materialClassCount
The theorem states that the finite type of material classes in the RS framework has exactly five elements, matching the configuration dimension D. Materials scientists using the Recognition Science model cite this when certifying the completeness of the canonical taxonomy of metals, ceramics, polymers, composites, and semiconductors. The proof is a direct decision procedure that evaluates the Fintype cardinality on the inductive enumeration.
claimThe cardinality of the finite type of material classes satisfies $Fintype.card(MaterialClass) = 5$, where MaterialClass is the inductive type with constructors metals, ceramics, polymers, composites, and semiconductors.
background
The Materials Science from RS module identifies five canonical material classes that equal the configuration dimension D equals five. Crystal symmetry groups map to Q₃ sublattices of cardinality eight, equal to two to the power D, with the cubic system having Oh symmetry of order 48 factored as six times eight. The upstream MaterialClass inductive supplies the DecidableEq, Repr, BEq, and Fintype instances required for the cardinality computation.
proof idea
The proof is a one-line wrapper that applies the decide tactic to compute Fintype.card on the inductive type MaterialClass, whose Fintype instance is derived automatically from its five constructors.
why it matters in Recognition Science
This supplies the five_classes field inside the materialsScienceCert definition, completing the RS materials taxonomy certification. It directly implements the module claim that five classes equal configDim D, and connects to the eight-tick octave via the relation |Q₃| = 2^D = 8. The result closes the E2/B10 section with zero sorry or axiom.
scope and limits
- Does not derive the five classes from the forcing chain T0-T8 or J-uniqueness.
- Does not prove the mapping from crystal symmetries to Q₃ sublattices.
- Does not compute physical properties or mass formulas for any class.
- Does not address the Berry creation threshold or phi-ladder structure.
Lean usage
def materialsScienceCert : MaterialsScienceCert where five_classes := materialClassCount oh_order := ohGroupOrder_eq_6_times_8formal statement (Lean)
26theorem materialClassCount : Fintype.card MaterialClass = 5 := by decide
proof body
Term-mode proof.
27
28/-- Oh group order = 48 = 6 × 2^3. -/