complex_hilbert_structure
The declaration proves that any Recognition Science Hilbert space extracted from the cost ledger yields normalized states with unit vector norm. Quantum modelers working from the functional equation would cite this when confirming that the derived Hilbert space obeys the basic normalization axiom. The term-mode proof is a direct one-line reduction that invokes the unit-norm property built into NormalizedState.
claimFor every type $H$ equipped with an instance of the Recognition Science Hilbert space structure, and for every normalized state $ψ$ in $H$, the norm satisfies $‖ψ.vec‖ = 1$.
background
The ComplexHilbertStructure module sits inside the Quantum namespace and imports HilbertSpace together with the cost algebra. Its central definition complex_hilbert_from_ledger asserts that every RSHilbertSpace instance produces states whose vector has norm exactly one. This rests on the shifted cost function $H(x) = J(x) + 1$ from CostAlgebra, which converts the Recognition Composition Law into the d'Alembert equation $H(xy) + H(x/y) = 2 H(x) H(y)$ as stated in the upstream doc-comment.
proof idea
The proof is a term-mode one-liner. It introduces the Hilbert space type $H$, the structure instance $hH$, and the normalized state $ψ$, then discharges the goal by exact application of $ψ.norm_one$, the unit-norm axiom supplied by the definition of NormalizedState.
why it matters in Recognition Science
This theorem anchors the quantum extraction step in the Recognition Science framework by confirming that the ledger-derived Hilbert space satisfies the normalization axiom required for any further state or operator construction. It closes the basic interface supplied by complex_hilbert_from_ledger and aligns with the T5 J-uniqueness and T6 phi fixed-point steps in the forcing chain. No downstream uses are recorded, indicating the result functions as an early foundational anchor before spectrum or dynamics are addressed.
formal statement (Lean)
17theorem complex_hilbert_structure : complex_hilbert_from_ledger := by
proof body
Term-mode proof.
18 intro H hH ψ
19 exact ψ.norm_one
20
21/-- Universe-stable extractor for complex-Hilbert structure. -/
22theorem complex_hilbert_implies_unit_norm.{u}
23 (h : @complex_hilbert_from_ledger.{u}) : @complex_hilbert_from_ledger.{u} :=
24 h
25
26end ComplexHilbertStructure
27end Quantum
28end IndisputableMonolith