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

Abstract: LLMs 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.