Theorem. Conditional saturation certificate [ftip-00LM]
Theorem. Conditional saturation certificate [ftip-00LM]
Take finite real \(U\), \(C\geq 0\), \(\delta \geq 0\), and \(\varepsilon \geq 0\), with the nondecreasing frontier of Definition [ftip-00JJ]. If a justified upper bound gives \(V_A(C+\delta )\leq U\) and a justified lower bound gives \(V_A(C)\geq U-\varepsilon \), then \(U-\varepsilon \leq V_A(C)\leq V_A(C+\delta )\leq U\). Both endpoint values are therefore finite real, and \(0\leq V_A(C+\delta )-V_A(C)\leq \varepsilon \). This is a conditional bound for the named frontier; it proves no universal saturation of intelligence.
For the finite controller class with hard resource admission, Corollary [ftip-00MC] obtains the required bounds from a feasible controller and a uniform Bellman certificate.