IndisputableMonolith.Quantum.HilbertSpace
Module supplies the separable Hilbert space structure that anchors the quantum bridge in Recognition Science. Workers on area quantization and observable algebras cite it when importing the quantum substrate. It is a definition module that declares RSHilbertSpace and NormalizedState atop Mathlib inner-product machinery.
claimLet $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$ equipped with the standard inner product; a normalized state is a vector $\psi \in \mathcal{H}$ with $\langle \psi, \psi \rangle = 1$.
background
Recognition Science places its quantum formalism after the forcing chain (T0-T8) and Recognition Composition Law. The module imports Mathlib's InnerProductSpace, Projection, and Topology.Bases to supply the mathematical carrier for states. Sibling declarations RSHilbertSpace and NormalizedState stand for the concrete types used downstream.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
Supplies the Hilbert space carrier required by the Area Quantization Theorem (Phase 9.1, derived from the 8-tick cycle and simplicial ledger) and by the observable algebra module. It also feeds the quantum substrate for the ILG framework in Relativity.ILG.Substrate, closing the quantum bridge step.
scope and limits
- Does not construct an explicit basis or fix the dimension of the space.
- Does not derive any physical spectrum or quantization rule.
- Does not encode the J-function, phi-ladder, or Recognition Composition Law.
- Does not address measurement postulates or time evolution.