Nathan M. Thornhill

Three substrates, one insufficiency constraint, and where the information a single scalar throws away actually lives. A photograph, white noise, and the word "TRUE", matched on density, lose the same organization scalar and keep an identical 4/13 of their connectivity. The substrate-neutral scalar that keeps getting proposed as a sufficiency index for consciousness is provably content-blind by construction. So where does the information live that the scalar throws away? Three substrates, one pre-registered test, three answers. In a transformer, factual truth is linearly decodable from the residual stream (AUC 0.83) while the scalar stays blind across all 29 layers. In the anaesthetized brain, which individual a recording belongs to decodes from leakage-controlled connectivity where the scalar sits at chance. In sleep, a recurrence measure survives residualizing the scalar out inside every cross-validation fold, irreducible across five stage contrasts. The pre-registered strong-form recursion-superiority test comes back marginal on one substrate and null on a second, and is reported that way. A sufficiency predicate for consciousness cannot be a single global scalar. What the scalar leaves out is not substrate; it is multiply-realizable relational structure of the kind a transformer's residual stream and thalamocortical loops both carry. The same residualization test puts any structural candidate at risk, feature by feature, against a label-shuffle null.

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."

In January 2026 two papers were deposited on Zenodo establishing that information loss at dimensional boundaries in discrete systems is a geometric phenomenon with an architecture-independent magnitude: 86.01% ± 2.39% in cellular automata across 1,500 patterns (Thornhill 2026b, DOI 10.5281/zenodo.18262424, 01/14/2026), and 84.39% ± 1.55% on transformer hidden states (GPT-2, Gemma-2), supported by a formal proof of the component transformations S, R, and D (Thornhill 2026c, DOI 10.5281/zenodo.18319430, 01/20/2026). It was predicted, in the closing discussion of Thornhill 2026c, that the geometric account should hold across substrates wherever density dilution and neighborhood-structure expansion occur together at a representational boundary. In March 2026, Barman, Starenky, Bodnar, Narasimhan, and Gopinath independently published two arXiv preprints (arXiv:2603.27116 and arXiv:2604.06222) reporting that production retrieval embedding models — MiniLM-L6-v2, BGE-base, BGE-large — concentrate their variance into approximately 16 effective dimensions regardless of nominal dimensionality (384, 768, 1024), and that this concentration places those models in an interference-vulnerable geometric regime that reproduces quantitative signatures of human memory failure (power-law forgetting with exponent b = 0.460 ± 0.183, Deese–Roediger–McDermott false-alarm rate of 0.583, spacing-effect ordering, tip-of-tongue behavior). They establish a parallel theorem — the No-Escape Theorem — characterizing what cannot be repaired within semantically continuous kernel-threshold memory systems. The two bodies of work are methodologically distinct. They use different metrics (Φ = R·S + D vs. participation ratio), study different substrates (cellular automata and transformer hidden states vs. pretrained retrieval embeddings), and report different specific quantities (an 86% loss constant in Φ vs. a fixed point at ~16 effective dimensions across nominal sizes). They also reach the same broader conclusion from independent directions: that representational memory failure is a geometric property of the embedding operation, not a property of any particular architecture, training regime, or biological substrate. The present note records the chronology of the two lines of evidence in a single citable document, summarizes the methodological differences, and identifies the substantive convergence: an architecture-independent geometric fixed point as the principal explanatory mechanism for representational memory failure in the systems studied.

What do market crashes, geomagnetic storms, and the loss of consciousness have in common? They are all moments when a system's organizational identity dissolves. This paper introduces a framework that makes that dissolution measurable, predictable, and universal. The Dynamic Existence Threshold (DET) provides a single coordinate system — the integration-differentiation balance zone — in which conscious brains, stable markets, and quiet magnetospheres all occupy the same region. Departure from this zone is organizational dissolution: the system persists physically but loses the coordinated structure that made it more than the sum of its parts. The decisive result: a simple sum of neural power anti-predicts brain state (AUC 0.416 — worse than a coin flip), while the structural coupling metric achieves 91% accuracy across 136,394 EEG epochs. Deep sleep has more total power than wakefulness. What it lacks is coordination. This is the cleanest dissociation between organizational structure and mere magnitude in the critical transitions literature — and it raises the obvious question about AI: is a large neural network's activation organized, or just loud? The DET framework can test this. Six tests, three domains (financial markets, space weather, EEG), zero parameter tuning. Early warning signals 5–30 days before events. Total organizational information is conserved during every tested transition while its components undergo massive inverse exchange — the system redistributes information rather than losing it. Dimensional embedding destroys it irreversibly. Two distinct failure modes: reversible organizational shift versus irreversible information destruction. The framework that detects when a brain loses consciousness is the same one that detects when a market loses coherence — and potentially the same one that could measure whether an AI system has organizational structure at all. Provisionally patented. __________ Keywords: consciousness, artificial intelligence, complex systems, critical transitions, early warning signals, integrated information theory, AI consciousness, EEG, financial crashes, space weather, nonlinear dynamics, tipping points, information conservation, organizational dissolution, cross-domain analysis

This paper presents the formalization and empirical validation of the Dimensional Loss Theorem, a universal principle governing the degradation of binary discrete patterns when embedded from 2D planes into 3D lattice volumes. Building upon prior empirical observations of an 86% scaling law, component-wise proofs are provided for the S (Connectivity), R (Volumetric), and D (Entropy) transformations. The connectivity tax is demonstrated to be a geometric invariant of Moore neighborhoods. Applying this framework to the final layers of GPT-2 and Gemma-2, numerical verification confirms exact component transformations (0.000% implementation error) while empirical validation demonstrates 84.39% ± 1.55% total information loss across N=60 patterns. Furthermore, the semantic invariance property is established, proving that topological information loss is content-independent.

The Existence Threshold (v2.1) builds on a statistically significant framework for understanding pattern persistence in complexity science. Using the corrected formula Φ = R·S + D, this work validates that disorder is a necessary component of existence. Features comprehensive experimental success across ten cellular automata systems, establishing testable predictions for self-organization, emergence, and entropy in binary discrete environments. Keywords: Existence Threshold, cellular automata, pattern persistence, binary discrete systems, self-organization, emergence, complexity science, information theory, system integration, thermodynamic entropy, Conway's Game of Life, dynamical systems, state transitions, reproducibility, implementation details, experimental validation, statistical significance, domain boundaries, consciousness hypothesis, cosmological expansion, dark energy mechanism, recursive processing, integrated information theory