MaterialClass
plain-language theorem explainer
MaterialClass enumerates the five canonical material classes in Recognition Science, identified with configuration dimension D = 5. Condensed-matter researchers applying the RS model to crystal symmetry and electronic structure would cite this enumeration when establishing cardinality or certification results. The declaration is a direct inductive definition that automatically derives DecidableEq, Repr, BEq and Fintype instances with no proof obligations.
Claim. The inductive type $MaterialClass$ consists of five constructors: metals, ceramics, polymers, composites, and semiconductors.
background
In the Recognition Science framework, materials science is obtained from configuration dimension D = 5, which produces exactly five material classes. Crystal symmetry groups map to Q₃ sublattices with |Q₃| = 8 = 2^D. The cubic system takes the Oh group of order 48, written as 6 × 2^3. The module states that Lean encodes these five classes directly, with zero axioms or sorry statements.
proof idea
This is a direct inductive definition that introduces five constructors and derives the listed type-class instances automatically.
why it matters
The definition supplies the five_classes field of MaterialsScienceCert and the left-hand side of the materialClassCount theorem. It realizes the E2 / B10 step that equates five material classes with configDim D = 5 and the eight-tick octave 2^3, thereby connecting the Recognition Composition Law and the D = 3 spatial dimension to concrete solid-state classification.
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