From Impossibility to Physical Necessity: The Mathematical and Engineering Foundations of Terminal AI Governance

This paper proves that no purely computational system can serve as its own final governance authority, and that the only stable resolution is a physical root. Part I establishes two formal barriers: GIT-I (self-governance cannot be proved from within, via Gödel's Second Incompleteness Theorem) and GIT-II (governance-violation reachability is undecidable, via reduction from the Halting Problem). An empirical barrier from Berdoz et al. (2026) shows that visible reasoning effort hides structurally different reasoning traces, so explanation length cannot be treated as governance evidence. Part II demonstrates that the impossibility of computational self-governance creates an infinite regress: every computational guardian needs another guardian. The regress terminates only when governance encounters an authority that the governed software cannot rewrite, bypass, or reinterpret through ordinary execution paths. The Hardware Governance Operator (HGO) is specified: a physically constrained mechanism placed on the causal path between autonomous decision and external consequence, enforcing bounded, pre-certified invariants. The architecture is aligned with ISO/IEC 42001, NIST AI RMF, IEC 61508, and FIPS 140-3. Keywords: Incompleteness, Undecidability, Reasoning Structure, Hardware Governance, Physical Root, Terminal Authority, AI Safety