Residual-Window Induced Dynamics in a Finite Non-Autonomous System-A Non-Reducible Formulation of KFX-R

This paper introduces KFX-R (Residual KFX), a finite discrete system in which macroscopic residuals are not treated as autonomous states. Instead, residual evolution is defined exclusively through a sliding-window–induced operator acting on finite trajectories. The system is constructed from a minimal local rule set over a finite state space, together with five-step and ten-step residual projections defined on length-10 trajectories. We prove that no autonomous evolution function depending solely on residual states can exist. Residual dynamics is necessarily non-autonomous, non-Markovian, and driven by boundary injections induced by window translation. This result is established both by formal derivation and by exhaustive finite-state enumeration. The original KFX system is defined within this framework as the zero-residual slice of KFX-R, yielding a unified formal description. By construction, the framework prevents reconstruction of micro-history from macroscopic quantities and enforces a strict separation between trajectory dynamics and residual evolution. KFX-R provides a minimal, non-reductionist template for coarse-grained modeling in automata theory, control systems, and non-equilibrium dynamics. 本文提出 KFX-R(Residual KFX),一种有限离散系统,其中宏观残差并不构成自治状态,其演化仅由固定长度滑动窗口在轨道空间中的平移所诱导。系统基于最小局部规则集与有限状态空间,通过对长度为 10 的轨道定义五步与十步残差,构造出一个由窗口边界驱动的残差演化算子(RSID)。 本文证明:不存在任何仅依赖残差态本身的自治演化函数;残差动力学必然是非自治、非马尔可夫的。该结论通过形式化推导与有限穷举计算双重验证。进一步地,本文将原 KFX 系统定义为 KFX-R 的零残差切片,从而在同一形式框架下统一描述两者。 该工作从结构上阻断了由宏观量反推微观历史的可能性,明确区分了轨道动力学与残差演化,为自动机理论、控制论及非平衡系统中的粗粒化建模提供了一个不可叙事化、不可还原的最小范式。