Set-Valued Causal Inheritance for Functional Self-Continuity: Representation Theorems and Persistent-Agent Benchmarks

Functional self-continuity is often modeled as if one present process must map to exactly one future process and as if each future candidate can be scored independently. These assumptions fail under copying, fission, fusion, distributed computation, and restoration. We develop a substrate-neutral computational framework in which current self-attribution, causal continuation, successor belief, post-branch self-location, and motivational stake are represented as distinct objects. The central temporal object is a set-valued inheritance function \[ \Gamma_t:2^{\mathcal C_{t+\Delta}}\rightarrow[0,1], \] defined from an intervention channel on a minimally sufficient endogenous functional state. The formal results establish several representation limits. Categorical successor variables cannot represent zero, one, and multiple successors without making sets themselves the state space; normalized successor-membership weights are incompatible with full co-successor preservation under fission; singleton successor marginals do not identify a successor-set distribution; and singleton inheritance scores do not identify joint inheritance. A secret-sharing construction yields \(\Gamma(B)=\Gamma(C)=0\) but \(\Gamma(\{B,C\})=1\). For successor-inclusion indicators \(X\), the KL-optimal independent-Bernoulli approximation incurs irreducible excess log loss equal to the total correlation \(TC(X)\). A matched-marginal construction further gives an unavoidable average decision regret of \(1/2\) for policies restricted to singleton successor marginals. We also establish representation constraints on the causal measure itself. Exact mutual-information inheritance is invariant under bijective recoding when intervention semantics are transported with the recoding, whereas coordinate-aligned estimators need not be. An intervention-response quotient defines a task- and intervention-relative minimal endogenous control state (MECS), and shared upstream information is shown to be insufficient for current-process lineage unless the intervention is anchored after causal separation. Controlled computational studies illustrate these results at increasing levels of realism: finite synthetic constructions, intervention-derived inheritance estimates, learned high-dimensional functional quotients, recurrent hidden-state quotients, reward-trained fork/restore/merge lineages, autonomous persistence-operation selection, and long-horizon persistence control. The recurrent lineage constructions are also repeated with three independently trained reward-driven GRUs and three independently trained merger networks. Absolute inheritance magnitudes vary with task learning quality, but the redundancy, complementarity, stale-restore, and merge structural signatures replicate in all three stacks. A six-seed contextual-bandit replication shows stable learning of all four tested persistence-operation classes, with mean regret 0.0085 (95% seed-level CI 0.0079 to 0.0091) and mean oracle-action agreement 81.6%. By contrast, an eight-seed replication of a process-aware versus context-only end-to-end recurrent control comparison does **not** confirm the earlier single-seed advantage: the mean paired return difference is -0.0011, with a 95% seed-level interval spanning zero. The computational evidence therefore supports learnability of persistence-operation selection in the controlled one-decision setting, while leaving a reliable process-aware advantage in end-to-end long-horizon recurrent control unestablished. The framework is explicitly functional. It does not establish numerical personal identity, phenomenal consciousness, moral status, or legal identity. Its narrower aim is to make branching and distributed continuation precise enough to support falsifiable computational tests and persistence engineering. TA-TR-2026-17, version 1.0. Human author of record and responsible depositor: Hongju Liu. Substantial ChatGPT assistance in literature retrieval, formalization, theorem checking, coding, benchmark design, replication analysis, drafting, editing and publication preparation under human direction. Not peer reviewed; no institutional endorsement is claimed. This is adjacent first-party non-amending scholarship. It neither defines nor validates nor changes the Trinity Accord. Controlled computational experiments do not establish human identity, phenomenal consciousness, moral status, legal identity, or a universal measure of survival. Negative and failed-replication results are retained. CC BY 4.0 applies to newly written material to the extent rights are held. Third-party sources retain their rights.