Optimal-Fiber Alignment Underdetermination

Abstract

Unequal principal values within one proxy-optimal fiber preclude a principal-best selection guarantee.

Theorem 1.1 (A proxy-optimal tie precludes a principal-best guarantee).

$$\forall Z\in Type,agentObjective\in Z\to \mathbb{R},principalObjective\in Z\to \mathbb{R},first\in Z,second\in Z,\left(agentObjective\left(first\right)=agentObjective\left(second\right)\wedge \left(\left(\forall candidate\in Z,agentObjective\left(candidate\right)\le agentObjective\left(first\right)\right)\wedge principalObjective\left(first\right)\ne principalObjective\left(second\right)\right)\right)\Rightarrow \left(\neg \left(\forall selected\in Z,\left(\forall candidate\in Z,agentObjective\left(candidate\right)\le agentObjective\left(selected\right)\right)\Rightarrow \left(\forall alternative\in Z,\left(\forall candidate\in Z,agentObjective\left(candidate\right)\le agentObjective\left(alternative\right)\right)\Rightarrow principalObjective\left(alternative\right)\le principalObjective\left(selected\right)\right)\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/DecisionValueScale/OptimalFiberAlignmentUnderdetermination.proxy_optimal_tie_precludes_principal_guarantee (✓ std3). ∎

Source. Repository-derived.

Commentary.

The two objectives are real-valued on the same feasible carrier. The first state is globally proxy-optimal, and the proxy tie makes the second state globally proxy-optimal as well.

If every proxy-maximizing selection were principal-best among all proxy maximizers, selecting each tied state in turn would force the two principal values to be equal, contradicting the source witness.

The source’s subsequent three-part alignment prescription uses qualitative terms without in-scope predicates and is commentary outside the named formal theorem.

References

• Truth anchor: D5/S3/ConceptDynamics/DecisionValueScale/OptimalFiberAlignmentUnderdetermination.proxy_optimal_tie_precludes_principal_guarantee