The Two Noethers: Not Conflating the First Theorem (Conservation) with the Second Theorem (Identities and Constraints).
The corpus used Noether’s theorem as “conservation is the shadow of symmetry” (Paper 80). But Emmy Noether’s 1918 paper had TWO theorems, and the corpus used only the first. This paper places the SECOND theorem and separates the two as DISTINCT ROOTS. First theorem: a GLOBAL (rigid, finitely many parameters) continuous symmetry -> a CONSERVATION LAW (energy/momentum/angular momentum, measurable). Second theorem: a LOCAL (parametrized by arbitrary functions of spacetime, infinite-dimensional) gauge symmetry -> NOT a conservation law but IDENTITIES / dependencies among the equations of motion (Noether identities). Examples: EM d_mu d_nu F^{mu nu} = 0 identically; GR the contracted Bianchi identity nabla_mu G^{mu nu} = 0 identically — these hold OFF-SHELL and mean the field equations are NOT all independent (gauge freedom). Core: even for “the same Noether,” first (global->conservation) and second (local->identities/constraints) are different theorems = distinct roots. This answers a puzzle: why in gauge theory the “charge” becomes a BOUNDARY TERM (Paper 96) — a local symmetry’s Noether current is identically conserved in the bulk, and the true charge is a surface integral at infinity; also why the 10 equations of G_{mu nu} hide 4 identities (general covariance = 4 arbitrary functions), the general backbone of Paper 72 (constraints/unimodular). Connections: 80 (first thm, the distinct-root twin), 96 (gauge), 72 (constraints), 86 (refines the “symmetry” root into global/local sub-roots), 92 (Bianchi dF=0), 85 (Galois->Noether lineage, two phases). IMPORTANT (scope): a Layer-2 structural interpretation of established mathematical physics; no new mathematics and no new prediction. The Universe OS is a Layer-3 non-science lens.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Two Noethers [Paper 107]. Zenodo.
DOI: 10.5281/zenodo.21837205
ORCID: 0009-0006-8587-3450