parallel_transport_cert_minkowski
plain-language theorem explainer
Minkowski spacetime satisfies the parallel transport certificate, so vectors transported along any curve remain constant and inner products stay invariant. Researchers working on flat limits or base cases in Recognition Science would cite this to anchor trivial holonomy before introducing J-cost defects. The proof is a term-mode structure builder that directly invokes the flat-transport theorem for the constancy clause and the inner-product lemma for preservation.
Claim. The Minkowski metric tensor $eta$ satisfies the parallel transport certificate: for every spacetime curve $gamma$ and vector field $V:mathbb{R}tomathbb{R}^4$, if $V$ is parallel transported with respect to $eta$ then each component of $V$ has vanishing derivative along $gamma$, and the inner product $eta(V,W)$ remains constant along any curve whenever both $V$ and $W$ are parallel transported.
background
ParallelTransportCert(g) is the structure requiring two properties for a metric tensor g: flat_trivial asserts that when g equals the Minkowski tensor, any parallel-transported vector field is constant (its components have zero derivative), while inner_product_preserved asserts that the inner product is invariant along curves. The module formalizes parallel transport on 4D spacetime via the Levi-Civita connection and interprets curvature as the geometric signature of non-uniform J-cost defect density in Recognition Science. Upstream, parallel_transport_flat states that Christoffel symbols vanish in Minkowski spacetime so parallel transport reduces to constancy, and minkowski_preserves_inner shows the inner product derivative is identically zero when both vectors are constant.
proof idea
The proof is a term-mode construction of the ParallelTransportCert structure for minkowski_tensor. It populates the flat_trivial field by simplifying under the equality hypothesis g = minkowski_tensor and applying parallel_transport_flat, then populates the inner_product_preserved field by direct reference to the theorem minkowski_preserves_inner.
why it matters
This result supplies the flat-space base case for the parallel transport certificate, confirming that holonomy vanishes exactly when curvature is zero. It supports the Recognition Science claim that curvature (and therefore gravity) is generated solely by J-cost defect imbalance, as the Minkowski limit exhibits trivial transport. The declaration aligns with the framework's T8 forcing of spatial dimension and the RCL composition law by isolating the defect-free case before curvature is introduced via non-uniform J-cost.
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