commutation_structure
plain-language theorem explainer
The theorem asserts that projector operators satisfy idempotence in any Recognition Science Hilbert space, which directly yields the commutation structure for quantum observables. Modelers of quantum algebra inside the Recognition framework cite this to confirm consistency of the projector ledger. The proof is a one-line term application of the idempotence axiom already present in the Projector definition.
Claim. In any Recognition Science Hilbert space, every projector operator $P$ satisfies $P^2 = P$.
background
The module states that its content is the structural commutation property that projector algebra is idempotent. The referenced definition commutation_from_ledger encodes this as the universal statement that every projector operator is idempotent under composition. Upstream CostAlgebra.H reparametrizes the J-cost as the shifted function $H(x) = J(x) + 1$, converting the Recognition Composition Law into the d'Alembert wave equation. The local theoretical setting is the quantum sector of the Recognition framework, where observables are realized via projectors on an RSHilbertSpace.
proof idea
Term-mode proof. The intro tactic binds the Hilbert space, its RSHilbertSpace instance, and the projector; the exact tactic then discharges the goal with the idempotence field of that projector.
why it matters
The result supplies the algebraic foundation for commutation in the Quantum module by deriving idempotence directly from the projector ledger. It sits inside the forcing chain after T5 J-uniqueness and supports later observable constructions. No downstream theorems are yet recorded as users of this declaration.
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