Riemann, Repulsion, and the Collapse of Treewidth on the Line

Read this as a measurement-facing companion. The paper does not prove the Riemann Hypothesis and does not assume it. It explains the graph-theoretic signature of zero repulsion: small, slow-growing local occupancy in one-dimensional threshold graphs.

Start with a set of points on a line. Put an edge between two points whenever they are within a fixed distance of each other. This construction sounds like graph theory, but in one dimension it has very little room to hide: the graph is a unit interval graph.

That fact pins the treewidth exactly. The treewidth is not measuring some mysterious global backbone. It is the largest number of points you can fit into any window of the chosen width, minus one. In symbols, for a point set X and threshold delta, tw(G_delta(X)) = M(X, delta) - 1, where M is maximal local occupancy.

Why This Matters for Spectra

Spectra are ordered point sets. Eigenvalues, zeta zeros, and random point processes all sit on a line after unfolding to unit mean spacing. Once you build a distance-threshold graph from them, interval-graph collapse applies. The only remaining question is how crowded the fullest local window becomes as the data set grows.

For an independent Poisson process, crowding grows like log n / log log n. Independent points sometimes clump by chance, and the largest clump grows slowly but visibly. For a repulsive sine-kernel process, the paper proves a much smaller upper bound: delta + O(sqrt(log n)). Repulsion keeps points apart, so the fullest window stays sparse.

The Riemann Connection

The Montgomery-Odlyzko law predicts that the unfolded nontrivial zeros of the Riemann zeta function have the same local statistics as eigenvalues from random matrix theory. If that prediction governs the relevant scale, then the zeros should sit on the repulsive side of the separator-growth divide.

The computation does exactly that. On the first 100,000 unfolded Odlyzko zeros, the measured treewidth at threshold one is 2. A Poisson sample of the same size is around 9; a simulated GUE spectrum is around 3. The zeros behave like a repulsive spectrum by this occupancy-derived treewidth meter.

What Is Proven

• Interval-graph collapse: every one-dimensional threshold graph is a unit interval graph, with treewidth equal to maximal window occupancy minus one.
• Poisson benchmark: independent points have maximal occupancy, hence treewidth, growing like log n / log log n.
• Sine-kernel upper bound: repulsive spectra have almost-sure treewidth at most delta + O(sqrt(log n)) for fixed threshold.
• Measured zeta behavior: the first 100,000 unfolded Riemann zeros land on the repulsive side of the benchmark.

What Remains Open

The sharp extreme law for the sine-kernel process is still conjectural. The upper bound is proved; the matching lower bound and the window-decoupling argument needed for a precise maximum law remain open. The transfer from random sine-kernel theory to the deterministic zeta zeros also rests on the Montgomery-Odlyzko statistical picture.

So the paper should be read as a correction and a measurement theorem, not as a resolution of RH. It identifies the right invariant for one-dimensional spectral threshold graphs and shows where the Riemann zeros sit under that invariant.