AI 中文总结
本文研究量子首次通过统计中例外点与Jordan链特征，证明阈值分解与传递分解，提出固定重置不可能定理，并构造最小三重置Lindblad见证模型，展示次主导阈值项与相干调谐下的多项式特征。
AI 中文摘要
首次通过可观测量中的例外点特征由两个独立的生存机制控制：非线性谱缺陷必须从单记录或传递描述传递到物理的单级首次通过阶梯，并且由此产生的阶梯Jordan模式必须与所选的制备和终端观测具有非零重叠。对于具有向上无跳计数的有限重置形式监控量子系统，我们证明了精确的阈值分解$H_N(s)=R_s^N$，识别了物理Perron分支，并推导了传递分解$Q_s(r)=B_s(r)(R_s-rI)$。余因子的可逆性给出了Smith数据的局部相等性，而奇异余因子可能贡献阶梯中不存在的传递重数。一个二阶Keldysh公式产生了一个独立的观测门。然后我们证明了一个固定重置的不可能定理：即使一个缺陷的全倾斜生成器，在后计数阶梯为标量时，也不能生成累积计数Jordan多项式。最后，我们构造了一个三重置监控Lindblad见证，在不可约非负阶梯中最小，具有真正的次主导$N\lambda^N$阈值项和一个相干调谐模型，其中二元终端效应在横向变换域例外点处保留了精确的多项式特征。该分析在精确变换域结构的层面上进行；时域渐近和微扰鲁棒性仍是独立的问题。
英文摘要
For cumulative-count first-passage problems, the Laplace transform of the time $\mathcal T_N$ of the $N$th count is naturally expressed through powers of a one-count kernel. A Jordan defect of that kernel does not, however, automatically survive the complete sum over terminal phases. We give an exact three-state Markov-renewal example in which it does. At $s=1$, the one-count kernel has spectrum $\{1/2,1/8,1/8\}$, with a one-dimensional eigenspace at $1/8$, and for initial phase $1$, \[ \mathbb E_1[e^{-\mathcal T_N}] =\frac56\,2^{-N}+\frac{N+1}{6}\,8^{-N}. \] Thus the ordinary, unconditioned cumulative-count transform contains the explicit Jordan contribution $N/(6\,8^N)$ at every threshold $N\ge1$. For the same fixed stochastic process, the two eigenvalues meeting at $1/8$ unfold as \[ λ_\pm(s)=\frac18\pm\frac1{96}\sqrt{s-1} -\frac{131}{2304}(s-1)+O((s-1)^{3/2}), \] and the characteristic discriminant has a simple zero at $s=1$. We place the example beside two structures that suppress such a marginal signature: fixed post-count reset, which gives scalar renewal, and a common total holding rate, for which terminal summation scalarizes the matrix power. The model also admits a monitored-Lindblad realization, but the mechanism is entirely classical. The result provides a small exact benchmark for how generalized spectral modes can remain visible in finite-threshold first-passage statistics.
CommentsSubstantially revised and narrowed. Corrects the earlier exceptional-point interpretation, clarifies counted self-transitions and terminal-summed visibility, and adds an exact three-state Markov-renewal example with a nonzero N8^{-N} Jordan term and square-root eigenvalue ramification. The mechanism is classical; the Lindblad model is an embedding
