ququartX
The ququartX declaration defines the canonical shift operator X on the four-dimensional ququart Hilbert space. Researchers modeling discrete Weyl algebras or coupled quantum cores would cite this when establishing algebraic relations. The definition constructs the linear map by shifting the index via the predecessor function modulo 4, with additivity verified by direct simplification.
claimThe linear operator $X : H_4 → H_4$ on the ququart state space satisfies $(Xψ)(m) = ψ(m-1 mod 4)$, where $H_4$ is the space of functions from Fin 4 to ℂ.
background
QuquartState is the type of states for a ququart system, realized as linear maps from Fin 4 to ℂ. The predecessor prev4 computes the index shift m ↦ m-1 mod 4. This sits inside the CoupledRecognitionCores module, which supplies the operator algebra for coupled recognition cores.
proof idea
The declaration is an abbreviation that reexports the definition of ququartX from CoupledRecognitionCores. The core construction uses a toFun clause that applies the state to the previous index and confirms the linear map property through simplification on prev4.
why it matters in Recognition Science
This operator is the direct input to the ququartWeyl_relation theorem establishing ZX = i XZ and to the supporting lemma ququartWeyl_relation_apply. It supplies the shift component of the discrete Weyl algebra used in the Recognition Science treatment of coupled cores.
scope and limits
- Does not compute eigenvalues or spectra of X.
- Does not generalize the shift to dimensions other than 4.
- Does not define or use the dual phase operator Z.
- Does not address continuous limits or embeddings into larger spaces.
formal statement (Lean)
12abbrev ququartX := IndisputableMonolith.Foundation.CoupledRecognitionCores.ququartX