---
slug: oip-node-c03-symmetry-conservation
title: Node C03: Symmetry ↔ Conservation
canonical: https://miscsubjects.com/a/oip-node-c03-symmetry-conservation
---

# Node C03: Symmetry ↔ Conservation

C03 — Symmetry ↔ Conservation
{
  "id": "C03",
  "claim": "Every continuous symmetry of a physical system's action corresponds to a conserved quantity; symmetries are the source of conservation laws.",
  "domain": ["classical mechanics", "quantum field theory", "particle physics", "crystallography", "cosmology", "aesthetics"],
  "pattern": ["symmetry", "conservation", "Noether_correspondence", "Lie_groups"],
  "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.",
  "scale": "quantum → cosmic",
  "claim_tier": "T0",
  "sources": [
    "Noether, E. (1918). 'Invariante Variationsprobleme.' Nachr. v. d. Ges. d. Wiss. zu Goettingen, 235-257.",
    "Weyl, H. (1928). Gruppentheorie und Quantenmechanik.",
    "Wigner, E. (1939). 'On Unitary Representations of the Inhomogeneous Lorentz Group.' Ann. Math., 40(1), 149-204."
  ],
  "dual": "Symmetry-breaking (C04) — the structure that voids symmetry produces the phenomenological world.",
  "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.",
  "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.",
  "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.",
  "pattern_type": "mathematical",
  "maps_to_axiom": ["A1", "A3"]
}

---

## Corpus map
- Same node, other planes: [Encyclopedia C03](/a/convergence-encyclopedia-c03) · [Inventory invariant](/a/oip-invariant-3-33-symmetry-conservation)
- Edges touching C03: [convergence edge 3](/a/oip-convergence-edge-3) · [disconfirming edge 4](/a/oip-disconfirming-edge-4)
- Catalogue hub: [Public Article](/a/oip-convergence-public-article) · [Schema](/a/oip-convergence-schema)
