Target-Relative Emergence Thresholds: A Minimal Transition-System Framework with a Petri Net Formulation

Claims about emergence are often stated without specifying what is to emerge, how its presence is recognized, or what would make its appearance probable rather than merely possible. This paper proposes a minimal target-relative formulation. A generating system is represented as a stochastic transition process over microstates; a target specification is represented by an admissible extraction rule together with a predicate on the resulting macrotrajectory. Emergence is then an event whose probability can be studied as system parameters vary. The framework separates four logically distinct questions: structural admissibility, dynamical accessibility, persistence, and concentration of successful formation opportunities. A Petri-net formulation is introduced as a concrete formal instantiation. Standard first- and second-moment arguments then yield a minimal threshold principle: emergence is unlikely when the expected number of persistent target witnesses vanishes, and becomes highly probable when that number diverges and is sufficiently concentrated. The paper also addresses the common claim that every sufficiently complex system can produce a given structure. Unrestricted total complexity is shown to be insufficient: arbitrarily complex causally irrelevant extensions need not change the probability of the target event. Meaningful thresholds therefore require target-relative realizational and dynamical complexity. The aim is not a universal theory of all emergence, but a compact formal problem statement from which domain-specific threshold theorems can be developed.