Inverse-Free Wilson Loops for Transformers: A Practical Diagnostic for Invariance and Order Sensitivity

Large language models can change answers under harmless edits that matter in practice: RAG outputs flip when passages are reordered, fine-tuning erodes invariances learned at pretraining, debate or chain-of-thought prompts take path-dependent routes, and compiler fusion or reordering perturbs logits near decision boundaries. These failures violate intended invariances, break continuous integration, and force teams to trade safety for speed. The effects are small yet distributed across layers and positions, sensitive to context length and evaluation order, and costly to repair with retraining or formal verification. We present WILSON, a minimal post-hoc diagnostic suite that converts simple loop and reordering checks on internal representations into system signals. WILSON combines an inverse-free curvature map over positions and layers, computed with JVPs and Hutchinson probes, with activation-level commutators that flag reorder risk. Signals are cheap to compute, model-agnostic for standard Transformers, and exported as thresholds and CSV artifacts for orchestrators. This enables concrete actions: guard RAG against order effects, catch fine-tuning regressions, stabilize debate pathways and long multi-turn contexts, and gate fusions or reorders in deployment. In short, WILSON helps anticipate failures and approve safe optimizations so reliability and throughput can improve together without changing model architecture or training.