Active Transaction Graphs: A Formal Framework for Transactional Interactive Systems

We introduce Active Transaction Graphs (ATGs), a formal framework for interactive systems in which persistent state, mediated interaction, execution traceability, and accounting consequences must be modeled simultaneously. The framework is based on five core commitments: participating entities are modeled as actors, actors may be recursively structured, formally relevant interaction is transactional, graph edges are first-class computational entities, and observational semantics is determined by the triple (result, trace, ledger).We distinguish explicitly between the minimal formal core of the framework, derived results, architectural principles, and conjectural extensions. Within the formal layer, we prove a noise projection theorem, show that finite Petri nets embed into ATGs, and show that CHAM-style systems arise as quotients of ATGs under suitable erasures. We further prove that explicit edge mediation is semantically non-eliminable in ledger-sensitive mediated systems. A worked example based on invoice approval and payment illustrates the formalism.The framework can be extended upward by a signal layer, whose projection onto the transactional layer explains the classification of noise as semantically irrelevant interaction. The resulting framework is intended not as a replacement for classical models of computation, but as a semantic envelope for transactional interactive systems, including workflow engines, enterprise platforms, and mixed human–AI operational environments.