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

TL;DR

This paper introduces a mathematical framework based on the golden ratio to balance prediction and surprise in adaptive systems, linking information theory, criticality, and antifragility.

Contribution

It proposes a novel information-theoretic balance function and a structural partition based on the golden ratio, connecting prediction, surprise, and antifragile adaptation.

Findings

Identifies a unique maximum of the balance function at p* ≈ 0.882.

Derives a golden-ratio-based partition p ≈ 0.618 for nested information scales.

Demonstrates that the proposed model exhibits criticality and antifragility in a dynamic loop.

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.