{"slug":"noether-1918","verification":{"valid":false,"broken_at":0,"reason":"prev mismatch"},"count":5,"sources":[{"id":"s1","type":"primary","url":"https://doi.org/10.1007/978-0-387-87868-3_1","title":"Invariante Variationsprobleme (1918) / Invariant Variational Problems","quote":"Wenn das Integral I invariant ist unter einer [ kontinuierlichen ] Gruppe von Transformationen mit ρ Parametern, so ergeben sich ρ linear unabhängige Kombinationen der Lagrangeschen Ableitungen, die Divergenzen sind. / If the integral I is invariant under a continuous group of transformations with ρ parameters, then there arise ρ linearly independent combinations of the Lagrangian derivatives which are divergences.","summary":"The original 1918 paper by Emmy Noether proving that continuous symmetries of a variational principle yield conserved currents. Contains both the first theorem (global symmetries → conservation laws) and the second theorem (local/gauge symmetries → constraint identities).","claim_ids":["c1","c2","c4","c6","c7"],"quality_score":1},{"id":"s2","type":"primary","url":"https://doi.org/10.1007/978-0-387-87868-3_1","title":"The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century (Springer, 2011)","quote":"","summary":"English translation and historical commentary by Yvette Kosmann-Schwarzbach, providing the authoritative rendering of Noether's 1918 paper into modern mathematical English.","claim_ids":["c1","c2","c4"],"quality_score":0.95},{"id":"s3","type":"adjacent","url":"https://miscsubjects.com/articles/convergence-c03","title":"C03 — Symmetry ↔ Conservation","quote":"","summary":"The convergence pattern node that this source instantiates. The theoretical convergence connecting symmetry and conservation across domains.","claim_ids":["c1","c2"],"quality_score":0.9},{"id":"s4","type":"adjacent","url":"https://miscsubjects.com/articles/convergence-c14","title":"C14 — Duality / Complementarity","quote":"","summary":"The strongest convergence edge in the GRAIN catalogue, linked through C03. Connects Noether (mathematics, 1918), Bohr (physics, 1928), Heraclitus (philosophy, ~500 BCE), Taoism (religion, ~6th c. BCE), and Jung (psychology, 1951).","claim_ids":["c1"],"quality_score":0.85},{"id":"s5","type":"rival","url":"https://miscsubjects.com/articles/convergence-c24","title":"C24 — Fine-Tuning","quote":"","summary":"The rival frame that asks whether symmetries are necessary or contingent. String theory's landscape of 10^500 vacua suggests symmetries may be accidental, making Noether's theorem a local feature rather than a universal necessity.","claim_ids":["c6"],"quality_score":0.8}]}