隐马尔可夫过程的热力学可实现性：可达性、可观测证书与架构代价
Thermodynamic Realizability of Hidden Markov Processes: Attainment, Observable Certificates, and the Price of Architecture
- Lomonosov Moscow State University（莫斯科国立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明隐马尔可夫过程在固定隐维数下精确重现的最小耗散可达，并引入架构代价相图，将剩余问题归结为增长架构能否使耗散趋于下确界。
AI中文摘要：
能够精确重现给定随机过程的最小耗散有限马尔可夫机器是什么？一个基本的紧致性担忧是，最小化序列可能仅通过将微观速率发送至无穷大来降低成本，而从不收敛到实际的有限速率机器。我们证明在固定隐维数下这种逃逸是不可能的。消失平稳占据的状态可以被追踪去除，无限快的传导类可以被收缩，而不会增加熵产生或失去极限观测路径定律。因此，精确定律成本 $V_n(P)$ 在每个固定状态数下均可达到且下半连续，对速率或平均活动没有上限。对每个使用的隐状态添加正价格 $\kappa$，可在所有有限维数下产生可达到的最优值，并通过由有限多个可观测路径期望构建的全局校准下证书进行精确表示。下包络 $G_\kappa(P)=\min_n[V_n(P)+\kappa n]$ 定义了一个架构相图，其斜率是所选状态数；当 $\kappa\downarrow0$ 时该状态数发散等价于任何有限机器无法达到全维数下确界 $V_\infty(P)$。对于两块更新族 $P_q$，我们推导出完整的有限维相型消去方程，给出一个显式的四相反例以反驳一种诱人的受限参数化，并证明对所有 $0<q<1$ 在每个可行有限维数下最小耗散严格为正。这些结果将剩余问题简化为一个尖锐的问题：增长的隐架构能否驱动 $P_q$ 的耗散达到其全维数下确界而无需任何有限优化器，并在最强情景下达到零？
英文摘要：
What is the least dissipative finite Markov machine that can reproduce a given stochastic process exactly? A basic compactness worry is that a minimizing sequence might lower its cost only by sending microscopic rates to infinity, never converging to an actual finite-rate machine. We prove that this escape is impossible at fixed hidden dimension. States of vanishing stationary occupation can be traced out, and infinitely fast conductance classes can be contracted, without increasing entropy production or losing the limiting observed path law. Hence the exact-law cost $V_n(P)$ is attained and lower semicontinuous at every fixed state count, with no cap on rates or mean activity. Adding a positive price $κ$ per used hidden state yields an attained optimum over all finite dimensions and an exact representation by globally calibrated lower certificates built from finitely many observable path expectations. The lower envelope $G_κ(P)=\min_n[V_n(P)+κn]$ defines an architecture phase diagram whose slope is the selected state count; divergence of that count as $κ\downarrow0$ is equivalent to failure of any finite machine to attain the all-dimension infimum $V_\infty(P)$. For the two-block renewal family $P_q$, we derive complete finite-dimensional phase-type cancellation equations, give an explicit four-phase counterexample to a tempting restricted parameterization, and prove strictly positive minimal dissipation at every feasible finite dimension for all $0<q<1$. These results reduce the remaining problem to one sharp question: can growing hidden architecture drive the dissipation of $P_q$ to its all-dimension infimum without any finite optimizer, and in the strongest scenario, to zero?
