MA02(01) Composable Mathematical Aggregates: Certified Interfaces, Feedback, and Minimal Observations

Status and claim discipline. This is a reproducible research note, not a claim of a new universal mathematical theory. Thesafety-kernel, set-cover, observation-compatibility and guarded-feedback principles are derived from established mathematicalideas. The contribution at this stage is a precise engineering-oriented integration and a checked example; substantive originalityrequires a stronger theorem or algorithm for previously untreated classes of aggregates.AbstractWe study a literal mathematical analogue of a sealed engineering component: an aggregate with typed externalports, internal transformations, a specified observational semantics, and a machine-checkable functional contract.Starting from a finite adversarial dynamical system, we compute its greatest robust controlled-invariant set and synthesize two different safe controllers. We then derive an exact characterization of the minimal number of abstractobservation symbols needed by a deterministic memoryless safety controller: it equals a set-cover number over locally admissible actions. A 35-state example has 20 initially safe states and an 18-state winning kernel; its controllerrequires exactly three abstract observation symbols, or two binary port bits, despite depending on a system with hidden thermal state. We give proof certificates, replacement conditions, counterexamples to unsafe sensor merging andunguarded feedback, and a precise publication novelty audit. All finite checks are independently repeatable with astandard-library Python implementation.