—including response-type derivatives with respect to admissible holonomy/twist parameters and heat/Mellin/trace projections—then yield quantities whose dimensionless content is fixed by invariance under the admissible equivalence relations, rather than by sector-specific normalization conventions. As a corollary, any remaining unfixed dimensionful relevant deformations appear as diagnostic signatures of incomplete closure. The proof is given by reduction to structural lemmas together with an explicit audit checklist specifying how, and only how, the criterion can fail. The result is a closure criterion rather than a microscopic model: it clarifies when constants, spectra, and global baselines are fixed by invariance rather than supplied by convention. License note: Distributed under CC BY-NC-ND 4.0.