DiffGeoCert
plain-language theorem explainer
The DiffGeoCert structure records that the recognition manifold admits exactly five canonical differential geometric structures and that spacetime has dimension four. Researchers deriving geometry from recognition axioms cite this certificate when confirming the manifold setup and Lorentzian extension. It is introduced as a record type whose two fields hold a cardinality equality from the Fintype instance and a direct dimension equality.
Claim. A certificate consisting of the assertions that the set of differential geometric structures on the recognition manifold has cardinality five and that spacetime dimension equals four.
background
The module treats the recognition manifold as a smooth 3-manifold with spatial dimension D=3. It defines five canonical structures via an inductive type: smooth manifold, Riemannian, pseudo-Riemannian, Kähler, and symplectic; this type carries a Fintype instance so its cardinality is computable. Spacetime dimension is obtained by adding one to the spatial dimension, producing the Lorentzian 4-dimensional case.
proof idea
The declaration is a pure structure definition with two fields. The first field records the cardinality of the inductive type of structures via its Fintype instance. The second field records the spacetime dimension equality. No lemmas or tactics are applied.
why it matters
This certificate is instantiated downstream to produce the concrete witness confirming five structures and four-dimensional spacetime. It supports the module's derivation of differential geometry from recognition principles, aligning with the forcing chain that yields D=3 spatial dimensions and spacetime as D+1. It closes the local setup before further geometric constructions.
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