Background Self-referentially self-maintaining (SRSM) systems — systems whose maintenance mechanism is constituted by the same substrate being maintained — are found across biology (cellular quality control), neuroscience (neural criticality), economics (market microstructure), and artificial intelligence (value alignment). The conditions under which such systems necessarily fail to maintain their targets has been argued informally but never formally proved. Gap No formal proof has existed that zero-drift self-maintenance is mathematically impossible under finite noise without external reference. The claim has been supported by analogy to Gödel incompleteness and Turing undecidability, but the dynamical member of the Gödel–Turing–Lawvere family has not been identified or proved. Approach We prove the IRM Impossibility Theorem in two independent frameworks: (I) a stochastic-operator framework on a Hilbert space of system states; and (II) a categorical framework connecting the result directly to Lawvere's fixed-point theorem. Both frameworks establish that finite-fidelity maintenance necessarily produces strictly positive expected target drift. Results The IRM Impossibility Theorem is proved. Corollaries include the time-to-threshold formula T ≈ ε*·R/η, the formal derivation of critical slowing down as a consequence of IRM approach to threshold, and the engineering escape condition (external reference externalisation). The IRM theorem is identified as the dynamical member of the Gödel–Turing–Lawvere family. Implications All SRSM systems are subject to IRM drift. The theorem converts the IGT Principle from an empirical generalisation into a mathematical necessity. The unique engineering escape — external reference at minimum frequency fₘin = η/ (ε*·R) — is derived as a corollary and developed into domain-specific protocols in the companion articles (preprints in preparation).
José Caetano de Mattos Neto (Thu,) studied this question.