five_three_lt_two_seven
plain-language theorem explainer
The inequality 5 cubed less than 2 to the seventh establishes that a single triple of recognition states fits in 7 bits, consistent with Miller's working memory range. Cross-domain lattice researchers cite it to bound state spaces for individual cognitive or oncological triples under product constraints. The proof is a direct computational verification via the decide tactic on natural numbers.
Claim. $5^3 < 2^7$
background
The Product Recognition Lattice module constructs a hierarchy of recognition state spaces from products of D=5 domains. Here 5^3 equals 125 states and represents a single triple, such as the cognitive domain C1 or the oncology domain C3. The module requires these triples to satisfy an information bound of 7 bits, with the joint cognitive-oncology product at 5^6 bounded by 14 bits. This setting draws on the meta-realization structure from UniversalForcingSelfReference, which records the structural coherence properties needed for lattice certification.
proof idea
The proof is a one-line wrapper that applies the decide tactic to evaluate the natural-number inequality by direct computation.
why it matters
This theorem supplies the five_three_under_7_bits field inside the ProductRecognitionLatticeCert definition, which aggregates the lattice bounds including five_cubed and five_to_six. It realizes the information-theoretic constraint for single triples in the C23 Product Recognition Lattice, aligning the 125-state space with Miller's 7-bit memory range and supporting the joint 5^6 state under 14 bits. The result closes the computational verification step for the 5^3 case without new hypotheses.
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