Quantum Gödel Machine

Herein a Quantum Gödel Machine is defined as a self-rewriting, proof-driven agent whose acceptance gate for self-modifications is allowed to depend on quantum witnesses and quantum program logic, rather than only on classical proof strings. The central move is to promote quantum proofs to first class internal objects, a candidate rewrite is executed only when there exists a quantum witness state that makes a quantum verifier accept a formal claim that the rewrite increases a specified quantum utility functional while preserving formally stated safety constraints. The paper introduces an operator-level decomposition of self-rewrite dynamics in which proof search, verifier checking, and environment interaction jointly induce a completely positive map on the agent’s internal state and policy representation. Within this representation, stability of iterative self-rewrites is controlled by a spectral radius condition on a linearized update operator, and achievable improvement rate is bounded by information throughput across the agent-environment channel. The resulting framework yields two kinds of technically sharp statements, conditional complexity leverage claims that separate quantum-witness verifiability from classical-certificate verifiability for natural classes of self-improvement obligations, and measurable rate constraints that tie certified self-improvement to channel capacities and verification costs. The definitions isolate what is genuinely quantum in the construction and provide a path to both theorem-driven and system-level empirical evaluation.