Introduction: descent before operational interpretation

A relational expression must first be a function of the equivalence class that it is intended to describe. A measurement interface cannot repair a failure of that set-theoretic requirement. Conversely, classical descent alone does not construct an operator, a quantum field apparatus, or a record.

Manuscript III separates four classical equation-space questions CheungIII. I-B supplies a finite experiment on a prescribed conical background CheungIB. This paper gives a precise local composition of those structures. Its new interface is deliberately finite: selected values of the proved classical functionals are encoded in finite matrices and read by a complete instrument.

The classical results are reproved for the declared laboratory rather than imported as global statements about the cone. The finite instrument results are also proved directly. Companion citations identify origins and comparisons; no unwritten companion theorem is a premise.