normalized_state_unit_norm
Normalized states in an RS Hilbert space satisfy the unit norm condition by construction. Quantum modelers in the Recognition Science setting cite this when verifying that state vectors produce valid probability measures under the inner product. The proof is a direct field projection onto the norm_one component of the NormalizedState structure.
claimLet $H$ be a type equipped with an RSHilbertSpace structure. For any normalized state $ψ$ in $H$, the Euclidean norm of its underlying vector equals one: $‖ψ.vec‖ = 1$.
background
RSHilbertSpace is a type class that extends a separable Hilbert space over ℂ with seminorm, inner product, completeness, and separability. NormalizedState is the structure that bundles a vector vec : H together with the explicit proof norm_one : ‖vec‖ = 1. The module ComplexHilbertStructure sits inside the Quantum namespace and imports the HilbertSpace definitions to build complex structures compatible with the Recognition Science ledger. Upstream, the H reparametrization from CostAlgebra and FunctionalEquation supplies the shifted cost H(x) = J(x) + 1 that converts the Recognition Composition Law into the d'Alembert equation, providing the algebraic backbone for the quantum embedding.
proof idea
The proof is a one-line wrapper that applies the norm_one field from the NormalizedState structure definition.
why it matters in Recognition Science
The declaration supplies the normalization guarantee required for any quantum state representation inside the Recognition Science framework. It directly supports sibling constructions such as complex_hilbert_structure and complex_hilbert_from_ledger by ensuring that vectors used for amplitudes remain on the unit sphere. In the larger T0–T8 chain it anchors the Hilbert-space layer that must sit atop the phi-ladder and eight-tick octave before mass formulas or Berry thresholds can be applied to quantum objects.
scope and limits
- Does not prove existence of normalized states beyond the empty or trivial cases.
- Does not extend the unit-norm property to non-separable or incomplete spaces.
- Does not derive the norm condition from the J-cost or Recognition Composition Law.
- Does not address time evolution or measurement postulates.
formal statement (Lean)
13theorem normalized_state_unit_norm {H : Type*} [RSHilbertSpace H]
14 (ψ : NormalizedState H) : ‖ψ.vec‖ = 1 :=
proof body
Term-mode proof.
15 ψ.norm_one
16