Six Birds: Foundations of Emergence Calculus
Ioannis Tsiokos
TL;DR
This work introduces a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent endomaps on coarse descriptions, and shows that when processes are composable but access is mediated by a bounded observational interface, six closure-changing primitives P1--P6 become canonical. It distinguishes completion (stable fixed points) from extension (non-definable predicates) and anchors directionality to auditable functionals that remain monotone under coarse observation, formalized through a three-certificate loop: Stability, Novelty, and Directionality. The main contributions include a data-processing inequality for path-reversal asymmetry, a protocol-trap audit to detect hidden clocks, and a finite forcing lemma that bounds the rarity of definable extensions; together these establish a robust, testable framework for open-ended theory growth. The framework is instantiated with a finite theory package (Z,f,Σ_f,E,A) and operationalized via E_{\tau,f} as a dynamics-induced endomap, a graph-1-form ACC for affinities, and path-space KL measures; the results are validated through reproducibility artifacts, toy examples, and a clear route to domain-specific instantiations. The work offers a principled approach to analyze and certify emergent structure across disciplines, with practical implications for understanding how stable objects arise, how novelty can be introduced without compromising audits, and how apparent irreversibility can be audited and distinguished from hidden clocks.
Abstract
We develop a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent operators acting on descriptions. We show that, once processes are composable but access to the underlying system is mediated by a bounded observational interface, a canonical toolkit of six closure-changing primitives (P1--P6) is unavoidable. The framework unifies order-theoretic closure operators with dynamics-induced endomaps $E_{τ,f}$ built from a Markov kernel, a coarse-graining lens, and a time scale $τ$. We introduce a computable total-variation idempotence defect for $E_{τ,f}$; small retention error implies approximate idempotence and yields stable "objects" packaged at the chosen $τ$ within a fixed lens. For directionality, we define an arrow-of-time functional as the path-space KL divergence between forward and time-reversed trajectories and prove it is monotone under coarse-graining (data processing); we also formalize a protocol-trap audit showing that protocol holonomy alone cannot sustain asymmetry without a genuine affinity in the lifted dynamics. Finally, we prove a finite forcing-style counting lemma: relative to a partition-based theory, definable predicate extensions are exponentially rare, giving a clean anti-saturation mechanism for strict ladder climbing.