cognitive_oncology_joint
plain-language theorem explainer
The theorem confirms that two 5³ state spaces multiply to a single 5⁶ space of cardinality 15625. Cross-domain modelers in Recognition Science cite the equality to bound the joint cognitive-oncology lattice. The proof is a one-line wrapper that invokes the ring tactic to discharge the power-multiplication identity.
Claim. The joint cognitive-oncology state space satisfies $5^{3} × 5^{3} = 5^{6}$.
background
The Product Recognition Lattice module constructs hierarchies of recognition state spaces from cross-domain products. C1 supplies the cognitive triple with 5³ states while C3 supplies the oncology triple with 5³ states; their Cartesian product yields the 5⁶ joint space. The module lists the lattice levels explicitly: 5² = 25 for domain pairs, 5³ = 125 for single triples, 5⁴ = 625 for four-fold products, up to 5⁸ = 390625, together with the information-theoretic constraint 5⁶ < 2¹⁴ = 16384.
proof idea
The proof is a one-line wrapper that applies the ring tactic, which automatically confirms the arithmetic identity by the rule for adding exponents under multiplication.
why it matters
The result instantiates the C23 claim of the Product Recognition Lattice by supplying the exact cardinality of the joint cognitive-oncology space. It anchors the 5⁶ level in the lattice hierarchy and supplies the concrete bound used for the 14-bit information limit on full patient state. No downstream theorems currently depend on it.
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