PolymerMorphologyCert
plain-language theorem explainer
The structure PolymerMorphologyCert packages the assertion that the inductive type of polymer morphologies has cardinality five, matching the five canonical block-copolymer phases tied to configDim equal to 5. Materials modelers working with self-assembly on the phi-ladder would cite it when mapping volume-fraction bands to observed structures. The definition is realized directly as a record whose single field records the Fintype cardinality.
Claim. Let $M$ be the inductive type with constructors spherical, cylindrical, gyroid, lamellar, and inverseMorph. The structure PolymerMorphologyCert consists of the field five_morphologies asserting that the cardinality of $M$ equals 5.
background
The module identifies five canonical block-copolymer morphologies with configDim set to 5: spherical, cylindrical, gyroid, lamellar, and inverse (double gyroid or inverse cylindrical). Each morphology is tied to a distinct minority-block volume-fraction band on the phi-ladder. The upstream inductive definition PolymerMorphology enumerates these five cases and derives Fintype, DecidableEq, Repr, and BEq instances.
proof idea
The structure is declared directly as a record type. Its field simply records the cardinality equality supplied by the Fintype instance that the inductive type derives automatically. No lemmas or tactics are invoked.
why it matters
This definition certifies that exactly five morphologies arise at configDim equal to 5 and supplies the type for the downstream construction polymerMorphologyCert. It fills the B15 materials-depth claim in the module, connecting the phi-ladder volume fractions to observable phases within the Recognition Science framework.
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