Sampling the Shell: From Exponential to Constant Sample Complexity in Negative-Sampling Anomaly Detection

Negative-sampling anomaly detection reduces unsupervised fault detection to binary classification: observed telemetry is labeled normal, synthetic negatives are drawn from a reference probability distribution, a binary classifier is trained, and Integrated Gradients converts its alarms into per-sensor symptoms for downstream root-cause analysis with language models. The recipe is deployed in **intelligent diagnostics**, yet no theory has said when, and at what dimension, it can work. We supply one, grounded in high-dimensional probability : concentration of measure, Le Cam's minimax theory, Markov semigroups, Poincar\'e inequalities, and Langevin dynamics. *Limits:* manufactured negatives are ancillary to learning decision boundaries; no negative-generation scheme adds boundary information, so $M\gtrsim \rho /\epsilon$ positive samples are necessary at boundary resolution $\epsilon$. And uniform sampling operates only inside a dimension window with closed-form optimum $D^{\star }$ obeying a $1/e$ occupancy law, having certified budgets that are linear in $1/\epsilon$ inside the window and exponential outside it. *Constructions:* shell-concentrated, importance-weighted samplers restore a dimension-free band-hit rate, an exponential-to-constant reduction in negative sample complexity, realized by data-seeded Langevin dynamics and dimension-reducing autoencoders with dimension-free mixing and finite-sample certificates. We present an algorithmic framework as a ladder of eight detectors, from the MADI baseline to certified latent pipelines, each licensed by a theorem. Experiments confirm the predictions: on synthetic suites ($D=128$–$1024$, one to five operating modes) detectors attain ROC-AUC of $1.00$ on box-structured modes at every dimension while the baseline method decays toward chance. On the multivariate track of the TSB-AD benchmark, under its official evaluation protocol and over fifteen series spanning the benchmark families, the strongest variants exceed the PCA, LSTMAD, and OmniAnomaly incumbents and match the top-ranked CNN (median AUC $\approx 0.72$ vs $0.73$). Finally, on real commercial flight-test data, certified and latent detectors reach $0.92$, and $1.00$ AUC on fan-blade-out and compressor-stall events and up to $1.00$ on vibration, against $0.40$-$0.63$ for the deployed baseline.