The TAP equation: evaluating combinatorial innovation in Biocosmology

TL;DR

The paper analyzes the TAP equation as a phenomenological model of adjacent possible-driven innovation, focusing on plateau-to-blow-up dynamics and extinction interactions. It derives analytic blow-up times for constant and power-law $\alpha_i$, characterizes a critical extinction rate $\mu_{critical}$, and introduces a two-scale TAP with an early exponential phase. A continuous-limit viewpoint yields a formal integral for blow-up time and clarifies the distinction between discrete-time and continuum predictions, including tetration in the late-time regime. The work broadens TAP's applicability to economics, environmental change, and the evolution of laws, while highlighting the practical challenges of predicting sudden transitions.

Abstract

We investigate solutions to the TAP equation, a phenomenological implementation of the Theory of the Adjacent Possible. Several implementations of TAP are studied, with potential applications in a range of topics including economics, social sciences, environmental change, evolutionary biological systems, and the nature of physical laws. The generic behaviour is an extended plateau followed by a sharp explosive divergence. We find accurate analytic approximations for the blow-up time that we validate against numerical simulations, and explore the properties of the equation in the vicinity of equilibrium between innovation and extinction. A particular variant, the two-scale TAP model, replaces the initial plateau with a phase of exponential growth, a widening of the TAP equation phenomenology that may enable it to be applied in a wider range of contexts.

Figures (6)

• Figure 1: Critical values of $\mu$ as a function of the parameters $M_0$ and $\alpha$. From top to bottom the contours are $\mu_{\rm critical}$ equal to $1$, $10^{-2}$, $10^{-4}$, $10^{-6}$. The shaded region is where extinction equilibrium is impossible as it would require $\mu > 1$.
• Figure 2: Varying $\alpha$ with no extinction and $M_0 = 2$ --- the number of steps to blow-up is inversely proportional to $\alpha$. From top to bottom $\alpha = 1$, $10^{-2}$, and $10^{-4}$. In all figures we omit the last-computed point before computational overflow, to avoid compression of the $y$-axis scale.
• Figure 3: Changing extinction at fixed $\alpha = 10^{-2}$ and $M_0 = 2$ --- large enough $\mu$ bends the curve down to zero. From top to bottom $\mu = 0$, $10^{-3}$, and $10^{-2}$.
• Figure 4: As the lower panels of Figure \ref{['noextinction']}, but now with $M_0 = 10$.
• Figure 5: Power-law combinatoric suppression at fixed $\alpha = 1$, $M_0 = 2$, and zero extinction. From top to bottom $a = 10$, $100$, and $1000$ (compare also to the top panel of Figure \ref{['noextinction']} which is the same with $a=1$). The time to blow-up is linear with $a$ (for $a > M_0$).