AI 中文总结
本文研究路径上图钉扎的支撑坍缩版本，证明可堆叠概率在固定偏移处发生相变，并给出精确的转变中心，方法包括递归得分和消息递推等。
AI 中文摘要
我们研究了路径上图钉扎问题的一种支撑坍缩版本。如果一个配置可以通过一系列合法的图钉扎移动产生一个支撑在单个顶点上的非零配置，则该配置是可堆叠的。在路径 P_n 上，我们从总数为 n 乘以 mu_n 的所有弱组合中均匀选择一个配置，其中 mu_n 是一个正整数。我们证明了对数密度的双侧固定偏移转变。该转变以 sqrt(log_2 n) - (1/2) log_2 log_2 n + log_2(3e) 为中心。对于任意固定的 epsilon 大于零，当 log_2 mu_n 最终至多为中心减去 epsilon 时，可堆叠概率趋于零；当 log_2 mu_n 最终至少为中心加上 epsilon 时，可堆叠概率趋于一。在零偏移处不做断言。证明使用了树上的精确递归可堆叠性得分、一维路径消息递推、罕见二元赤字偏移的二元划分渐近、常数成本再生论证以及精确深消息必要性定理。将独立的几何占用条件化在其总和上，得到均匀的固定总模型。有限确定性必要性定理及其精确固定总推论在 Lean 中形式化并在 Palomar 中注册；完整的概率渐近定理不包含在该注册中。
英文摘要
We study a support-collapse version of graph pebbling on paths. A configuration is stackable if a sequence of legal pebbling moves can produce a nonzero configuration supported on a single vertex. On the path P_n, we choose a configuration uniformly from all weak compositions of total n times mu_n, where mu_n is a positive integer. We prove a two-sided fixed-offset transition for the logarithmic density. The transition is centred at sqrt(log_2 n) - (1/2) log_2 log_2 n + log_2(3e). For every fixed epsilon greater than zero, the stackability probability tends to zero when log_2 mu_n is eventually at most the centre minus epsilon, and tends to one when it is eventually at least the centre plus epsilon. No assertion is made at zero offset. The proof uses an exact recursive stackability score on trees, a one-dimensional path-message recurrence, binary-partition asymptotics for rare dyadic deficit excursions, a constant-cost regeneration argument, and an exact deep-message necessity theorem. Conditioning independent geometric occupancies on their sum returns the uniform fixed-total model. The finite deterministic necessity theorem and its exact fixed-total corollary are formalised in Lean and registered with Palomar; the full probabilistic asymptotic theorem is not part of that registration.
Comments16 pages, 0 figures. A finite deterministic deep-message necessity theorem and its exact fixed-total corollary are formalised in Lean and registered with Palomar (PALOMAR-2026-09-30-000023, version 1); the full probabilistic asymptotic theorem is not part of that registration
