Example. Finite observations need not identify a frontier [ftip-00LN]
AGENTDRAFTED
Let \(S\subset \mathbb R_{\geq 0}\) be a finite set of measured costs, and
choose \(C_0>0\) larger than every element of \(S\). Suppose the exact frontier
value observed at each cost in \(S\) is zero. For finite budgets \(C\geq 0\),
suppose the declared class of possible frontiers permits both
\[
V_0(C)=0,
\qquad
V_1(C)=
\begin {cases}
0,&0\leq C<C_0,\\
1,&C\geq C_0.
\end {cases}
\]
These nondecreasing frontiers with scores in \([0,1]\) agree on every observed
cost and differ at \(C_0\). The observations alone do not distinguish them.
Both possibilities have finite realizations in the framework of
Definition [ftip-00JJ]: allow two interventions with costs \(0\) and \(C_0\).
Give the first score zero and the second score \(\theta \in \{0,1\}\).
For example, on a single deterministic evaluation task with utility equal
to the output bit, let the two resulting protocols return \(0\) and
\(\theta \). The two possible choices of \(\theta \) give \(V_0\) and \(V_1\),
respectively, under the same intervention and cost specification. Every
score is finite, and the feasible score maximum is attained at each budget.
This example does not supply a strictly better agreeing frontier for every
possible data set or every admissible class. If a proved global score bound
is \(1\) and an intervention attains it at a finite cost \(C_*\geq 0\),
monotonicity forces \(V_A(C)=1\) for every \(C\geq C_*\); no higher value is
admissible. A singleton class of possible frontiers can also identify the
frontier without such an alternative. A conditional certificate such as
Theorem [ftip-00LM] therefore requires its stated upper-bound evidence; that
evidence does not follow merely from the absence of an observed improvement.