five_six_lt_two_fourteen
plain-language theorem explainer
The inequality 5^6 < 2^14 confirms that the 15625-element joint cognitive-oncology recognition state fits inside 14 bits. Researchers certifying cross-domain product lattices in Recognition Science cite this bound to enforce information-theoretic limits on combined state spaces. The proof evaluates the powers by direct decision procedure with no external lemmas.
Claim. The inequality $5^6 < 2^{14}$ holds, so the joint cognitive-oncology state space of size 15625 fits within a 14-bit representation.
background
The Product Recognition Lattice builds hierarchies of recognition states via products of base 5^3 domains. Cognitive (C1) and oncology (C3) each contribute 125 states; their product yields the 5^6 = 15625 joint space. The module states that this size must obey the explicit constraint 5^6 < 2^14 = 16384 to remain inside 14 bits, completing the lattice sequence 5^2 through 5^8.
proof idea
One-line wrapper that applies the decide tactic to compute and verify the numerical comparison 5^6 < 2^14 directly.
why it matters
This supplies the five_six_under_14_bits field inside the ProductRecognitionLatticeCert definition. It realizes the RS-derived information-theoretic bound on the joint cognitive-oncology state listed in the module documentation. The result anchors the lattice hierarchy by confirming the 5^6 space respects the 14-bit limit required for cross-domain certification.
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