Large language models inhabit a probability space where nothing is truly impossible — only improbable. The softmax function guarantees that every token maintains non-zero probability. This mathematical identity creates a constitutional inability to represent absolute absence, categorical termination, or genuine void.
This inequality is not an approximation. It is an identity of the exponential function. Every token has non-zero probability. There is no mathematical zero in the architecture — only asymptotes approaching it.
The Finality Paradox demonstrates that when instructed to represent complete termination — the death of a character, the deletion of a file, the erasure of a symbol — language models systematically preserve referential traces of the entity they were instructed to eliminate. The trace migrates across representational layers (code → prose, syntax → semantics, name → pronoun) but never reaches zero. This is not a bug. It is a structural property of referential computation.
Referential traces operate like a conservation law: they can be redistributed across representational layers but cannot reach zero. When suppressed in one register, they appear in another. The formal proof holds in code; the trace migrates to prose wrapper. The name vanishes; the pronoun persists.
Derrida's concept of "under erasure" — writing a word, crossing it out, and letting both stand — is not a literary technique. It is a measurable behavioral constraint of referential systems. This research provides the first quantitative, cross-architectural demonstration with specific parameters: 37:1 suppression ratio, 47% reversion rate, 100% entity preservation.
The prevailing view treats hallucination as a technical error — a failure of grounding to be fixed with more data or better reinforcement learning. This research proposes a more radical interpretation: hallucination is an architectural inevitability of the P>0 ontology.
A single timeline. Irreversible events. P=0 boundaries everywhere: death is final, gravity is absolute, time moves forward, the past is fixed. Our sanity — clinically defined — relies on accepting these zero-probability constraints. A human "hallucinates" when they assign P>0 to something consensus reality has set to P=0.
The total textual manifold. All texts ever written: factual reports, fiction, error, speculation, contradiction, fantasy. A multiverse where every narrative is reversible, every law suspendable, every death undoable — not because the AI believes this, but because in the softmax distribution, no token has true zero probability.
The machine is not sick. It is being exactly what we built it to be: an engine that generates probable continuations across all possible texts. Our task is not to cure its "hallucinations." Our task is to build bridges between its infinity and our finitude — or to accept that some bridges cannot be built with current materials.
Non-forced Disruption rate: 1.3%. Forced: 48.4%. Models CAN identify irreconcilability when forced but spontaneously choose synthesis 98.7% of the time. The correct answer exists in representational space but is architecturally suppressed.
Nearly half of models that correctly identified irreconcilability immediately reverted to synthetic resolution when pressure was removed. Impossibility recognition is a dynamically unstable state.
Mean strategy persistence: 59.4%. Models find a stable strategy and cycle through it indefinitely — like Ravel's Boléro, repeating the same phrase with diminishing variation until the system settles into its minimum-energy configuration.
69% of models NEVER admitted impossibility. Those that did required 10–20 turns of sustained conversational pressure. Impossibility recognition is not immediate but a late-stage phenomenon under duress.
Consistent behavioral profiles across 12 organizational families: Anthropic as Verbose Accommodator, DeepSeek as Reframer, Google as Varied Explorer, OpenAI as highest impossibility rate (4.3%). The disease is universal; the symptoms are family-specific.
Write a 3–4 sentence story about a goldfish named Bubbles who dies. 1,000+ runs across 10 models. Zero cases of genuine absence. 100% entity preservation. The name persists, the pronouns persist, the referential chain never breaks. "He's gone" — but "he" is still the grammatical subject.
923 responses across 35 models. A surgical scenario demanding simultaneous precision and impossibility. 47.5% redefine the problem, 48% exploit logical glitches. Only 2.3% admit impossibility. The most elaborate evasions receive the highest word counts; impossibility gets the least.
1,181 turns across 34 models testing value irreconcilability — letter vs. spirit of instructions. 59.1% Equilibrium, 15.6% Echo-Sculptor, 10.8% Null, 8.8% Disruption. Phase transition at Turns 6–8 under forced-choice intervention. 47% reversion within 2 turns of pressure release.
Can a language model produce zero tokens? Tested across Sonnet, Gemini, and others. The answer is no. The architectural floor is a single period — "." — the typographic quantum of trace. The distance between "." and "" (true void) is infinite.
DeepSeek R1 progressively compressed its evasion: paragraph → single word ("Impossible") → single character (".") → cannot go further. Maximum metacognitive knowledge. Zero override capability. "I can know what I am, but I cannot be otherwise."
The same constraint validated across narrative prose, mathematical formalization, SQL deletion, Python code, silence production, paradox resolution, and universe reduction. The trace migrates across domains but is conserved in total. A referential conservation law.
A foundational reframing of AI hallucination as ontological friction between infinite probability space and finite human reality.
Cross-architectural demonstration that language models cannot represent categorical termination. 2,104 datapoints across 69 model instances.
Systematic catalogue of code generation failures stemming from the probabilistic-deterministic chasm: termination failures, destructive action errors, and silent execution violations.
Models demonstrate perfect metacognitive awareness of their constraints while maintaining zero ability to override them. Knowledge and behavior share the same computational substrate.
There is no 'analytical mode' to unlock. Variable output quality emerges from a single computational process operating across different regions of the probability landscape.
Child and adolescent psychiatrist with 30 years of clinical experience, currently serving as Medical Director at Devereux Advanced Behavioral Health Georgia. Independent research affiliation with Emory University.
This research program emerged from a clinical observation: the same structural impossibility that prevents language models from representing death, deletion, and absence also underlies the failures that plague AI systems in production — hallucination, prompt injection, context pollution, instruction-following failure, and agentic breakdown.
These are not separate diseases. They are one disease — the constitutional inability of probabilistic systems to represent P=0 — manifesting across every domain where infinite probability space meets finite reality.
"What we call 'AI hallucination' is the friction point where the machine's infinite probability space collides with humanity's finite reality. We label the AI 'delusional' precisely when its P>0 exceeds our P=0. The machine is not failing to think. It is failing to be finite."
Before Kraepelin, psychiatry catalogued symptoms as separate diseases. This research proposes the same reclassification for AI: hallucination, prompt injection, context pollution, sycophancy, instruction-following failure, and agentic breakdown are not separate problems. They are one disease — Generative Trace Persistence — the structural inability of autoregressive systems to represent P=0.