research/graph-theory

• Essay
  Ball-Occupation Certificates under Coarse Graph Projections 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 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.

  research/mathematics
• Essay
  The Annealed Critical Window for Growing-Radius Domination in Random Regular Graphs 24 July 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 result again strengthens 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.

  Superseded by a proof of the full bounded critical window and its universal scaling function; retitled and restructured to match the new preprint.

  research/mathematics
• Essay
  From Path Tubes to a Near-Critical Domination Bound 22 July 2026

  A first-person, code-heavy companion to the preprint Near-Critical First-Moment Lower Bounds 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.

  research/mathematicstech/python
• Essay
  Branch-Tube Persistence and Static Coverage in Tree-Ball Geometry 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.

  research/mathematics