Backtracking-weighted PageRank is not coordinatewise monotone: minimal counterexamples, a certified census and endpoint tests

Unrefereed research candidate. Backtracking-weighted PageRank runs a restarted random walk on the directed edges of a graph, giving an immediate return along the edge just used weight μ and every other continuation weight one: μ = 0 is non-backtracking PageRank, μ = 1 is ordinary PageRank and μ → ∞ has a cheap closed form. Glover, Jones, Kempton and Oveson (Applied Mathematics and Computation 531 (2026) 130182; arXiv:2211.13353v2) conjectured that every vertex score is monotone in μ (Conjecture 3.1), which would let the two cheap scores predict the side of the non-backtracking score, and that a regularly varying degree law forces scale-free infinite-backtracking scores (Conjecture 3.2).

This package refutes both. On the five-vertex house graph at damping 1/2 the roof vertex has exact scores 638/3615, 22/125 and 8/45 at μ = 0, 1, ∞, so it first loses and then regains score. A closed cubic formula shows that the roof is monotone iff the damping is at most ε_c ≈ 0.2407, with one strict minimum otherwise, and identifies exactly when the endpoint shortcut points the wrong way (ε > 0.3291). Among graphs of minimum degree two, five vertices is the smallest order for every damping value (house and gem); with leaves allowed the four-vertex paw fails for damping outside an explicit interval [ε₁, ε₂] ≈ [0.078, 0.436], so the smallest order is four there and five on that interval (Corollary 5.4), and even a five-vertex tree has a nonmonotone leaf (Proposition 5.5). Failures lift through graph covers to arbitrarily large graphs. An exact census of all 995 connected graphs on 2–7 vertices at five damping values (exact rational interpolation and exact Sturm root counts on open intervals, no floating point) finds a nonmonotone vertex in 675 of the 853 seven-vertex graphs; it is certified at those five damping values only. Both readings of the conjecture's spread clause fail exactly, and the reversed reading fails even when the extreme vertices are unique (Proposition 8.2). A two-step criterion explains small-damping dips, an exact slope formula at μ = 1 gives three cheap sufficient tests, and the full score vector is invariant iff P²v = v. Conjecture 3.2 fails for every tail index above one, by a clique-chain construction adapted from Henning, van der Hofstad and Litvak. Numerical experiments find 10–41% of vertices nonmonotone on standard networks, with small excursions and stable rankings.

The archive contains the 17-page manuscript (PDF and LaTeX), Python verification code (python-flint, sympy, networkx), exact JSON results, a separate numerical prevalence study, semantic negative controls, CI and SHA-256 sums. Assurance boundary: written proofs; exact certificates; a byte-identical producer-side replay of all exact results from the frozen archive; internal AI-assisted review, AI-assisted adversarial audits by a different model before and after revision (the second found and led to the repair of a census root-counting error), and a verification re-review. This is not peer review, independent reproduction or formal verification. No proof assistant was used. The census covers five damping values; the conjecture wording was checked against arXiv v2 only because the journal text was inaccessible; the configuration-model tail transfer is left open; prevalence results are numerical. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.