To Chart a Stone with Six Birds: Emergence Phase Diagrams for Effective Theories in Control Space

Abstract

Scientists routinely draw maps of macroscopic behavior: phase diagrams, stability charts, bifurcation diagrams. Emergence is rarely mapped in the same way. Usually one asks for the best coarse-graining or the best score. We propose to ask instead where a good macroscopic description exists. An emergence atlas is a region, or a family of scalar fields, over a space of control variables (the microscopic law, the coarse-graining, the time scale, the mechanism, and a budget), marking the points at which a coarse description is closed, nontrivial, honest about the arrow of time, and affordable. We build such atlases for finite Markov chains and prove the facts that make them readable. Exact closure at one stage descends to all multiples of that stage. With the partition, lift, and stage fixed, every closure diagnostic is Lipschitz in the microscopic kernel. In a benchmark family, the closure locus is the same set, δ = 0, at every positive stage. Observing a chain through a coarse-graining can hide its arrow of time but cannot create one, whereas a Markov model fitted to the coarse states can invent an arrow that is not there. The resulting maps show that closure, nontriviality, effective information, retained directionality, novelty slack, and cost pull in different directions: some partitions are perfectly closed yet trivial, and some carry the most effective information while failing to be objects. Familiar measures of emergence then appear as slices of one atlas rather than as rival definitions. We also apply the method to PICA, a stochastic mechanism algebra, and test whether a compact six-cell mechanism yields geometry-like, nearly reversible macroscopic dynamics. Once geometry and directionality are measured on one well-defined conditioned process, the evidence does not support that claim. In the primary test, fixed in advance, the compact mechanism passes the criterion in none of four held-out validation cases, while the mechanism-free control passes in three, and a structureless reversible matrix passes as well. The atlas therefore separates what is established (closure loci, reversible lines, tradeoffs, and the no-fake-arrows bound) from what the mechanism experiments do not establish.