Recursive Golden‑Ratio Decision Systems

Introduction

The golden ratio φ (approximately 1.618) appears in contexts ranging from quasicrystals and phyllotaxis to algorithmic search and control theory. Its continued fraction representation [1;1,1,1,…] makes it the "most irrational" number, which prevents simple rational resonances. This property underpins many stability and optimisation results. Here we synthesise the mathematical, economic, engineering and biological appearances of φ, and build a unified frequency‑state decision model that generalises golden‑ratio search. We then compare this Golden Gradient approach with traditional optimisers across five comprehensive benchmark dimensions.

Mathematical Foundations

Hurwitz's theorem shows that φ is the worst approximable number, giving it a special place in Diophantine approximation. Zeckendorf's theorem states that every integer can be uniquely represented as a sum of non‑consecutive Fibonacci numbers. These two results link φ to optimal search: golden‑section search divides an interval in the ratio 0.618 to 0.382 at each step and minimises the worst‑case number of evaluations for unimodal functions.

In control theory, Benavoli et al. proved that the steady‑state Kalman gain of a scalar random‑walk system with equal noise variances converges exactly to hi=0.618, with the corresponding error covariance approaching φ. This implies that optimal linear fusion weights between prior estimate and new measurement follow the golden ratio. In linear– quadratic control and inventory smoothing, φ arises as the optimal adjustment factor.

Theoretical Snapshot

We anchor the golden‑ratio dynamics with \( \varphi = \frac{1 + \sqrt{5}}{2} \), and its reciprocal \( \varphi^{-1} \approx 0.618 \). Each state \( s \) maintains a Laplace‑smoothed success rate \( r_s = \frac{a_s + 1}{a_s + b_s + 2} \), which determines whether we follow or flip the nominal golden‑section move.

The canonical step update for a bounded one‑dimensional slice uses a signed direction \( \sigma_t \in \{-1, +1\} \) and adaptive scale \( \eta_t \):

\[ x_{t+1} = x_t + \sigma_t \, \varphi^{-1} (b-a) \, \eta_t \]

To detect regime changes, we compare recent and baseline state frequencies with a Wasserstein distance:

\[ p_{\mathrm{drift}} = W_1(\text{recent}, \text{baseline}) \]

Applications Across Domains

Finance. Fibonacci retracements are widely used in technical analysis to draw support and resistance levels at 23.6%, 38.2%, 61.8%, etc., though evidence for predictive power is mixed. More compelling are φ‑based allocation strategies: portfolios with asset weights in 1:φ ratios show robust performance over decades, and corporate capital structures near φ proportion outperform random allocations.

Machine learning. Modern hyperparameter and neural architecture search methods implicitly partition spaces into "good" and "bad" regions. Golden‑ratio proximal algorithms achieve large step sizes and provable convergence. Golden Gradient extends this by treating the entire search process as a Markov decision problem with φ‑based priors.

Signal processing. φ appears in optimal Kalman gains and in the organisation of neuronal oscillations. Frequency ratios separated by φ minimise cross‑frequency interference. Fibonacci lattice sampling yields near‑uniform point distributions on the sphere, reducing error versus latitude–longitude grids.

Game theory and biology. Ultimatum game experiments show that offers around 38.2% maximise acceptance probability—matching the golden split. Phyllotaxis patterns in sunflowers, pine cones and cacti place successive leaves at 137.5°, the "golden angle", optimising sunlight and packing efficiency. Self‑organising dynamical systems recreate these patterns from simple rules.

Computer science and operations research. Fibonacci heaps achieve O(1) decrease‑key operations with tree heights bounded by log_φ(n). Multiplicative hashing uses φ to distribute keys uniformly. In supply chains, the golden smoothing rule suggests adjusting inventory by 61.8% of the discrepancy each period to minimise variance.

The Frequency‑State Model

Golden Gradient generalises golden‑section search by framing optimisation as navigation on a frequency‑state graph. At each iteration, a decision is made to cut the search region in a φ ratio (hi≈0.618 vs lo≈0.382). The algorithm records a history of long (L) or short (S) decisions, forming a discrete state sequence. Each state s has an associated success probability r_s, estimated from past outcomes. When facing a new decision, if r_s≥0.5 the algorithm follows the nominal golden‑section recommendation; if r_s<0.5 it flips direction. Periodic verification steps re‑evaluate previously discarded options to detect drift. A Wasserstein distance between recent and historical state distributions flags regime changes and triggers extra verification.

This frequency‑state approach retains the interval‑reduction optimality of golden‑section search while learning from context. It naturally incorporates additional optimisers (CMA‑ES, Nelder–Mead, Bayesian optimisation, Particle Swarm, Differential Evolution) by allowing them to operate as modules that propose candidate points. At each step(), external proposal generators are called and their candidate positions are immediately clamped to the active bounds. These external candidates then pass through the same frequency‑state key construction, state‑probability lookup, and Aharonov‑Bohm phase adjustment as native moves, with a source label attached for debugging.

Theoretical Perspectives on GoldenGradient through Measure Theory and Duality

Introduction

GoldenGradient can be reframed as a probabilistic decision process that evolves over a structured state space. Measure theory offers a clean vocabulary for describing how state frequencies, update rules, and drift detection behave under uncertainty.

Algorithm Overview

The algorithm maintains a history of frequency‑state outcomes and updates its policy based on observed success rates. Each iteration chooses a golden‑ratio cut, optionally flips direction, and performs verification checks when drift is detected.

• State tracking: encode recent frequency decisions into discrete states.
• Bayesian updates: update success likelihoods for each state.
• Drift sensing: compare recent vs. baseline distributions.

Measure‑Theoretic Framing

The frequency‑state space can be treated as a measurable space (𝒮, 𝔽), where 𝒮 collects all discrete states and 𝔽 is the σ‑algebra over those states. Each iteration induces a probability measure μ over 𝒮, capturing how often the algorithm visits each state.

• State space: 𝒮 as the set of all frequency‑state labels.
• σ‑algebra: 𝔽 as all subsets of 𝒮 (discrete measurable structure).
• Measure: μ(A) as the empirical mass of states in A.

Probability & Integration

Success probabilities are updated with a Beta prior and a Laplace‑smoothed posterior mean, so integration over μ becomes a simple weighted average of state outcomes.

• Prior/posterior: Beta(1,1) updated by successes a_s and failures b_s.
• Estimator: r_s = (a_s + 1)/(a_s + b_s + 2).
• Expected utility: integrate rewards over μ to choose accept/flip actions.

Wasserstein Distance

Drift detection compares recent and historical measures using the 1‑Wasserstein distance, highlighting transport cost between two frequency distributions. Large distances trigger verification actions and additional sampling.

• Recent measure: μ_recent from a sliding window of states.
• Baseline: μ_base from long‑run frequencies.
• Signal: W₁(μ_recent, μ_base) as drift magnitude.

Duality & Representation

Riesz representation links linear functionals on continuous functions over 𝒮 to measures, providing a principled way to interpret score functions as integrals. This viewpoint aligns with MDP duality, where value functions and occupation measures describe the same policy in primal and dual forms.

• Riesz: scoring functionals correspond to integration against μ.
• MDP duality: value‑function updates ↔ state‑action occupancy measures.
• Policy view: accept/flip rules encode a dual feasible measure.

Computational Implications

The measure‑theoretic framing clarifies why small updates remain stable and why Wasserstein drift detection is robust to noisy transitions. It also motivates efficient summaries of μ, such as bucketed histograms and sparse state visitation.

• Stable updates: Beta/Laplace smoothing dampens volatility.
• Fast drift checks: compute W₁ on coarse buckets.
• Scalable storage: sparse state tables approximate measures.

Metaheuristic Positioning

Golden Gradient sits between deterministic interval methods and population‑based metaheuristics. It preserves golden‑section optimality for unimodal slices while delegating exploration to external proposal modules (CMA‑ES, Nelder–Mead, Bayesian optimisation, Particle Swarm, Differential Evolution), then adjudicates their candidates using the learned frequency‑state policy.

• Proposal modules generate candidate points (search diversity and exploration).
• Frequency‑state gate accepts, flips, or verifies proposals based on r_s.
• Verification hooks re‑sample discarded regions when drift is detected.

This positioning allows Golden Gradient to remain sample‑efficient on smooth objectives while still leveraging broader exploration when the proposal modules detect non‑convex structure.

Multi‑Objective Variants

Multi‑objective objectives are handled through either scalarization or Pareto‑based selection. Scalarization compresses multiple criteria into a single target, while the Pareto‑based path retains a Pareto set of non‑dominated proposals for downstream MCDA selection.

• Scalarization: weighted sums or Chebyshev metrics to reuse single‑objective updates.
• Pareto path: maintain a Pareto set hook and filter dominated proposals.
• Hybrid: alternate scalarization weights while keeping a running Pareto archive.

The Pareto set hook stores non‑dominated proposals emitted by the proposal modules, enabling later MCDA scoring without discarding trade‑off structure.

MCDA Selection Strategies

When multiple objectives are active, the system applies MCDA scoring to choose among candidates in the Pareto set. The MCDA hook is deliberately modular so new scoring rules can be added without altering proposal generation.

• Weighted utility: aggregate normalized criteria into a single score.
• Outranking: pairwise comparisons to rank Pareto alternatives.
• Reference point: select candidates closest to a target aspiration vector.

MCDA scoring consumes the Pareto set hook and produces the final accepted candidate, which then feeds back into the frequency‑state update for the next iteration.

Multi‑Dimensional Applications

Golden Gradient can be extended to multiple variables by treating each dimension as its own search axis. At each iteration the algorithm cuts along each dimension in the golden ratio and records whether the move was on the high side (L) or low side (S). The collection of decisions across all dimensions forms a discrete joint state vector represented by a tuple (s₁, s₂, …, s_d). Each state vector has an associated success probability r_(s1,…,sd) estimated from past outcomes, and future decisions can either follow or flip the golden‑ratio recommendation based on these probabilities.

Dimensional Reduction Formulas

The algorithm's complexity and power emerge from the recursive structure across dimensions. We express the core formulas for each dimensional layer, showing how n-dimensional search decomposes into lower-dimensional cross-sections. Let φ = (1+√5)/2 denote the golden ratio.

Phase Dynamics (Aharonov-Bohm Formulation)

Explicit Formulas for Dimensions 1–5

To understand decision‑making in truly multidimensional settings, consider a 4D hypercube where each axis represents a distinct marginal state distribution. Each marginal applies Golden Section Search (GSS)—the frequency distribution itself is a golden‑ratio distribution where the center holds the most probability mass. Every vertex in the hypercube corresponds to a unique joint state vector across four dimensions—for instance, (L,S,L,S) indicates upper-partition decisions on dimensions 1 and 3, and lower-partition decisions on dimensions 2 and 4. Each vertex carries an empirical probability mass derived from historical observations.

In higher‑dimensional spaces, dimensional cross‑sections reveal a crucial phenomenon: rather than a single upper-tail regime and a single lower-tail regime, the joint distribution exhibits multiple upper-tail regimes and multiple lower-tail regimes, each with different empirical probabilities. This multiplicity is what enables informed choice—the algorithm selects among different upper-tail regimes based on which offers the highest likelihood of success. Joint-state memory is crucial because heavy‑tailed phenomena mean not all upper-tail regimes are equal. Some combinations of upper/lower partition moves across different dimensions lead to large improvements, whereas others do not.

At joint-state convergence points—where multiple Golden Section Search applications from different marginal distributions intersect—the algorithm evaluates which combination of upper or lower partition moves yields the highest expected improvement. These convergence points form a joint probability density field: regions of the hypercube with consistently high success rates are high‑density (favorable state configurations), while underperforming combinations are low‑density (unfavorable configurations). The algorithm preferentially samples high‑density regions, yet periodically verifies low‑density zones to detect regime shifts.

Crucially, each marginal distribution maintains its own state-transition history—a record of past transitions and their outcomes. In multidimensional space, marginal state frequencies form a probability metric space where distances between marginal distributions carry meaningful information. The Wasserstein distance (earth‑mover distance) measures the minimum "transport cost" required to reshape one probability distribution into another, making it ideal for comparing distances between marginal state distributions. When the Wasserstein distance between a marginal's current distribution and its baseline exceeds a threshold, the system flags a potential drift and triggers additional verification steps. This metric is particularly powerful in multidimensional settings because it captures the geometric structure of the probability metric space—how probability mass is distributed across the hypercube and how it shifts over time. In one‑dimensional settings, Wasserstein distance offers little advantage for understanding local probability structure, but in higher dimensions it becomes essential for tracking which joint state configurations are drifting and which remain stable.

Aharonov-Bohm Inspired Phase Dynamics

The algorithm incorporates insights from the Aharonov-Bohm effect in quantum mechanics, where charged particles acquire phase shifts from electromagnetic potentials even in field-free regions. This quantum phenomenon reveals that potentials—not just their gradients (fields)—have physical significance. We apply this insight to optimization:

• Potential over Gradient: Rather than following objective gradients alone, we learn a potential field V(x) representing the expected objective value at each region. Candidates in lower-potential regions receive probability bonuses, even if the local gradient is unfavorable.
• Path-Dependent Phase: Like the AB phase Φ = ∮ A·dl accumulated along a particle's path, we accumulate a topological phase based on the search trajectory. This phase encodes directional bias using the golden angle 2π/φ² ≈ 137.5°, preventing resonance patterns.
• Winding Number: When the search path completes a cycle (returns near a previous position), we compute a winding number analogous to the topological invariant in AB experiments. Productive cycles (improvement during the loop) increase the winding number; unproductive cycles decrease it.
• Non-Local Information: The AB effect shows that enclosed flux affects particles that never enter the flux region. Similarly, our potential field and winding number provide information about unexplored regions, enabling more informed exploration.

MCDA Weighting Defaults

Candidate selection uses a multi-criteria decision analysis (MCDA) score that blends state probability, predicted improvement, step size penalties, novelty, and momentum alignment. The defaults are chosen to balance exploitation (probability + improvement) with gentle exploration (novelty + momentum) while discouraging overly aggressive jumps.

• State probability (0.30): prioritises learned high-success states.
• Predicted improvement (0.30): rewards candidates that estimate stronger objective drops.
• Step size penalty (0.15): tempers large moves to maintain stability.
• Novelty bonus (0.15): favours candidates distant from the recent path history.
• Momentum alignment (0.10): encourages moves consistent with successful recent direction.

Comprehensive Benchmark Suite

We evaluate Golden Gradient across five critical optimization dimensions, comparing it against state-of-the-art algorithms including CMA-ES, Nelder–Mead, Bayesian Optimization, Particle Swarm Optimization (PSO), and Differential Evolution (DE). Each dimension tests different aspects of optimization performance:

References

For background, see Kantorovich–Rubinstein duality (Wasserstein-1) and the Aharonov–Bohm effect.

• Serge Lang, Real and Functional Analysis. Springer
• Kantorovich–Rubinstein duality (Wasserstein-1). Wikipedia
• Aharonov–Bohm effect. Wikipedia