Varifold Lifts of Visibility Graphs: Beyond Fractality and the Geometry of Safe Haven Decoupling in Commodity and Currency Markets

Visibility graphs map time series to networks whose combinatorial structure encodes fractality, recovering the Hurst exponent of self-affine processes. We ask what the visibility construction carries beyond this fractal content. We lift the visibility graph to a 1-varifold, a measure on position and direction space from geometric measure theory, and equip it with a multiscale positive definite kernel. The lift embeds visibility graphs of unequal size in a common Hilbert space and yields a channel-resolved measure of cross-series geometric alignment. On a 25.8-year daily panel of thirteen commodity and currency layers, we define a relative alignment contrast that compares commodity currencies and safe haven currencies in their geometric alignment with the commodity complex. During global risk-off episodes the contrast is large and positive: commodity currencies import commodity shock geometry far beyond a time-shift independence benchmark, while the Japanese yen remains near geometric independence and the franc is confounded by a managed regime. The contrast is significant under three stress definitions with autocorrelation robust inference, holds as a continuous dose response, survives the removal of any single crisis, withstands moment, fractal, and topological controls, is direction-consistent across sixteen specifications, and collapses under a time-shift placebo. Detrended fluctuation analysis explains only two percent of it, so the reconfiguration is geometric information beyond fractality at this horizon, and a scaling exponent of the kernel mass separates a fractal-free component from a fractal-driven one. For investors, financial institutions, and policymakers, the contrast is a real-time structural diagnostic of flight to safety: it marks when commodity currencies stop diversifying the commodity complex while genuine safe havens still do, signaling through a channel that second-moment risk measures are built to miss.