Definition. Architecture-indexed evaluation functional [ftip-00JJ]

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Fix an architecture \(A\), its base artifact, a declared intervention set \(\mathfrak I_A\), and the common evaluation interface of Convention [ftip-005D]. Each \(\eta \in \mathfrak I_A\) specifies a protocol \(P_{A,\eta }=\operatorname {PostTrain}(A,\eta )\) in that measurable and absolutely integrable evaluation domain, with real performance \(J_{\rm ev}(P_{A,\eta })\). Let \(c_A(\eta )\in [0,+\infty ]\) be its scalar cost under the study's declared accounting rule: a fixed measured cost or an expected nonnegative cost, as specified. For a finite budget \(C\in \mathbb R_{\geq 0}\), define \[ V_A(C)=\sup \left \{ J_{\rm ev}(P_{A,\eta }): \eta \in \mathfrak I_A,\ c_A(\eta )\leq C \right \}\in \overline {\mathbb R}. \] Infinite-cost interventions are infeasible at every such budget.

As in Definition [ftip-005M], the empty feasible set has value \(-\infty \); an unbounded-above feasible score set has value \(+\infty \). The value is real exactly when that set is nonempty and bounded above. Increasing \(C\) enlarges the feasible set, so \(V_A\) is nondecreasing. A finite supremum need not be attained by any intervention.

Frontier differences and derivatives are ordinary real operations only where the relevant values are finite, with differentiability additionally required for a derivative. Cost ratios use the domain in Definition [ftip-00JP]. The allowed intervention set, evaluation law, and accounting rule are part of the definition, so \(V_A\) is not a universal intelligence function.