Noether and Galois as Branches of Conservation of Coherence

Noether’s theorem and Galois theory are usually treated as belonging to different mathematical worlds. Noether’s theorem belongs to variational mechanics, dynamical systems, and field theory, where continuous symmetries generate conserved quantities. Galois theory belongs to algebra, field extensions, automorphism groups, and fixed structures, where algebraic symmetries reveal the solvability and invariant structure of equations. This paper proposes that these two regimes are not merely separate uses of symmetry, but distinct branches of a deeper closure principle. A transformation becomes a symmetry only when it preserves the governing coherence relation of the system. In Noetherian mechanics, the preserved coherence relation is variational or dynamical closure. In Galois theory, the preserved coherence relation is algebraic closure under admissible automorphism. Between these two regimes stands differential Galois theory, which studies automorphism structures of differential equation solution spaces and provides a mathematical bridge between algebraic symmetry and dynamical systems. The central claim is that Noetherian, differential Galois, and Galois symmetries are regime-specific expressions of conservation of coherence under admissible transformation. Closure Mathematical Physics therefore unifies them under the hierarchy: conservation of coherence -> closure-preserving transformation -> symmetry -> invariant or conserved structure. Noether conserves variational closure. Galois conserves algebraic closure. Differential Galois reveals their bridge through solution-space symmetry. Keywords: Noether’s theorem; Galois theory; differential Galois theory; Lie symmetry; conservation of coherence; closure mathematical physics; symmetry; invariance; algebraic closure; variational closure