The TAP Equation

In The You On AI Field Guide

The paper builds on Kauffman's decades of work on the adjacent possible — the space of configurations accessible one step from any current configuration — and on Smolin's framework for understanding genuine novelty as emergence that is not predetermined by prior states. The specific contribution is the TAP (Theory of the Adjacent Possible) equation, which provides a mathematical treatment of how new possibilities open up as systems explore their current configurations.

The key insight is that the space of the possible is not fixed. In most combinatorial analyses, the elements are given and the combinations are bounded by the number and structure of the elements. If there are N elements, there are 2^N possible combinations, and that set is the universe of what can exist. The TAP framework challenges this assumption. As combinations are actualized, new elements become possible that were not available before. Metallurgy made possible the steam engine; the steam engine made possible industrial civilization; industrial civilization made possible the electronic computer; the electronic computer made possible AI. Each stage opened possibilities that were not adjacent to the prior stage.

Applied to AI, the framework offers a specific way to think about whether current systems participate in genuine category-creation or perform sophisticated exploration within a fixed space. Large language models explore vast spaces of linguistic combination. The spaces are large enough to produce outputs that are genuinely surprising to human observers. But the spaces are defined by the training data and the architecture; they do not grow as the model explores them. The model finds new arrangements within an existing space rather than creating new categories that expand the space.

Whether future AI systems could participate in category-creation is an open question. The framework does not rule it out — it simply distinguishes between the two kinds of combinatorial process and provides mathematical tools for recognizing the difference. The distinction matters for how to think about the current AI moment. If AI can contribute to category-creation (perhaps through collaboration with humans in ways that neither could accomplish alone), then it is an amplifier of genuine novelty. If AI can only explore existing spaces, then the category-creation must come from elsewhere — from conscious creatures engaged in the thick present in ways that current AI architectures do not replicate.

Origin

The paper was published in the European Economic Review in 2025, the culmination of sustained collaboration between Smolin and Kauffman on the mathematics of genuine novelty. It draws on Kauffman's decades of work on the adjacent possible and Smolin's framework for temporal naturalism and the reality of time.

Key Ideas

Category-creating combinations. Certain combinations — fire and metal producing metallurgy — create new categories of possibility rather than merely rearranging existing elements.

Expanding possibility space. As combinations are actualized, new elements become possible that were not available before.

Mathematical treatment. The TAP framework provides tools for analyzing how the space of the possible grows through exploration.

Distinction from recombination. Category-creation is qualitatively different from finding new arrangements within a fixed space.

Application to AI. Whether current AI participates in category-creation or only in sophisticated recombination is an open question the framework makes tractable.