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.

Figures (2)

• Figure 1: The curve represents the informational balance function $f(p) = -(1-p)\ln(1-p) + \ln p$, where $p \in (0,1)$ denotes the proportion of variance explained by an internal model (prediction) and $1-p$ the residual unexplained component (novelty). The function quantifies the net informational imbalance between the expected surprise arising from the unknown component and the verification cost associated with confirming what is already predicted. The unique maximum at $p^* \approx 0.882$ identifies the regime in which residual uncertainty exerts the greatest asymmetric informational influence, corresponding to a state of high confidence coupled with maximal vulnerability to being wrong. Independently, the self-similar partition $p_\phi = 1/\varphi \approx 0.618$, derived from the relation $p:(1-p)=1:p$, marks a structurally distinguished informational split in which the ratio of known to unknown equals the ratio of known to the whole. The interval between $p_\phi$ and $p^*$ defines an informational corridor in which predictive structure and novelty coexist, providing the theoretical basis for adaptive operation near criticality, where systems remain coherent yet sensitive to perturbations.
• Figure 2: textbfSelf-similar informational partition and the $\varphi$-balance. This diagram illustrates the self-similar partition defined by the relation $p:(1-p)=1:p,$ whose solution $p_\phi = \frac{1}{\varphi} \approx 0.618$ identifies a structurally distinguished informational split between explained and unexplained variance. Unlike the maximum of the balance function $f(p)$, this partition does not correspond to an extremum of informational imbalance, but to a recursive proportionality: the ratio of known to unknown equals the ratio of known to the whole. This self-similar structure preserves coherence under hierarchical recursion and provides a formal model of balanced predictive coding, in which internal models retain sufficient structure to guide inference while maintaining openness to residual novelty. The $\varphi$-partition therefore represents a candidate organizational principle for multiscale systems operating near criticality, where stability and variability coexist in scale-invariant proportions. For a deeper discussion see supplementary materials.

Theorems & Definitions (4)

• Proposition 1: Strict concavity
• proof
• Proposition 2: Existence and uniqueness of the maximizer
• proof