Projector
plain-language theorem explainer
A projector is an observable on an RSHilbertSpace whose operator satisfies idempotence under composition. Quantum researchers using the Recognition Science bridge cite this structure when deriving measurement algebra from ledger properties. The definition is a direct extension of Observable that asserts the idempotent axiom as a field.
Claim. Let $H$ be a space equipped with the Recognition Science Hilbert structure. A projector $P$ is a self-adjoint bounded linear operator $P:H→H$ satisfying $P∘P=P$.
background
The module supplies observable algebra for the Recognition Science quantum mechanics bridge. Observable is the structure consisting of a bounded linear operator op together with the self-adjointness condition ⟨op x, y⟩_ℂ = ⟨x, op y⟩_ℂ for all vectors. Projector extends this base by adding the idempotent requirement on operator composition.
proof idea
The declaration is a structure definition that extends Observable and introduces the single field idempotent : op.comp op = op. No tactics or lemmas are applied; the property is part of the type definition itself.
why it matters
This supplies the idempotent operators required by the downstream commutation_from_ledger result, which extracts structural commutation content from the ledger. It completes the projector algebra inside the observable layer of the Recognition Science quantum bridge, linking cost-algebra composition to measurement operators.
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