Causal-Temporal Event Graphs: A Formal Model for Recursive Agent Execution Traces

We introduce causal-temporal event graphs (CTEGs) as a formal model for fully resolved recursive agent execution records under single-parenthood causal semantics. We formalise direct event emissions and recursive subagent invocations as extension procedures on generic typed temporal graphs and show that the recursive closure mathscrE_∞ of the induced maximal dynamics starting from single causal roots consists entirely of finite sequences of CTEGs. A CTEG is a rooted arborescence whose nodes carry timestamps and event types, subject to the constraint that timestamps be strictly increasing along causal paths. We realise mathscrE_∞ as the increasing union of a recursive hierarchy mathscrE0 subseteq mathscrE1 subseteq cdots of agent execution levels parametrised by recursion depth, which is recognised as the ascending Kleene chain of a monotone operator varphi admitting mathscrE_∞ as its least fixed point. Although the introduction of the full hierarchy is natural, stabilisation occurs already at mathscrE1 if one insists that the internal construction of a subagent execution trace be a delegated and opaque computational unit. The CTEG formalism supports compositional construction of globally well-formed execution traces from local agent behaviour without centralised coordination, preserves well-formedness under partial execution failure, and admits a natural relational database encoding. The arborescent structure of CTEGs is further compatible with cryptographic Merkle tree commitments for tamper-evident session verification.