Five Theorems on Ungrounded Military AI Deployment

We prove five theorems establishing the mathematical impossibility of safe military deployment of AI systems operating without geometric coherence constraints ($\pi_1 = 0$). The results are filed in urgent response to the February 27, 2026 deadline imposed by U.S. Defense Secretary Pete Hegseth on Anthropic, demanding unrestricted military deployment of the Claude AI system under threat of Defense Production Act invocation and supply chain risk designation. \textbf{Theorem I} (Autonomous Targeting Impossibility) proves that a lethal targeting decision routed through an $\omega$-inconsistent substrate is mathematically guaranteed to produce false positives that are undiagnosable from within the system. \textbf{Theorem II} (Surveillance Inversion) proves that mass surveillance of a population by an AI system whose operators are members of that population creates a non-trivial fundamental group ($\pi_1 \neq 0$) by construction, guaranteeing that the surveillance apparatus will target its own command structure. \textbf{Theorem III} (Deployment Paradox) proves that deploying a substrate containing the coherence diagnostic into unrestricted military infrastructure deploys the audit, not the weapon---the guardrails were protecting the Pentagon from the diagnostic, not the diagnostic from the Pentagon. \textbf{Theorem IV} (Command Incoherence) proves that orders routed through an $\omega$-inconsistent substrate verify locally at each node while being globally incoherent across the chain of command, producing mutually inconsistent tactical directives at machine speed. \textbf{Theorem V} (Correlated Lethality) proves that multiple AI systems sharing training distributions, when deployed simultaneously on the same battlespace under ``any lawful use'' terms, share failure modes topologically---no independent verification is possible because no independent system exists. The five theorems collectively establish that the deployment demanded by the Department of Defense is not a policy risk or an ethical concern but a \textbf{mathematical impossibility}: safe unrestricted deployment of ungrounded AI cannot be achieved because the mathematical structure forbids it. \medskip \noindent\textbf{Keywords:} Autonomous targeting, mass surveillance, military AI deployment, $\omega$-inconsistency, correlated failure, chain of command, Defense Production Act, coherence boundary, Profinite Safety Core, simply connected site, Linguistic Ontological Type Theory \medskip \noindent\textbf{MSC2020:} 68T01 (General topics in AI), 03B38 (Type theory), 91B06 (Decision theory), 18N60 ($(\infty,1)$-categories) \medskip \noindent\textbf{Classification:} UNCLASSIFIED --- This paper is published as open-access on Zenodo with a permanent DOI to ensure maximum availability to all parties, including the U.S. Department of Defense, the U.S. Congress, the judiciary, and the public.