five_pow_8
plain-language theorem explainer
Five to the eighth power equals 390625 in the natural numbers. Researchers enumerating state-space sizes in the cross-domain product recognition lattice cite this value to close the hierarchy of 5-powers. The proof is a direct arithmetic evaluation via the decide tactic.
Claim. In the natural numbers, $5^8 = 390625$.
background
The module constructs a lattice of product recognition state spaces obtained by combining domains such as cognitive (C1) and oncology (C3) triples. Each 5^k counts the size of a k-fold product: 5^3 for a single domain triple, 5^6 for the joint cognitive-oncology state, and 5^8 for the next level in the hierarchy. The setting imposes the explicit bound 5^6 < 2^14 so that the full patient state fits inside 14 bits.
proof idea
The proof is a one-line term proof that applies the decide tactic to evaluate the power expression directly.
why it matters
This theorem supplies the concrete value 390625 for the 5^8 entry in the enumerated lattice, completing the sequence listed in the module documentation. It thereby supports the structural claim that cross-domain products remain inside the information-theoretic bound derived from the 5^3 domain size. No downstream theorems depend on it in the current graph.
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