Actor Graphs: Triple-Identity Accountable Mediation and Coinductive Disclosure

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 LinkID, each recognised occurrence carries a TransactionID, and the reusable mediator carries an ActorID. Link and transaction references preserve Local versus OpaqueExternal constructors. Identity-faithful representation is defined by recoverability of this sorted signature, and a non-collapse theorem formalises the independent discriminators required for links, occurrences, and mediators. Account mutation is transaction-sourced. Typed account state is a dependent sum; every account type has a deterministic effect action; each conserved type additionally has an abelian effect group and equivariant valuation; and a successful commit couples canonical record creation, typed history append, account postings, and causal events. This yields deterministic state, order independence for commuting effects over causal linear extensions, closed-boundary conservation, and an explicit open-boundary flow identity. Recursive structure is given by a finitary polynomial functor and its final coalgebra. Raw finite observations are adequate for identity-preserving bisimulation under C0–C7, while registry-backed back-reference compression is proved adequate under S1–S4. At the resolved accountability layer, ActorID, LinkID, TransactionID, AccountCoordinate, and PostingOccurrence sorts form an ID-normal skeleton. Its instance category is a presheaf topos with pointwise limits and colimits, adhesive structural gluing, and a categorical obstruction to reconstruction after identity collapse. Identity-preserving adjacency stores (LinkID, ActorID) pairs; mediator-only and numeric matrices are explicit lossy projections. Closure under graph-valued continuation is proved at the signature level and supports an interpretation as structural self-similarity. Geometric fractal properties lie outside the theorem’s stated scope. Programming-language macro-expressibility and relative maximality remain open programmes.