{"slug":"oip-node-c03-symmetry-conservation","title":"Node C03: Symmetry ↔ Conservation","body":"# Node C03: Symmetry ↔ Conservation\n\nC03 — Symmetry ↔ Conservation\n{\n  \"id\": \"C03\",\n  \"claim\": \"Every continuous symmetry of a physical system's action corresponds to a conserved quantity; symmetries are the source of conservation laws.\",\n  \"domain\": [\"classical mechanics\", \"quantum field theory\", \"particle physics\", \"crystallography\", \"cosmology\", \"aesthetics\"],\n  \"pattern\": [\"symmetry\", \"conservation\", \"Noether_correspondence\", \"Lie_groups\"],\n  \"mechanism\": \"Noether's theorem: if the action S is invariant under a continuous transformation parameterized by ε, then the Noether current j^μ satisfies ∂_μ j^μ = 0, yielding a conserved charge Q = ∫ j^0 d³x. The mechanism is pure mathematics applied to physical Lagrangians.\",\n  \"scale\": \"quantum → cosmic\",\n  \"claim_tier\": \"T0\",\n  \"sources\": [\n    \"Noether, E. (1918). 'Invariante Variationsprobleme.' Nachr. v. d. Ges. d. Wiss. zu Goettingen, 235-257.\",\n    \"Weyl, H. (1928). Gruppentheorie und Quantenmechanik.\",\n    \"Wigner, E. (1939). 'On Unitary Representations of the Inhomogeneous Lorentz Group.' Ann. Math., 40(1), 149-204.\"\n  ],\n  \"dual\": \"Symmetry-breaking (C04) — the structure that voids symmetry produces the phenomenological world.\",\n  \"falsifier\": \"For the theorem: proof error (none found in 107 years). For the mapping to physical reality: a conserved quantity in nature with no underlying symmetry of the action; or a symmetry with no corresponding conservation law.\",\n  \"rival_frame\": \"Noether's theorem is a mathematical identity, not a physical claim. It says nothing about WHY nature has symmetries — it only tells us that IF a symmetry exists, a conservation law follows. The symmetries themselves remain unexplained.\",\n  \"independence_check\": \"MATHEMATICAL PROOF — universally applied, not independently derived. However: the applications span classical mechanics (Noether), quantum theory (Weyl), particle physics (Wigner's classification), and cosmology — each domain found the theorem independently useful without borrowing from another domain's application.\",\n  \"pattern_type\": \"mathematical\",\n  \"maps_to_axiom\": [\"A1\", \"A3\"]\n}\n\n---\n\n## Corpus map\n- Same node, other planes: [Encyclopedia C03](/a/convergence-encyclopedia-c03) · [Inventory invariant](/a/oip-invariant-3-33-symmetry-conservation)\n- Edges touching C03: [convergence edge 3](/a/oip-convergence-edge-3) · [disconfirming edge 4](/a/oip-disconfirming-edge-4)\n- Catalogue hub: [Public Article](/a/oip-convergence-public-article) · [Schema](/a/oip-convergence-schema)","register":"oip_protocol","tags":["philosophy","oip","convergence-catalogue","node","systems-theory"],"category":null,"style":{},"claims":[{"id":"c1","text":"Every continuous symmetry of a physical system's action corresponds to a conserved quantity; symmetries are the source of conservation laws.","section":"# Node C03: Symmetry ↔ Conservation","tier":"mechanistic","source_ids":[],"source_status":"unsourced","why_material":"Core assertion linking symmetry to conservation via Noether correspondence across physics domains."},{"id":"c2","text":"If the action S is invariant under a continuous transformation parameterized by ε, then the Noether current j^μ satisfies ∂_μ j^μ = 0, yielding a conserved charge Q = ∫ j^0 d³x.","section":"# Node C03: Symmetry ↔ Conservation","tier":"mechanistic","source_ids":[],"source_status":"unsourced","why_material":"Formal mathematical mechanism of Noether's theorem applied to physical Lagrangians."},{"id":"c3","text":"Noether's theorem is a mathematical identity that applies universally when a symmetry exists but does not explain the origin of symmetries in nature.","section":"# Node C03: Symmetry ↔ Conservation","tier":"mechanistic","source_ids":[],"source_status":"unsourced","why_material":"Distinguishes the theorem's scope as formal implication rather than ontological claim."},{"id":"c4","text":"Applications of the theorem span classical mechanics, quantum theory, particle physics, and cosmology, each domain deriving independent utility from it.","section":"# Node C03: Symmetry ↔ Conservation","tier":"anecdotal","source_ids":[],"source_status":"unsourced","why_material":"Documents cross-domain independence of the mathematical result."}],"sources":[],"prov":{"model":"Fable 5 (Claude Code)","action":"write"}}