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/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.

💡 Research Summary

The paper proposes a novel information‑theoretic foundation for discrete dynamical systems built around a “cost of comparison”. The authors start by formalising the primitive act of comparing two positive quantities a and b via their ratio x = a/b. They postulate a cost function F(x) that measures the informational penalty for deviating from perfect balance (x = 1). Three axioms are imposed: (A1) Normalisation F(1)=0, (A2) Coherent composition under multiplicative chaining, expressed as a d’Alembert‑type functional equation
 F(xy)+F(x/y)=2 F(x) F(y)+2 F(x)+2 F(y),
and (A3) Quadratic calibration at the unit ratio, i.e. the second derivative of G(t)=F(e^t) at t=0 equals 1. Under these constraints the cost is uniquely determined (Theorem T5, proved in a companion paper) and takes the reciprocal form

 J(x)=½(x+x⁻¹)−1.

J(x) is non‑negative, vanishes only at x=1, diverges as x→0⁺ or ∞, and satisfies J(x)=J(1/x). This cost becomes the keystone input for the rest of the development.

Using J(x) the authors construct a discrete ledger model that records “recognition events” on a directed graph. The ledger is not an ad‑hoc addition; it emerges from three explicit design principles: (L1) deterministic state updates S_{t+1}=U(S_t,σ_t), (L2) minimality – no intra‑tick ordering metadata, and (L3) conservation of total quantity (no external sources or sinks). From (L1) and (L2) they prove an atomic‑tick theorem (Theorem T2): at most one event can occur per tick. Each event is a signed increment ±δ applied to exactly two nodes, thereby enforcing a balanced double‑entry posting reminiscent of bookkeeping. Quantisation (δ∈ℤ) forces the ledger to operate in discrete units (Proposition T8).

In graphs containing cycles, a single‑edge impulse would normally generate circulating flow, preventing a scalar potential representation. To restore path‑independence the authors introduce a “time‑aggregated cycle‑closure” (no‑arbitrage) hypothesis: over a fixed clearing window the cumulative flow around every directed cycle sums to zero. Under this hypothesis they prove (Theorem T3) that cycle closure ⇔ path independence, and consequently (Theorem T4) that the cleared flow admits a unique scalar potential φ on each connected component, unique up to an additive constant. This discrete Poincaré lemma mirrors continuous potential theory but explicitly accounts for the impulsive, atomic nature of updates.

The framework is then instantiated on hypercube graphs Q_d. Atomicity forces a minimal schedule length of 2^d ticks to visit all vertices without repetition (Theorems T6–T7). For d=3 the authors exhibit an explicit Gray‑code Hamiltonian cycle achieving the 8‑tick lower bound. They argue that d=3 is distinguished when additional synchronization/linking hypotheses (the “gap‑45” or golden‑angle motivation) are imposed, suggesting a special role for three‑dimensional structures.

Overall, the paper weaves together several strands: (1) a functional‑equation derivation of a unique reciprocal cost from coherence, (2) a cost‑first construction of a lossless, deterministic, double‑entry ledger, (3) a clearing‑based bridge from discrete flows to scalar potentials, and (4) combinatorial lower bounds on periodicity in hypercubic networks. By grounding dynamics in an information‑cost that is forced by the very requirement of coherent comparison, the authors provide a fresh “cost‑first” perspective on discrete dynamical systems, with potential implications for statistical physics, network economics, and the foundations of computation.