When Is One Number Enough? A Containment Theorem for Scalar Comparison in Staged Decision Architectures

Abstract

A recurring temptation in decision systems is to compress a problem into a single score and optimize it. Sometimes that compression is sound; often it is not. This paper makes the boundary precise for one well-defined case. Working inside the Productive Value-Productive Power (PV-PP) framework—a staged decision architecture that filters candidates by governing-domain viability before ranking—we establish a containment theorem showing that scalar comparison can be recovered as a restricted, certifiable special case. On a named structural subclass, an unweighted-mean comparator recovers exactly the same maximal set as the staged procedure. The result is deliberately local, conditional, and set-valued. It identifies the structural conditions under which scalar compression is valid while making explicit the assumptions required for that recovery. The formal theorem authority and supporting proof materials are maintained in the public PV-PP Scalar Reduction Proof Program repository.