Reciprocal Cost Functional

• Reciprocal Cost Functional is a unique formulation that assigns a cost to scalar ratio deviations, ensuring compositional coherence through a hyperbolic cosine-based calibration.
• It is strictly positive away from equilibrium (x = 1) and diverges at boundaries, thereby excluding degenerate cases and maintaining robust comparison metrics.
• The functional underpins applications in discrete dynamics, optimal transport, and multi-marginal frameworks, offering practical insights for ledger constructions and correlated physics models.

A reciprocal cost functional quantifies the deviation—or "defect"—in the comparison between two positive scalar quantities, assigning a cost to their ratio in a manner that is information-theoretically coherent and robust to composition. The canonical reciprocal cost is the unique functional satisfying strict requirements of compositional coherence, normalization at equilibrium, and quadratic calibration in logarithmic coordinates, with applications spanning discrete dynamical ledgers, optimal transport problems, and multi-marginal cost formulations in mathematical physics.

1. Foundational Principles and Compositional Coherence

Central to the construction of the reciprocal cost functional is the demand for coherence under multiplicative composition of ratio comparisons. Considering three positive quantities $a$, $b$, $c$, there are two routes to compare $a$ and $c$: directly via $a/c$, and indirectly via chaining $a$ vs $b$ and $b$ vs $c$. The cost functional $F$ must satisfy the d'Alembert functional equation: $$F(xy) + F(x/y) = 2 F(x) F(y) + 2 F(x) + 2 F(y), \qquad x, y > 0.$$ This constraint arises from requiring that the total cost associated with comparing via intermediate steps (and their ratios) matches the cost of direct comparison, with the only symmetric, bilinear composition law compatible with the multiplicative structure of ratio chaining. This principle is further sharpened by normalization ($F(1)=0$ for equilibrium ratios) and a quadratic calibration at unity—expressing local information-theoretic consistency—yielding the second-order expansion in logarithmic coordinates with unit curvature (Pardo-Guerra et al., 17 Jan 2026).

2. Unique Formulation and Derivation

The functional equation is solved by symmetry and log-coordinates substitution, leading to a hyperbolic cosine-based solution. Let $t = \ln x$ and $G(t) = F(e^{t})$; the functional equation is translated into an additive d'Alembert equation for $G$, whose unique normalized and calibrated solution is $G(t) = \cosh(t)-1$. Thus, reverting to the original variable, the reciprocal cost functional is: $J(x) = F(x) = \frac{1}{2}(x + x^{-1}) - 1,$ where $J(x)$ is manifestly symmetric ($J(x) = J(x^{-1})$), vanishes only at $x=1$, and diverges at the "nothingness" boundaries $x \to 0^+, x \to \infty$ (Pardo-Guerra et al., 17 Jan 2026).

3. Mathematical Properties and Domain Exclusion

The reciprocal cost functional $J(x)$ possesses several critical mathematical properties:

• Reciprocity: $J(x) = J(x^{-1})\; \forall x > 0$. This ensures invariance under exchange of the compared quantities.
• Strict Positivity and Zero at Equilibrium: $J(x) = 0$ iff $x=1$; everywhere else $J(x)>0$.
• Divergence at Boundaries: $$\lim_{x \to 0^+} J(x) = \lim_{x \to \infty} J(x) = +\infty$$, enforcing infinite cost for ratios approaching zero or infinity and excluding degenerate ("nothingness") cases.
• Quadratic Calibration: For small deviations $x = e^t$, $J(e^t) = \cosh(t)-1 = \frac{1}{2} t^2 + O(t^4)$, aligning with second-order expansions in classical divergence measures.

This structure prevents recognition or comparison events in the mathematical framework from occurring if either quantity is vanishingly small, consistent with information-theoretic exclusion of null events (Pardo-Guerra et al., 17 Jan 2026).

4. Information-Theoretic Interpretation

Analogous to conventional divergences (e.g., Kullback–Leibler, Rényi) defined on log-ratios of probability distributions, $J(x)$ quantifies informational cost for scalar ratio comparisons. For deviations near equilibrium ($x \approx 1$), $J(x)$ locally matches the Fisher information metric through its quadratic expansion, providing a natural cost geometry for discrete recognition events. The infinite cost at degenerate boundaries ensures that "nothing cannot recognize itself"—empty configurations are inherently excluded by assigning them infinite cost.

This quantification generalizes standard divergence concepts to contexts—such as discrete ledgers and recognition events—where pairwise comparison of non-negative scalars (rather than full distributions) is primary (Pardo-Guerra et al., 17 Jan 2026).

5. Applications in Discrete Dynamics and Optimal Transport

With $J(x)$ as core input, a ledger framework for discrete dynamical event encoding is constructed:

• Ledger Construction: Each atomic recognition event corresponds to a cost posting derived from $J(x)$, yielding a minimal and lossless encoding without intra-tick ordering metadata (maximal one event per tick).
• Graph and Potential Theory: Under conservation, pairwise locality, and quantized update assumptions, the ledger formalism enforces balanced double-entry recordings and yields scalar potentials on cycles through time-aggregated closure (no-arbitrage over finite windows), with cycle closure guaranteeing path-independence by discrete Poincaré lemma (Pardo-Guerra et al., 17 Jan 2026).
• Hypercube Graph Periodicity: For $Q_d$, atomicity via $J$ imposes a $2^d$-tick minimal period, with explicit Gray-code cycles at $d=3$.

In mathematical physics, reciprocal cost functionals arise as kernels in pairwise comparison-based transport problems. For instance, Coulomb cost $1/|x-y|$ (a reciprocal-type cost function) underpins strictly correlated electron models in density functional theory; its integrability, regularity, and existence/uniqueness properties are developed for multi-marginal optimal transport (Cotar et al., 2011, Gerolin et al., 2018, Bouchitté et al., 2019). Duality frameworks and stratification formulas for the cost and its relaxations admit concrete characterization of optimal plans and mass quantization phenomena.

6. Reciprocity and Repulsive Cost Kernels in Optimal Transport

Reciprocal functionals also encompass a class of repulsive cost kernels in optimal transport, notably $h(r) = 1/r$. For marginal distributions over metric spaces, these costs satisfy key continuity and coercivity properties, resulting in well-posed multi-marginal optimization problems:

• Duality Theory: The Kantorovich dual, with potentials constrained by the cost structure, attains supremum and admits Lipschitz regularity (Gerolin et al., 2018, Bouchitté et al., 2019).
• Relaxations: Weak* lower-semicontinuous relaxations (sub-probabilities on compactifications) and stratified cost decompositions further generalize the reciprocal cost context, and dual formulations are explicitly characterized (Bouchitté et al., 2019).
• Uniqueness and Regularity: For two marginals and reciprocal costs in dimension $d$, optimal transport maps are unique and cyclically monotone; for higher multi-marginal cases, stratification into clusters is explicit but uniqueness can fail (Cotar et al., 2011, Gerolin et al., 2018).

These reciprocal cost frameworks rigorously exclude degenerate pairings and encode exchange-anticorrelation explicitly, forming the basis for new approaches in strongly-correlated many-body systems and robust discrete event allocations.

7. Physical Significance and Future Directions

The reciprocal cost structure encodes a unique coherent penalty per scalar comparison, connecting information geometry, ledger minimality, conservative flows on graphs, and multi-marginal transport. A plausible implication is that atomicity and exclusion principles in recognition systems or quantum analogs are universally governed by reciprocal kernel costs, suggesting avenues for constructing lossless encodings, optimization strategies with integrated exclusion, and quantized mass rules in transport-driven systems.

The synthesis of coherent multiplicative comparison, symmetry, equilibrium calibration, and exclusion creates a cost paradigm for discrete dynamics and correlated physics, with substantial implications for information-based formalizations beyond classical divergence measures.