IndisputableMonolith.Crystallography.SelectionRules
The module Crystallography.SelectionRules supplies the definitional core for selection rules governing triads in Recognition Science lattices. It introduces Triad as the basic triple, legalTriad as the predicate enforcing neutrality and RCL compliance, and countLegal together with neutral8 to enumerate valid configurations under eight-tick periodicity. The module contains only declarations and no theorems, establishing the combinatorial language needed for crystal classification derived from the phi-ladder.
claimA triad $t$ is legal when $J(t_1 t_2) + J(t_1/t_2) = 2J(t_1)J(t_2) + 2J(t_1) + 2J(t_2)$ holds with zero defect distance; neutral8 counts the fixed points of the period-$8$ action; countLegal$(n)$ returns the cardinality of legal triads of rung $n$.
background
Recognition Science crystallography arises from the discrete structures forced by T5 (J-uniqueness), T7 (eight-tick octave) and T8 (D=3). The module introduces Triad as an ordered triple drawn from the phi-ladder, legalTriad as the predicate that the Recognition Composition Law is satisfied with vanishing defectDist, and neutral8 as the count of configurations invariant under the 2^3-periodic symmetry. These objects sit between the mass formula (yardstick times phi to a rung offset) and the Berry creation threshold, supplying the allowed combinatorial units for lattice construction.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the primitive objects that later theorems on lattice stability and selection rules will invoke. It directly encodes the neutrality condition required by the eight-tick octave and the RCL, thereby closing the combinatorial gap between the J-function and physical crystal enumeration. No downstream theorems are yet recorded, indicating the module functions as foundational scaffolding.
scope and limits
- Does not prove existence or uniqueness of any physical lattice.
- Does not link definitions to experimental diffraction data.
- Does not import or define the phi-ladder or J-cost primitives.
- Does not address continuous limits or macroscopic crystals.