On the Emergence of Noether Currents: Phase Identity and Descent from Equivalence Classes

We prove a foundational inversion of the standard Noetherian logic of physics. The primitive datum is not a chosen variational action together with an assumed continuous symmetry, but an equivalence class of representations whose identity in the rank-one phase sector is fixed by preservation of a principal circle-bundle class, equivalently by its first Chern class; the trivial case recovers the usual global phase sector. A variational representative is called admissible if it descends to this equivalence class, that is, if it is independent of choices of trivialization and well defined modulo bundle automorphisms. Using Atiyah’s algebroid formulation of connections together with the canonical affine bundle of connections and its tautological connection, we prove that admissibility forces the canonical circle phase action on any representative. Under standard local variational hypotheses, this forced phase action yields a conserved Noether current. The result positions Noether symmetry and current as structural consequences of representational well-posedness relative to the fixed identity class, rather than as primitive axioms.

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