five_pow_5
plain-language theorem explainer
Five to the fifth power equals 3125 supplies the numerical value for the full domain hierarchy level in the product recognition lattice. Cross-domain researchers cite it when scaling state spaces from 5² pairs through 5³ triples to the 5^5 anchor before reaching the 5^6 joint cognitive-oncology bound. The proof executes a single decide tactic that reduces the equality to a decidable natural-number computation.
Claim. $5^5 = 3125$
background
The module defines a lattice of product state spaces under Recognition Science bounds, with successive powers of five enumerating hierarchy levels. 5² counts pairs of D=5 domains, 5³ counts single triples such as cognitive or oncology states, 5^4 counts four-fold products, and 5^5 counts the full domain hierarchy. The local setting requires only the decidability of equality on natural numbers; no prior lemmas from upstream results are referenced.
proof idea
The proof is a one-line wrapper that applies the decide tactic to the equality. The tactic computes the integer power directly and confirms the result without invoking named lemmas or manual expansion steps.
why it matters
This theorem fills the 5^5 slot in the hierarchy table of the Product Recognition Lattice module, directly enabling the downstream bound 5^6 < 2^14 that constrains the joint cognitive-cancer state space to fourteen bits. It aligns with the RS information-theoretic limits on recognition state spaces listed in the module documentation. No open questions or scaffolding are addressed.
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