Fibonacci Ensembles: Golden Ratio Models

• Fibonacci Ensembles are a framework that leverages the recursive Fibonacci sequence and golden ratio to design ensemble models with enhanced expressivity and stability.
• They employ normalized Fibonacci weighting and second-order recursive dynamics to reduce variance and optimize prediction performance in overcomplete and correlated model settings.
• The methodology extends to nonequilibrium statistical physics and quasicrystalline lattice design, offering practical scaling laws, spectral stability, and experimentally verifiable diffraction patterns.

Fibonacci Ensembles are a mathematical and algorithmic framework for model aggregation, inference, or structural design in which the recursive, second-order relations of the Fibonacci sequence—and its limiting ratio, the golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$—play a foundational role. This concept is notably realized in ensemble learning, dynamical universality classes in nonequilibrium physics, and the structure of quasicrystalline lattices, where the Fibonacci sequence structures weights, exponents, or spatial order, yielding harmonies in expressivity, stability, and scaling.

1. Fibonacci Ensembles in Ensemble Learning

Fibonacci Ensembles, as introduced in "On Fibonacci Ensembles: An Alternative Approach to Ensemble Learning Inspired by the Timeless Architecture of the Golden Ratio" (Fokoué, 25 Dec 2025), comprise two intertwined methodological innovations:

(a) Normalized Fibonacci weighting for base learners: Given $M$ base predictors $h_1, ..., h_M$, raw and normalized Fibonacci weights are defined by

$$F_m = F_{m-1} + F_{m-2},\quad F_1=1,\;F_2=1,\qquad \alpha_m = \frac{F_m}{\sum_{j=1}^M F_j}$$

The ensemble predictor is then: $$\widehat f_{\mathrm{Fib}}(x) = \sum_{m=1}^M \alpha_m h_m(x)$$ To enhance variance reduction, the base learners are orthogonalized via the Gram matrix $G$ and Rao–Blackwellization, giving minimum-variance aggregates.

(b) Second-order Fibonacci-type recursive ensemble dynamics: Extending beyond simple stagewise boosting, the ensemble evolves by

$$F_m = \beta F_{m-1} + \gamma F_{m-2} + \Delta F_{m-1},\quad m\geq 3$$

with $(\beta,\gamma)$ selected (Fibonacci case $\beta=\gamma=1$) so that the homogeneous part is governed by the golden ratio. This introduces spectral and expressive control via the golden ratio eigenmodes of the associated companion matrix.

The explicit incorporation of Fibonacci weights and recursions provides variance suppression, expressive expansion (via the FibCone—Editor's term—the set of conic combinations controlled by the golden ratio), and stable ensemble growth, as formally established in variance and generalization bounds.

2. Theoretical Properties and Generalization

Variance and Bias Control

The variance of the Fibonacci-orthogonalized ensemble, $V_{\mathrm{Fib}}$, satisfies

$$V_{\mathrm{Fib}} = \sum_{m=1}^M \alpha_m^2 \sigma_m^2 \leq c_{\mathrm{var}} V_{\text{unif}}$$

for sufficiently large $M$ and $c_{\mathrm{var}} < 1$, establishing systematic variance reduction relative to uniform weighing. However, in strongly orthogonal bases with invertible Gram structure and available oracle inverse-variance weights, Rao–Blackwell (RB) weighting remains optimal (Fokoué, 25 Dec 2025).

Expressive Power: Fibonacci Conic Hull

The convex hull is strictly contained in the Fibonacci conic hull,

$\FibCone(h_1, ..., h_M) = \left\{ \sum_{m=1}^M c_m h_m: c_m \geq 0, \frac{c_{m+1}}{c_m} \leq \varphi + o(1) \right\}$

This admits more expressive ensemble elements than convex combinations, but constrains them via the exponential tail of the Fibonacci progression.

Spectral Stability

Stability of the recursive flows is ensured whenever $\beta^2 + 4\gamma < 4$, with the eigenvalues of the recursion matrix bounded in magnitude. At $\beta=\gamma=1$, the “golden critical” regime is accessed, dominated by the golden ratio.

Generalization Bound

For 1-Lipschitz loss and bounded base learners, the empirical Rademacher complexity of the Fibonacci ensemble class is upper-bounded linearly in the base learner complexity, with the constant depending essentially on the golden ratio: $$\widehat{\mathcal{R}}_n(\mathcal{H}_{\mathrm{Fib}}) \leq K \widehat{\mathcal{R}}_n(h_1)$$

3. Empirical Evaluation and Regimes

Experiments on univariate regression (targets $f_{\sin}, f_{\sinc}$) with random Fourier feature ensembles and polynomial ensembles demonstrate:

• In rich, redundant dictionaries (e.g., overcomplete RFFs), Fibonacci weighting achieves lowest integrated squared error and matches or surpasses both uniform and RB schemes.
• In strongly ordered, orthogonal bases (e.g., polynomials), RB weighting is superior, with Fibonacci weighting yielding excessive emphasis on high-degree components, resulting in ISE inflation (Runge phenomenon).

Representative results:

Model	Test MSE (sinc)	ISE (sinc)
Fibonacci	0.0921±0.0039	0.0028±0.0007
Orthogonal RB	0.0959±0.0042	0.0066±0.0015
Uniform	0.0963±0.0039	0.0070±0.0014

This suggests that Fibonacci ensembles are optimally positioned between uniform and RB weighting for correlated, overcomplete settings, but are suboptimal in strictly orthogonalized environments (Fokoué, 25 Dec 2025).

4. Fibonacci Flows in Nonequilibrium Statistical Physics

Fibonacci structures also appear as a family of dynamical universality classes in systems governed by nonlinear fluctuating hydrodynamics (NLFH) (Popkov et al., 2015). In such systems, the dynamic exponent $z_\alpha$ of the $\alpha$th normal mode is recursively defined by

$$z_1 = 3/2,\quad z_\alpha = 1 + 1/z_{\alpha-1},\quad \alpha \geq 2$$

The closed-form solution is

$z_\alpha = \frac{F_{\alpha+3}}{F_{\alpha+2}}$

with $F_n$ the Fibonacci sequence, and the sequence converges to the golden ratio $\varphi$. If no KPZ or diffusive modes are present, all exponents collapse to $\varphi$.

The universal two-point scaling functions for density fluctuations $S_\alpha(x,t)$ are fully determined Lévy $\alpha$-stable (maximally skewed) distributions. Their parameters depend only on the macroscopic current-density relation and compressibility matrix, rendering the scaling forms fully experiment-accessible. Mode identification proceeds by examining the self- and cross-coupling coefficients of normal modes, and the exponents and scaling functions can be directly measured from the asymptotic width, peak shift, and full rescaled shape of the two-point correlations.

5. Fibonacci Structures in Quasiperiodic Lattices

The architecture of Fibonacci lattices—aperiodic one-dimensional structures defined by a substitution rule on two intervals with lengths $L = 1 + 1/\eta$ and $S=1$—offers a further archetype of “Fibonacci Ensemble” behavior (Gullo et al., 2016). Each such lattice can be transformed into another by a composition operator $\mathcal{C}$, mapping the spacing parameter as $\eta \mapsto \eta_1 = 1 + 1/\eta$. Iterating this operation generates equivalence classes of Fibonacci lattices, all converging under composition to the canonical case $\eta = \tau$ (the golden ratio).

The diffraction spectra of these lattices exhibit “skeletons” of bright Bragg peaks at wavevectors

$Q(h, h') = \frac{2\pi}{d}(h + h'/\tau),$

with brightest peaks satisfying $h/h' \approx 1 + 1/\eta$, and the spectra for all members of an equivalence class are related by a predicted scaling in both peak positions and intensities. These scaling laws have been experimentally verified in photonic quasicrystal gratings (Gullo et al., 2016).

6. Outlook and Connections

Fibonacci Ensembles provide a universal, deterministic aggregation principle rooted in the golden ratio, applicable across statistical machine learning, dynamical systems, and structural physics. In ensemble methods, they offer a midpoint between uniform and learning-based weighting; in hydrodynamics, they characterize a whole family of superdiffusive universality classes; in quasicrystal theory, they underlie the equivalence classes with invariant diffraction skeletons. Critical limitations include suboptimality in bases amenable to risk-optimal weighting and algorithmic challenges in high-dimensional and tree-based learners. Extensions include hybridizing with data-driven stacking, adapting to diverse bias–variance landscapes, and deploying in high-dimensional models.

Further research directions include exploration of effective dimensionality control under the Fibonacci conic hull constraint, hybridization with Bayesian model averaging, and application in random forests or neural architectures (Fokoué, 25 Dec 2025).

• "On Fibonacci Ensembles: An Alternative Approach to Ensemble Learning Inspired by the Timeless Architecture of the Golden Ratio" (Fokoué, 25 Dec 2025)
• "Fibonacci family of dynamical universality classes" (Popkov et al., 2015)
• "Equivalence classes of Fibonacci lattices and their similarity properties" (Gullo et al., 2016)