Byzantine-Robust Aggregation of Uncertainty Sets, Decision Regret and Closed-Loop Resilience: A Robust Optimization Analysis of Multi-Agent Supply Chain Planning

This paper is a mathematical study of robust optimization under aggregated, partially adversarial uncertainty. The setting is a planning problem in which m independent estimators each emit a candidate uncertainty set for the same unknown demand vector, up to f of the emitted sets are arbitrary (Byzantine), and one here-and-now decision must be taken and then corrected by a delayed replenishment loop. No language model is executed anywhere in this work, and none is required by any statement in it: an estimator is abstracted as a set-valued map with a prescribed marginal coverage level α, so the objects analyzed are an aggregation rule, a robust program, and a delay difference recursion. Three groups of results are obtained. First, for the depth-k vote with f+1≤k≤m−f, the aggregate U(k) is shown to cover the realized demand with probability at least 1−(m−f)α/((m−f)−k+1) and to satisfy the two-sided endorsement sandwich UR(k)⊆U(k)⊆UR(k−f), with both halves being available exactly when m≥2f+1; the upper inclusion is proven to be attained, so faulty estimators can inflate the aggregate by at most, and sometimes exactly, f depth levels. The probability of an empty aggregate is bounded, an exact finite representation is given together with its complexity in m and the dimension p, and the bounding-box relaxation consumed by a solver is shown to retain validity and the monotone form of the endorsement bound while losing the raw form. Second, the regret bound 2LudH is proven false once the robust feasible set depends on the uncertainty set; a counterexample is given and the corrected bound (2Lu+LxLgdiam(X)/γ)dH is established under a robust Slater margin γ, together with a one-sided safety bound and a split-conformal certificate that converts calibration data into a computable set distance guarantee. Third, the time-varying lead time recursion is replaced by a material-conserving arrival law, under which the stability threshold κ<1/ℓ¯ and the classical constant-delay boundary 2sin(π/(2(2ℓ+1))) are recovered with the delay indexed by arrival rather than by placement; when orders cross or bunch, exact recovery is shown to fail and only a bounded-pipeline estimate survives. A simulation study implements split-conformal calibration, exact aggregation, robust solving over a union of blocks, and forward integration of the delayed, capacity-saturated network, and the paper reports confidence intervals throughout. A final experiment transmits a climatology predictor, a linear model, and a recurrent neural network through the identical pipeline and finds the coverage guarantee indifferent to the choice while the cost of the induced decision varies by 62%, which is the sense in which aggregation and forecasting are orthogonal.