IndisputableMonolith.Quantum.HilbertSpace
The module establishes a separable Hilbert space structure as the quantum state space foundation in Recognition Science. Researchers building the quantum bridge to gravity and observables cite it for the basic setup. It is a definition module that imports Mathlib inner-product machinery to declare RSHilbertSpace and NormalizedState without theorems.
claimA separable Hilbert space $H$ over $mathbb{C}$ equipped with inner product $langle cdot, cdot rangle$ satisfying the standard axioms of a complete inner-product space, serving as the state space for Recognition Science quantum formalism.
background
Recognition Science places its quantum sector on a Hilbert space that must later respect the J-uniqueness functional and Recognition Composition Law derived from the T0-T8 forcing chain. This module supplies the minimal separable Hilbert space structure by importing Mathlib's InnerProductSpace.Basic, Projection, and Topology.Bases to guarantee completeness and separability. The local setting is the quantum domain, with sibling declarations RSHilbertSpace and NormalizedState providing the concrete type and normalization predicate.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the Hilbert space substrate that directly feeds the Phase 9.1 Area Quantization Theorem (predicting area in integer units of ell_0^2 from the 8-tick cycle), the ComplexHilbertStructure, the Observables algebra, and the ILG Substrate for Induced Lattice Gravity. It occupies the foundational position in the quantum bridge, preparing the ground for the eight-tick octave and D=3 spatial constraints of the Recognition framework.
scope and limits
- Does not derive the inner product from the J-cost functional equation.
- Does not embed the phi-ladder or mass formula into the space.
- Does not enforce the Recognition Composition Law as an axiom.
- Does not prove separability or completeness beyond the declaration.
- Does not connect to Berry creation threshold or Z_cf = phi^5.