Optimality is relational
A system is optimal only relative to a defined context and objective. Adding a more accurate measurement of an already relevant consequence may separate alternatives that previously appeared tied.
If a refinement changes what is valued rather than measuring the original criterion more precisely, it creates a new objective and domain.
More complete context can narrow accidental ties, but it does not guarantee uniqueness.
Scores are not functional equivalence
Functional equivalence must be defined through prespecified properties, invariances and thresholds rather than final score alone. Different encodings may be equivalent, while systems with the same score may fail differently under intervention.
The descriptive level matters: outcomes may converge while algorithms or architectures remain distinct.
The geometry near the frontier
Strong IAH requires the functional diameter of the entire near-optimal set to approach zero and a common limiting class to exist in the stated space.
A unique exact optimum does not by itself prove that every sequence of near-optimal systems converges to it. Conversely, a non-attained supremum may still permit concentration near a limit.
Multiplicity can falsify the strong claim
Symmetries, plateaus, specialisation, indifference under the objective or unresolved trade-offs may preserve several equally optimal functional classes.
In a finite or exhaustively explored space, two globally optimal systems with nonzero functional distance refute unique optimal functional class for that domain. An arbitrary preference added after observing the tie cannot remove the counterexample.