# The Deterministic Horizon: Impossibility Results as Design Specifications for Trustworthy AI Systems

arXiv: https://arxiv.org/abs/2605.23024 · HTML: https://mlsanity.com/paper/2605.23024/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 197 references that are in the map.
- First posted on arXiv 21 May 2026.

## Builds on (most cited first, 18 of 18)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Constitutional AI: Harmlessness from AI Feedback](https://mlsanity.com/paper/2212.08073/) (2022)
- [Retrieval-Augmented Generation for Large Language Models: A Survey](https://mlsanity.com/paper/2312.10997/) (2023)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Show Your Work: Scratchpads for Intermediate Computation with Language Models](https://mlsanity.com/paper/2112.00114/) (2021)
- [Measuring Faithfulness in Chain-of-Thought Reasoning](https://mlsanity.com/paper/2307.13702/) (2023)
- [Measuring short-form factuality in large language models](https://mlsanity.com/paper/2411.04368/) (2024)
- [AdaLoRA: Adaptive Budget Allocation for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2303.10512/) (2023)
- [Personalized Soups: Personalized Large Language Model Alignment via Post-hoc Parameter Merging](https://mlsanity.com/paper/2310.11564/) (2023)
- [AI Alignment: A Comprehensive Survey](https://mlsanity.com/paper/2310.19852/) (2023)
- [Algorithmic Collusion by Large Language Models](https://mlsanity.com/paper/2404.00806/) (2024)
- [COCO-DR: Combating Distribution Shifts in Zero-Shot Dense Retrieval with Contrastive and Distributionally Robust Learning](https://mlsanity.com/paper/2210.15212/) (2022)
- [On the Fundamental Limits of LLMs at Scale](https://mlsanity.com/paper/2511.12869/) (2025)
- [Fundamental Safety-Capability Trade-offs in Fine-tuning Large Language Models](https://mlsanity.com/paper/2503.20807/) (2025)
- [Sharp Generalization Bounds for Foundation Models with Asymmetric Randomized Low-Rank Adapters](https://mlsanity.com/paper/2506.14530/) (2025)
- [On the Fundamental Impossibility of Hallucination Control in Large Language Models](https://mlsanity.com/paper/2506.06382/) (2025)

## Abstract (arXiv)

Large language models now write software, draft legal documents, and produce clinical notes, yet fundamental limits, from Turing and Arrow to the No Free Lunch theorems, shape what computation can do. This thesis turns such impossibility results from curiosities into design rules. Its flagship result proves an accuracy ceiling set by architecture alone: past a critical reasoning depth, no amount of training moves it, at any adapter rank, sample size, or loss function. Computable before deployment from layer count and embedding width, this Deterministic Horizon is measured between nineteen and thirty-one across twelve transformer architectures, and fine-tuning on optimal-length traces recovers under four percentage points. The mechanism is a capacity invariant of the residual stream, and an information-theoretic conversion yields super-exponential accuracy decay past the horizon. An unconditional circuit-complexity lower bound for modular exponentiation against constant-depth prime-modulus circuits complements this result. The same argument recasts across subfields: preference learning under any misspecified model jumps discontinuously in sample complexity; multi-stage retrieval pipelines require at least as many independent metrics as stages; standard truthful auctions fail for agents with prompt-dependent valuations; and zero-knowledge verification of neural inference pays a measured overhead of one hundred ten to one hundred ninety times per non-linear activation. Together these form a catalogue of sixteen specifications, each pairing a computable boundary, a quantified violation cost, and a constructive design rule: two compositions are proved, one pairing is an honest obstruction, and four remain open. The impossibility-specification methodology is offered for the generative research programme that trustworthy AI may need. Every fundamental limit of AI is also a design rule.