Oliver López-Corona

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.

Understanding how ancient Mesoamerican societies integrated mathematical ideas into calendrical design and monumental architecture requires approaches that acknowledge their distinct epistemological frameworks. While explicit textual evidence for concepts such as $π$ or the golden ratio $φ$ is absent, numerical patterns embedded in Mesoamerican calendars, iconography, and ritual architecture reveal a coherent system of proportional reasoning grounded in simple integer ratios. Here we show that the numbers 5 and 8, central to Venus and solar calendrical relations and widely represented in Mesoamerican cosmology, generate rational approximations that reproduce, within known construction tolerances, the geometric relations associated with decagonal layouts. Using high-resolution measurements of the Iguana structure at Guachimontones, we demonstrate that its proportions align with integer ratios consistent with those found in the calendrical system and with the practical geometry of the regular decagon, without requiring knowledge of irrational constants. These findings suggest that Mesoamerican builders employed stable proportional modules that harmonized astronomical cycles, cosmological symbolism, and architectural design. This should not be interpreted as a lack of mathematical sophistication; rather, the material record reveals a distinct mathematical tradition in which number, measure, and cosmology were mutually reinforcing elements of cultural knowledge.