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

Ioannis Tsiokos

Abstract: Many sciences rely on maps of macroscopic behavior - phase diagrams, stability charts, bifurcation sets, and feasibility regions - but emergence itself is rarely treated as a mapping problem. We introduce emergence atlases: subsets and scalar fields on control space indicating where a macro-description is feasible. In the finite-state setting used here, feasibility means that a packaged description is simultaneously closed, nontrivial, audit-safe, and budget-feasible. We compute such atlases in three settings: analytic law families, full packaging spaces of small systems, and the PICA mechanism algebra. The resulting maps recover exact closure loci and reversible lines, expose multi-objective tradeoffs among closure, directionality, effective information, novelty slack, and budget, and identify mechanism-space regimes together with their sensitive ablations and conservative compact supports. These computations support a simple reframing: emergence is usefully treated as a phase-diagram problem for effective theories in control space. In that view, effective information, irreversibility, coarse-graining quality, and mechanism-space analyses appear not as rival definitions of emergence, but as slices of a larger atlas.