IndisputableMonolith.Quantum.HilbertSpace
The module defines a separable Hilbert space structure to underpin the quantum bridge in Recognition Science. It is imported by area quantization, observables, and ILG substrate modules. This is a definition module importing Mathlib inner-product machinery with no proofs.
claimA separable Hilbert space $\mathcal{H}$ over $\mathbb{C}$ equipped with inner product $\langle \cdot | \cdot \rangle$.
background
The module sits in the quantum domain and supplies the Hilbert-space foundation for Recognition Science. It imports Mathlib's inner-product-space libraries to guarantee separability and the standard projection and reflection operations. No upstream theorems are referenced; the setting is the minimal structure needed before observables or area quantization are introduced.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
This module feeds the Area Quantization Theorem (Phase 9.1), which derives integer quantization of spatial area in units of $\ell_0^2$ from the 8-tick cycle and simplicial ledger. It also supplies the space for the Observable Algebra and the quantum substrate used by the ILG framework.
scope and limits
- Does not derive the concrete dimension or basis from RS axioms.
- Does not encode the phi-ladder, J-cost, or defectDist.
- Does not contain any measurement or observable algebra.
- Does not address finite-dimensional truncations or lattice embeddings.