Definition. Iso-quality cost ratio [ftip-00JP]
AGENTDRAFTED
For a finite real target \(q\in \mathbb R\) and the frontier of
Definition [ftip-00JJ], define the inverse cost
\[
C_A(q)=\inf \{C\in \mathbb R_{\geq 0}:V_A(C)\geq q\}
\in [0,+\infty ],
\qquad \inf \varnothing =+\infty .
\]
The iso-quality ratio is defined only on the domain
\[
\rho _{A/B}(q)=\frac {C_A(q)}{C_B(q)},
\qquad
0\leq C_A(q)<+\infty ,
\quad 0<C_B(q)<+\infty .
\]
The architectures must use the same target, evaluation interface, cost units,
and comparison arm. The ratio compares efficiency under these choices.
An inverse cost is a threshold infimum, not an executable minimum.
The infimum over budgets may be unattained; even if a budget satisfies
\(V_A(C)\geq q\), its performance supremum may be unattained at \(q\).
An actual target-achieving intervention requires a separate witness.
Positive individual costs do not guarantee a positive inverse cost.
For interventions \(\eta _n\), \(n\geq 1\), with score \(1\) and cost \(1/n\),
the target \(q=1\) has \(C_A(1)=0\), although every intervention costs more
than zero. If both architectures have this family, the putative ratio is
\(0/0\) and is excluded by the displayed domain.