Explanatory Engagement Under Rare Anomalous Failure: Asymptotic Rarity in Model Behavior (or: The Asymptotic AI)

Prior work on LLM behavior under anomalous conditions asks whether a model notices anomalies. We ask a narrower question: once a model sits in a workflow with a low, controllable failure rate, does its explanatory engagement - length, specificity, self-reported confidence - change as failure grows asymptotically rarer? We built a local, zero-cost harness on three open-weight models (qwen3:8b, llama3.1:8b, mistral:7b) running a repeated tool-call task where one call fails at probability p, swept across eight rates from 0.2 to 0.0001, under five elicitation conditions from immediate prompting to none. We hypothesized a rise in engagement as failures grew rarer, then a collapse near a detectability threshold. Pooled across conditions this appeared false: length fell in a flat, monotonic pattern. Splitting by condition overturned that. Under immediate_forced, where the model must explain every failure instantly, the predicted rise is confirmed but followed by a plateau, not a collapse: length peaks at 28.4 words at p=0.05, settles to 17.4-19.0 words at the rarest rates, and confidence rises unevenly from about 53% to the 70s-90s. Under grouped_runs, explanation batched to run-end, no collapse appears. Under passive_unprompted, aggregate magnitude is a floor artifact, but a recovered logging gap revealed real, model-specific self-monitoring: llama3.1:8b volunteers structured confidence reports unprompted, sometimes eroding its own confidence as trials accumulate; the other two do so only once, as boilerplate. Elicitation structure is a first-class moderator of collapse observability. A companion guaranteed-failure run (72 cells, backfilling rates where random sampling gave zero real failures) shows models differ in whether they recognize an anomaly, distinct from engagement once recognized. Limitation: discrete rate points cannot capture behavior between them, a direction for future work.