Symmetry and Invariance in Mathematical Physics From Transformation to Preserved Coherence

This paper is developed from an earlier prepaper on the ontology of mathematical physics. The source manuscript included a substantial thread on symmetry, invariance, Lorentz symmetry, Lorentz invariance, symmetry reduction, coherence preservation, and the relation between mathematical transformation and physical law. The present paper narrows that material into a focused bridge paper. Its purpose is to reinterpret symmetry and invariance through the closure-resonance framework without replacing their established mathematical and physical meanings. The guiding claim is that symmetry and invariance are two faces of closure. Symmetry names the admissible transformation grammar of a coherent domain. Invariance names what remains preserved through those transformations. Closure names the deeper coherence-stability that allows transformation and preservation to belong to one intelligible structure. This paper should be read as an ontological clarification, not as a replacement for group theory, gauge theory, relativity, Noether's theorem, or established mathematical physics.