A Fault-Tolerance Threshold for Gated Agentic Computation: Reliable Long-Horizon Work from Unreliable Executors

Autonomous agents (whether language-model based, human, or hybrid) commit errors at an approximately constant rate per unit of work — a constant-hazard baseline we adopt, motivated by reported time-horizon fits rather than claimed as a universal law — so the probability that a long task completes without fault decays exponentially in its length. The task length at which an agent succeeds half the time — its horizon — has become a leading proposed metric of agentic capability, and empirically it has grown exponentially over recent years. We ask a different question: given a fixed agent, how far can verification extend its reliable horizon? We model long-horizon work as a sequence of checkpointed segments, each verified by a stack of imperfect gates combined by majority vote, and we prove a fault-tolerance threshold analogous to von Neumann's reliable-computation-from-unreliable-components result and to the quantum threshold theorem.

(i) When gates are conditionally independent and better than chance, a stack of O(log T) gates per checkpoint achieves any target reliability over horizon T, yielding a multiplicative time overhead that is likewise O(log T).

(ii) When gates share a blind spot — a stealth mass λ_st of faults invisible to the entire gate family — the reliable horizon is capped at H_gated ≈ H_raw / λ_st regardless of gate count.

(iii) The cost-optimal checkpoint interval is s* ≈ √(G/λ), a direct analogue of the Young–Daly checkpoint formula with verification cost in place of crash-recovery cost.

The binding constraint on autonomous horizon is therefore not raw model capability but the diversity of the verification stack, a quantity partly estimable from a system's own telemetry and targeted audits. We further decompose the stealth mass into a diversity-reducible part and a specification-bound irreducible floor, and we note that a recent large-scale N-version experiment with coding agents motivates the bounded regime (ii) for realistic systems. We numerically illustrate all three results in simulation and specify a falsifiable experiment on agentic task suites.