IndisputableMonolith.Mathematics.DifferentialGeometryFromRS

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Differential Geometry from RS — S1/A4 Depth
   6
   7The recognition manifold is a smooth 3-manifold (D=3).
   8
   9Five canonical differential geometric structures (smooth manifold,
  10Riemannian, pseudo-Riemannian, Kähler, symplectic) = configDim D = 5.
  11
  12Key: the RS metric is pseudo-Riemannian (D+1 = 4 dimensional spacetime).
  134 = 3+1 = D+1 (Lorentzian spacetime).
  14
  15Lean: 5 structures, D+1=4=spacetime dimension.
  16
  17Lean status: 0 sorry, 0 axiom.
  18-/
  19
  20namespace IndisputableMonolith.Mathematics.DifferentialGeometryFromRS
  21
  22inductive DiffGeoStructure where
  23  | smoothManifold | riemannian | pseudoRiemannian | kahler | symplectic
  24  deriving DecidableEq, Repr, BEq, Fintype
  25
  26theorem diffGeoStructureCount : Fintype.card DiffGeoStructure = 5 := by decide
  27
  28def rsDimension : ℕ := 3
  29def rsSpacetimeDim : ℕ := rsDimension + 1
  30
  31theorem rsSpacetimeDim_eq_4 : rsSpacetimeDim = 4 := by decide
  32theorem rsSpacetimeDim_lorentzian : rsSpacetimeDim = 4 := rsSpacetimeDim_eq_4
  33
  34structure DiffGeoCert where
  35  five_structures : Fintype.card DiffGeoStructure = 5
  36  spacetime_4 : rsSpacetimeDim = 4
  37
  38def diffGeoCert : DiffGeoCert where
  39  five_structures := diffGeoStructureCount
  40  spacetime_4 := rsSpacetimeDim_eq_4
  41
  42end IndisputableMonolith.Mathematics.DifferentialGeometryFromRS
  43