AI 中文总结
针对带泊松重置边界和死时间的不相关类瞬发单事件与关联瞬发-延迟源，本文推导了精确有序符合计数率，验证了结果并给出有限待处理总体的截断误差界，可用于延迟符合搜索的本底建模。
AI 中文摘要
延迟符合搜索用于识别罕见的瞬发-延迟信号，但其偶然本底并非边缘计数率的简单乘积：μ子 veto（μ子否决）、事件死时间以及在当前可记录事件序列的窗口条件之前产生的延迟事件都会影响结果。我们在给定的随机模型中，推导了受泊松重置边界和事件死时间约束的、由不相关类瞬发单事件与关联瞬发-延迟源组成的已记录流的精确有序符合计数率。该计算分为两个任务：马尔可夫历史链将重置边界处仍待处理的延迟事件跨窗口传递，而当前窗口传播子则使用块矩阵指数（应用概率的矩阵分析工具包）以闭式形式计算窗口内有序积分。该构造可通过增加块链深度扩展至任意规定的有限多重度。在此，我们报告由不相关单事件、关联瞬发事件和已记录延迟事件构成的所有有序一重、二重和三重计数率，以及瞬发-延迟对的真/偶然拆分、多重度效率和四重及以上多重度的总计数率。对于默认窗口关闭约定，我们推导了有限待处理总体截断的三项后验误差界。独立的流式玩具蒙特卡罗模拟验证了有序计数率、二重时间密度、匹配的死时间约定以及高多重度总和。在给定假设下，该构造在保留的有限状态空间上是精确的，并提供了显式、可计算的截断误差界。
英文摘要
Delayed-coincidence searches identify a rare prompt-delayed signal, but their accidental background is not a simple product of marginal rates: muon vetoes, event dead time, and delayed events created before the current window condition which event sequences can be recorded. We derive exact ordered coincidence rates, within a stated stochastic model, for a recorded stream of uncorrelated prompt-like singles and correlated prompt-delayed sources subject to Poisson reset boundaries and event dead time. The calculation separates two tasks: a Markov history chain carries the delayed events still pending at reset boundaries across windows, and a current-window propagator evaluates the ordered within-window integrals in closed form with block-matrix exponentials, the matrix-analytic toolkit of applied probability. The construction extends to any prescribed finite multiplicity by increasing the block-chain depth. Here we report every ordered one-, two-, and three-fold rate formed from uncorrelated singles, correlated prompts, and recorded delayed events, together with the genuine/accidental split of prompt-delayed pairs, multiplicity efficiencies, and the aggregate rate for multiplicity four or more. For the default window-close convention, we derive a three-term a posteriori error bound for the finite pending-population truncation. An independent streaming toy Monte Carlo validates the ordered rates, two-fold time densities, matched dead-time conventions, and aggregate high multiplicity. Within the stated assumptions, the construction is exact on the retained finite state spaces and provides explicit, computable truncation-error bounds.
