RSJoint
plain-language theorem explainer
The joint state type is the Cartesian product of two 5³ recognition states. Cross-domain researchers cite it when bounding the combined cognitive-oncology state space at 15625 elements. The declaration is a direct type abbreviation that composes the triple state to enable subsequent cardinality results.
Claim. Let $S$ be the 5³ recognition state space. The joint state space is the Cartesian product $S × S$.
background
The module develops the product recognition lattice for cross-domain applications. It combines two 5³ domains to form a 5^6 state space of 15625 elements, representing the full cognitive-oncology state of a cancer patient. The hierarchy runs from 5² = 25 for domain pairs through 5^8 = 390625, subject to the bound 5^6 < 2^14.
proof idea
This is a direct type abbreviation that constructs the joint state as the Cartesian product of two 5³ recognition states.
why it matters
This definition supplies the type for the product recognition lattice certificate, which records the joint size 15625 and the inequality under 2^14. It also supports the joint cardinality theorem and the 14-bit bound theorem. In the framework it realizes the 5^6 joint state for cognitive-oncology products, consistent with the 5³ triple and the information-theoretic bounds on state spaces.
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