Actor Graphs: Definition, Anatomy, Topology, Recursion, and Fractality

This paper defines the Actor Graph (AG) as a named mathematical object with pluggable execution semantics. Its central contribution is triple-identity accountable mediation: a persistent structural relation carries a link_id; each recognised occurrence through that relation carries a transaction_id; and the reusable actor performing the relation carries an edge_actor_id. The three identity sorts are pairwise disjoint. An identity-faithful representation is defined by recoverability of this sorted signature, and a non-collapse theorem proves that no faithful representation of the admitted model class may merge the three sorts. The mediator is not a passive label: it carries the same contract as every other actor — identity, accounts, transactions, events, and process — and may disclose into another Actor Graph while preserving identity and accountable continuity. Accounts are typed state attributable to stable ActorIDs; mutation is transaction-sourced, conserved quantitative types use the group-of-differences algebra of double-entry bookkeeping, and balance is stated relative to an explicit boundary. Identity-referenced specifications are mapped canonically to coalgebras of a finitary polynomial signature functor; the unique final morphism supplies their coinductive denotation. The paper establishes the final coalgebra, order-independent finite-depth observations, a full adequacy theorem for identity-preserving bisimulation, and structural fractality as closure of the accountable signature under disclosure. Cyclic ActorID references are admissible. The paper also gives faithful typed incidence encodings and identity-, occurrence-, and posting-preserving matrix families, with explicit novelty boundaries against RDF 1.2, TypeDB, metagraphs, Reo, DEMO/Enterprise Ontology, REA, double-entry algebra, the Actor Model, and Petri nets. Following Felleisen, it distinguishes computational equivalence from native expressibility and proposes Actor Graphs as a candidate limit-expressive representation that absorbs classical programming as an internal process layer. This expressiveness programme is interpretive, not a theorem.