The Interval-Graph Collapse: Separator Growth in One-Dimensional Point Processes

We prove a collapse theorem: for any locally finite point set X ⊂R and any δ > 0, the distance-threshold graph Gδ(X) is a unit interval graph whose treewidth, pathwidth, and clique number are all determined by a single quantity — the maximal number of points in a window of width δ. On the line, separator width is extreme local occupancy; no graph-decomposition parameter of a one-dimensional point set carries information beyond its counting statistics. The theorem is elementary, but it does structural work in both directions. Downward, it explains why no literature exists on width parameters of spectra, zeros, or eigenvalue sequences: the object degenerates before it becomes graph theory. Upward, it converts separator growth into a well-posed question about extreme counting statistics, where it cleanly separates universality classes: for unit-intensity Poisson processes, tw(Gδ) = Θ(log n/log log n) by classical extreme-value theory, while for the determinantal sine-kernel process the Bernoulli decomposition of counts together with the Landau–Widom eigenvalue plunge gives tw(Gδ) ≤δ+O(√log n) almost surely. Spectral repulsion is visible as a polynomial slowdown in separator growth: occupancy-derived treewidth growth distinguishes the Poisson and sine-kernel universality classes. Numerics on 106 Poisson points, 105 unfolded GUE bulk eigenvalues, and (as a natural-data validation set) Odlyzko’s first 105 Riemann zeros confirm the separation. The collapse is strictly one-dimensional: in the plane, threshold graphs are unit-disk graphs whose treewidth is not a window statistic, and we close by posing separator growth for hyperuniform planar processes as the first setting where the question becomes genuinely graph-theoretic.