commutation_from_ledger
plain-language theorem explainer
The definition states that projectors on Recognition Science Hilbert spaces are idempotent under operator composition. Quantum structure work in the framework cites it to ground observable algebra from ledger cost. It is introduced as a bare proposition that directly matches the built-in idempotent field of any projector.
Claim. For every separable Hilbert space $H$ equipped with the Recognition Science structure and every projector $P$ on $H$, the operator of $P$ composed with itself equals the operator of $P$.
background
Recognition Science places quantum observables on separable Hilbert spaces. The RSHilbertSpace class extends seminormed additive commutative groups, inner-product spaces over the complexes, completeness, and separability. Projector is a structure extending Observable that carries an explicit idempotent field on its operator under composition. The comp operation is taken from CostAlgebra, where it composes J-automorphisms, and from ArithmeticOf for homomorphism composition. The local module imports Observables and CostAlgebra to connect the shifted cost function H(x) = J(x) + 1, which satisfies the d'Alembert form of the Recognition Composition Law, to projector algebra.
proof idea
The declaration is a definition whose body is the quantified statement itself. No lemmas or tactics are invoked; the content is simply the assertion that projector operators satisfy op.comp op = op. This reproduces the idempotent field already present in the Projector structure definition.
why it matters
It supplies the proposition invoked by the commutation_structure theorem, which discharges the claim by extracting the idempotent field from any projector. The sibling commutation_implies_projector_idempotent likewise depends on it. The definition closes the structural link between cost-algebra composition and quantum projector idempotence, consistent with the Recognition Composition Law and the eight-tick octave periodicity.
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