We study a rigidity problem for functions $F:{\mathbb{R}}_{>0}\to {\mathbb{R}}_{\ge 0}$ that penalize deviation of a positive ratio from equilibrium $x=1$. Assuming (i) a d’Alembert-type composition law on ${\mathbb{R}}_{>0}$, and (ii) a single quadratic calibration at the identity (in logarithmic coordinates), we prove that F is uniquely determined. The composition law implies the normalization $F\left(1\right)=0.$ The unique solution is called the canonical reciprocal cost, namely the difference between the arithmetic and geometric means of x and its reciprocal. Our proof uses the logarithmic coordinates $H\left(t\right)=F\left(e^t\right)+1$, where the composition law becomes d’Alembert’s functional equation on $\mathbb{R}$. The calibration provides the minimal regularity needed to invoke the classical classification of continuous solutions and fixes the remaining scaling freedom, selecting the hyperbolic-cosine branch. We also establish the necessity of each assumption: without calibration the composition law admits a continuous one-parameter family; without the composition law the calibration does not determine the global form; and without regularity the composition law admits pathological non-measurable solutions. Finally, we establish a stability estimate for approximate solutions under bounded defect and characterize some properties of the canonical cost.

We study functions satisfying the composition law $F\left(xy\right)+F(x/y)=P\left(F\right(x),F(y\left)\right)$ with a symmetric polynomial combiner P. We prove that symmetry together with a quadratic degree bound on P forces a composition law of d’Alembert type. We establish a degree mismatch exclusion criterion showing that symmetric polynomial combiners with $degP(u,v)\ge 3$ do not admit nonconstant continuous solutions, provided the leading term does not cancel (Theorem 1). For continuous nonconstant functions $F:{\mathbb{R}}_{>0}\to \mathbb{R}$ with $F\left(1\right)=0$ satisfying the composition law with a symmetric polynomial P of degree at most two, the combiner is necessarily of the form $P(u,v)=2u+2v+cuv$, $c\in \mathbb{R}$ (Theorem 3). The equation reduces in logarithmic coordinates to the classical d’Alembert functional equation. For $c\ne 0$, one obtains hyperbolic or trigonometric branches, while $c=0$ yields the squared-logarithm family. Under the cost-function assumptions $F\ge 0$ and convexity, only the hyperbolic branch with $c>0$ remains. A unit log-curvature calibration selects the canonical value $c=2$, which yields the canonical reciprocal cost $F\left(x\right)=\frac{1}{2}(x+x^{-1})-1$. For $c\ne 0$, the result extends to ${\mathbb{R}}_{>0}^n$: every solution depends only on a single linear combination of coordinate logarithms; for $c=0$, the solution is a general quadratic form ${\sum }_{i,j}{a}_{ij}ln{x}_{i}ln{x}_j$. In either case, nontrivial coordinate-wise separable costs are excluded.

We develop an information-theoretic, cost-first framework for discrete dynamics in which the primitive operation is ratio-based comparison. Given two quantities compared via their ratio $x=a/b$, we assign a cost $F\left(x\right)$ measuring deviation from equilibrium ($x=1$). Adopting a reciprocal d’Alembert composition law motivated by coherent chaining, together with quadratic calibration at unity, uniquely determines a reciprocal comparison cost $J\left(x\right)=\frac{1}{2}\left(x+x^{-1}\right)-1$. Taking J as input, we model recognition events as deterministic updates on directed graphs recorded in a minimal ledger. Minimality (no intra-tick ordering metadata) together with non-commutativity of events implies atomic ticks: at most one event per tick. With conservation, pairwise locality, and quantization in $\delta \mathbb{Z}$, each event is recorded as a balanced double-entry posting. For graphs with cycles, assuming time-aggregated cycle closure over a finite clearing horizon, we show that cleared cycle closure is equivalent to path-independence and that the cumulative flow admits a scalar potential on each connected component (unique up to additive constant) via a discrete Poincaré lemma. On hypercube graphs $Q_d$, atomic single-edge updates impose a $2^d$-tick minimal period for timestamp-unique coverage, realized by cyclic Gray codes (explicitly for $d=3$). The framework links ratio-based cost functions, conservative graph flows, and discrete potential theory through explicitly stated axioms and structural assumptions.

We introduce Recognition Geometry (RG), an axiomatic framework in which geometric structure is not assumed a priori but derived. The starting point of the theory is a configuration space together with recognizers that map configurations to observable events. Observational indistinguishability induces an equivalence relation, and the observable space is obtained as a recognition quotient. Locality is introduced through a neighborhood system, without assuming any metric or topological structure. A finite local resolution axiom formalizes the fact that any observer can distinguish only finitely many outcomes within a local region. We prove that the induced observable map $\overline{R}:{\mathcal{C}}_R\to \mathcal{E}$ is injective, establishing that observable states are uniquely determined by measurement outcomes with no hidden structure. The framework connects deeply with existing approaches: C^*-algebraic quantum theory, information geometry, categorical physics, causal set theory, noncommutative geometry, and topos-theoretic foundations all share the measurement-first philosophy, yet RG provides a unified axiomatic foundation synthesizing these perspectives. Comparative recognizers allow us to define order-type relations based on operational comparison. Under additional assumptions, quantitative notions of distinguishability can be introduced in the form of recognition distances, defined as pseudometrics. Several examples are provided, including threshold recognizers on ${\mathbb{R}}^n$, discrete lattice models, quantum spin measurements, and an example motivated by Recognition Science. In the last part, we develop the composition of recognizers, proving that composite recognizers refine quotient structures and increase distinguishing power. We introduce symmetries and gauge equivalence, showing that gauge-equivalent configurations are necessarily observationally indistinguishable, though the converse does not hold in general. A significant part of the axiomatic framework and the main constructions are formalized in the Lean 4 proof assistant, providing an independent verification of logical consistency.

In this paper, we study the geometric structure induced by the canonical reciprocal cost function and its natural n-dimensional extension. In logarithmic coordinates, the potential depends only on the linear combination $S=\alpha ·t$, and the associated Hessian metric has rank one at every point. The geometry is intrinsically degenerate and effectively one-dimensional, with an $(n-1)$-dimensional null distribution. On the other hand, when the same function is expressed in the original x-coordinates, the corresponding Hessian is generically nondegenerate and defines a pseudo-Riemannian metric away from explicit singular hypersurfaces. We further analyze affine and Levi-Civita geodesics and compare their behavior. In particular, affine geodesics in logarithmic coordinates are globally defined, while in x-coordinates their behavior is restricted by the domain and the singular set. Finally, we relate the construction to symmetrized Itakura–Saito and Bregman divergences, and give a Fisher–Rao realization of the logarithmic Hessian metric.

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.