CoupledCoreSpace
The abbreviation supplies the concrete Hilbert carrier for N coupled recognition cores as the function space from a finite index set of size 4^N into the complex numbers. Researchers building basis states, operator lifts, and exact embeddings of finite-dimensional dynamics inside the Recognition Science framework cite this carrier when moving between abstract linear maps and explicit matrix representations on ququart indices. The declaration is a one-line type alias that re-exports the upstream definition without adding computational content or
claimFor each natural number $N$, let $H_N$ denote the vector space of all functions from the coupled-core index set of cardinality $4^N$ to the complex numbers.
background
The CoupledRecognitionCores module realizes each recognition core as a ququart (four-level system). The joint carrier for N such cores is therefore indexed by a set of cardinality $4^N$ and realized concretely as maps from that index set into ℂ. This supplies the standard finite-dimensional Hilbert space once an inner product is introduced downstream. The upstream definition states that the space is the concrete Hilbert carrier of N coupled recognition cores and records that its dimension is exactly $4^N$.
proof idea
The abbreviation is a direct one-line re-export of the upstream definition CoupledCoreSpace (N : ℕ) := CoupledCoreIndex N → ℂ. No lemmas or tactics are invoked.
why it matters in Recognition Science
This carrier is the common domain for the standard basis kets, the Hilbert-Schmidt-style operator inner product, state embeddings, and operator lifts defined in the same module. It is the arena in which the finite-dimensional exact embedding theorem realizes arbitrary linear dynamics on a d-dimensional space inside a coupled-core space of dimension 4^d. In the Recognition Science setting the construction discretizes operator evolution on the phi-ladder while preserving the exact dimension count required by the eight-tick octave.
scope and limits
- Does not equip the space with an inner product or norm.
- Does not restrict N to any particular value or impose growth conditions.
- Does not encode dynamics, operator actions, or time evolution.
- Does not relate the space to J-cost, defect distance, or mass formulas.
formal statement (Lean)
15abbrev CoupledCoreSpace := IndisputableMonolith.Foundation.CoupledRecognitionCores.CoupledCoreSpace