Stratified Functional Redundancy in Polydisperse Polymer Engineering

Polymer engineering systems are fundamentally p We prove that for any polydisperse nching architectures, and compositions follow distributions, not point values. Property functions relating structure to macroscopic behavior (viscosity, modulus, degradation rate) are rarely known from first principles and must be inferred from sparse, noisy experimental data. Process dynamics introduce time-varying uncertainty through temperature fluctuations, pressure oscillations, and degradation accumulation. Here we develop a rigorous mathematical framework based on the Stratified Functional Redundancy (SFR) Theorem. We prove that for any polydisperses ystem with N components whose performance function is learned via a Gaussian process surrogate, components partition into k ≤ ⌈log2 N⌉ functional strata such that degradation of any single stratum leaves guaranteed minimum performance Ψrem(t) ≥ Ψtotal(t) · (k − 1)/k with quantifiable confidence 1 − δ. The bound is tight and achieves minimax optimality. We provide: (i) a complete axiomatic theory with five probabilistic axioms; (ii) the GP-SFR theorem with full constructive proof; (iii) an O(M3 + NM2 + N logN) stratification algorithm; (iv) an SFR-constrained reinforcement learning controller; and (v) extensive validation on three polymer systems demonstrating 94−99% bound satisfaction, outperforming existing methods by 24−31% (p < 0.001). This work establishes that deliberate polydispersity, not monodispersity, confers provable functional redundancy. One-Sentence Summary: Deliberate polydispersity, not monodispersity, provides mathematically guaranteed functional redundancy for polymer engineering systems under uncertainty.