diff --git a/content/about.md b/content/about.md
index af37dbb..ecfc4fa 100644
--- a/content/about.md
+++ b/content/about.md
@@ -32,7 +32,7 @@ Computer vision and security thread through all three but do not stand on their
### Published / In Submission
- **Shu L, Neuwirth L†, Wang X†, Zheng H†.** *Beyond Comorbidity Indices: An Order-Invariant ICD-10-CM Embedding for Readmission and Mortality Prediction.* Under review at the *Journal of the American Medical Informatics Association* (JAMIA), 2026. [Preprint](/essays/beyond-comorbidity-indices/) · [Calculator](https://levineuwirth.github.io/icd_embeddings/) · [Code](https://github.com/levineuwirth/icd_embeddings)
-- **Neuwirth L.** *Branch-Based Local Capture in Tree-Ball Geometry: Sharp Positive and Negative Results.* Preprint, arXiv (forthcoming), May 2026. [Preprint](/essays/branch-based-local-capture-in-tree-balls/)
+- **Neuwirth L.** *Branch-Tube Persistence and Static Coverage in Tree-Ball Geometry.* Preprint, July 2026. [Preprint](/essays/branch-based-local-capture-in-tree-balls/)
- **Neuwirth L.** *Where Does SIMD Help Post-Quantum Cryptography? A Micro-Architectural Study of ML-KEM on x86 AVX2.* Technical report, Brown University Department of Computer Science, April 2026. [Report](/essays/where-does-simd-help-post-quantum-cryptography/) · [Artifact](https://git.levineuwirth.org/neuwirth/where-simd-helps)
### In Preparation / In Progress
diff --git a/content/essays/branch-based-local-capture-in-tree-balls/index.md b/content/essays/branch-based-local-capture-in-tree-balls/index.md
index 4b10492..ab39cc8 100644
--- a/content/essays/branch-based-local-capture-in-tree-balls/index.md
+++ b/content/essays/branch-based-local-capture-in-tree-balls/index.md
@@ -1,8 +1,22 @@
---
-title: "Branch-Based Local Capture in Tree-Ball Geometry: Sharp Positive and Negative Results"
-date: 2026-05-06
+title: "Branch-Tube Persistence and Static Coverage in Tree-Ball Geometry"
+date: 2026-07-21
abstract: >
- 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.
+ We analyze exhaustive static coverage by path tubes indexed by length-$t$
+ nonbacktracking robber paths from $v_0$ in a finite-horizon local chase on a
+ $d$-regular graph. An endpoint-sensitive geodesic lemma shows that, among
+ possibly infinite $d$-regular graphs, radius $R+t$ is sharp for arbitrary
+ pairs in $B_R(v_0)\times B_t(v_0)$, while synchronized witnesses along a
+ prescribed path require only a radius-$R$ tree-ball. If every tube is
+ occupied, each surviving round along every path in this class ends in
+ capture, blockage, or branch-load support on at least two branches. The
+ tubes partition the outer ball, so deterministic coverage has minimum cost
+ $N_t=d(d-1)^{t-1}$; conditional on a specified root, i.i.d. uniform
+ coverage has threshold $\Theta(N_t\log N_t)$ for fixed $d$. An augmented
+ prefix-depth profile that retains the complete shallow configuration still
+ need not determine later support. The result is local and root-dependent:
+ it treats neither arbitrary robber walks nor a robber-independent cop
+ strategy, and it gives no cop-number bound.
tags:
- research
- research/mathematics
@@ -20,549 +34,344 @@ evidence: 5
peer-status: unreviewed
result-shape: mixed
further-reading:
- - NowakowskiWinkler
- Quilliot
+ - NowakowskiWinkler
- AignerFromme
- Frankl
- LuPeng
- ScottSudakov
- FriezeKrivelevichLoh
- PralatWormald
+ - BradshawHosseiniMoharStacho
+ - HMG
---
-## Introduction
+# Introduction
-### Motivation
+## Motivation
-The cops-and-robbers game is a [pursuit-evasion](https://en.wikipedia.org/wiki/Pursuit-evasion) game on a graph in which $k$ cops attempt to capture a single robber. The minimum $k$ for which the cops have a winning strategy is the *cop number* $c(G)$. [*Meyniel's conjecture*](https://en.wikipedia.org/wiki/Meyniel%27s_conjecture) (Frankl, 1987) asserts that $c(G) = O(\sqrt{n})$ for every connected graph on $n$ vertices, and remains the central open problem in the area. The current best upper bound is $c(G) \leq n / 2^{(1-o(1))\sqrt{\log n}}$ (Lu–Peng; Scott–Sudakov; Frieze–Krivelevich–Loh).
+The cops-and-robbers game, introduced independently in its one-cop form by Quilliot and by Nowakowski–Winkler and developed in its multiple-cop form by Aigner–Fromme [@Quilliot; @NowakowskiWinkler; @AignerFromme], is a pursuit-evasion game on a graph in which $k$ cops attempt to capture one robber. The minimum such $k$ is the *cop number* $c(G)$. *Meyniel's conjecture*, attributed to Henri Meyniel and appearing in the early literature in Frankl's work [@Frankl], asserts that $c(G)=O(\sqrt n)$ for every connected graph on $n$ vertices and remains a central open problem in the area. The best general upper bound, $$c(G) \leq \frac{n}{2^{(1-o(1))\sqrt{\log_2 n}}},$$ was proved independently by Lu–Peng, Scott–Sudakov, and Frieze–Krivelevich–Loh [@LuPeng; @ScottSudakov; @FriezeKrivelevichLoh].
-A natural approach toward Meyniel's conjecture, in the spirit of Prałat–Wormald's resolution for random graphs and recent partial progress on expanders and high-girth graphs[@BradshawHosseiniMoharStacho; @HMG], is to combine random cop placement with a deterministic local chase. The bottleneck of this approach is the *local team-chase problem*: assuming several cops have been placed within a small distance $r$ of the robber, can they coordinate their gradient-descent moves to capture the robber within $O(r \log n)$ rounds?
+Prałat–Wormald's random-graph argument combines an initial random cop placement with assignments made after the robber's start is revealed: in different regimes, cop teams fully occupy a neighborhood or densely cover a sphere [@PralatWormald]. Tree-like geometry enters the literature in several other ways. Aigner–Fromme show that one moving cop can guard a fixed isometric path [@AignerFromme]. Frankl established a high-girth lower bound [@Frankl], and Bradshaw–Hosseini–Mohar–Stacho refine that line through a branch-weight argument in which unique local geodesics let the robber choose a sufficiently lightly controlled forward branch [@BradshawHosseiniMoharStacho]. Related bounded-degree reductions provide broader extremal context [@HMG]. These mechanisms motivate a path-conditioned local question: whether cops near the robber can coordinate their distance-decreasing moves along every prescribed length-$t$ nonbacktracking path starting at the robber's initial vertex.
-In [high-girth](https://en.wikipedia.org/wiki/Girth_(graph_theory)) regular graphs, where the local geometry is tree-like, the difficulty of this problem is structural: a single cop is always evaded (Aigner–Fromme), so capture requires multi-cop coordination, and the right coordination invariant is not obvious. In this paper, we work in the cleanest possible local geometry — a $d$-regular tree-ball — and identify the right invariant, prove that it works for one round, prove that it generalizes to $t$ rounds, and prove a matching barrier.
+The present paper instead isolates one rigid, root-dependent certificate: every length-$t$ nonbacktracking path from $v_0$ indexes an occupied descendant tube. Paths here index a static partition of the outer ball rather than a route guarded by one moving cop or an adaptively selected escape branch. We determine this certificate's guarantee and cost. Arbitrary robber walks, including stationary moves and reversals, are not analyzed, and exact tree geometry is not asserted to be necessary.
-### Setup
+The uniform sampling below is conditional on the fixed root $v_0$: the root is specified before cop positions are sampled from $B_R(v_0)$. This measures the local cost of the certificate, not a legal initial placement in the standard game, where cops choose positions before the robber chooses its start. A global strategy would need comparable coverage simultaneously or adaptively for an unknown robber position. Accordingly, the results neither construct a robber-independent global strategy nor improve the cop-number bound.
-Throughout, $G$ denotes a $d$-[regular graph](https://en.wikipedia.org/wiki/Regular_graph) with $d \geq 3$. We fix a vertex $v_0 \in V(G)$ and a radius $R \geq 1$. The fundamental hypothesis is:
+## Setup and endpoint-sensitive tree-ball geodesics
-::: {.annotation .annotation--static #def-tree-ball}
-
-Definition 1.1
-Tree-ball
-
-
-The ball $B_R(v_0)$ is a *tree-ball* if the induced subgraph on $B_R(v_0)$ is a tree. Equivalently, $\mathrm{girth}(G) > 2R$.
-
+Graphs are simple and undirected. The local results allow finite or infinite graphs unless finiteness is stated explicitly; discussion of the cop number and all $n$-vertex asymptotics concerns finite connected graphs. Throughout, $G$ denotes a $d$-regular graph with $d\geq3$, and $v_0\in V(G)$ is the robber's initial position in the rooted local experiment. All radii and time horizons are nonnegative integers, and $t,r\geq1$ whenever length-$t$ or order-$r$ objects are used. All unadorned logarithms are natural. We write $$B_R(v)=\{x\in V(G):\operatorname{dist}(x,v)\leq R\},
+\qquad
+S_j(v)=\{x\in V(G):\operatorname{dist}(x,v)=j\}.$$
+
+::: {#def-tree-ball .definition}
+**Definition 1** (Tree-ball). The ball $B_R(v_0)$ is a *tree-ball* if the induced subgraph $G[B_R(v_0)]$ is a tree.
:::
-In a tree-ball, $v_0$ has $d$ children (its neighbors), and every interior vertex has exactly one parent and $d-1$ children. We write $S_j(v) = \{x : \operatorname{dist}(x, v) = j\}$ for the depth-$j$ shell at $v$.
+This is a local condition at the specified center. The global condition $\operatorname{girth}(G)>2R+1$ implies that every radius-$R$ ball is a tree-ball. Conversely, if every radius-$R$ ball is a tree-ball, then $\operatorname{girth}(G)>2R+1$.
-::: {.annotation .annotation--static #def-cone}
-
-Definition 1.2
-Geodesic cone
-
-
-For $v \in V(G)$ with $B_r(v)$ a tree-ball and $u \in N(v)$, the *[geodesic](https://en.wikipedia.org/wiki/Distance_(graph_theory)) cone* through $u$ at radius $r$ is
+An induced tree-ball does not control shortest paths between arbitrary pairs of its vertices: a competing path may leave the ball and re-enter. The required radius depends on the endpoint depths and on the length of their path inside the tree-ball.
-$$C_u(v, r) := \{x \in B_r(v) \setminus \{v\} : \text{the unique geodesic from } x \text{ to } v \text{ has penultimate vertex } u\}.$$
-
-The *truncated cone at depth $\geq k$* is
-
-$$C_u^{\geq k}(v, r) := \{x \in C_u(v, r) : \operatorname{dist}(x, v) \geq k\}.$$
-
+::: {#lem-buffer .lemma}
+**Lemma 2** (Endpoint-sensitive tree-ball geodesics). *Suppose $B_q(v_0)$ is a tree-ball, let $x,y\in B_q(v_0)$, and let $L$ be the length of the $x$–$y$ path in $G[B_q(v_0)]$. If $$\operatorname{dist}(x,v_0)+\operatorname{dist}(y,v_0)+L\leq 2q,$$ then every ambient $x$–$y$ geodesic lies in $B_q(v_0)$. Consequently, that geodesic is unique and equals the $x$–$y$ path in $G[B_q(v_0)]$.*
:::
-::: {.annotation .annotation--static #def-tube}
-
-Definition 1.3
-Path tube
-
-
-For $v_0 \in V(G)$ with $B_R(v_0)$ a tree-ball, and a nonbacktracking path $\sigma = (v_0, v_1, \ldots, v_t)$ from $v_0$, and a depth threshold $k$, the *$t$-tube at depth $\geq k$* is
-
-$$\begin{gathered}
-T_\sigma^{\geq k}(R) := \bigl\{x \in B_R(v_0) : \operatorname{dist}(x, v_0) \geq k, \\
-\text{the unique geodesic from } x \text{ to } v_0 \text{ visits } v_t, v_{t-1}, \ldots, v_1 \text{ in order}\bigr\}.
-\end{gathered}$$
-
+::: proof
+*Proof.* Let $P$ be an ambient $x$–$y$ geodesic of length $\ell$. The path in $G[B_q(v_0)]$ gives $\ell\leq L$. For $z\in P$, write $h=\operatorname{dist}_P(x,z)$. Then $$\operatorname{dist}(z,v_0)
+\leq \min\{\operatorname{dist}(x,v_0)+h,\ \operatorname{dist}(y,v_0)+\ell-h\}
+\leq \frac{\operatorname{dist}(x,v_0)+\operatorname{dist}(y,v_0)+\ell}{2}
+\leq q.$$ Thus $P\subseteq B_q(v_0)$. Since the induced graph on this ball is a tree, $P$ is its unique $x$–$y$ path. □
:::
-The cops execute *gradient descent*: at each cop turn, every cop moves to a neighbor minimizing distance to the current robber position, breaking ties by an arbitrary rule (deterministic or randomized). The order of play is: cops move, then robber moves. A robber move to $w \in N(v(t))$ is *safe* if no cop occupies $w$ at the start of the robber's turn (i.e., after the cops' move).
+::: remark
+**Remark 3** (Uniform buffers versus chase endpoints). For arbitrary $x\in B_R(v_0)$ and $y\in B_t(v_0)$, the walk from $x$ to $y$ through $v_0$ has length at most $R+t$. If $B_{R+t}(v_0)$ is a tree-ball, the path between $x$ and $y$ in the rooted tree $G[B_{R+t}(v_0)]$ is no longer than this walk, so Lemma [2](#lem-buffer) applies. For $R,t\geq1$, this uniform radius cannot in general be reduced. Put $L=R+t$, begin with the rooted $d$-regular tree truncated at depth $L-1$, and choose vertices $x$ and $y$ at depths $R$ and $t$ in distinct root branches. Choose descendants $a$ of $x$ and $b$ of $y$ in $S_{L-1}(v_0)$, allowing equality, add a new vertex $z$, and join $z$ to $a$ and $b$.
-For the robber's position $v$ and a neighbor $u' \in N(v)$, we say a cop $c$ *contributes to branch $u'$ at $v$* if $c \neq v$ and the geodesic from $c$ to $v$ has penultimate vertex $u'$. The *branch-load support* at $v$ is the set of $u' \in N(v)$ that receive at least one contributing cop.
+Complete the graph explicitly: for each remaining degree deficit at a vertex of $S_{L-1}(v_0)\cup\{z\}$, attach by one edge a separate infinite rooted $(d-1)$-ary tree. Every vertex then has degree $d$, and the resulting graph is infinite, simple, and connected. The induced ball $B_{L-1}(v_0)$ is still the original tree. The path from $x$ to $y$ through $v_0$ has length $L$, while the path through $z$ has length $$\operatorname{dist}(x,a)+2+\operatorname{dist}(b,y)=(t-1)+2+(R-1)=L.$$ Each attached infinite tree meets the displayed core at only one vertex, so it creates no additional route between $x$ and $y$. Thus the two displayed paths are distinct geodesics. This proves sharpness at radius $R+t-1$ among possibly infinite $d$-regular graphs.
-### Results
-
-The paper has four main results, prefaced by a sharp counting lemma.
-
-::: {.annotation .annotation--static #lem-shell}
-
-Lemma 1.4
-Sharp shell and cone counts
-
-
-Let $G$ be $d$-regular with $d \geq 3$, and suppose $B_r(v)$ is a tree-ball. Then for every $u \in N(v)$ and $1 \leq j \leq r$,
-
-$$|C_u(v, r) \cap S_j(v)| = (d-1)^{j-1}, \qquad |S_j(v)| = d(d-1)^{j-1}.$$
-
-Hence
-
-$$|C_u(v, r)| = \sum_{j=1}^{r} (d-1)^{j-1} = \frac{(d-1)^r - 1}{d-2}, \quad |B_r(v)| = 1 + d \cdot \frac{(d-1)^r - 1}{d - 2}.$$
-
-For the truncated cone,
-
-$$|C_u^{\geq k}(v, r)| = \sum_{j=k}^{r} (d-1)^{j-1}, \quad 1 \leq k \leq r.$$
-
+The endpoint pairs used along a prescribed length-$t$ nonbacktracking path are more restricted. Their rooted-tree path and endpoint depths satisfy Lemma [2](#lem-buffer) with $q=R$, so the persistence result requires no tree-ball beyond the cop-placement radius. This does not assert that the radius-$R$ tree-ball hypothesis is necessary. Controlled cyclic geometry is not addressed here, and no cyclic extension is proved.
:::
-The first main result is a probabilistic deep-load corollary.
+Whenever $B_q(v_0)$ is a tree-ball, we root $G[B_q(v_0)]$ at $v_0$. Every vertex other than $v_0$ has a unique parent, and every vertex of depth less than $q$ has $d-1$ children.
-::: {.annotation .annotation--static #cor-occupancy}
-
-Corollary 1.5
-Deep-load occupancy
-
-
-Let $G$ be $d$-regular with $B_r(v)$ a tree-ball, and let $X_1, \ldots, X_m$ be [i.i.d.](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables) uniform samples from $B_r(v)$. Set
-
-$$p_{d,r}^{\geq k} := \frac{|C_u^{\geq k}(v, r)|}{|B_r(v)|},$$
-
-which by [Lemma 1.4](#lem-shell) does not depend on the choice of $u \in N(v)$, since all branches in a tree-ball have equal size. Then
-
-$$\Pr[\exists u \in N(v): C_u^{\geq k}(v, r) \text{ receives no sample}] \leq d \cdot (1 - p_{d,r}^{\geq k})^m.$$
-
+::: {#def-cone .definition}
+**Definition 4** (Geodesic cone). Suppose $B_r(v)$ is a tree-ball and $u\in N(v)$. The *geodesic cone* through $u$ at radius $r$ is $$C_u(v,r):=\{x\in B_r(v)\setminus\{v\}:\text{the path from $x$ to $v$ in $B_r(v)$ has penultimate vertex $u$}\}.$$
:::
-The second is the universal $t$-round persistence theorem with the correct depth budget.
-
-::: {.annotation .annotation--static #thm-persistence}
-
-Let $G$ be $d$-regular with $d \geq 3$, $v_0 \in V(G)$, and assume $B_R(v_0)$ is a tree-ball for some $R \geq 2t+1$. Suppose that for every nonbacktracking path $\sigma = (v_0, v_1, \ldots, v_t)$ of length $t$ from $v_0$, there exists at least one cop in $T_\sigma^{\geq 2t+1}(R)$.
-
-Then for every nonbacktracking robber path $(v_0, v_1, \ldots, v_t)$ and every $s \in \{1, \ldots, t\}$, after the cops' gradient-descent step at round $s$, exactly one of the following holds:
-
-- **(a)** a cop occupies $v_s$, so the robber's intended move to $v_s$ is illegal; or
-- **(b)** the move to $v_s$ is legal, and after the robber moves to $v_s$, the branch-load support at $v_s$ has size at least $2$.
-
+::: {#def-tube .definition}
+**Definition 5** (Path tube). Let $R\geq t$, suppose $B_R(v_0)$ is a tree-ball, and let $$\sigma=(v_0,v_1,\ldots,v_t)$$ be a length-$t$ nonbacktracking path starting at $v_0$: consecutive vertices are adjacent and $v_{i+1}\neq v_{i-1}$ whenever the latter condition is defined. The *length-$t$ tube* associated with $\sigma$ is $$T_\sigma(R):=\{x\in B_R(v_0):\text{the rooted path from $x$ to $v_0$ contains $v_t,v_{t-1},\ldots,v_1$ in order}\}.$$ Equivalently, $T_\sigma(R)$ is the set of descendants of $v_t$ lying in $B_R(v_0)$.
:::
-The third is the sampling barrier.
+## Game conventions
-::: {.annotation .annotation--static #thm-barrier}
-
-Theorem 1.7
-Barrier on branch-tube certificates
-
-
-Fix $d \geq 3$. In the $d$-regular tree-ball model, any argument that places $m$ cops by uniform sampling from $B_R(v_0)$ and relies on occupancy of every length-$t$ deep tube $T_\sigma^{\geq 2t+1}(R)$ requires
+A *cop configuration* $X$ is a finite multiset of vertices, represented by its finitely supported multiplicity function $X(v)\in\mathbb Z_{\geq0}$. For $A\subseteq V(G)$, set $$X(A):=\sum_{v\in A}X(v),
+\qquad
+|X|:=X(V(G)),
+\qquad
+\mathop{\mathrm{supp}}(X):=\{v:X(v)>0\}.$$ Thus counts of cops include multiplicity, several cops may occupy one vertex, and $A$ is occupied exactly when $X(A)>0$. We write $X=[x_1,\ldots,x_k]$ for a multiset of cop positions, with repeated entries allowed. Independent uniform sampling is with replacement and produces the multiset of sampled positions.
-$$m = \Omega\!\left((d-1)^{t-1} \cdot t\right)$$
+Capture occurs whenever a cop and the robber occupy the same vertex, including before the first cop move; once capture occurs, no further move is made. Conditional on no capture, the cops move first in each round, simultaneously, and each cop moves to a neighboring vertex minimizing its distance to the robber's current vertex. Ties may be broken arbitrarily. Some formulations also allow cops to pass; the results remain valid there by having the selected witnesses make the prescribed distance-decreasing moves. Under the tree-ball hypotheses below, Lemma [2](#lem-buffer) shows that each witness selected in our proofs has a unique distance-decreasing move.
-cops. In particular, for $m = \mathrm{polylog}(n)$ and fixed $d$, this method certifies only $t = O(\log\log n)$ rounds, not $t = \Theta(\log n)$.
-
+We analyze only length-$t$ nonbacktracking robber paths starting at $v_0$. In the full game a robber walk may reverse an edge or, under common conventions, remain stationary. Neither behavior is covered here. Fix a prescribed path $$\tau=(v_0,v_1,\ldots,v_t).$$ It survives through round $0$ precisely when no initial capture occurs. If it has survived through round $s-1$, then the robber is at $v_{s-1}$ when round $s$ begins. After the cops move toward $v_{s-1}$, the robber is captured if a cop occupies $v_{s-1}$. Otherwise the prescribed move $v_{s-1}\to v_s$ is *legal* if no cop occupies $v_s$ after the cop move. The path survives through round $s$ if no capture or blockage has occurred through that move.
+
+For a robber position $v$ and a neighbor $u\in N(v)$, an individual cop at $c\neq v$ whose $c$–$v$ geodesic is unique *contributes to branch $u$ at $v$* if that geodesic has penultimate vertex $u$. The *branch-load support* at $v$ is the set of branches receiving at least one contributing cop; its size counts branches, not cops.
+
+## Main results and scope
+
+Although $R+t$ is the sharp uniform radius for arbitrary endpoint pairs in $B_R(v_0)\times B_t(v_0)$, the synchronized pairs arising along a prescribed length-$t$ nonbacktracking path satisfy the endpoint-sensitive criterion already at radius $R$. Accordingly, the persistence theorem assumes only that the cop-placement ball $B_R(v_0)$ is a tree-ball.
+
+Exact rooted-tree counts identify first-level cones as the $t=1$ path tubes and show that the length-$t$ tubes partition the outer ball. Conditional on the prescribed path continuing, a witness's initial depth determines a potential capture or blockage time within the horizon or certifies descendant pressure throughout it. This yields a persistence theorem uniform over all length-$t$ nonbacktracking paths starting at $v_0$, and only over that class.
+
+The next results quantify the price of exhaustive static coverage. Exactly $$N_t=d(d-1)^{t-1}$$ cops are necessary and sufficient to occupy every tube at a fixed root. Along fixed-degree sequences with $t\to\infty$, conditional uniform sampling has a threshold of order $N_t\log N_t$; its leading factor depends explicitly on $R-t$ and tends to $1$ when $R-t\to\infty$.
+
+Finally, Section [5](#sec-profile-loss) records information lost by finite-order tube profiles. Even after retaining every shallow cop multiplicity, such a profile need not determine branch support after $r+1$ rounds. This exact nondeterminacy does not rule out one-sided certificates that reject ambiguous profile fibers.
+
+# Cone and Tube Counts
+
+::: {#lem-shell .lemma}
+**Lemma 6** (Sharp shell and cone counts). *Let $G$ be $d$-regular with $d\geq3$, and suppose $B_r(v)$ is a tree-ball. Then for every $u\in N(v)$ and $1\leq j\leq r$, $$|C_u(v,r)\cap S_j(v)|=(d-1)^{j-1},
+\qquad
+|S_j(v)|=d(d-1)^{j-1}.$$ Hence $$|C_u(v,r)|=\sum_{j=1}^r(d-1)^{j-1}
+=\frac{(d-1)^r-1}{d-2},$$ and $$|B_r(v)|=1+d\frac{(d-1)^r-1}{d-2}.$$*
:::
-The fourth is a complementary degeneration barrier showing that bounded-order compressions of the path-tube data are themselves insufficient. [Theorem 4.6](#thm-finite-order-barrier) (stated and proved in [Section 4.5](#a-finite-order-potential-degeneration-barrier)) constructs, for every $r \geq 1$, two cop configurations $X$ and $Y$ that agree on all order-$r$ tube data but differ in their post-$(r+1)$-round branch-load support: $X$ has support $\geq 2$ while $Y$ has support exactly $1$. Hence no certificate depending only on order-$r$ data can certify universal $(r+1)$-round persistence.
-
-### Strategic significance
-
-[Theorems 1.6](#thm-persistence), [1.7](#thm-barrier), and [4.6](#thm-finite-order-barrier) together capture the precise reach of branch-tube methods. The natural iteration of the one-round argument extends to $t$ rounds with the correct depth budget $2t+1$ ([Theorem 1.6](#thm-persistence)), but the path-entropy cost is exponential in $t$ ([Theorem 1.7](#thm-barrier)), and any compression to bounded-order data fails dynamically ([Theorem 4.6](#thm-finite-order-barrier)). Together, the two barriers tighten this trade-off: 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.
-
-The barriers rule out a substantial family of proof techniques but are silent on what could replace them. The pivots in [Section 5](#discussion-pivots-and-open-directions) sketch the most natural escape routes: epoch refresh (which probably needs a global progress measure to succeed), soft potentials that necessarily break locality or symmetry, and a non-local spectral approach via nonbacktracking-walk concentration.
-
-## Sharp Counts
-
-### Proof of Lemma 1.4
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Sharp shell and cone counts" data-exhibit-type="proof" data-exhibit-caption="Counting tree-ball shells, cones, and truncated cones via the (d−1)-ary branch structure."}
-
-:::: exhibit-body
-Since $B_r(v)$ is a tree-ball and $G$ is $d$-regular, $v$ has exactly $d$ children in the rooted tree (one per neighbor in $N(v)$), and every non-root vertex at depth $< r$ has exactly $d-1$ children. Fix $u \in N(v)$. The branch rooted at $u$ is a $(d-1)$-ary rooted tree of depth $r - 1$; its level $j-1$ has $(d-1)^{j-1}$ vertices, so the depth-$j$ shell of the branch (relative to $v$) has $(d-1)^{j-1}$ vertices.
-
-Hence $|C_u(v, r) \cap S_j(v)| = (d-1)^{j-1}$ for $1 \leq j \leq r$. Summing over the $d$ branches gives $|S_j(v)| = d(d-1)^{j-1}$, and summing $j$ from 1 to $r$ gives the cone size. Including $v$ itself yields the ball size, and restricting $j \geq k$ yields the truncated cone count. [□]{.proof-qed}
-::::
-
+::: proof
+*Proof.* The branch rooted at $u$ is a $(d-1)$-ary rooted tree of depth $r-1$. Its level $j-1$ contains $(d-1)^{j-1}$ vertices, giving the first formula. Summing over the $d$ branches gives the shell count, and the remaining identities follow by summing the geometric series. □
:::
-::: {.annotation .annotation--static #rem-ratio}
-
-Remark 2.1
-Asymptotic branch ratio
-
-
-The ratio $|C_u^{\geq k}(v, r)| / |B_r(v)|$ tends to $1/d$ as $r \to \infty$ for fixed $k$. Thus each branch asymptotically owns a $1/d$ fraction of the ball, and depth-$k$ truncation costs only $O(d^k)$ vertices out of $\Theta(d^r)$, which is negligible at large $r$.
-
+For $t=1$, geodesic cones are exactly the path tubes. We now count the general length-$t$ objects that drive the persistence argument.
+
+::: {#lem-tube-partition .lemma}
+**Lemma 7** (Tube count and partition). *Let $R\geq t$, and suppose $B_R(v_0)$ is a tree-ball. For every length-$t$ nonbacktracking path $\sigma$ starting at $v_0$, $$|T_\sigma(R)|
+=\sum_{j=t}^R(d-1)^{j-t}
+=\frac{(d-1)^{R-t+1}-1}{d-2}.$$ The length-$t$ tubes are pairwise disjoint and form a partition $$B_R(v_0)\setminus B_{t-1}(v_0)
+=\bigsqcup_{\sigma}T_\sigma(R),$$ where $\sigma$ ranges over all length-$t$ nonbacktracking paths from $v_0$. Their number is $$N_t=d(d-1)^{t-1}.$$*
:::
-### Proof of Corollary 1.5
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Deep-load occupancy" data-exhibit-type="proof" data-exhibit-caption="Union bound over the d branches at v."}
-
-:::: exhibit-body
-For a fixed branch $u \in N(v)$, the probability that no $X_i$ lies in $C_u^{\geq k}(v, r)$ is $(1 - p_{d,r}^{\geq k})^m$. Union bounding over the $d$ choices of $u$ gives the claim. [□]{.proof-qed}
-::::
+::: proof
+*Proof.* At total depth $j\geq t$, the descendants of the terminal vertex $v_t$ form the depth-$(j-t)$ level of a $(d-1)$-ary rooted tree and hence contribute $(d-1)^{j-t}$ vertices. Summing gives the tube size.
+Every vertex of depth at least $t$ has a unique rooted prefix of length $t$, so it lies in exactly one tube. The first edge of such a prefix has $d$ choices, and every later edge has $d-1$ choices, giving $N_t=d(d-1)^{t-1}$. □
:::
-For $k = 3$ and $r$ large, $p_{d,r}^{\geq 3} \to 1/d - O(d^{-r})$, so $m = \Theta(d \log d)$ samples suffice to hit every truncated branch with high probability. For fixed $d$, $m = O(\log n)$ drives the failure probability polynomially small in $n$.
+For a fixed rooted tree-ball $(G,v_0,R)$, after the root has been specified, let $C$ be uniform on $B_R(v_0)$. The probability that $C$ lies in a prescribed length-$t$ tube is $$q_{t,R}:=\Pr(C\in T_\sigma(R)\mid G,v_0,R)
+=\frac{|T_\sigma(R)|}{|B_R(v_0)|}.$$ The partition gives the exact identity $$
+N_tq_{t,R}
+=1-\frac{|B_{t-1}(v_0)|}{|B_R(v_0)|}.$$ Put $h=R-t$ and $b=d-1$. Lemmas [6](#lem-shell) and [7](#lem-tube-partition) give $$q_{t,R}=\frac{b^{h+1}-1}{(b+1)b^{t+h}-2}$$ and $$\frac{1}{N_tq_{t,R}}
+=\frac{b^{h+1}-2/((b+1)b^{t-1})}{b^{h+1}-1}
+=\frac{1}{1-b^{-(h+1)}}+O(b^{-t}),$$ uniformly for $h\geq0$. Thus $q_{t,R}=\Theta(N_t^{-1})$ uniformly over $R\geq t$, but its leading constant depends on $R-t$ when that difference remains bounded.
-## One-Round and $t$-Round Persistence
+# Path-Tube Persistence
-### The one-round case
+## The interception schedule
-We first prove the case $t = 1$ separately, both for clarity and because the argument is the model for the multi-round induction. [Lemma 3.1](#lem-oneround) below is the $t = 1$ specialization of [Theorem 1.6](#thm-persistence), presented here in self-contained form.
+::: {#lem-interception .lemma}
+**Lemma 8** (Interception schedule along a tube). *Let $R\geq t$, suppose $B_R(v_0)$ is a tree-ball, and let $$\sigma=(v_0,v_1,\ldots,v_t)$$ be a length-$t$ nonbacktracking path starting at $v_0$. Let a cop start at $c\in T_\sigma(R)$ at depth $j=\operatorname{dist}(c,v_0)$. Suppose the robber follows $\sigma$ for as long as the path survives. For every $s\in\{1,\ldots,t\}$ such that the path has survived through round $s-1$, after the cop move in round $s$:*
-::: {.annotation .annotation--static #lem-oneround}
-
-Lemma 3.1
-One-round persistence
-
-
-Let $G$ be $d$-regular with $d \geq 3$, $v_0 \in V(G)$, and assume $B_3(v_0)$ is a tree-ball. Suppose that for every $u \in N(v_0)$, there is at least one cop in $C_u^{\geq 3}(v_0, R)$ for some $R \geq 3$.
+1. *if $j=2s-1$, the cop reaches $v_{s-1}$ and captures the robber;*
-Then for every $w \in N(v_0)$, after the cop gradient-descent step, exactly one of the following holds:
+2. *if $j=2s$, the cop reaches $v_s$ and blocks the intended move;*
-- **(a)** a cop occupies $w$, so the robber's intended move to $w$ is illegal; or
-- **(b)** the move to $w$ is legal, and after the robber moves to $w$, the branch-load support at $w$ has size at least $2$.
-
+3. *if $j\geq2s+1$, the cop remains a strict descendant of $v_s$.*
+
+*The alternative $j\leq2s-2$ cannot occur: in that case the path was already captured or blocked by the end of round $s-1$.*
:::
-::: {.exhibit .exhibit--proof data-exhibit-name="One-round persistence" data-exhibit-type="proof" data-exhibit-caption="Descendant witness from the cop in C_w; parent witness from a cop in C_u with u ≠ w."}
-
-:::: exhibit-body
-Fix $w \in N(v_0)$. After the cop gradient-descent step, if some cop occupies $w$ then case (a) holds and we are done. Otherwise we exhibit both a descendant witness and a parent witness at $w$, establishing case (b).
-
-*Descendant witness.* By hypothesis there is a cop $c^\downarrow \in C_w^{\geq 3}(v_0, R)$. Then $\operatorname{dist}(c^\downarrow, v_0) = j \geq 3$ with the geodesic from $c^\downarrow$ to $v_0$ passing through $w$. Gradient descent toward $v_0$ moves $c^\downarrow$ to its parent in the tree rooted at $v_0$, which is at depth $j - 1 \geq 2$, still in the $w$-subtree. After this move, $c^\downarrow$ is at distance $j - 1 - 1 = j - 2 \geq 1$ from $w$. The geodesic from this new position to $w$ passes through some child of $w$ in the tree, so $c^\downarrow$ contributes to a descendant branch at $w$.
-
-*Parent witness.* Pick $u \in N(v_0)$ with $u \neq w$ (possible since $d \geq 3 \geq 2$). By hypothesis there is a cop $c^\uparrow \in C_u^{\geq 3}(v_0, R)$. After gradient descent, $c^\uparrow$ remains in the $u$-subtree at depth $\geq 2$ from $v_0$. Relative to $w$, the geodesic from $c^\uparrow$ to $w$ passes through $v_0$ (the unique tree path between distinct subtrees), so $c^\uparrow$ contributes to the parent branch at $w$ through $v_0$.
-
-*Combining witnesses.* The descendant and parent witnesses lie in distinct branches at $w$, so the branch-load support at $w$ has size at least $2$, establishing case (b). [□]{.proof-qed}
-::::
+::: proof
+*Proof.* Set $s_*=\lceil j/2\rceil$. For every reached round $k\leq\min\{s_*,t\}$, an induction shows that before the cop move the cop is a descendant of $v_{k-1}$ and has depth $j-k+1$. This is true at $k=1$. At round $k$, the endpoint-depth sum for the cop and $v_{k-1}$ is $j$, while their rooted-path length is $j-2k+2$. Hence the left side of Lemma [2](#lem-buffer)'s inequality, with $q=R$, is $$j+(j-2k+2)=2(j-k+1)\leq2j\leq2R.$$ The rooted path is therefore the unique ambient geodesic, and the cop moves one step toward $v_0$, to depth $j-k$. If $k s$, so the cop remains *strictly below* $v_s$: it is in the $v_s$-subtree at depth $\geq s + 1$ from $v_0$, equivalently at distance $\geq 1$ from $v_s$ within that subtree. Hence the geodesic from the cop's current position to $v_s$ has penultimate vertex equal to some child of $v_s$, and the cop contributes to a descendant branch at $v_s$.
-
-*Parent witness at $v_s$.* Choose any neighbor $u_0 \in N(v_0)$ with $u_0 \neq v_1$ (possible since $d \geq 3$). Extend $u_0$ to any nonbacktracking length-$t$ path $\sigma' = (v_0, u_0, \ldots)$. By hypothesis there is a cop $c^\uparrow \in T_{\sigma'}^{\geq 2t+1}(R)$.
-
-Initially $c^\uparrow$ is in the $u_0$-subtree at depth $\geq 2t+1$. At round $1$, gradient descent toward $v_0$ moves $c^\uparrow$ to depth $2t$, still in the $u_0$-subtree. At round $2$, gradient descent toward $v_1$ has target in a different subtree from $c^\uparrow$, so the cop's geodesic to $v_1$ goes up through $v_0$. The unique distance-decreasing neighbor is the cop's parent in the $v_0$-rooted tree, so the cop moves to depth $2t - 1$, still in the $u_0$-subtree. The same argument applies at every round $k$ with $1 \leq k \leq s$: from the cop's position in the $u_0$-subtree at depth $2t + 2 - k$, the robber's current position $v_{k-1}$ is reached only via $v_0$ (since $u_0 \neq v_1$), so the unique distance-decreasing direction is up to the parent.
-
-By induction, after $s$ rounds, $c^\uparrow$ is at depth $2t + 1 - s$ from $v_0$, still in the $u_0$-subtree. Since $u_0 \neq v_1$, the entire $u_0$-subtree is disjoint from the $v_1$-subtree, and since each $v_j$ for $j \geq 1$ lies in the $v_1$-subtree, the cop $c^\uparrow$ at depth $2t + 1 - s \geq t + 1 \geq 1$ from $v_0$ in the $u_0$-subtree is on the parent side of $v_s$. For $s \geq 1$, the geodesic from $c^\uparrow$ to $v_s$ passes through $v_0$, so the penultimate vertex of that geodesic (the one adjacent to $v_s$) is $v_{s-1}$. So $c^\uparrow$ contributes to the parent branch of $v_s$ through $v_{s-1}$.
-
-*Combining witnesses.* The descendant witness contributes to a non-parent branch at $v_s$, and the parent witness contributes to the parent branch at $v_s$. These are distinct branches, so the branch-load support at $v_s$ is at least $2$, establishing case (b). [□]{.proof-qed}
-::::
-
+::: remark
+**Remark 9**. Odd and even initial depths encode potential capture and blockage times, respectively. If the corresponding time is at most $t$ and the prescribed play reaches it, the event occurs then. If $\lceil j/2\rceil>t$, that time lies beyond the horizon and the cop remains descendant pressure throughout the first $t$ reached rounds.
:::
-### The depth budget is sharp
+## Persistence along nonbacktracking paths
-The hypothesis $j_0 \geq 2t + 1$ in [Theorem 1.6](#thm-persistence) is necessary, not merely sufficient. We illustrate.
+::: {#thm-persistence .theorem}
+**Theorem 10** (Nonbacktracking path-tube persistence). *Let $d\geq3$ and $1\leq t\leq R$, suppose $B_R(v_0)$ is a tree-ball, and let $X_0$ be any initial cop configuration satisfying $$X_0(T_\sigma(R))\geq1$$ for every length-$t$ tube. Cops outside $B_R(v_0)$ are permitted; the proof ignores them unless they cause capture, blockage, or additional branch support. If $X_0(\{v_0\})>0$, the robber is captured before any move. Otherwise, let $$\tau=(v_0,v_1,\ldots,v_t)$$ be any length-$t$ nonbacktracking path starting at $v_0$. At every round $s\in\{1,\ldots,t\}$ for which the robber has followed $\tau$ through round $s-1$, at least one of the following occurs during round $s$:*
-::: {.annotation .annotation--static #rem-sharpness}
-
-Remark 3.2
-Sharpness of the depth budget
-
-
-Consider $j_0 = 2t$. After $s = t$ rounds, the cop is at depth $j_0 - t = t$ from $v_0$, with the robber at $v_t$ (also depth $t$). The cop's geodesic to $v_0$ contains $v_t, v_{t-1}, \ldots, v_1$, so the cop is at $v_t$ itself, not at a strict descendant. This blocks the robber's move (case (a)) but does not provide a descendant branch at $v_t$ for the post-move analysis. Reducing further to $j_0 < 2t$ would place the cop on an ancestor of $v_t$, and the cop would contribute to the parent branch only — destroying the support-2 claim.
-
+1. *on the cops' move, a cop captures the robber at $v_{s-1}$;*
+
+2. *after the cops' move, the robber is not captured, but a cop occupies $v_s$, so the intended move is illegal;*
+
+3. *the move to $v_s$ is legal, and after that move the branch-load support at $v_s$ has size at least $2$.*
:::
-## The Barrier
+::: proof
+*Proof.* Assume there was no initial capture, and fix a round $s$ reached along $\tau$. If a cop reaches $v_{s-1}$, then (a) holds. If no cop captures the robber but a cop occupies $v_s$, then (b) holds. Assume instead that the move to $v_s$ is legal and produce two branch witnesses.
-### Tube density
+*Descendant witness.* Choose a cop $c^\downarrow$ from $T_\tau(R)$ and let $j_\downarrow$ be its initial depth. Survival through round $s-1$ rules out $j_\downarrow\leq2s-2$, while the failure of alternatives (a) and (b) rules out $j_\downarrow\in\{2s-1,2s\}$. Hence $j_\downarrow\geq2s+1$, and Lemma [8](#lem-interception) implies that after the round-$s$ cop move it is a strict descendant of $v_s$, at depth $j_\downarrow-s$. Its rooted path to $v_s$ has length $j_\downarrow-2s$, so the endpoint-depth sum plus this length is $$(j_\downarrow-s)+s+(j_\downarrow-2s)
+=2(j_\downarrow-s)\leq2R.$$ Lemma [2](#lem-buffer), with $q=R$, makes this the unique ambient geodesic. Thus the cop contributes to a child branch of $v_s$.
-::: {.annotation .annotation--static #lem-density}
-
-Lemma 4.1
-Asymptotic tube density
-
-
-For fixed $d \geq 3$ and a length-$t$ nonbacktracking prefix $\sigma$ in a $d$-regular tree-ball $B_R(v_0)$ with $R \geq 2t+1$,
+*Parent witness.* Choose $u_0\in N(v_0)\setminus\{v_1\}$, extend $(v_0,u_0)$ to a length-$t$ nonbacktracking path $\sigma'$, and choose a cop $c^\uparrow\in T_{\sigma'}(R)$ of initial depth $j$. Then $R\geq j\geq t\geq s$. We prove by induction on $k\in\{1,\ldots,s\}$ that before its move in round $k$, the cop has depth $j-k+1$ and lies in the root branch through $u_0$. This is immediate for $k=1$. At round $k$, the rooted path from the cop to $v_{k-1}$ passes through $v_0$; both its length and the sum of the endpoint depths equal $$(j-k+1)+(k-1)=j.$$ Lemma [2](#lem-buffer), with $q=R$, therefore forces one step toward $v_0$. If $k
+The two witnesses contribute to distinct branches, proving (c). □
:::
-::: {.exhibit .exhibit--proof data-exhibit-name="Asymptotic tube density" data-exhibit-type="proof" data-exhibit-caption="Both numerator and denominator are geometric sums dominated by their top shells; the ratio of top shells gives the limit."}
-
-:::: exhibit-body
-Each vertex in $T_\sigma^{\geq 2t+1}(R) \cap S_j(v_0)$ is a descendant of $v_t$ at depth $j$ from $v_0$, hence at depth $j - t$ within the $v_t$-rooted subtree. The latter is a $(d-1)$-ary tree (since $v_t$ has $d - 1$ children in the tree-ball), so its depth-$(j - t)$ shell has $(d-1)^{j-t}$ vertices for $j \geq t + 1$. Restricting to $j \geq 2t + 1$, summing, and substituting $\ell = j - t$ gives the closed form above.
-
-For the asymptotic density: both $|T_\sigma^{\geq 2t+1}(R)|$ and $|B_R(v_0)|$ are geometric sums dominated by their top shells. The top shell of the tube at $j = R$ has $(d-1)^{R-t}$ vertices; the top shell of the ball at $j = R$ has $d(d-1)^{R-1}$ vertices. Hence the ratio tends to
-
-$$\frac{(d-1)^{R-t}}{d(d-1)^{R-1}} = \frac{1}{d(d-1)^{t-1}},$$
-
-as claimed. [□]{.proof-qed}
-::::
+## The one-round case
+::: {#cor-one-round .corollary}
+**Corollary 11** (One-round persistence). *Let $R\geq1$, suppose $B_R(v_0)$ is a tree-ball, and let $X_0$ be a cop configuration satisfying $$X_0(C_u(v_0,R))\geq1
+\qquad\text{for every }u\in N(v_0).$$ For every proposed neighboring first move $v_0\to w$, the robber is captured at $v_0$ (initially or on the first cop move), blocked at $w$, or, after a legal move to $w$, faces branch-load support of size at least two.*
:::
-### Number of length-$t$ prefixes
-
-::: {.annotation .annotation--static #lem-prefixes}
-
-Lemma 4.2
-Number of length-$t$ prefixes
-
-
-The number of nonbacktracking length-$t$ paths from $v_0$ in a $d$-regular tree-ball is
-
-$$N_t = d(d-1)^{t-1}.$$
-
+::: proof
+*Proof.* Initial capture gives the first alternative. Otherwise apply Theorem [10](#thm-persistence) with $t=1$. □
:::
-::: {.exhibit .exhibit--proof data-exhibit-name="Number of length-$t$ prefixes" data-exhibit-type="proof" data-exhibit-caption="d choices for the first step, d−1 choices thereafter."}
+# The Cost of Exhaustive Tube Coverage
-:::: exhibit-body
-The first step has $d$ choices (any neighbor of $v_0$). Each subsequent step has $d - 1$ choices (any neighbor of the current vertex except the immediately previous one). So $N_t = d(d-1)^{t-1}$. [□]{.proof-qed}
-::::
+Theorem [10](#thm-persistence) uses a strong static hypothesis: every length-$t$ nonbacktracking path starting at $v_0$ indexes an occupied tube. We now quantify the exact deterministic cost and the conditional random-sampling threshold of this local certificate. These costs are not cop-number bounds and do not establish a robber-independent global placement.
+::: {#prop-deterministic-cost .proposition}
+**Proposition 12** (Deterministic coverage cost). *Suppose $B_R(v_0)$ is a tree-ball with $R\geq t$, and let $X$ be any cop configuration. Then $X$ occupies every length-$t$ tube if and only if $$X(T_\sigma(R))\geq1$$ for every member of the family of $N_t=d(d-1)^{t-1}$ pairwise disjoint tubes. Consequently:*
+
+1. *every such configuration satisfies $|X|\geq N_t$, counting cops with multiplicity;*
+
+2. *equality suffices for this occupancy property, by choosing one cop position from each tube.*
:::
-### Proof of Theorem 1.7
-
-::: {.exhibit .exhibit--proof data-exhibit-name="Barrier on branch-tube certificates" data-exhibit-type="proof" data-exhibit-caption="Coupon-collector-style union bound: covering N_t = d(d−1)^{t−1} tubes each with density 1/[d(d−1)^{t−1}] requires Ω((d−1)^{t−1} · t) cops."}
-
-:::: exhibit-body
-Suppose $m$ cops are sampled i.i.d. uniformly from $B_R(v_0)$, and the proof relies on every length-$t$ tube being occupied. By [Lemma 4.1](#lem-density), for each fixed prefix $\sigma$,
-
-$$\Pr\!\left[T_\sigma^{\geq 2t+1}(R) \text{ is empty}\right] = (1 - q_t(R))^m \leq \exp(-m \cdot q_t(R)).$$
-
-Union bounding over the $N_t$ prefixes,
-
-$$\Pr[\exists \sigma : T_\sigma \text{ empty}] \leq N_t \cdot \exp(-m \cdot q_t(R)).$$
-
-For this to be $o(1)$, we require
-
-$$m \geq \frac{\log N_t + \omega(1)}{q_t(R)}.$$
-
-Using $N_t = d(d-1)^{t-1}$ and $q_t(R) \sim 1/[d(d-1)^{t-1}]$ from Lemmas [4.2](#lem-prefixes) and [4.1](#lem-density),
-
-$$m \gtrsim d(d-1)^{t-1} \cdot \log\!\bigl[d(d-1)^{t-1}\bigr].$$
-
-For fixed $d$, this is $m = \Omega((d-1)^{t-1} \cdot t)$.
-
-*Inverting.* Suppose $m = \mathrm{polylog}(n)$. From $(d-1)^{t-1} \leq m$, taking logarithms gives
-
-$$t - 1 \leq \frac{\log m}{\log(d-1)} = \frac{O(\log\log n)}{\log(d-1)},$$
-
-so $t = O(\log\log n)$.
-
-In particular, the certificate cannot certify $t = \Theta(\log n)$ rounds with $m = \mathrm{polylog}(n)$ cops. To certify $t = \Theta(\log n)$ rounds, the certificate requires $m = (d-1)^{\Theta(\log n)} \cdot \Theta(\log n) = n^{\Theta(\log(d-1))} \cdot \Theta(\log n)$ cops, which is polynomial in $n$ with exponent depending on $d$. [□]{.proof-qed}
-::::
-
+::: proof
+*Proof.* This is immediate from Lemma [7](#lem-tube-partition). □
:::
-### What the barrier says, precisely
+::: {#thm-sampling-threshold .theorem}
+**Theorem 13** (Conditional uniform-sampling threshold). *Fix $d\geq3$. For each positive integer $t$, let $G_t$ be a $d$-regular graph with distinguished vertex $v_{0,t}$, and let $R_t\geq t$ be such that $B_{R_t}^{G_t}(v_{0,t})$ is a tree-ball. Let $\mathcal P_t$ be the set of length-$t$ nonbacktracking paths from $v_{0,t}$, and let $T_{\sigma,t}$ be the tube of $\sigma\in\mathcal P_t$ in that ball. Set $$N_t:=|\mathcal P_t|=d(d-1)^{t-1},
+\qquad
+q_t:=\frac{|T_{\sigma,t}|}{|B_{R_t}^{G_t}(v_{0,t})|}.$$ For each $t$, after the rooted ball has been fixed, sample $m_t$ positions independently and uniformly with replacement from $B_{R_t}^{G_t}(v_{0,t})$. Let $\mathcal C_t$ be the event that every $T_{\sigma,t}$ contains at least one sample. For every fixed $\varepsilon$ with $0<\varepsilon<1$:*
-The barrier rules out a specific proof technique: it does not say that the $\Theta(\log n)$-round local team-chase is impossible, only that it cannot be obtained by occupying every length-$t$ deep tube uniformly. The barrier identifies the cost as exponential in $t$, with the exponent base $d - 1$ — the local branching factor of the tree-ball.
+1. *if $$m_t\geq(1+\varepsilon)\frac{\log N_t}{q_t},$$ then $\Pr(\mathcal C_t)\to1$;*
-::: {.annotation .annotation--static #rem-robust}
-
-Remark 4.3
-Robustness of the barrier
-
-
-The barrier is robust to mild relaxations of the certificate. For example, requiring *most* (rather than all) tubes to be occupied still requires $\Omega((d-1)^{t-1})$ cops in expectation: the expected number of empty tubes is $N_t (1 - q_t)^m$, and bounding this even by $1$ gives $m \geq \log N_t / q_t$, which is the same order. Similarly, weakening the depth threshold from $2t + 1$ to any $\Theta(t)$ does not change the exponential dependence on $t$, only the constant in the exponent.
-
+2. *if $$m_t\leq(1-\varepsilon)\frac{\log N_t}{q_t},$$ then $\Pr(\mathcal C_t)\to0$.*
+
+*Writing $h_t=R_t-t$, one has, uniformly over $h_t\geq0$, $$\frac{\log N_t}{q_t}
+=\left(\frac{1}{1-(d-1)^{-(h_t+1)}}+o(1)\right)N_t\log N_t.$$ If $h_t=k$ eventually, the leading factor relative to $N_t\log N_t$ is $[1-(d-1)^{-(k+1)}]^{-1}$; if $h_t\to\infty$, it tends to $1$. Uniformly over $R_t\geq t$, $$\frac{\log N_t}{q_t}
+=\Theta(N_t\log N_t)
+=\Theta\bigl((d-1)^{t-1}t\bigr).$$*
:::
-### A finite-order potential-degeneration barrier
+::: proof
+*Proof.* For a fixed $\sigma\in\mathcal P_t$, the probability that $T_{\sigma,t}$ receives no sample is $(1-q_t)^{m_t}$. Hence $$\Pr(\mathcal C_t^{\mathsf c})
+\leq N_t(1-q_t)^{m_t}
+\leq N_te^{-m_tq_t}.$$ Under (i), this is at most $N_t^{-\varepsilon}\to0$.
-[Theorem 1.7](#thm-barrier) shows that occupying every length-$t$ deep tube is exponentially expensive in $t$ under uniform local sampling. A natural response is to compress the local state: instead of tracking every length-$t$ tube, one might try to use an invariant that remembers only bounded-order tube data, aggregating all information below a fixed depth. The next theorem shows that this cannot certify persistence beyond one additional round.
-
-::: {.annotation .annotation--static #def-order-r-profile}
-
-Definition 4.4
-Order-$r$ tube profile
-
-
-Fix a rooted tree-ball $B_R(v_0)$ in a $d$-regular graph. For a cop configuration $X$ and an integer $r \geq 1$, the *order-$r$ tube profile* of $X$ is the family
-
-$$\Pi_r(X) := \bigl\{N_X(\sigma, j)\bigr\}_{\sigma, j},$$
-
-where $\sigma = (v_0, v_1, \ldots, v_r)$ ranges over all nonbacktracking paths of length $r$ from $v_0$, and $j \geq r + 1$, and
-
-$$N_X(\sigma, j) := \#\bigl\{x \in X : \operatorname{dist}(x, v_0) = j, \text{ and the unique geodesic from } x \text{ to } v_0 \text{ begins with } \sigma\bigr\}.$$
-
-Thus $\Pi_r(X)$ records the exact depth histogram inside every length-$r$ tube, but forgets how the mass in that tube splits below level $r$.
-
+For (ii), let $$Z_t:=\sum_{\sigma\in\mathcal P_t}I_{\sigma,t},
+\qquad
+I_{\sigma,t}:=\mathbf 1_{\{T_{\sigma,t}\text{ receives no sample}\}}.$$ Then $\mathbb E Z_t=N_t(1-q_t)^{m_t}$. For distinct $\sigma,\tau\in\mathcal P_t$, disjointness gives $$\mathbb E[I_{\sigma,t}I_{\tau,t}]
+=(1-2q_t)^{m_t}
+\leq(1-q_t)^{2m_t}
+=\mathbb E I_{\sigma,t}\,\mathbb E I_{\tau,t}.$$ Thus $$\operatorname{Var}(Z_t)
+\leq\sum_{\sigma\in\mathcal P_t}\operatorname{Var}(I_{\sigma,t})
+\leq\mathbb E Z_t.$$ Since $q_t=\Theta(N_t^{-1})$, one has $q_t\to0$, and under (ii), $$\mathbb E Z_t
+\geq N_t(1-q_t)^{(1-\varepsilon)(\log N_t)/q_t}
+=N_t^{\varepsilon+o(1)}\longrightarrow\infty.$$ Chebyshev's inequality therefore yields $$\Pr(\mathcal C_t)
+\leq\frac{\operatorname{Var}(Z_t)}{(\mathbb E Z_t)^2}
+\leq\frac1{\mathbb E Z_t}
+\longrightarrow0.$$ Finally, the exact calculation preceding the theorem, with $R=R_t$, gives $$\frac1{N_tq_t}
+=\frac1{1-(d-1)^{-(h_t+1)}}+O((d-1)^{-t})$$ uniformly for $h_t\geq0$, proving the remaining assertions. □
:::
-::: {.annotation .annotation--static #def-order-r-invariant}
-
-Definition 4.5
-Order-$r$ invariant
-
-
-A local invariant $F$ on cop configurations in a rooted tree-ball is *order-$r$* if it factors through $\Pi_r$, i.e. if $F(X) = F(Y)$ whenever $\Pi_r(X) = \Pi_r(Y)$. It is *neighbor-symmetric* if it is invariant under rooted automorphisms of the tree-ball fixing $v_0$.
-
+::: {#cor-polylog-horizon .corollary}
+**Corollary 14** (Finite horizon from polylogarithmic conditional sampling). *Fix $d\geq3$. Let $(G_n)$ be a sequence of $n$-vertex $d$-regular graphs with distinguished vertices $v_{0,n}$ and integers $1\leq t_n\leq R_n$ such that $B_{R_n}^{G_n}(v_{0,n})$ is a tree-ball. After each rooted ball is fixed, sample $m_n$ positions independently and uniformly with replacement from it, and let $\mathcal C_n$ be the event that every length-$t_n$ tube rooted at $v_{0,n}$ is occupied.*
+
+*Call $(m_n)$ *polylogarithmic* if $m_n=O((\log n)^C)$ for some fixed $C>0$. If $(m_n)$ is polylogarithmic and $\Pr(\mathcal C_n)\to1$, then $t_n=O(\log\log n)$. If $t_n=\Theta(\log n)$ and $\Pr(\mathcal C_n)\to1$, then $$m_n=\Omega(N_{t_n}\log N_{t_n}),
+\qquad
+N_{t_n}=d(d-1)^{t_n-1};$$ in particular, $m_n\geq n^c$ eventually for some $c>0$.*
:::
-::: {.annotation .annotation--static #thm-finite-order-barrier}
-
-Fix $d \geq 3$ and $r \geq 1$. Let $B_R(v_0)$ be a $d$-regular tree-ball with $R \geq 2r + 3$. Then there exist two cop configurations $X$ and $Y$ in $B_R(v_0)$ such that:
+::: proof
+*Proof.* If $t_n\neq O(\log\log n)$, pass to a subsequence on which $t_n/\log\log n\to\infty$, and then to a further subsequence on which $t_n$ is strictly increasing. Apply the proof of Theorem [13](#thm-sampling-threshold) along this subsequence, with $t$ replaced by $t_n$. Now $t_n\to\infty$ and $N_{t_n}=(\log n)^{\omega(1)}$. Because the threshold is $\Theta(N_{t_n}\log N_{t_n})$ uniformly in $R_n\geq t_n$, polylogarithmic $m_n$ satisfies $$m_n\leq\frac12\frac{\log N_{t_n}}{q_n}$$ eventually on this subsequence, where $q_n$ is the common tube mass. Part (ii) of Theorem [13](#thm-sampling-threshold), with $\varepsilon=1/2$, then gives $\Pr(\mathcal C_n)\to0$, a contradiction. Hence $t_n=O(\log\log n)$.
-- **(i)** $\Pi_r(X) = \Pi_r(Y)$; in particular, every order-$r$ invariant takes the same value on $X$ and $Y$.
-- **(ii)** There exists a nonbacktracking path
-
- $$(v_0, v_1, \ldots, v_{r+1})$$
-
- such that this robber path is safe in both configurations, but after $r+1$ rounds of cop gradient descent followed by robber motion along that path:
-
- - in configuration $X$, the branch-load support at $v_{r+1}$ has size at least $2$;
- - in configuration $Y$, the branch-load support at $v_{r+1}$ has size exactly $1$.
-
-Consequently, no certificate whose hypotheses depend only on order-$r$ data can correctly distinguish $\geq 2$-branch support from $1$-branch support after $r+1$ rounds.
-
+If $t_n=\Theta(\log n)$ and $m_n=o(N_{t_n}\log N_{t_n})$ along a subsequence, pass if necessary to a further subsequence with strictly increasing horizons and apply the same subsequence argument. This contradicts $\Pr(\mathcal C_n)\to1$. Thus $m_n=\Omega(N_{t_n}\log N_{t_n})$. Since $N_{t_n}=n^{\Omega(1)}$, this is at least $n^c$ eventually for some $c>0$. □
:::
-::: {.exhibit .exhibit--proof data-exhibit-name="Finite-order potential-degeneration barrier" data-exhibit-type="proof" data-exhibit-caption="Two two-cop configurations agree on order-r tube data but differ on which children of v_{r+1} they cover after r+1 forced upward steps."}
-
-:::: exhibit-body
-Fix any nonbacktracking path
-
-$$(v_0, v_1, \ldots, v_{r+1})$$
-
-inside the tree-ball. Since $d \geq 3$, the vertex $v_{r+1}$ has at least two distinct children in the rooted tree. Choose two such children and call them $a$ and $b$.
-
-We now define two cop configurations, each consisting of exactly two cops, both at depth $2r + 3$ from $v_0$.
-
-*Configuration $X$.* Place one cop in the subtree rooted at $a$ and one cop in the subtree rooted at $b$, both at total depth $2r + 3$ from $v_0$.
-
-*Configuration $Y$.* Place both cops in the subtree rooted at $a$, at distinct vertices, again at total depth $2r + 3$ from $v_0$. (Since the depth-$(r+1)$ shell of the $a$-subtree has $(d-1)^{r+1} \geq 2$ vertices, this is possible.)
-
-In both configurations, every cop lies in the same length-$r$ tube determined by the prefix
-
-$$\sigma_0 = (v_0, v_1, \ldots, v_r),$$
-
-and every cop lies at the same total depth $2r + 3$ from $v_0$. Therefore the order-$r$ tube counts are identical:
-
-$$N_X(\sigma_0, 2r+3) = N_Y(\sigma_0, 2r+3) = 2,$$
-
-and $N_X(\sigma, j) = N_Y(\sigma, j) = 0$ for all other $(\sigma, j)$. Hence $\Pi_r(X) = \Pi_r(Y)$. The order-$r$ invariant cannot see how the two cops in $\sigma_0$ split between the children $a$ and $b$ of $v_{r+1}$.
-
-Now consider the robber path
-
-$$(v_0, v_1, \ldots, v_{r+1}).$$
-
-We claim it is safe in both configurations. Each cop starts at depth $2r + 3$ from $v_0$. In a tree-ball, gradient descent toward any robber position on the cop's geodesic to $v_0$ is forced: each round the cop moves exactly one step upward toward $v_0$. Hence after $s$ rounds, each cop is at depth $2r + 3 - s$ from $v_0$. After the cop's move at round $s$, the robber moves $v_{s-1} \to v_s$. The cop's distance to the robber's destination $v_s$ is
-
-$$(2r + 3 - s) - s = 2r + 3 - 2s.$$
-
-For every $1 \leq s \leq r + 1$ this is at least $1$, so no cop ever lands on $v_s$ during these $r + 1$ rounds. Thus the robber path is safe in both configurations.
-
-Finally, inspect the cop locations after $r + 1$ rounds. Each cop has moved upward by exactly $r + 1$ steps, so each is now at depth
-
-$$2r + 3 - (r + 1) = r + 2$$
-
-from $v_0$, while $v_{r+1}$ has depth $r + 1$. Hence each cop is now a child of $v_{r+1}$.
-
-In configuration $X$, the two cops lie on two distinct children of $v_{r+1}$, namely $a$ and $b$. Therefore the branch-load support at $v_{r+1}$ has size at least $2$.
-
-In configuration $Y$, both cops climbed within the $a$-subtree and now sit on its root, namely $a$ itself (since gradient descent in a tree-ball is forced upward). Therefore both cops contribute to the same branch at $v_{r+1}$, and the branch-load support has size exactly $1$.
-
-This proves (ii). Since $\Pi_r(X) = \Pi_r(Y)$ but the outcomes differ, no certificate depending only on order-$r$ data can correctly classify both configurations. [□]{.proof-qed}
-::::
-
+::: {#rem-partial-coverage .remark}
+**Remark 15** (Partial coverage is cheaper). At one fixed, known root, the expected fraction of length-$t$ tubes occupied by $m$ independent uniform samples is $$1-(1-q_{t,R})^m.$$ Since $q_{t,R}=\Theta(N_t^{-1})$, $m=\Theta(N_t)$ samples occupy a fixed positive fraction of the tubes in expectation, while $m=\omega(N_t)$ occupies a $1-o(1)$ fraction in expectation. Under uniform sampling, reducing the expected number of empty tubes to $O(1)$ requires $\Theta(N_t\log N_t)$ samples; the same order is required to leave at most $O(1)$ tubes empty with probability tending to one. These are occupancy statements only: they give neither an adversarial guarantee over all paths from that root nor a root-independent placement, and they do not show that partial coverage supports an adaptive chase.
:::
-::: {.annotation .annotation--static #cor-first-order-insufficient}
-
-Corollary 4.7
-First-order data is insufficient
-
-
-Any invariant depending only on first-level branch data, even with exact depth histograms inside each first-level branch, cannot certify universal $2$-round persistence in the tree-ball model.
-
+# Information Loss in Tube Profiles {#sec-profile-loss}
+
+Fix integers $Q\geq r\geq1$ and $k\geq0$, and suppose $B_Q(v_0)$ is a tree-ball. Let $\mathcal P_r(v_0)$ be the set of length-$r$ nonbacktracking paths starting at $v_0$, and let $$\mathcal C_{Q,k}(v_0)
+:=\{X: X\text{ is a $k$-cop multiset with }\mathop{\mathrm{supp}}(X)\subseteq B_Q(v_0)\}.$$ For $X\in\mathcal C_{Q,k}(v_0)$, $\sigma\in\mathcal P_r(v_0)$, and $r\leq j\leq Q$, set $$N_X(\sigma,j)
+:=\sum_{\substack{x\in S_j(v_0)\\
+\text{the rooted path from $v_0$ to $x$ begins with }\sigma}}X(x).$$ The *augmented order-$r$ tube profile* is $$\Pi_{r;Q,k}(X)
+:=\left(
+\bigl(X(x)\bigr)_{x\in B_{r-1}(v_0)},
+\bigl(N_X(\sigma,j)\bigr)_{\substack{\sigma\in\mathcal P_r(v_0)\\ r\leq j\leq Q}}
+\right).$$ It records the complete configuration through depth $r-1$ and the exact depth histogram in every length-$r$ tube; the $j=r$ coordinates also recover the multiplicities at depth $r$. It forgets only how cops at a fixed greater depth split among descendants below the terminal vertex of a length-$r$ prefix. For an admissible profile $P$, define its fixed-universe fiber by $$\mathcal F_{r;Q,k}(P)
+:=\{X\in\mathcal C_{Q,k}(v_0):\Pi_{r;Q,k}(X)=P\}.$$
+
+::: {#prop-profile-nondeterminacy .proposition}
+**Proposition 16** (Augmented tube profiles do not determine later support). *Fix $d\geq3$ and $r\geq1$, put $Q=2r+3$, and suppose $B_Q(v_0)$ is a tree-ball. There exist $X,Y\in\mathcal C_{Q,2}(v_0)$ and a length-$(r+1)$ nonbacktracking path $\tau=(v_0,\ldots,v_{r+1})$ such that $$\Pi_{r;Q,2}(X)=\Pi_{r;Q,2}(Y).$$ The path survives all $r+1$ rounds from both configurations, with all relevant cop moves unique, but immediately after the robber's legal move to $v_{r+1}$ the branch-load supports in $X$ and $Y$ have sizes $2$ and $1$, respectively. Hence, for this fixed path $\tau$, the predicate that the branch-load support after $r+1$ safe rounds has size at least $2$ does not factor through $\Pi_{r;Q,2}$.*
:::
-::: {.exhibit .exhibit--proof data-exhibit-name="First-order data is insufficient" data-exhibit-type="proof" data-exhibit-caption="Apply the finite-order barrier with r = 1."}
-
-:::: exhibit-body
-Take $r = 1$ in [Theorem 4.6](#thm-finite-order-barrier). [□]{.proof-qed}
-::::
+::: proof
+*Proof.* Choose $\tau$ and two distinct children $a,b$ of $v_{r+1}$. Choose descendants $x_a$ of $a$ and $x_b$ of $b$ at total depth $Q=2r+3$ from $v_0$, and choose two distinct descendants $y_a,y_a'$ of $a$ at the same total depth. Such choices exist because the vertices at descendant-distance $r+1$ below $a$ number $(d-1)^{r+1}\geq2$. Define $$X=[x_a,x_b],
+\qquad
+Y=[y_a,y_a'].$$ Both configurations have zero multiplicity on $B_{r-1}(v_0)$, and their only nonzero tube-depth coordinate is the cell indexed by $((v_0,\ldots,v_r),Q)$, where both have value $2$. Thus their augmented profiles agree.
+Every robber position on $\tau$ is an ancestor of every cop in the construction. Immediately before the cop move in round $s$, each cop has total depth $Q-s+1$, while the robber is at $v_{s-1}$. Their rooted path has length $Q-2s+2$, and $$(Q-s+1)+(s-1)+(Q-2s+2)=2Q-2s+2\leq2Q.$$ Lemma [2](#lem-buffer), with $q=Q$, therefore makes this path the unique ambient geodesic and forces one step upward. After that move, the cop's distances to $v_{s-1}$ and $v_s$ are $$2r+4-2s\geq2
+\qquad\text{and}\qquad
+2r+3-2s\geq1$$ for $1\leq s\leq r+1$. Thus no capture or blockage occurs. After $r+1$ cop moves, the configurations are $[a,b]$ and $[a,a]$, so the branch-load support sizes after the robber moves to $v_{r+1}$ are $2$ and $1$. □
:::
-::: {.annotation .annotation--static #rem-complementary}
-
-Remark 4.8
-Complementarity of the two barriers
-
-
-Theorems [1.7](#thm-barrier) and [4.6](#thm-finite-order-barrier) are complementary. [Theorem 1.7](#thm-barrier) shows that full order-$t$ path information is sufficient but exponentially expensive to certify by uniform random local sampling. [Theorem 4.6](#thm-finite-order-barrier) shows that any bounded-order compression of that information is dynamically insufficient beyond one additional round. Together they imply that, in the tree-ball model, there is no bounded-order shortcut around path entropy: certifying $t$-round persistence by a local invariant requires at least order-$t$ resolution, and order-$t$ resolution requires $\Omega((d-1)^{t-1})$ cops to occupy. Together they rule out order-$r$ branch-aggregated potentials for every fixed $r$ as a route to $\Theta(\log n)$-round chase from polylogarithmic cops.
-
+For $r=1$, the profile includes the multiplicity at $v_0$ as well as the depth histogram in every first-level tube, yet it still does not determine the two-round outcome. Proposition [16](#prop-profile-nondeterminacy) identifies one ambiguous fiber, not a barrier to every one-sided certificate: a sound certificate may reject that fiber.
+
+# Limitations and Further Directions {#sec-open}
+
+The present arguments do not address five natural directions; no claim of novelty is made for the questions themselves.
+
+## Arbitrary robber walks
+
+Theorem [10](#thm-persistence) treats only length-$t$ nonbacktracking paths starting at $v_0$. Stationary moves and reversals destroy the monotone depth evolution used by the interception schedule.
+
+::: question
+**Question 17**. *Can a static or adaptive local certificate give an analogue of Theorem [10](#thm-persistence) for arbitrary length-$t$ robber walks from $v_0$, including stationary moves and reversals? How do repeated vertices and reversed edges change the required witnesses and coverage cost?*
:::
-## Discussion: Pivots and Open Directions
+## Partial and adaptive coverage
-The barriers of Theorems [1.7](#thm-barrier) and [4.6](#thm-finite-order-barrier) together rule out a substantial family of approaches but leave several escape routes open.
+Remark [15](#rem-partial-coverage) separates partial from exhaustive conditional coverage at one fixed root. Along a prescribed nonbacktracking path, the persistence proof uses its occupied tube for the descendant witness and a tube with a different first edge for the parent witness; exhaustive coverage supplies both uniformly. A large expected covered fraction alone gives no persistence or adaptive-chase guarantee.
-### Pivot A: Epoch refresh
+::: question
+**Question 18**. *For a fixed horizon $t$, is there a condition strictly weaker than occupancy of every length-$t$ tube that guarantees capture or a quantified decrease in the number of compatible future nonbacktracking continuations? Can it be updated after each move by reusing witnesses among tubes with a common shorter prefix?*
+:::
-Run epochs of length $t_0 = O(\log\log n)$, for which the tube certificate is affordable with $\mathrm{polylog}(n)$ cops, then re-randomize and restart. The barrier does not preclude this: it constrains a single epoch but not a sequence of epochs.
+## Uniformly favorable augmented-profile fibers
-The crucial open question is whether one $t_0$-round epoch produces *permanent* progress (a coarser robber freedom parameter shrinks by a constant factor). In a tree-ball, no such parameter is obvious: the robber's territory is unbounded by hypothesis, and after one epoch the robber can simply retreat to a fresh subtree. This suggests that epoch refresh in the pure tree-ball model is unlikely to succeed, and the program must combine the local tree-ball analysis with a global structural bound that limits the robber's total territory.
+Fix integers $Q\geq r+1\geq2$ and $k\geq1$, suppose $B_Q(v_0)$ is a tree-ball, and fix a length-$(r+1)$ nonbacktracking path $\tau=(v_0,\ldots,v_{r+1})$. The graph, root, radius, cop number, allowed support region, and allowed distance-minimizing tie-breaks are thereby fixed.
-### Pivot B: Soft potentials must break locality or symmetry
+::: question
+**Question 19**. *For which admissible profiles $P$ do both of the following hold?*
-[Theorem 4.6](#thm-finite-order-barrier) rules out any *order-$r$* potential as a route to $(r+1)$-round persistence. To escape the barrier via a soft potential, the potential must therefore depend on at least one of:
+1. *Some $X\in\mathcal F_{r;Q,k}(P)$ and some allowed sequence of cop moves let $\tau$ survive through round $r+1$.*
-- *Unbounded local order:* information that depends on tube data of order growing with $t$. This re-introduces path-entropy costs and hits [Theorem 1.7](#thm-barrier).
-- *Non-local data:* information that depends on the global graph structure beyond the local tree-ball, such as cycle structure at radius $> R$, or spectral data of the full graph.
-- *Asymmetric data:* information that breaks the rooted-tree symmetry, such as a fixed orientation, an external labeling, or a privileged subset of vertices not invariant under tree automorphisms.
+2. *For every $X\in\mathcal F_{r;Q,k}(P)$ and every allowed sequence of cop moves, the robber is captured or blocked by round $r+1$, or $\tau$ survives and its final branch-load support has size at least $2$.*
+:::
-The most promising candidate among these is the second: *[nonbacktracking-walk](https://en.wikipedia.org/wiki/Non-backtracking_random_walk) concentration* of cop mass on the full graph. If cops are placed by a global averaging that respects the nonbacktracking spectrum, the resulting cop distribution at time $t$ may concentrate on the robber's location at exponential rate $\rho(B)^{-1}$, where $B$ is the Hashimoto nonbacktracking transition matrix of the full graph. The local tree-ball view of such a placement is no longer order-$r$ for any fixed $r$; it inherits global spectral information that is invisible to any local invariant. This suggests that purely local tree-ball analysis is fundamentally insufficient.
+## Global overlap of tube systems
-### Pivot C: Hierarchical packets
+Fix a graph $G$, integers $R\geq t\geq1$, a set $A$ of admissible roots whose radius-$R$ balls are tree-balls, and a root-independent set $W$ of allowed cop locations. For $v\in A$ and a length-$t$ nonbacktracking path $\sigma$ from $v$, write $T_\sigma^v(R)$ for its rooted tube. Define the global tube hypergraph $$\mathcal H_{R,t}(G,A;W)
+:=\left(W,\{T_\sigma^v(R)\cap W:v\in A,\ \sigma\text{ is a length-$t$ nonbacktracking path from }v\}\right).$$ A root-independent set of cop positions supplies the exhaustive tube certificate simultaneously for every $v\in A$ exactly when it is a transversal of $\mathcal H_{R,t}$. This is the local-to-global gap absent from the conditional sampling theorem; multiplicity at an already selected vertex does not improve coverage.
-Use cop packets at multiple depth scales. A packet at scale $j$ guarantees persistence for $2^j$ rounds inside a tube of depth $\Theta(2^j)$. Packets are activated sequentially, with later packets covering longer ranges. This is a multiscale variant of refresh.
+::: question
+**Question 20**. *How do girth, expansion, and nonbacktracking path counts bound the transversal number, fractional transversal number, and codegrees of $\mathcal H_{R,t}$? Can these estimates produce a root-independent placement with controlled cost?*
+:::
-A naive cost analysis is discouraging: each scale costs $(d-1)^{2^j}$ cops by the barrier of [Theorem 1.7](#thm-barrier), and at the top scale $j = \log\log n$ this is already $(d-1)^{\log n} = n^{\log_2(d-1)}$, which is polynomial in $n$. So the straightforward hierarchical-packet scheme does not deliver polylogarithmic cop count. This matches the cost of [Theorem 1.7](#thm-barrier) inverted at $t = \Theta(\log n)$: the multi-scale scheme inherits the same fundamental cost as the single-scale scheme it tries to circumvent. To salvage the idea one would need a sub-exponential variant in which packets at scale $j$ exploit structure (epoch refresh between scales, partial coverage of tubes, or shared cops across scales) to evade the barrier. We do not pursue this here.
+## Beyond exact tree-balls
-### Beyond the tree-ball model
+The persistence proof uses only the unique geodesics of synchronized witnesses inside $B_R(v_0)$. With cycles, rooted branches would have to give way to a shortest-path directed acyclic graph in which paths may split or merge. No persistence theorem or coverage bound is proved in that setting.
-The deepest limitation of the present results is the tree-ball hypothesis. Real [expanders](https://en.wikipedia.org/wiki/Expander_graph) — including [Ramanujan graphs](https://en.wikipedia.org/wiki/Ramanujan_graph) of girth $\Theta(\log n)$ — are tree-balls only up to radius $\Theta(\log n)$, but a chase argument typically needs to reason about radii $r \approx \log n / \gamma$ where $\gamma$ is the [spectral gap](https://en.wikipedia.org/wiki/Spectral_gap). For $\gamma$ bounded below by a constant, these match up to constants; but for $\gamma$ small, the chase exits the tree-ball and must contend with cycles. In the cyclic regime, the geodesic cone $C_u(v, r)$ is no longer a clean object (vertices may have multiple shortest paths to $v$), and the entire branch decomposition becomes ill-defined.
+::: question
+**Question 21**. *For a radius-$R$ ball obtained from a tree by adding one edge, can geodesic tubes and branch-load support be replaced by notions for which an analogue of Theorem [10](#thm-persistence) holds? More generally, how do the answer and the minimum exhaustive-coverage cost depend on bounded tree excess?*
+:::
-A full local team-chase theorem on expanders likely requires either (i) a generalization of the present analysis to graphs with bounded but nontrivial girth, where "tubes" become equivalence classes of approximately-tree-like geodesics, or (ii) a fundamentally different invariant — spectral, fence-based, or potential-theoretic — that does not rely on tree branch decomposition at all.
+# Conclusion
-## Conclusion
+We analyzed a local certificate indexed by length-$t$ nonbacktracking robber paths starting at a fixed root $v_0$ in radius-$R$ tree-ball geometry. The larger radius $R+t$ is sharp, among possibly infinite $d$-regular graphs, for uniform control of arbitrary endpoint pairs, but the synchronized chase witnesses require only $B_R(v_0)$. The length-$t$ tubes partition the outer ball into $N_t=d(d-1)^{t-1}$ parts, and occupying every tube gives the stated capture, blockage, or two-branch-support alternative along every prescribed path in this class.
-We have proved a sharp local persistence theorem in the tree-ball model and two matching barriers: a sampling barrier showing that the natural iteration of the one-round argument is exponentially expensive in time, and a finite-order potential-degeneration barrier showing that any bounded-order compression of the path-tube certificate is dynamically insufficient. Together they tighten the trade-off: 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. Together, they rule out order-$r$ branch-aggregated potentials (for any fixed $r$) as a route to $\Theta(\log n)$-round chase from polylogarithmic cops in the tree-ball model. A structural new ingredient — non-local information, broken symmetry, or escape from the tree-ball regime entirely — is required.
-
-The four main results — deep-load occupancy, $t$-round persistence with depth budget $2t+1$, the sampling barrier, and the finite-order potential-degeneration barrier — together with the supporting tube-counting lemma form a self-contained module that we hope will be useful as one block in any future proof of Meyniel's conjecture via local chase arguments. The pivots in [Section 5](#discussion-pivots-and-open-directions) suggest several directions in which the program could be continued, though we leave their analysis to future work.
-
-## Acknowledgments
-
-I warmly thank Eric Jovinelly, who led me through Graph Theory at Brown and greatly assisted me in building both graph theoretic and broader mathematical skills and intuition. I also thank my peers in the Spring 2026 S02 of Graph Theory at Brown.
+At one fixed root the deterministic occupancy cost is $N_t$. Along fixed-degree rooted sequences with $t\to\infty$, the conditional uniform-sampling threshold is $$\left(\frac{1}{1-(d-1)^{-(h_t+1)}}+o(1)\right)N_t\log N_t,
+\qquad h_t=R_t-t.$$ Because the sampling distribution is chosen after the root, it is not a legal standard-game initial placement and gives no cop-number bound. The arbitrary-walk, partial/adaptive, global, augmented-profile, and cyclic directions above are not a
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