Introduction

The preceding parts established a progression: a learner that holds a level as a goal and reads its own drift from it; a physics derived from the structure of a graph-structured lambda theory, in which holding that level turns out to have a price; and an argument that agents in massive populations are structurally prevented from seeing the true ontology of their GSLT.

The present part turns to a question that is in some sense prior to all three: how does life arise at all? We connect the framework to two of the most mathematically serious contemporary proposals about the origins of life. The first is Assembly Theory, due to Cronin and Walker [19], which assigns to each physical object an assembly index and a copy number and proposes that their joint magnitude is a biosignature. The second is Eric Smith’s proposal [20] that life constitutes a fourth geosphere, emerging from prebiotic chemistry through a phase transition driven by autocatalytic self-reinforcement.

The connections we draw are not analogies. Assembly index is identified precisely with causal depth in the synchronization tree of a GSLT. Copy number is given a rigorous definition in terms of the Rho calculus namespace structure, filling a gap in Assembly Theory that has attracted criticism. The phase transition is identified with the crossing of a basin of attraction boundary in the logical topology of Part II, with the attractor being the concurrent fixed point — the Rho calculus Y combinator — that is the abstract core of hereditary replication.

As throughout this volume, the arguments are presented as mathematical skeletons with gaps identified explicitly.

47.1 Organization

Chapter 48 develops the connection to Assembly Theory: assembly index as causal depth, copy number in a namespace, and the biosignature criterion as a well-posed mathematical statement. Chapter 49 identifies replication as a fixed point of the interaction dynamics and notes the proximity of the comm rule to a genetic crossover operator — the same proximity Chapter 23 exploited to give a lineage of learners a genetics. Chapter 50 develops the connection to Smith’s phase transition via the density of recursive attractors in the logical topology. Chapter 51 treats metabolism and the ecosystem as an account-flow network, which is the ecology of Part Part II seen from the outside rather than from inside one of its learners. Chapter 52 brings the two together: it bounds the height of the medium tower by the assembly index, and thereby bounds what an ecology-as-mind can tell apart. Chapter 53 turns the bound around. It supplies copy number with the region and the observer it has been missing, finds that a coarse instrument does not miss copies but manufactures them, gives six cheap floors on what a mind must be assembled from, and shows that the two parameters of the biosignature lie on a frontier rather than varying freely. Chapter 54 catalogs the open gaps.