Axiomatic Atlas: A Prescriptive Framework for Neural Architecture Design

Neural architecture design lacks first principles: innovations are discovered empirically and justified post-hoc, with no systematic way to diagnose *why* an architecture fails or derive *what* repair will succeed. We introduce the *Axiomatic Atlas*, encoding requirements as composable axioms over graph connectivity, operator contracts, numerical stability, and information preservation. Given an operator library and wiring conventions, the Atlas constructs certificates lower-bounding output variation via min-cut analysis and diagnoses failures by locating axiom violations. Crucially, the framework is prescriptive: each violation implies a targeted repair, reducing architecture design to constraint satisfaction. We prove variation bounds under exact and finite-precision arithmetic, enabling modular verification across transformers, MoEs, SSMs, and GNNs. Four Atlas-derived interventions validate the approach: +46 percentage points on GNN bottlenecks, $3\times$ robustness to MoE quantization, 83\% gap closure with adaptive expert budgets, and 0\%$\to$100\% retrieval via orthogonal keys---each against matched negative controls.