shiftedConfig
plain-language theorem explainer
The definition shiftedConfig produces a new coupled-core configuration by subtracting a displacement vector a from an existing configuration s, componentwise modulo 4. Researchers modeling tensor-Weyl operators on finite ququart systems would cite this when composing displacements or recovering original states. The definition is a direct pointwise application of the mod-4 subtraction sub4 with no additional computation.
Claim. Let $N$ be a natural number. For a displacement $a$ and configuration $s$ in the space of maps from finite set of size $N$ to the set of four elements, the shifted configuration is the map $xmapsto s(x)-a(x) mod 4$.
background
CoupledCoreIndex N is the type Fin N → Fin 4, the finite-site configuration space of coupled ququart cores. The sibling definition sub4 implements subtraction modulo 4 on Fin 4 via ⟨(a.val + (4 - b.val)) % 4, by omega⟩. The module CoupledRecognitionCores builds the algebraic structure for these cores on top of the operator-core version of the same index type, importing only Mathlib.
proof idea
The definition is a one-line wrapper that applies sub4 pointwise: for each site x it returns sub4 (s x) (a x). No tactics or lemmas are invoked beyond the functional abstraction itself.
why it matters
shiftedConfig supplies the inverse displacement operation used by the recovery theorems addedConfig_shiftedConfig and shiftedConfig_addedConfig, and it appears directly in the definition of tensorWeylMonomial. It therefore supports the concrete Hilbert-space action of coupled recognition operators on finite ququart carriers, consistent with the Recognition Science modeling of discrete configuration spaces.
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