research/mathematics

• Essay
  Index-Period Normal Forms for Monoid-Aggregated Recursive Summaries 16 May 2026

  A monoid-aggregated summary evaluates a finite rooted cop-labeled tree bottom-up through a finite state set and a finite commutative child-aggregation monoid. Once the multiplicity observation map and the monoid are fixed, context equivalence has finite index and is exactly equality of a finite behavior vector. This note sharpens the resulting pumping and normal-form theory: the crude pigeonhole bound in the product monoid is replaced by an exact index–period bound on each behavior type’s child contribution, isolating support, modular, and saturation counting in the Boolean, cyclic, and threshold families. Combining exact sibling pumping with a size-minimality argument — no behavior vector may repeat along a root-to-leaf path — yields a finite universe of normal representatives, and an external tie-break selects one canonical representative per class. Worked computations for one-node trees, stars, unary chains, and split-versus-concentrated examples make the bounds concrete.

  research/algebraresearch/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/graph-theory