ParallelTransportIC
This structure records the starting parameter value and 4-component vector for parallel transport along a spacetime curve. It is referenced when constructing solutions that must match these data at the initial point. The definition is a direct record introduction with no computational content or lemmas required.
claimInitial conditions for parallel transport consist of a real parameter value $λ_0$ together with an initial vector $V_0 ∈ ℝ^4$.
background
The module formalizes parallel transport on 4D spacetime using the Levi-Civita connection derived from a metric tensor. Parallel transport moves a vector along a curve while keeping it as parallel as possible, with failure around closed loops measuring integrated curvature. In Recognition Science this curvature arises from non-uniform J-cost defect density along the curve.
proof idea
This is a structure definition that directly introduces the record type with fields lam0 and V0.
why it matters in Recognition Science
The structure supplies the initial data required by ParallelTransportSolution, which asserts a vector field satisfying the parallel transport equation together with these conditions. It supports the module's development of holonomy as the geometric signature of J-cost imbalances, linking to the forcing chain where curvature is forced once D=3 and the eight-tick octave are in place.
scope and limits
- Does not encode the parallel transport differential equation.
- Does not reference any metric tensor or connection.
- Does not impose normalization or orthogonality on the initial vector.
- Does not address closed loops or holonomy computation.
formal statement (Lean)
69structure ParallelTransportIC where
70 lam0 : ℝ
71 V0 : Fin 4 → ℝ
72
73/-- A parallel-transported vector field satisfying initial conditions. -/