coupledCoreEquivFin
plain-language theorem explainer
The declaration supplies a canonical bijection between the configuration space of N coupled ququart cores and the finite set of size equal to its cardinality. Researchers modeling finite-dimensional states in coupled recognition systems cite this when embedding or projecting vectors onto the ququart basis. The proof is a direct one-line wrapper invoking the library construction Fintype.equivFin on the index type.
Claim. For any natural number $N$, the type of functions from Fin $N$ to Fin $4$ is in bijection with Fin $(4^N)$.
background
CoupledCoreIndex N is the abbreviation Fin N → Fin 4, the finite-site configuration space of coupled ququart cores that serves as the concrete Hilbert carrier for N coupled recognition cores. The module Foundation.CoupledRecognitionCores builds finite-dimensional linear maps over this space, with upstream carrier definitions fixing the frequency at 5φ Hz in both neuromodulation and phantom GW antenna contexts.
proof idea
The definition is a one-line wrapper that applies Fintype.equivFin directly to CoupledCoreIndex N. No additional tactics or reductions are required; the library lemma supplies the equivalence to Fin of the cardinality of the finite type.
why it matters
This definition is invoked by embedState and projectState to perform zero-padding and coordinate restriction on coupled-core states, and it is the key step in the round-trip theorem projectState_embedState. It supplies the finite basis enumeration needed for the Recognition framework's treatment of coupled ququart systems at the engineering interface.
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