research/graph-theory

• Essay
  Branch-Based Local Capture in Tree-Ball Geometry: Sharp Positive and Negative Results 6 May 2026

  We study a local team-chase problem in a $d$-regular graph whose ball of large radius around the robber is a tree. We isolate the right local invariant — the deep load along a nonbacktracking path tube — and prove a coordinated package of positive and negative results: (i) sharp counts of geodesic cones and their truncations, (ii) a one-round universal branch-persistence lemma, (iii) a $t$-round generalization with depth budget $2t+1$, (iv) a sampling barrier showing that any proof relying on the path-tube certificate requires $\Omega((d-1)^{t-1} \cdot t)$ cops, and (v) a finite-order potential-degeneration barrier showing that any local invariant depending only on order-$r$ tube data is strictly insufficient to certify even $(r+1)$-round persistence. Together, these results form a double pincer: certifying $t$-round persistence by an order-$r$ local invariant requires $r \geq t$, and order-$t$ resolution requires $\Omega((d-1)^{t-1})$ cops to occupy by uniform sampling. This rules out order-$r$ branch-aggregated potentials (for any fixed $r$) as a route to $\Theta(\log n)$-round chase from polylogarithmic cops, and identifies the precise structural reason why the natural iteration of the one-round argument fails.

  research/mathematics