research/mathematics

• Essay
  Timed Guarding in Degree-Reduction Clouds 9 September 2026

  The actual Hosseini–Mohar–Gonzalez Hermosillo de la Maza stopping tower admits a square-root cop bound in the order of its base whenever the base satisfies the fixed multistage hypotheses used by the earlier occupation transfer. Every ancestry cloud is a retract, so a region that would cost many cops to occupy costs one cop to guard. Fixed port anchors pay for timed retraction setup within the original five-layer accessibility margin. Synchronized setup leaves at most two exceptional clouds, handled by a reusable reserve. The resulting bound $c(H)\le C\sqrt N$ has no multiplicative dependence on cloud order or tower scale. A 104-vertex stopped tower shows that the timing gain is a hypothesis and not a consequence of adjacency, and a Hall-capacity inequality shows why full remaining-ball demand stays incompatible with slow growth. The theorem retains the full remaining-ball Hall demand and does not give an all-graphs Meyniel bound.

  research/graph-theory
• Essay
  Ball-Occupation Certificates under Coarse Graph Projections 9 September 2026 ·Revised from 27 July 2026

  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 a quantitative cop-number bound from uniform growth on the base. Applied to the Hosseini-Mohar-Gonzalez Hermosillo de la Maza degree-reduction construction, this gives the known hard family an upper bound within a polylogarithmic factor of the square-root scale. 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. The adaptive multistage strategy of Prałat and Wormald does transfer when the base has sphere growth and reservoirs accessible five layers early: for fixed degree, every $(\lambda,P)$-occupation projection satisfies $c(H)\le C_dP\sqrt N$, with no logarithm or scale dependence in the cop count. Fixed-degree random regular bases satisfy the strengthened hypothesis with high probability. A companion paper now removes the cloud-order factor for the actual stopping tower under these same fixed multistage hypotheses, using timed cloud guards rather than full-fiber occupation.

  Linked the companion timed-guarding theorem, which removes the cloud-order factor for the actual stopping tower under the same fixed multistage hypotheses; reconciled the scope and remaining-demand question.

  research/graph-theory
• Essay
  Branch-Tube Persistence and Static Coverage in Tree-Ball Geometry 5 September 2026 ·Revised from 21 July 2026

  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.

  Corrected the root-degree statement and clarified admissible-root coverage with a counterexample to residual domination.

  research/graph-theory
• Essay
  From Path Tubes to a Near-Critical Domination Bound 5 September 2026 ·Revised from 22 July 2026

  A first-person, code-heavy companion tracing the near-critical bound that led to The Annealed Critical Window for Growing-Radius Domination in Random Regular Graphs. Rather than reproducing its theorem-and-proof form, this page traces where the problem came from — a cops-and-robbers hypergraph question that collapsed into a domination bound — why the answer carries an unnecessary coupon-collector logarithm, and how a chain of computational detours (a failed concavity conjecture, a catastrophic cancellation, an independent audit that caught a stale constant) repeatedly redirected the proof before it reached its final shape.

  Clarified the scope of the tube interpretation, linked the corrected current preprint, and recorded the resolved bounded critical window.

  research/graph-theorytech/python
• Essay
  The Annealed Critical Window for Growing-Radius Domination in Random Regular Graphs 5 September 2026 ·Revised from 22 July 2026

  A complete determination of the bounded annealed critical window for growing-radius domination in random regular graphs, resolving the open problem left by an earlier near-critical bound. Writing L_h = log B_h for the radius-h tree-ball volume, the annealed exponent obeys a universal scaling law (B_h/L_h^2) Psi_{d,h}((L_h - 2 log L_h + s)/B_h) -> 1 - e^{-s}, and its lower zero is pinned to within B_h^{-1/7+o(1)} of the scalar coupon-collector root H(C/B_h) = (1-C/B_h)^{B_h}, giving a complete fixed-order inverse-logarithmic expansion. The proof adds a quantitative reverse-transfer error analysis and an explicit capped-free profile whose only entropy loss is the cost of one terminal nonemptiness event, sharpening the earlier annealed lower bound into a two-sided window theorem. The lower bound also applies to internally two-path domination, a graph-general parameter motivated by exhaustive static tube coverage in pursuit-evasion games. Quenched matching and the direct two-branch leading constant remain open.

  Completed the relative overlap estimate, corrected the root-message bound and endpoint domains, and separated the domination consequence from restricted-root tube coverage. Corrected the Duckworth–Mans attribution and clarified the earlier radius-two label representation.

  research/graph-theory