IndisputableMonolith.Relativity.Geometry.Manifold

   1import Mathlib
   2
   3/-!
   4# SCAFFOLD MODULE — NOT PART OF CERTIFICATE CHAIN
   5
   6**Status**: Scaffold / Placeholder
   7
   8This file provides a minimal typed manifold structure for the Relativity geometry
   9infrastructure. It is **not** part of the verified certificate chain.
  10
  11The definitions here are structural placeholders to enable downstream modules to
  12type-check. They do not constitute rigorous differential geometry formalization.
  13
  14**Do not cite these definitions as proven mathematics.**
  15
  16For the verified RS formalization, see:
  17- `IndisputableMonolith/Verification/` — verified certificate infrastructure
  18- `IndisputableMonolith/URCGenerators/` — proven generator certificates
  19
  20---
  21
  22# Manifold Structure for ILG (Scaffold)
  23
  24This module provides a minimal typed manifold structure for differential geometry.
  25We work with smooth manifolds equipped with coordinate charts.
  26-/
  27
  28namespace IndisputableMonolith
  29namespace Relativity
  30namespace Geometry
  31
  32/-- A smooth manifold with dimension and coordinate system. -/
  33structure Manifold where
  34  dim : ℕ
  35  deriving Repr
  36
  37/-- A point on the manifold (coordinates). -/
  38def Point (M : Manifold) := Fin M.dim → ℝ
  39
  40/-- A vector at a point (tangent space). -/
  41def TangentVector (M : Manifold) := Fin M.dim → ℝ
  42
  43/-- A covector at a point (cotangent space). -/
  44def Covector (M : Manifold) := Fin M.dim → ℝ
  45
  46/-- Standard 4D spacetime manifold. -/
  47def Spacetime : Manifold := { dim := 4 }
  48
  49/-- Coordinate indices for spacetime. -/
  50abbrev SpacetimeIndex := Fin 4
  51
  52/-- Time coordinate (index 0). -/
  53def timeIndex : SpacetimeIndex := 0
  54
  55/-- Spatial indices (1, 2, 3). -/
  56def spatialIndices : List SpacetimeIndex := [1, 2, 3]
  57
  58/-- Check if an index is spatial. -/
  59def isSpatial (μ : SpacetimeIndex) : Bool := μ ≠ 0
  60
  61/-- Kronecker delta for indices. -/
  62def kronecker {n : ℕ} (μ ν : Fin n) : ℝ := if μ = ν then 1 else 0
  63
  64theorem kronecker_symm {n : ℕ} (μ ν : Fin n) :
  65  kronecker μ ν = kronecker ν μ := by
  66  by_cases h : μ = ν
  67  · simp [kronecker, h]
  68  · have h' : ν ≠ μ := by
  69      intro hνμ
  70      exact h hνμ.symm
  71    simp [kronecker, h, h']
  72
  73theorem kronecker_diag {n : ℕ} (μ : Fin n) :
  74  kronecker μ μ = 1 := by
  75  simp [kronecker]
  76
  77theorem kronecker_off_diag {n : ℕ} (μ ν : Fin n) (h : μ ≠ ν) :
  78  kronecker μ ν = 0 := by
  79  simp [kronecker, h]
  80
  81/-- Partial derivative of a scalar function using a directional derivative along the basis vector. -/
  82noncomputable def partialDeriv {M : Manifold} (f : Point M → ℝ) (μ : Fin M.dim) (x : Point M) : ℝ :=
  83  deriv (fun t => f (fun i => if i = μ then x i + t else x i)) 0
  84
  85
  86end Geometry
  87end Relativity
  88end IndisputableMonolith
  89