Web extra · not in the printed book · Part IV territory

The Loop That Remembers

Picture a curved landscape — one hill, one hollow — and a tiny arrow lying flat on the ground. You walk it around a loop, and at every step you press it back down onto the surface so it stays flat, but you never once twist it by hand. This step-and-flatten walk is called parallel transport. Do it honestly and the arrow still comes home rotated — the angle it turned by is the curvature the loop enclosed, and this page computes both sides of that equation independently: the transport by walking, the integral by integrating, and shows you the receipt.

The whole terrain above lives in GIGI as data. The surface is one hill and one hollow — a bump and a dip in an otherwise flat sheet — and every point of it becomes one record, indexed by its (x, z) footprint. Each record carries the fields the picture is painted from: h (height above the sheet), kappa (curvature — positive orange on the hilltop, negative violet on the skirt and saddle), normal (which way the ground faces at that spot), and, once you drag a loop over it, holonomy_debt (how far a carried arrow rotates by the time it walks back to where it started).

Because GIGI is a fiber-bundle database, the curvature (κ) and its confidence ride along with every read — no sidecar service to compute them. Walking the loop and integrating the curvature underneath it are two different calculations that share no code; the queries below show both sides of that equation printing the same number, which is what Gauss–Bonnet has always promised.

surface color = Gaussian curvature · orange positive (hilltop) · violet negative (the skirt & saddle) · drag the white loop to move it
drag empty space to orbit · wheel to zoom

Levers & receipts

transport the frame to print the receipt — the Gauss–Bonnet check that walked rotation equals integrated curvature

Two things to try. First: shrink the loop onto flat ground far from the hill — the deficit dies, because flat transport is trivial (the first of Chapter 11's five named call sites). Second, the sneaky one: grow the loop until it swallows the hill entirely, skirt and all. The deficit heads back toward zero — a bump's positive summit and negative skirt cancel exactly, so a loop that encloses all of it remembers nothing. The arrow’s come-home rotation has a name — holonomy, the memory a loop keeps of the curvature it swept over. Holonomy is not "did I walk near something curved"; it is a signed ledger of exactly what the loop surrounds. In the engine the loop is a cycle of shards and this ledger is holonomy_debt — same math, less scenery.

The same thing in GQL — run it against an engine

Holonomy rides along per bundle; CURVATURE prints the κ ledger the deficit is built from. The tetmesh bundle is live on the public instance right now.

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