We develop an information-theoretic framework for discrete dynamics grounded in a comparison-cost functional on ratios. Given two quantities compared via their ratio \(x=a/b\), we assign a cost \(F(x)\) measuring deviation from equilibrium (\(x=1\)). Requiring coherent composition under multiplicative chaining imposes a d'Alembert functional equation; together with normalization (\(F(1)=0\)) and quadratic calibration at unity, this yields a unique reciprocal cost functional (proved in a companion paper): \[ J(x) = \tfrac{1}{2}\bigl(x + x^{-1}\bigr) - 1. \] This cost exhibits reciprocity \(J(x)=J(x^{-1})\), vanishes only at \(x=1\), and diverges at boundary regimes \(x\to 0^+\) and \(x\to\infty\), excluding ``nothingness'' configurations. Using \(J\) as input, we introduce a discrete ledger as a minimal lossless encoding of recognition events on directed graphs. Under deterministic update semantics and minimality (no intra-tick ordering metadata), we derive atomic ticks (at most one event per tick). Explicit structural assumptions (conservation, no sources/sinks, pairwise locality, quantization in \(δ\mathbb{Z}\)) force balanced double-entry postings and discrete ledger units. To obtain scalar potentials on graphs with cycles while retaining single-edge impulses per tick, we impose time-aggregated cycle closure (no-arbitrage/clearing over finite windows). Under this hypothesis, cycle closure is equivalent to path-independence, and the cleared cumulative flow admits a unique scalar potential on each connected component (up to additive constant), via a discrete Poincaré lemma. On hypercube graphs \(Q_d\), atomicity imposes a \(2^d\)-tick minimal period, with explicit Gray-code realization at \(d=3\). The framework connects ratio-based divergences, conservative graph flows, and discrete potential theory through a coherence-forced cost structure.

翻译：我们建立了一个基于比值比较成本泛函的离散动力学信息论框架。给定通过比值 \(x=a/b\) 进行比较的两个量，我们分配一个成本 \(F(x)\) 来衡量其偏离平衡状态 (\(x=1\)) 的程度。要求在乘法链下具有相干复合性，这施加了一个达朗贝尔函数方程；结合归一化条件 (\(F(1)=0\)) 和在单位点处的二次校准，我们得到了一个唯一的互易成本泛函（在配套论文中已证明）：\[ J(x) = \tfrac{1}{2}\bigl(x + x^{-1}\bigr) - 1. \] 该成本具有互易性 \(J(x)=J(x^{-1})\)，仅在 \(x=1\) 时为零，并在边界区域 \(x\to 0^+\) 和 \(x\to\infty\) 处发散，从而排除了“虚无”构型。以 \(J\) 作为输入，我们引入了一种离散账本，作为对有向图上识别事件的最小化无损编码。在确定性更新语义和最小性（无内部时间刻度排序元数据）条件下，我们推导出原子时间刻度（每个刻度至多一个事件）。明确的结构假设（守恒性、无源/汇、成对局部性、在 \(δ\mathbb{Z}\) 上的量子化）强制了平衡的双重记账分录和离散账本单位。为了在具有环路的图上获得标量势，同时保持每个时间刻度的单边冲量，我们施加了时间累积环路闭合条件（在有限时间窗口内无套利/清算）。在此假设下，环路闭合等价于路径无关性，并且经过清算的累积流量在每个连通分量上（相差一个加性常数）允许存在唯一的标量势，这通过一个离散的庞加莱引理实现。在超立方图 \(Q_d\) 上，原子性施加了一个 \(2^d\) 时间刻度的最小周期，并在 \(d=3\) 时给出了明确的格雷码实现。该框架通过一个由相干性强制产生的成本结构，将基于比值的散度、保守图流和离散势理论联系起来。