A golden-ratio partition of information and the balance between prediction and surprise: a neuro-cognitive route to antifragility

Abstract: Adaptive systems must strike a balance between prediction and surprise to thrive in uncertain environments. We propose an information-theoretic balance function, $ f(p) = -(1 - p)\ln(1 - p) + \ln p $, which quantifies the net informational gain from contrasting explained variance $p$ with unexplained novelty $(1 - p)$. This function is strictly concave on $(0,1)$ and reaches its unique maximum at $ p^* \approx 0.882$, revealing a regime where confidence is high but the residual uncertainty carries a disproportionate potential for surprise. Independently of this maximum, imposing a self-similarity condition between known, unknown and total information, $p : (1-p) = 1 : p$, leads to the golden-ratio reciprocal $p = 1/\varphi \approx 0.618$, where $ \varphi$ is the golden ratio. We interpret this value not as the maximizer of $f$, but as a structurally privileged \emph{partition} in which known and unknown are proportionally nested across scales. Embedding this dual structure into a Compute-Inference-Model-Action (CIMA) loop yields a dynamic process that maintains the system near a critical regime where prediction and surprise coexist. At this edge, neuronal dynamics exhibit power-law structure and maximal dynamic range, while the system's response to perturbations becomes convex at the level of its payoff function-fulfilling the formal definition of antifragility. We suggest that the golden-ratio partition is not merely a mathematical artifact, but a candidate design principle linking prediction, surprise, criticality, and antifragile adaptation across scales and domains, while the maximum of $f$ identifies the point of greatest informational vulnerability to being wrong.

• The paper presents an information-theoretic function f(p) with a maximum at p* ≈ 0.882 to quantify the trade-off between prediction and surprise.
• It derives a golden-ratio partition (pφ ≈ 0.618) that models self-similarity between known and unknown information, providing structural stability insights.
• By linking criticality with antifragility in neural and computational frameworks, the study suggests new design principles for robust, adaptive systems.

A Golden–Ratio Partition of Information and the Balance between Prediction and Surprise

Introduction

The paper "A Golden–Ratio Partition of Information and the Balance between Prediction and Surprise: A Neuro–Cognitive Route to Antifragility" (2602.15266) introduces a theoretical framework that explores how adaptive systems can balance prediction and surprise to thrive in uncertain environments. It focuses on an information-theoretic balance function, $f(p) = -(1-p)\ln(1-p) + \ln p$, which quantifies the informational gain from the explained variance versus the unexplained novelty. This function's unique maximum at $p^* \approx 0.882$ highlights a regime where systems maintain high confidence but are vulnerable to unexpected events.

The Informational Balance Function

The information-theoretic balance function $f(p)$ is a central contribution of the paper, capturing the net informational imbalance between prediction and surprise. The function's strict concavity across $(0,1)$ yields a maximum at $p^* \approx 0.882$, indicating the point where residual uncertainty most significantly impacts the system's informational vulnerability.

Figure 1: The curve represents the informational balance function $f(p) = -(1-p)\ln(1-p) + \ln p$, highlighting critical informational imbalances.

The balance function can be broken into two components: the expected surprise from unexplained variance and the discounted surprise upon confirming expected predictions. The sign of $f(p)$ offers insights into potential vulnerabilities or stability of adaptive systems: positive values suggest potential liability due to surprise from unknowns, while negative values indicate dominant surprise from confirming expectations.

Golden Ratio and Informational Partitioning

Beyond identifying the maximum of $f(p)$, the authors propose a golden-ratio partition ($p_\phi \approx 0.618$), deriving this via a self-similarity condition that mirrors the golden ratio phenomenon present in various natural contexts. This partition represents a structurally stable point where the proportion of known to unknown information recursively mirrors itself.

Figure 2: Self-similar informational partition and the $\varphi$-balance.

This ratio allows for a harmonious coexistence of known and unknown phenomena, providing the theoretical basis for operating near criticality—a state that confers both coherence and sensitivity to environmental perturbations.

Adaptive Systems at Criticality

The paper draws connections between its theoretical constructs and empirical evidence of criticality in complex systems, particularly in neurological contexts. The identified interval between $p_\phi$ and $p^*$ defines a critical zone where systems operating at the edge benefit from maximal dynamic range and flexibility—cornerstones of antifragility.

Adaptations near criticality, where prediction and surprise coexist optimally, allow systems to leverage both structure and novelty for functional gain. Such a regime supports scale-invariant dynamics seen in physiological and computational systems ranging from $1/f$ noise in neural circuits to fractal brain structures.

Implications and Future Perspectives

The framework posits that understanding and manipulating the informational partition could inform the design of antifragile systems, both biological and artificial, that thrive under uncertainty. This theoretical insight aligns with the principle of dynamic criticality, suggesting self-organization around critical points as a natural evolutionary adaptation for optimizing computational and adaptive capabilities.

The recursive golden-ratio partition could serve as a guiding principle not only in understanding brain function but as a foundational aspect for crafting robust yet adaptable AI systems designed under uncertainty. By positioning systems near criticality, they may harness perturbations as opportunities for growth and adaptation.

Conclusion

In conclusion, the paper provides a compelling argument that adaptive systems, through an optimal balance between prediction and surprise, can achieve antifragility. The golden-ratio partition offers a structural model for navigating criticality, and the maximum of $f(p)$ delineates the point of greatest informational susceptibility. Together, these constructs form a coherent theoretical framework suggesting that adaptive cognition and self-organizing systems naturally gravitate toward criticality, enabling them to convert environmental volatility into informational and operational advantage.