Axiom of Rational Convergence

• The Axiom of Rational Convergence is a principle stating that iterative processes based on rational structures consistently converge in various mathematical frameworks.
• It applies across fields such as model reduction, complex dynamics, random matrix theory, decision theory, and q-deformations to ensure analytical stability.
• The principle guarantees explicit convergence properties, linking theoretical analysis with practical algorithms and geometric insights in multidisciplinary settings.

The Axiom of Rational Convergence refers to a family of principles and results that assert some form of regular, stable, or locally attractive convergence involving rational structures—often rational functions, sequences, or models—in diverse mathematical and applied domains. It appears in a variety of contexts, including model reduction, complex dynamics, random matrix theory, decision theory, number theory, measure theory, and geometry. Despite differences in technical settings, the unifying theme is that under rational or structurally coherent rules, approximations or iterative processes reliably converge, often with explicit guarantees or characterizations.

1. Rational Convergence in Model Reduction and Linear Systems

In optimal model reduction for linear systems, rational convergence acquires a precise and algorithmic form. The Iterative Rational Krylov Algorithm (IRKA) is designed to compute reduced-order models that are $\mathcal{H}_2$-optimal interpolants of high-order rational transfer functions. IRKA operates by iteratively updating interpolation points to coincide with the reflected poles of the current reduced model:

$$s_i \leftarrow -\tilde{\lambda}_i, \quad i=1,\ldots,r,$$

where $\tilde{\lambda}_i$ are the poles of the reduced transfer function

$$H_r(s) = \boldsymbol{c}_r^T(s\boldsymbol{I}_r-\boldsymbol{A}_r)^{-1}\boldsymbol{b}_r.$$

This mapping defines a fixed-point iteration targeting stationary points of the $\mathcal{H}_2$ error norm.

The convergence analysis for IRKA, when applied to state-space symmetric systems ($\boldsymbol{A} = \boldsymbol{A}^T$, $\boldsymbol{b} = \boldsymbol{c}$), demonstrates that the iteration is locally attractive: the Jacobian $J = -\boldsymbol{S}_c^{-1} \boldsymbol{K}$ at a fixed point (a local minimizer) has spectral radius $\rho(J) < 1$. The matrix $\boldsymbol{S}_c$ is constructed from the poles of the reduced model, and $\boldsymbol{K}$ encodes second derivatives of the error system. The result:

• Any fixed point (corresponding to locally optimal reduced models) attracts small perturbations in the interpolation points.
• This ensures practical, robust convergence for large-scale symmetric systems, including those in electrical circuits or certain PDE discretizations (1107.5363).

2. Rational Convergence in Complex Dynamics

In complex dynamics, rational convergence surfaces in the context of the landing of parameter rays with rational arguments for families of entire or polynomial maps. For a one-parameter family, such as $f_a(z)=a [e^z (z-1) + 1]$, and similarly structured polynomial families, Carathéodory convergence topology is used to rigorously analyze the limiting behavior of marked Fatou components and Böttcher coordinates as parameters approach the boundary of the main hyperbolic component.

A central result is the landing theorem: every internal parameter ray with rational argument lands at a unique point on the boundary of the main hyperbolic component. More precisely, for angles $\theta$ with eventually periodic binary expansion, the limiting parameter falls into one of a finite collection of dynamical types determined by combinatorics. The proof uses the convergence properties of Böttcher coordinates (holomorphic invariants) under parameter variation, along with explicit geometric estimates (e.g., using Green functions and hyperbolic metrics). This framework generalizes to many other families, expressing an "axiom of rational convergence": under rational combinatorics, convergence phenomena in both the analytic and geometric senses align to yield unique limiting behaviors (Deniz, 2014).

3. Rational Convergence in Random Matrices and Free Probability

In the probabilistic analysis of large random matrices, rational convergence refers to the convergence in distribution of functions of random matrices that are rational (i.e., constructed from the variables with addition, multiplication, and inversion). Extending classical free probability—which typically handles noncommutative polynomials—recent work has proven that if a sequence $(X_n)$ of random matrices converges strongly in *-distribution to a free limit, then for any non-degenerate noncommutative rational function $R$, the evaluation $R(X_n)$ has a well-defined empirical spectral distribution converging to that of $R(x)$ in the limiting operator variables, provided singularities are avoided.

Key tools include formal linearization: each rational function $R(X)$ admits a presentation $u \cdot A(X)^{-1} \cdot v$, with $A(X)$ an affine linear pencil. Provided this pencil is invertible for sufficiently large $n$, convergence results analogous to those for polynomials hold. This extension answers an open question and shows how rational operations (including inverses) do not break convergence in spectral laws under suitable hypotheses (Collins et al., 2021).

4. Axiom of Rational Convergence in Decision Theory and Probability

A distinct manifestation of rational convergence arises in stochastic choice and decision theory. In the axiomatization of probabilistic rationality, one finds that:

• Any random choice rule satisfying Luce's Choice Axiom can be decomposed into a deterministic stage (choosing optimal supports according to the Weak Axiom of Revealed Preference, WARP) and a stochastic tie-breaking stage (obeying Rényi's Conditioning Axiom).
• As the tie-breaking "noise" diminishes, the random choice converges to the deterministic choice prescribed by the underlying preference or utility, thereby effecting a rational convergence.
• Even in boundedly rational or nearly rational settings, such as those characterized by bounded arbitrage (payoffs from any normalized bet bounded by $\varepsilon$), aggregate behavior lies within $\varepsilon$ of a fully rational model (e.g., linear opinion aggregation or random utility maximization). As $\varepsilon \to 0$, convergence to the canonical rational benchmark is assured (Cerreia-Vioglio et al., 2020, Nascimento, 2022).

5. Rational Convergence in Number Theory and q-Deformations

In the theory of $q$-rational and $q$-irrational numbers—particularly $q$-metallic numbers (including $q$-deformed versions of golden and silver ratios)—rational convergence concerns the power series expansion in $q$. For such numbers defined by $q$-deformed continued fraction expansions, there exist explicit sequences of $q$-rational approximants converging to the $q$-metallic number, with the stability of the coefficients ensuring a well-defined power series.

A central result is a lower bound for the radius of convergence of these $q$-series. Specifically, for all $n\geq3$, the radius of convergence $R_n$ of the $q$-metallic number $[n, n, \ldots]_q$ satisfies $R_n \ge (3-\sqrt{5})/2$, and for $n=3,4$, truncated finite $q$-continued fractions have strictly larger radius. These estimates involve discriminants of polynomials associated with the recurrence relations of the $q$-deformed numbers and are central to ensuring analytic control in applications to quantum algebra and combinatorics (Ren, 2021).

6. Foundational and Geometric Perspectives

At a foundational level, rational convergence connects with the completeness properties of ordered fields and measure theory. In real analysis, Littlewood's three principles—expressing "almost" approximation for measurable sets and functions—are shown to be equivalent to the completeness (Cut) axiom, which in turn is equivalent to the Axiom of Rational Convergence: every Cauchy (or "rationally convergent") sequence has a limit in the field. Incomplete fields obstruct such approximations, highlighting that rational convergence is necessary for the regularity phenomena underpinning measure-theoretic real analysis (Cantuba, 2023).

In ordered geometry, rational convergence is encoded axiomatically. For instance, in ordered affine geometry, a complex axiom concerning the "convergence" of three lines is shown to be equivalent to a simple symmetry property among lines and their direction reversals, streamlining reasoning about geometric convergence and demonstrating the utility of automated theorem-proving in verifying and simplifying such axioms (Li, 2023).

7. Geometric Structures: Rational Curves and the Formal Principle

In complex geometry, a recent direction proposes a conjectured "formal principle with convergence" for bracket-generating families of rational curves: any formal isomorphism between the germs of such curves in a complex manifold is convergent, i.e., realized by a genuine biholomorphic map. The conjecture is proven for families of Goursat type—those with minimally positive normal bundles (specifically, $N_{C/X} \simeq O(1)\oplus O^{\oplus d-2}$)—by associating canonical Cartan connections to natural ODE-structures on the deformation space. The analytic proof leverages classical results on the convergence of formal isomorphisms when the underlying Cartan connection is preserved, yielding a powerful geometric instantiation of the rational convergence principle (Hwang, 9 Apr 2024).

Conclusion

The axiom (or principle) of rational convergence underlies a striking spectrum of convergent behavior in mathematical systems that are rationally structured, whether by explicit rational maps, symmetry, combinatorial or algebraic rationality, or the presence of rationally convergent sequences. It serves as a criterion for local or global convergence in iterative algorithms, a regularity condition ensuring analyzability and computational stability, a tool in establishing foundational properties in analysis and geometry, and a framework generalizing classical results to noncommutative, stochastic, or deformed settings. Its diverse technical realizations are united by the reliability and stability imparted by rational structure in the underlying mathematical systems.