Foundations of the Existence Threshold

The Existence Threshold introduces the metric and locks the boundary between persistence and dissolution in discrete systems. Pattern Loss at Dimensional Boundaries measures what crosses when geometry changes — the 86% Scaling Law, reported with specimen-level reproducibility across cellular automata of every rule and scale. The Dimensional Loss Theorem supplies the proof behind the empirical regularity and finds the same conservation inside transformer attention and real neural networks. The Dynamic Existence Threshold moves the framework forward in time, where the balance of integration and differentiation predicts the state of a system — financial markets, space weather, the human brain across sleep and wake. Consciousness turns out to obey the same conservation as everything else. The papers are reproduced as submitted to the archive, with their arguments, data, and figures intact.

Three bridge essays, written for this edition, join the papers into one continuous argument. The Existence Threshold is not a boundary a pattern merely sits above. It is one the pattern continuously defends — at every transition, every dimensional shift, every tick of time.

"To exist is to continually overcome loss."

Nathan M. Thornhill is a complexity science researcher working in information theory and pattern persistence. The four papers collected here develop a single instrument - a scalar metric for pattern persistence and the cost of dimensional change - validated across cellular automata, transformer attention, financial markets, space weather, and the human brain in sleep and wake.He came to this field from outside the traditional academy. Years in healthcare administration - nursing homes first, then ICU admissions - put him across from coordinated systems succeeding, failing, and rearranging themselves on a nightly basis. Computers and technology have been a lifelong passion; AI, when it arrived, read not as the latest instrument but as the most natural subject the research could be pointed at. The four papers come out of the three threads running together.He holds provisional patents on measurement systems derived from this work, with applications in artificial intelligence, artificial general intelligence, and AI/AGI safety. His research is archived through the Institute for Complexity Science and Advanced Computing.He lives in Fort Wayne, Indiana, where he also runs a technology consultancy. He enjoys aviation and flight, camping, plays guitar, and spends too much time trying to maintain a fairly acceptable lawn and landscaping, but family always comes first. The lab may just be a spare room, but the hours - they are late, and the work must continue.