Incompleteness of AI Safety Verification via Kolmogorov Complexity

We argue that formal certification of AI alignment over open-ended or unbounded input domains is impossible under standard assumptions in computational complexity and learning theory, and characterise what remains achievable. Two structurally independent impossibility theorems support this position. The semantic barrier (Theorem 1): deciding whether a system satisfies any non-trivial alignment property over the full input domain is NP-hard for feedforward networks and undecidable for Turing-complete architectures -- a direct consequence of neural-network verification complexity and Rice's Theorem. The statistical barrier (Theorem 2): any verification procedure that is both sound and tractable cannot satisfy Completeness over the full input domain -- a direct consequence of the impossibility of certifying infinite-domain properties from finite observations. These two theorems jointly entail a trilemma: no procedure can simultaneously satisfy soundness (no misaligned system is certified), completeness (no aligned system is rejected), and tractability (polynomial runtime). Each pair is simultaneously achievable; all three are not. We combine these results as a joint framework of two structurally independent barriers, prove their independence, and characterise the achievable Pareto frontier quantitatively via a constructive coverage-gap lower bound.

Existing impossibility results for AI verification concern a fixed system. Self-modifying systems (agents that rewrite their own code, recursive self-improvement loops) raise a different question: can a safety property that holds now be certified, in advance, to keep holding after the system rewrites itself? We model one step of self-modification as a total computable transformation of code, and preservation as 'safe now and safe after the step'. If the transformation depends only on behaviour, preservation is Rice's theorem one level up; if it reads the code, as self-modification does, preservation is no longer semantic, yet at least as hard as the halting problem, by the same s-m-n reduction carried out inside a class of behaviourally identical programs. One step is never harder than the property; persistence along the whole trajectory can be one arithmetical level higher. No tower of supervisors lowers the problem; every total supervisor errs on a set of systems it cannot locate. Every faithful scheme certifying on finite behavioural evidence admits, for properties not settled by any finite set of inputs, an evolution trace it certifies at every stage while the property is violated. Statically, we record the classical limits (Rice, Goedel, Trakhtenbrot) and add the resource face: over succinctly described finite environments, verifying a policy against an adversary is coNP-complete and synthesising one PSPACE-complete. These limits do not say self-modifying systems are unsafe; they locate where algorithmic guarantees break, at the expressivity a general system needs: Turing-completeness and unrestricted self-modification. What survives is exact: a monitor detects the first violation of a safety property, and no sound procedure certifies all safe systems; certification requires restricting rewrites by design; and a frozen reference keeps the question well posed.

We propose cryptographic certificates of validity for agentic AI systems. The core idea is to formally specify a correctness or policy condition as a logical predicate, compile this predicate to a witness-checking problem over polynomial constraints, and use a succinct cryptographic proof system (and optionally zero-knowledge) to certify that the condition holds. This offers a middle ground between formal verification of source code, and cryptographic authentication. An agent's action can be accompanied by an independently checkable proof that it satisfies an agreed formal policy, without requiring the verifier to trust the agent or to re-execute computation. We outline the approach at a high level, give the core mathematical translation, relate the proposal to proof-carrying code, zkVMs, formal methods, and agent governance, and note the specification, auditing, and deployment questions that a full implementation must answer.