diff --git a/content/essays/ball-occupation-under-coarse-projections.md b/content/essays/ball-occupation-under-coarse-projections.md
index 27a23de..a90f9c6 100644
--- a/content/essays/ball-occupation-under-coarse-projections.md
+++ b/content/essays/ball-occupation-under-coarse-projections.md
@@ -3,28 +3,21 @@ title: "Ball-Occupation Certificates under Coarse Graph Projections"
 subtitle: "Degree Reduction, Square-Root Hard Families, and Toroidal Barriers"
 date: 2026-07-27
 abstract: >
-  We isolate an abstract strategy-transfer principle for Cops and Robber: if a
-  surjection from H onto G has fibers of bounded order P and distances
-  between distinct fibers stretched by a scale factor lambda with bounded
-  additive slack, then a Hall-type assignment occupies a lifted macro-ball
-  before the robber can leave it, giving cop number O(P(d^3 + log(ePN))
-  sqrt(N)) and capture time at most lambda R -- independent of the tower
-  depth lambda itself. Applied to the iterated degree-reduction construction
-  of Hosseini-Mohar-Gonzalez Hermosillo de la Maza, this shows the known
-  polylog-degree hard family for Meyniel's conjecture already meets the
-  square-root exponent up to polylogarithmic factors, sharper than the
-  M^{1/2+o(1)} notation suggests. A separate counting argument then proves a
-  sharp limitation of the one-shot occupation strategy class itself: for
-  every fixed k, Cartesian tori of k cycles have bounded metric doubling,
-  exact cop number k+1, and linear one-shot occupation cost at every radius,
-  so polynomially weak expansion alone cannot certify a universal robustness
-  theorem -- the obstruction in the universal problem is adaptive reuse
-  across many weak-growth layers, not degree reduction itself.
+  We isolate an abstract strategy-transfer principle for Cops and Robber: a
+  coarse graph projection with bounded fibers and bounded distance distortion
+  lets cops occupy a lifted macro-ball before the robber escapes, giving cop
+  number O(sqrt N) up to polylogarithmic factors. Applied to the
+  Hosseini-Mohar-Gonzalez Hermosillo de la Maza degree-reduction
+  construction, this shows the known hard family for Meyniel's conjecture
+  already meets the square-root exponent, sharper than the usual notation
+  suggests. A counting argument then proves a sharp limit on the strategy
+  class itself: Cartesian tori of cycles have bounded doubling and constant
+  cop number but linear occupation cost at every radius, so weak expansion
+  alone cannot certify a universal robustness theorem.
 tags:
   - research
   - research/mathematics
   - research/graph-theory
-keywords: [cops and robber, meyniel's conjecture, graph products, degree reduction, coarse graph projections, expansion, occupation certificates]
 authors:
   - "Levi Neuwirth | /me.html"
 affiliation:
diff --git a/content/essays/near-critical-growing-radius-domination.md b/content/essays/near-critical-growing-radius-domination.md
index bc56953..fa7c391 100644
--- a/content/essays/near-critical-growing-radius-domination.md
+++ b/content/essays/near-critical-growing-radius-domination.md
@@ -197,7 +197,7 @@ with $0\log 0=0$; for a scalar $a\in[0,1]$, $H(a)$ denotes the binary entropy $H
 | $z=e^\theta$ | grand-canonical activity |
 | $b=d-1,\ D=d/(d-2)$ | recurring degree constants |
 
-::: {#def-two-path .exhibit .exhibit--definition data-exhibit-type="definition" data-exhibit-name="Definition 1 (Internally two-path $(h,2)$ domination)"}
+::: {#def-two-path .exhibit .exhibit--definition data-exhibit-type="definition" data-exhibit-name="Definition 1 (Internally two-path (h,2) domination)"}
 **Definition 1** (Internally two-path $(h,2)$ domination). A set $S\subseteq V(G)$ is *internally two-path $(h,2)$ dominating* if every vertex $v\notin S$ has two $v$–$S$ paths of length at most $h$ whose only common vertex is $v$. In particular, the paths begin through distinct neighbors of $v$. Let $\gamma_{h,2}^{\mathrm{int}}(G)$ denote the minimum size of such a set.
 :::