度量树的精确随机覆盖：平衡舍入、对偶性与尖锐阈值
Exact random covers of metric trees: balanced rounding, duality, and sharp thresholds
AI总结：
本文证明了Norin和Turcotte提出的有限度量树精确随机覆盖猜想，通过重铸自举计算、结合分解与递归构造精确覆盖，还建立了对偶准则并拓展了均匀范围、推导了相关估计。
AI中文摘要：
Norin和Turcotte针对图燃烧问题得出了渐近尖锐的界[《组合论杂志B辑》168卷（2024），208--235页]，这促使他们提出了有限度量树的精确随机覆盖猜想。设U[0,r]为区间[0,r]上的均匀概率测度，他们猜想，每条长度L≥2r的有限度量树T，都存在一个0-好球覆盖上的概率测度，其期望半径测度至多为(L/r)U[0,r]。我们证明了该猜想对所有有限度量树均成立。我们将Norin和Turcotte的自举计算重铸为零误差替换证书，所得的局部尺度缩减结合三分分解与宏递归，生成了具有精确半径预算的分数标记球覆盖；随后通过紧致舍入论证，将分数覆盖转化为随机有限覆盖。对于度量树球，Tamir的平衡定理与标准平衡矩阵理想性提供了有限维完整性输入。我们还证明了任意预算的对偶准则：若0<R≤L且β是[0,R]上的有限正Borel测度，则β支配随机0-好覆盖的期望半径测度当且仅当对T上的每个有限正Borel测度σ，都有σ(T)≤∫_{[0,R]}max_{v∈T}σ(B_T(v,s))dβ(s)，且只需检验有限原子测度即可。我们利用该准则将均匀范围扩展至所有r≤L−diam(T)/2，确定了等臂度量星的精确范围，并推导了确定性界、区间刚性与直径缺陷稳定性估计。
英文摘要：
Norin and Turcotte's asymptotically sharp bound for graph burning [J. Combin. Theory Ser. B 168 (2024), 208--235] led them to an exact random-cover conjecture for finite metric trees. Let $U[0,r]$ be the uniform probability measure on $[0,r]$. They conjectured that every finite metric tree $T$ of length $L\ge2r$ admits a probability measure on $0$-good ball covers whose expected radius measure is at most $(L/r)U[0,r]$. We prove the conjecture for every finite metric tree. We recast the bootstrapping calculation of Norin and Turcotte as a zero-error replacement certificate. The resulting local scale reduction, together with a three-piece decomposition and a macro-recursion, produces a fractional marked-ball cover with the exact radius budget. We then pass from the fractional cover to random finite covers by a compact rounding argument. For metric-tree balls, Tamir's balancedness theorem and standard balanced-matrix ideality provide the finite-dimensional integrality input. We also prove an arbitrary-budget duality criterion. If $0<R\le L$ and $β$ is a finite positive Borel measure on $[0,R]$, then $β$ dominates the expected radius measure of a random $0$-good cover if and only if $σ(T)\le\int_{[0,R]}\max_{v\in T}σ(B_T(v,s))\,dβ(s)$ for every finite positive Borel measure $σ$ on $T$; it is enough to test finite atomic measures. We use this criterion to extend the uniform range to every $r\le L-\operatorname{diam}(T)/2$, determine the exact range for equal-arm metric stars, and derive deterministic bounds, interval rigidity, and a diameter-defect stability estimate.
