Guide to ProductRecognitionLattice

Purpose

This module establishes the structural cardinality bounds for cross-domain recognition spaces. In Recognition Science, specific domains (like C1 cognitive or C3 oncology) are modeled definitionally (MODEL) as $5^3$ state spaces. This module proves the exact sizes and information-theoretic bit-bounds of taking Cartesian products of these domains to form joint clinical states.

Main Declarations

1. Concrete Types and Cardinalities

• The base domain state is defined as RSTriple (a Cartesian product of three Fin 5 spaces). Its size is proved to be $125$ in rsTriple_card.
• The joint state for two domains (e.g., cognitive and oncology) is RSJoint, proved to contain exactly $15625$ states in rsJoint_card.
• Larger combinations are also computed, such as a three-domain joint yielding $\sim 1.95$ million states in triple_joint.

2. Information-Theoretic Bounds

A major focus is proving that these nested dimensionalities fit within constrained bit-widths:

• A single domain ($5^3$) fits strictly under 7 bits ($2^7 = 128$), proved by five_three_lt_two_seven. This bounds a single domain to Miller's working memory range.
• The cross-domain joint state ($5^6$) fits strictly under 14 bits, proved mathematically via five_six_lt_two_fourteen and formally applied to the type in rsJoint_fits_14_bits.

3. Certification

All key dimensional bounds are bundled into productRecognitionLatticeCert to provide a single verified interface.

Fit in the RS Framework

This module is downstream of the fundamental forcing chain. It does not force fundamental physics (THEOREMs T0–T8), but rather provides guaranteed static sizing bounds for clinical runtime environments (e.g., Wave 64). It formally guarantees that a full joint state tracking a patient's cognitive and oncological profile fits strictly within a 14-bit bandwidth constraint.

What Remains Outside

This module solely formalizes the static structural bounds of the state space. It does not define dynamic transitions between these states, interaction potentials, or how specific clinical data maps onto the $15625$ elements (these remain OPEN or belong to application-layer modules).