FROST: Fractal of Stationary Transformations

• FROST is a unified framework that formalizes how stationary, shape-preserving transformations maintain fractal geometry and scale consistency across diverse systems.
• It employs techniques like Taylor expansion, fractal conjugacy, and code-space correspondence to predict embedded Multibrot subfractals and transfer spectral properties between self-similar sets.
• FROST also drives adaptive halting and robust representation learning in latent state-space models, enhancing computational efficiency and geometric convergence in neural networks.

The Fractal of Stationary Transformations (FROST) is a general theoretical and computational framework unifying invariance, self-similarity, and scale-consistent dynamics across fractal geometry, spectral analysis on self-similar sets, and, more recently, deep latent state-space models. FROST formalizes how stationary (shape-preserving) transformations and fractal conjugacy enable both geometric invariance in escape-time fractals and stable, scale-consistent representation learning in neural networks. Its mathematical infrastructure ensures that transformations or iterative updates commute with scaling, ensuring intrinsic consistency at all scales or depths. FROST arises in three principal domains: (1) complex dynamical systems and generalizations of Mandelbrot-like sets, (2) fractal transformations and operator theory on attractors of iterated function systems (IFS), and (3) scale-consistent latent dynamics in high-dimensional learning systems, particularly state-space models.

1. Stationary Transformations in Dynamical Systems

Let $z_{n+1} = f(z_n, c)$ denote an escape-time iteration in the complex plane. FROST defines a stationary (shape-preserving) transformation $T: \mathbb{C} \to \mathbb{C}$ for this map if $T(f(z, c)) = f(T(z), c)$ for all $z$ and $c$. For the polynomial family $f(z, c) = P(z) + c = \sum_{p \ge 2} a_p z^p + c$, $P$ must be homogeneous of degree $n$, or all significant terms must transform with the same global factor under $T$. The structure of stationary transformations on the Multibrot family $z \mapsto z^n + c$ is rigorously characterized by Mondal & Ghosh (Mondal et al., 2012):

• Translation: $T_t(z) = z-b$ rigidly carries the fractal set.
• Rotation: $T_r(z) = e^{i\theta} z$ rotates the Multibrot by $\theta/(n-1)$ about the origin.
• Scaling: $T_s(z) = a z$ scales the fractal by $a^{1/(1-n)}$.
• General: Combinations of translation, rotation, and scaling preserve the set's shape.

For truncated polynomials $P(z)$ without a linear term, these transformations generate an invariance group under which fractal geometry is preserved.

2. Embedded Fractals and the FROST Prediction Principle

FROST provides a predictive mechanism for locating inner Multibrot subfractals in generalized maps. The key is the dominance principle: if $G(z)$ locally satisfies $G(z) \in \Theta(g(z))$ (i.e., $G(z)$ is asymptotically comparable to $g(z)$ near $z_d$), then the filled-in Julia or Mandelbrot set generated by $G+c$ near $z_d$ recapitulates that of $g+c$ (Mondal et al., 2012). Practically, the method involves:

• Taylor/Laurent expansion: Decompose $G(z)$ near $z_d$; monomial dominant term $z^k$ guides subsequent scaling.
• Local multibrot extraction: Apply the change of variables $z \mapsto (z-z_d)/\epsilon$ and $c \mapsto c/\epsilon^k$ to resolve miniature Multibrot sets of exponent $k$.

Polynomial and transcendental maps—including $z^2+z^3+c$, $z^4+z^7-z^{10}+c$, $F(z)=\cos z - 1 + c$, $F(z)=(\tan z)^2 + c$—demonstrate nested classical fractals corresponding to the dominant local monomial.

3. Fractal Transformations on Self-Referential Spaces and Operator Conjugacy

In the context of iterated function systems (IFS), FROST, as formalized in Barnsley et al. (Bandt et al., 2014), provides a rigorous framework for constructing stationary (invariant) measures and inducing unitary transformations between Hilbert spaces associated with self-similar sets. For IFS $F = \{X; f_1, ..., f_N\}$, the attractor $A_F$ supports a stationary $p$-measure $\mu_F$ uniquely defined via

$\mu_F(B) = \sum_{i=1}^N p_i \mu_F(f_i^{-1}(B))$

for Borel sets $B$. Given two such IFSs $F$ and $G$ with an identical number of maps and the same probability vector $p$, the fractal transformation $T_{FG}$—constructed via code-space correspondence—pushes $\mu_F$ to $\mu_G$ and is continuous $\mu_F$-almost everywhere.

This induces:

• Unitary conjugacy between $L^2(A_F, \mu_F)$ and $L^2(A_G, \mu_G)$,
• Transport of orthonormal bases and flows: Classical basis functions or dynamics can be synthesized into their fractal analogs (e.g., fractal sines, wavefront fractals),
• Preservation of spectral and ergodic properties under fractal transformation.

This principle enables a "fractal Fourier analysis" generalizing classical harmonic analysis to non-Euclidean self-similar domains.

4. FROST in State-Space Dynamics and Deep Learning

FROST has been extended to latent state-space models in high-dimensional learning, enforcing a fractal self-similar inductive bias within the dynamics (Yu et al., 27 Jan 2026). In these models:

• The update map $T: \mathbb{R}^d \to \mathbb{R}^d$ is applied iteratively: $x_{t+1} = (1-\alpha)x_t + \alpha F(x_t, u_t)$, with $F$ stationary and $\alpha \in (0,1)$ learnable.
• Self-similarity is encoded by requiring $T(S_\lambda(x)) = S_\lambda(T(x))$, $S_\lambda(x) = \lambda^H x$ for $H \in (0,1]$, so every intermediate latent $x_t$ is an exact scaled version of the same representation manifold.
• Banach contraction mapping ensures unique fixed points and geometric convergence: Choosing $L = (1-\alpha) + \alpha L_A < 1$ for the Lipschitz constant $L_A$ of $A$ secures stability and scale-consistent refinement.

Empirically, this yields trajectories whose geometry is fractally self-similar, verified by:

• Smooth, monotonic increase in representation alignment (cosine similarity approaching 0.99),
• Higher Minkowski dimension ($D \approx 1.295$ vs. $D \approx 1.262$ for baselines), indicating greater information density across iterations.

5. Adaptive Halting via Fractal Geometry

A direct consequence of FROST's scale-consistency is a geometry-driven halting mechanism in iterative inference (Yu et al., 27 Jan 2026). For each batch and step, per-sample losses are ranked and a halting score $s_{i,t}$ is predicted. The learning objective combines relative ranking loss (optimizing correct ordering of stops) and absolute anchoring loss (pushing "easy" stops toward 1, "hard" toward 0). The method achieves monotonic accuracy versus depth and significant computational savings (e.g., halving GFLOPs at target quantile $q=0.5$). Unlike non-fractal recurrent baselines, the FROST model remains stable under this loss and produces well-separated, interpretable scores suitable for real-time adaptive computation.

6. Limitations and Scope of FROST Invariance

FROST invariance results depend on several key conditions:

• Homogeneity: Rotation and scaling require the dominant term in the dynamics or IFS to be homogeneous. If the main term is non-integer or inhomogeneous (e.g., $z^{p/q}$), shape-preserving conjugacies generally fail.
• Linear terms: Presence of a $z^1$ term disrupts the invariance due to singularities in denominators for the rotation/scaling formulae.
• Singularities in transcendental maps: For transcendental $G(z)$, only regions where Taylor or Laurent expansions converge inherit the inner Multibrot fractals; global self-similarity may be lost outside those neighborhoods.
• Numerical/finite-iteration issues: For computational implementations, escape radii and numerical artifacts may mask or erase fine structure in the tiniest embedded fractals.

7. Representative Illustrations and Applications

Across mathematical and computational experiments, FROST provides a unified lens for:

• Locating and predicting inner self-similar structures in polynomial and transcendental fractals (Mondal et al., 2012)
• Constructing spectral dualities, orthonormal bases, and flows between fractal geometries (Bandt et al., 2014)
• Enforcing scale-consistent representations and adaptive halting in deep learning architectures (Yu et al., 27 Jan 2026)

Key examples include the identification of "baby" Mandelbrot sets in $z^2+z^3+c$, fractal sine bases on non-linear attractors, and large efficiency gains with stable adaptive depth in neural networks via scale-consistent state-space dynamics. This demonstrates FROST's generative and analytic power across mathematical, physical, and data-driven systems.