τ-Noether Theorem
τ-Noether's theorem (`IV.R180`) is a corollary of the categorical structure of τ: in Category τ, the natural transformations are determined by the structure, each automatically satisfies naturality, and reading naturality on E₁ produces a conservation law while reading the same naturality from a different angle produces a symmetry. Conservation and symmetry are two faces of one categorical fact (`IV.T247` Conservation as Naturality).
τ-Definition
τ-Noether's theorem (`IV.R180`) is a corollary of the categorical structure of τ: in Category τ, the natural transformations are determined by the structure, each automatically satisfies naturality, and reading naturality on E₁ produces a conservation law while reading the same naturality from a different angle produces a symmetry. Conservation and symmetry are two faces of one categorical fact (`IV.T247` Conservation as Naturality).
Categorical invariant. Every conservation law in τ is a naturality square; every continuous symmetry is the dual reading of the same square (IV.R180 / IV.T247).
Primary registry anchor:
IV.R180
Supporting items:
IV.T247
τ-Derivation Chain
SI Translation
Calibration anchor: PG-P01-neutron
Calibration chain:
- natural transformations on Category τ
- E₁-readout for conservation currents
- SI bridge via m_n anchor for the conserved-quantity units (energy, momentum, etc.)
Manuscript reference: manuscript-sources/book-04/part08/ch72-laws-as-structure.tex
Lean Coverage
Status: Formalized
Module: TauLib.BookIV.Coda.LawsAsStructure
Lean kind: theorem
Lean symbol: Tau.BookIV.Coda.NoetherTheoremAsCorollary