RSTriple
plain-language theorem explainer
RSTriple supplies the type of a generic 5³ recognition state as the Cartesian product of three 5-element sets. Cross-domain researchers cite it when constructing the joint cognitive-oncology state space of size 15625. The abbreviation directly encodes the D-cubed structure without additional structure or proof.
Claim. The type of a generic recognition state in three dimensions is the Cartesian product $Fin(5) × Fin(5) × Fin(5)$.
background
The Product Recognition Lattice models recognition states as finite sets whose cardinalities are powers of 5. A single domain (cognitive or oncological) occupies a 5³ space of 125 states. The module builds a lattice hierarchy: 5² pairs, 5³ triples, up to the 5⁶ joint state of 15625 elements, constrained by 5⁶ < 2¹⁴ to fit in 14 bits.
proof idea
The declaration is a one-line type abbreviation that constructs the product of three copies of the finite set with five elements.
why it matters
This type definition underpins the ProductRecognitionLatticeCert structure, which records the equalities (5)^3 = 125 and (5)^6 = 15625 together with the bit-length bounds. It also supports the RSJoint abbreviation for the product of two such triples. In the Recognition Science framework it instantiates the D=3 spatial dimensions at the level of state spaces, closing the cross-domain product construction in C23.
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