The Geometry of Dissolution and Fragility: Predicting Catastrophic Failure from Structural Analysis

A pilot with ten thousand hours flies competently without conscious attention to hundreds of sub-tasks. Airspeed, altitude, heading, engine instruments, radio calls, traffic scanning — each one was once a demanding task requiring full concentration. Through years of repetition in consistent conditions, each collapsed into structure that produces correct results automatically. The pilot's conscious processing pipeline is almost entirely free, available for weather assessment, route planning, communication with passengers, or handling the unexpected. A bee enters the cockpit. The pilot swats at it, loses visual reference for two seconds, and suddenly must consciously re-acquire altitude, heading, and attitude simultaneously. Three tasks that cost zero conscious effort moments ago now each demand full processing. The pilot's pipeline handles one task at a time. Three are competing. Performance degrades — not because the pilot forgot anything, but because the context shifted outside the conditions under which those skills became automatic. This paper concerns the geometry of that fragility. The vocabulary is small and builds in order. Processing is what any system does when it must act on information. A CPU executing instructions, a surgeon operating, a pilot flying, a manager deciding — each is a processor acting on elements. The irreducible unit of processing is the **op**: one transformation by one processor. A diagnostic question is one op. A mirror glance is one op. A cache lookup is one op. Ops are countable, observable, and universal across domains the way bits are universal across communication channels. When a processor performs the same operation repeatedly under consistent conditions, the op count required decreases toward zero over time. The mirror check that cost a new driver six ops — locate mirror, focus, scan image, identify objects, assess threat, return gaze — eventually costs zero. The processing chain collapses into structure that produces the correct result without consuming the processor's scarce sequential pipeline. This is **dissolution**. A dissolved task costs zero ops. The processor's total capacity is bounded by one inequality: total ops multiplied by average op duration must not exceed the available time budget. The time budget is fixed by domain physics — lane tolerance before the guardrail, anesthesia window before risk, request timeout before the client disconnects. Dissolved tasks don't count against the budget. This is what makes expertise powerful: the expert's budget is mostly free because most routine processing has dissolved. But dissolution has conditions. It occurred under specific circumstances — specific visibility, specific aircraft behavior, specific cockpit configuration. When circumstances change beyond what the dissolution can accommodate, the task promotes back from dissolved to active. It costs ops again. When many tasks promote simultaneously, total ops spike past the time budget and the processor fails. This is a **cascade**. The number of simultaneous promotions is the severity. The bee is tiny. The cascade is large. The event's magnitude and the cascade's severity are unrelated quantities — severity depends on how many dissolved tasks break, not on what broke them. The question this paper answers: given knowledge of what a processor has dissolved and under what conditions, can we predict where cascades will be severe before they occur? The answer is yes, and the tool is geometry.