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