The defect of transport fixes the curvature

Transport moves on the path, a tree. A link between two sites closes a cycle exactly when the sites are not neighbours, and every cycle carries one quantum δ∞ = C∞ − 1. Two links make one handle: +2δ∞ of defect and, by Gauss–Bonnet, −4π of curvature. The new cycle is felt only inside the light cone, one site per step.

Handles on the path

genus g1

The new cycle stays inside the cone

steps k0

cycle rank = b₁

defect Ω(Σ_g) = 2g·δ∞

curvature ∫K dA = 2π(2 − 2g)

max |difference| outside the cone

Left: the path on n + 1 sites with the 2g links {0, j + 2} of handleGraph; same colour, same handle. Lean: cycleRank_handleGraph, handleGraph_defect, handle_quanta (Gauss–Bonnet is a declared hypothesis). Right: |(L^k − T^k)ψ| on 41 sites for the transport L with one link {17, 23} and a random state ψ; grey cells are exactly zero, as link_inside_cone proves. Values on this page are computed in your browser. The entropy of a cut, and the entropy of a surface as its curvature, are on the entropy page. source.