AlgebraicGeometryObject
plain-language theorem explainer
The declaration enumerates five canonical algebraic geometry objects as an inductive type corresponding to the configuration dimension of five in Recognition Science. Researchers examining the Hodge connection between RS lattices and Calabi-Yau manifolds cite this enumeration when linking polynomial varieties to the Q₃ lattice over F₂. The definition introduces the five constructors and derives decidable equality, representation, boolean equality, and finite type instances directly.
Claim. Let $O$ be the inductive type whose constructors are the affine variety, the projective variety, the Calabi-Yau threefold, the K3 surface, and the elliptic curve, equipped with decidable equality, a representation instance, boolean equality, and finite type structure.
background
Algebraic geometry studies varieties defined by polynomial equations. In Recognition Science the recognition lattice Q₃ is an algebraic variety over F₂. The module sets five canonical objects (affine variety, projective variety, Calabi-Yau, K3 surface, elliptic curve) equal to configuration dimension D = 5. This supplies the Hodge numbers h^{p,q} for Q₃ as the five canonical Hodge types and predicts mirror symmetry of Q₃ as a Calabi-Yau threefold at D = 3.
proof idea
The declaration is an inductive definition with five explicit constructors that automatically derives the DecidableEq, Repr, BEq, and Fintype instances.
why it matters
This enumeration supplies the five objects required by agObjectCount (Fintype.card = 5) and by the AlgebraicGeometryCert structure (five_objects and cy_dim = 3). It realizes the module's claim that the five objects equal configDim D = 5 and supports the Hodge connection to the Calabi-Yau threefold at D = 3. The definition closes the enumeration step in the C Mathematics section without introducing axioms.
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