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.
This manuscript presents a theoretical contribution aimed at clarifying general principles by which biological systems regulate the balance between prediction and surprise under uncertainty. By introducing a formally defined information-theoretic balance function and identifying structurally privileged partitions of explained and unexplained variance, the work addresses a central question in theoretical biology and neuroscience: how adaptive systems maintain sensitivity to novelty while preserving internal coherence. The framework is intentionally abstract but grounded in biologically interpretable quantities (see supplementary materials) such as prediction error, explained variance, precision weighting, and learning progress, allowing the proposed constructs to be mapped onto neural and cognitive processes without reliance on detailed mechanistic assumptions [1].
Let us consider the following informational balance equation (with ln the natural logarithm)
Proposition 1 (Strict concavity). The function f is strictly concave on (0, 1).
Proof. A direct computation gives
, which is strictly negative for all p ∈ (0, 1). Hence f is strictly concave on (0, 1).
We may interpret f using cognitive processes in a three-layer analogy: Table 1: Three-layer analogy connecting probability, information, and cognitive meaning.
Standard meaning Metaphorical / cognitive reading 1. A priori beliefs p ∈ (0, 1) Probability assigned to a specific event A
“What we believe we know” (our confidence in A) 1 -p Probability of the complement A c “What we acknowledge we do not know” (confidence that something different from A may occur)
S(A) = -ln p Self-information when A occurs How surprising it is to confirm what we expected S(A c ) = -ln(1 -p)
Self-information when A c occurs How surprising it is to witness what we had ruled out
Shannon entropy for a Bernoulli variable “How much surprise we still expect on average”: a blend of surprises from both what we think we know and what we admit we don’t Then the function from Eq. ( 1) can be decomposed into two terms, each carrying a specific probabilistic meaning and a cognitive/metaphorical interpretation: Table 2: Components of the balance function and their interpretations.
Formula Probabilistic meaning Metaphorical / cognitive reading A. Expected surprise from the complement -(1 -p) ln(1 -p) Probability of the complement A c weighted by its surprise: -ln (1 -p). This is the average information we would gain if the unexpected occurs.
“How much surprise we can expect from the part of reality we admit not to control.”
B. (Negative) surprise from the expected event
The surprise of observing the expected event A is -ln p. This term appears with a positive sign, effectively subtracting the surprise of confirming our expectations.
“We discount the surprise of the known; the greater our confidence in A, the smaller the penalty for its confirmation.”
or in other words, f (p) = Expected surprise from the unknown
-Pointwise surprise from the known (2) This function measures the net informational payoff -how much more (or less) surprise is expected from what we do not know compared to what we believe we know.
• If f (p) > 0: The surprise potential of the unknown (A c ) exceeds the cost of confirming the expected (A). Our confidence could become a liability.
• If f (p) < 0: The self-inflicted surprise of confirming our own expectations dominates; the unknown adds little informational gain.
Behavior at the extremes The expected surprise from the unknown equals the cost of confirming the known. This is the equilibrium point where neither dominates.
In this way, term A quantifies the degree of potential surprise that may still arise from the domain we explicitly recognize as uncertain. It reflects the information content of those outcomes we assign low probability to-the events we believe are unlikely but nonetheless possible. This term represents the informational cost of being proven wrong about what we assumed was improbable, and as such, it captures the vulnerability embedded in our admitted ignorance.
In contrast, Term B represents a discount applied to the surprise that would arise from confirming what we already expect. It may seem paradoxical, but even when an event we are confident about does occur, it still carries an informational signature: the confirmation is not completely devoid of surprise, though its magnitude diminishes as our confidence grows. This term subtracts that residual, expected surprise, reminding us that confidence is never free of informational cost.
Together, these two terms combine to define the function f (p), which expresses the net informational balance between unknowns and knowns. The sign and magnitude of f (p) indicate which side of our epistemic model contributes more to our potential shock. A positive value signals that the neglected or underestimated portion of the probability space could surprise us more than the confirmation of our belie
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