Remark: Exactly what J does not establish

Theorem thm:J is a compactness theorem with finitely many constraints at each stage. Its unknowns are still functions on ray spaces. It is not a finite-dimensional LP.

Sequential probabilities contain products and evaluation at unknown residual states. A grid discretization must prove an interpolation or extension theorem, error bounds, and preservation of all required constraints before its optimum can be identified with ${\epsilon }_k$. Neither effective computability nor a convergence rate follows from compactness alone.

If ${\epsilon }_{\infty }=0$, every positive target is eventually excluded by a sufficiently small upper bound. This supports rigidity; it does not “refute rigidity.” Finite tests of one candidate do not optimize over $K_k$. Without Assumption ass:selective-modulus, this theorem does not settle Open Problem op:G.