The Unverifiability of Artificial General Intelligence (AGI) Alignment, Static and Dynamic: From Trakhtenbrot's Wall to the Safety-Generality Tension

We establish the mathematical limits of AGI safety in two forms: verifying a fixed system, and verifying that a certified safety property persists once the system self-modifies. In the static case, no algorithm can certify a highly expressive AGI's safe behaviour infallibly, completely and tractably, whether over unbounded input domains (blocked by Rice's and Godel's theorems) or over all finite hardware configurations (blocked by Trakhtenbrot's theorem, which splits into a PSPACE-hardness barrier and a co-RE-completeness barrier), forcing a Soundness-Completeness-Tractability Trilemma as a structural, not statistical, necessity. In the dynamic case, we formalise self-modification as a computable transition operator and prove that no algorithm can determine, from a system's current certified safety, whether safety survives its next self-modification step: a result that reduces to Rice's Theorem one level up, making the static and dynamic barriers two faces of one obstruction. This forces an exclusive dichotomy: persistent certification is attainable only for systems that have stopped evolving semantically, i.e. only for narrow, not general, systems. Nor can the obstruction be delegated: any supervisor adequate to audit a general AGI is itself a general AGI, so the supervisory regress never terminates. Three practical risks (finite test coverage, bounded deliberation time, restricted observation) are one phenomenon: every bounded scheme that does not reject correct evidence admits an evolution trace it certifies at every stage while the property is persistently violated. These results give formal content to the unverifiability of AI, showing it is not an engineering target deferred by current limits but a structural tension, an Expressivity Invariant governed by the same computational laws as the Halting Problem and Rice's Theorem.

Intervention Complexity as a Canonical Reward and a Measure of Intelligence

The Legg--Hutter universal intelligence measure provides a rigorous scalar assessment of general intelligence as expected reward across all computable environments, weighted by simplicity. However, the measure presupposes an externally specified reward function, raising the question of whether the reward primitive is inherently arbitrary or whether a canonical choice exists. We propose a new measure, called intervention complexity, that has five natural properties: environment-derivedness, universality, minimality, sensitivity, and achievement preference. Given a resource function rho encoding an inductive bias (such as program length, execution time, or energy), rho-intervention complexity is a universal reward. The result yields a family of canonical rewards indexed by resource bias, providing a principled completion of the Legg--Hutter framework that does not require external normative input. We further propose a two-dimensional characterisation of intelligence: agent competence (how well the agent performs relative to the oracle optimum) and learning efficiency (how quickly this competence improves with experience). A separation theorem establishes that the choice of resource bias determines the computability of the resulting measure: action-count IC is computable in polynomial time, while program-length IC without oracle access is uncomputable, with the gap between oracle and bare IC precisely quantifying the information-theoretic content of learning. We discuss implications for superintelligence and for pre-training universal agents.

Primitive Recursion without Composition: Dynamical Characterizations, from Neural Networks to Polynomial ODEs

What do recurrent neural networks, polynomial ODEs, and discrete polynomial maps each bring to computation, and what do they lack? All three operate over the continuum--real-valued states evolved by real-valued dynamics--even when the target functions are discrete. We study them through primitive recursion. We prove that primitive recursion admits equivalent characterizations in all three frameworks: bounded iteration of a fixed recurrent ReLU network, robust computation by a fixed polynomial ODE, and iteration of a fixed polynomial map with an externally supplied step-size parameter. In each, the time bound is itself primitive recursive, composition emerges from the dynamics rather than as a closure rule, and inputs are raw integer vectors. Every primitive recursive function is first compiled into bounded iteration of a single threshold-affine normal form, then interpreted as a ReLU computation and as a polynomial ODE. The equivalences expose a structural asymmetry: no fixed polynomial map can round uniformly to the nearest integer or realize exact phase selection--operations polynomial ODEs perform robustly via continuous-time flow. Each formalism compensates for a limitation the others lack: the ReLU gate provides exact branching, continuous time provides autonomous rounding and control, and the step-size parameter recovers both at the cost of discretization precision. This opens dynamical characterizations of subrecursive hierarchies and complexity classes by restricting time bounds, polynomial degrees, or discretization resources within one framework. More broadly, these models do not compute by composing subroutines: they shape the trajectory of a dynamical system through clocks, phase selectors, and error correction built into the dynamics. This differs structurally from symbolic programming, and our theorem gives a precise framework to study the difference.