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

Abstract: 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.