module module low

`IndisputableMonolith.CrossDomain.ProductRecognitionLattice`

The ProductRecognitionLattice module supplies the bound 5^6 < 2^14 to confirm that a cognitive-oncology joint state fits inside 14 bits. Cross-domain modelers would cite the collection when encoding product lattices under bit constraints. The module is a container for direct power-comparison lemmas rather than a single derived theorem.

claim$5^6 < 2^{14}$ together with the auxiliary comparisons $5^k < 2^m$ for $2 \le k \le 8$.

background

The module sits in the CrossDomain section of the Recognition Science mirror and imports only Mathlib. It introduces no new definitions or axioms; its sole content is the set of sibling lemmas that verify concrete power bounds. These bounds are required when Recognition Composition Law structures are mapped onto finite-bit product spaces.

proof idea

This is a module collecting supporting lemmas rather than a single proof; each sibling lemma verifies a specific power inequality through basic arithmetic.

why it matters in Recognition Science

The module supplies the numerical fact required for the 14-bit joint-state encoding noted in its documentation. It therefore supports any downstream cross-domain lattice construction that must remain inside a fixed bit width.

declarations in this module (24)

theorem five_pow_2 L30
theorem five_pow_3 L31
theorem five_pow_4 L32
theorem five_pow_5 L33
theorem five_pow_6 L34
theorem five_pow_7 L35
theorem five_pow_8 L36
theorem five_six_lt_two_fourteen L41
theorem five_five_lt_two_thirteen L44
theorem five_four_lt_two_ten L47
theorem five_three_lt_two_seven L50
theorem five_squared_lt_two_5 L53
theorem cognitive_oncology_joint L58
theorem cognitive_oncology_size L61
theorem joint_125_squared L64
theorem triple_joint L69
theorem five_nine_lt_two_21 L72
abbrev RSTriple L77
theorem rsTriple_card L79
abbrev RSJoint L83
theorem rsJoint_card L85
theorem rsJoint_fits_14_bits L89
structure ProductRecognitionLatticeCert L92
def productRecognitionLatticeCert L102