Gandy Direct-Limit Theorem: Polynomial-Time Prediction on Operator-Generated Networks

Abstract

Predictive analytics increasingly runs over structures that grow without bound, such as logistics networks, digital twins, and knowledge graphs, where queries carry hard latency budgets yet the deployed guarantees are only statistical. We develop a worst-case alternative from computable model theory. We model the evolving structure as a Gandy direct limit, the limit of a chain generated by a fixed-point operator rather than by Fraïssé amalgamation, and prove the Gandy direct-limit theorem: if a polynomially computable chain is generated by such an operator, every operation returns the canonical code of its value, and a functional boundary condition holds, then membership, predicates, operations, and equality are all decidable in polynomial time. The countable atomless Boolean algebra and the unit-free Ershov algebra are presented this way. The applied payoff is an operator-generated logistics network—the universal envelope of all admissible consolidations, which subsumes any particular deployment rather than recording one—on which every bounded prediction is decidable in time polynomial in the queried code, with a degree fixed by the query rather than the horizon. Here prediction means an emergence or reachability decision against a deterministic generator, not statistical forecasting; probabilistic rules add a provable confidence floor. A reproducible simulation on synthetic instances confirms this cost model: the work a bounded query does is polynomial in the length of its input—the code that names the target—and does not grow with the size of the network.