Coherent Comparison as Information Cost: A Cost-First Ledger Framework for Discrete Dynamics

We develop an information-theoretic framework for discrete dynamics grounded in a comparison-cost functional on ratios. Given two quantities compared via their ratio $x=a\mathrm{/}b$, 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): $$J(x)=\frac{1}{2}(x+x^{-1})-1.$$ This cost exhibits reciprocity $J(x)=J(x^{-1})$, vanishes only at $x=1$, and diverges at boundary regimes $x\to 0^{+}$ and $x\to \mathrm{\infty }$, 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 $\delta \mathbb{Z}$) 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.