cyDimension
The definition assigns the natural number 3 to the Calabi-Yau threefold dimension inside the Recognition Science algebraic geometry module. Researchers examining the Hodge connection between the recognition lattice Q₃ and Calabi-Yau mirror symmetry would cite this constant when certifying the five canonical objects. It is supplied as a direct constant definition that downstream structures reference without further reduction.
claimThe dimension of the Calabi-Yau threefold is $3$.
background
The module treats algebraic geometry as the study of varieties defined by polynomial equations, with the recognition lattice Q₃ realized as an algebraic variety over F₂. Five canonical objects (affine variety, projective variety, Calabi-Yau, K3 surface, elliptic curve) are identified with configDim D = 5. The Calabi-Yau threefold connection follows because RS predicts mirror symmetry of Q₃ realized as a Calabi-Yau threefold precisely at D = 3, with the five canonical Hodge types h^{p,q} for Q₃ supplying the corresponding Hodge numbers.
proof idea
Direct definition that binds the constant 3 to cyDimension. No lemmas or tactics are invoked; the value is introduced by fiat for use in certification structures.
why it matters in Recognition Science
This definition supplies the cy_dim field required by the AlgebraicGeometryCert structure, which also records that exactly five algebraic geometry objects exist. It implements the framework landmark T8 that forces D = 3 spatial dimensions and closes the Calabi-Yau threefold prediction for the recognition lattice Q₃. No open questions remain inside the current certification.
scope and limits
- Does not derive the value 3 from the forcing chain T0-T8.
- Does not compute explicit Hodge numbers h^{p,q}.
- Does not address varieties outside the five canonical objects.
- Does not connect to the mass formula or RS-native constants.
Lean usage
AlgebraicGeometryCert.mk (by decide) rflformal statement (Lean)
31def cyDimension : ℕ := 3