MMP.Preface:3.2 - Represent admissible objects and construct missing relations

MMP.10 starts from intended candidate objects and their requirements. It chooses a representation, derives the conditions that make the representation valid and expresses the required answer. These steps matter for sets, sequences, functions and quantitative objects alike. If several records describe one object, a count or probability over records can require correction before it answers the subject question.

MMP.11 starts with supported relations and an unknown contribution among them. It constructs adjustable families that retain the needed properties and inserts that contribution into the connected model. A known total, a monotone response and a normalized probability law require different mathematical constructions. Their common modeling question is how to permit the unknown variation while retaining what is supported.

Choose the form from the next operation. Direct constraints may already describe all useful candidates. A parameterization can make changes preserve the constraints automatically, but may introduce duplicate descriptions or omit parts of the allowed family. Retain that difference when interpreting a fitted value, an impossibility result or an observed agreement.