Remark: Family 245. The β -Barendregt–Geuvers–Klop conjecture

Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a $\beta$-normal form, every $\beta$-reduction sequence terminates. This resolves the $\beta$-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.