NoetherCharge
plain-language theorem explainer
NoetherCharge packages any real-valued function on the state space X that stays constant under the one-parameter flow of G. Researchers deriving conservation laws from symmetry in Recognition Science cite this definition to bridge cost stationarity to standard Noether charges. The definition is a direct subtype construction requiring only the IsConservedAlong predicate.
Claim. For a one-parameter group $G$ acting on a space $X$, a Noether charge is any function $Q : X → ℝ$ that remains invariant along the flow generated by $G$.
background
In the QFT module, Noether's theorem is recast using cost stationarity from Recognition Science. A one-parameter group consists of a flow map satisfying the identity and composition properties, modeling continuous symmetries such as time translations. Upstream results include the topological version of NoetherCharge, which requires integer values and variational successors, and constants like G derived from the phi-ladder.
proof idea
This is a definition that directly constructs the subtype of functions satisfying the conservation condition along the group flow. It relies on the IsConservedAlong predicate defined in sibling declarations.
why it matters
This definition supplies the target type for the theorem invariant_is_noether_charge, which shows that any invariant function under the symmetry is a Noether charge. It connects to the topological conservation certificate in the Foundation module, where charges are shown to be topological rather than necessarily Noetherian, and supports the paper proposition on Noether from ledger structure.
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