ParallelTransportIC
plain-language theorem explainer
ParallelTransportIC is the record type holding the starting parameter λ₀ and initial 4-vector V₀ for a parallel-transported field along a spacetime curve. Workers on holonomy and curvature in Recognition Science cite it when initializing solutions to the transport equation. As a bare structure definition it introduces the data type with no proof obligations.
Claim. The initial conditions for parallel transport consist of a real parameter $λ_0$ and an initial vector $V_0 : ℝ^4$ at that parameter value.
background
The module formalizes parallel transport along curves in 4D spacetime using the Levi-Civita connection from Curvature.christoffel. Parallel transport moves a vector while keeping it as parallel as possible with respect to the connection; on a curved manifold the failure around closed loops yields a rotation proportional to the integrated Riemann tensor. In Recognition Science this failure is the geometric signature of non-uniform J-cost defect density that sources gravity. The structure ParallelTransportIC supplies the starting data consumed by the downstream ParallelTransportSolution record.
proof idea
This declaration is a structure definition that introduces the record type with two fields; no lemmas or tactics are applied.
why it matters
ParallelTransportIC supplies the initial data consumed by ParallelTransportSolution. It supports the module results on holonomy vanishing iff Riemann vanishes and on curvature as infinitesimal holonomy. The definition operationalizes the geometric manifestation of ledger imbalance described in the module documentation, connecting to the Recognition Science account of curvature forced by J-cost defect density.
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