Representation Sufficiency in Prospective Health

This paper develops the representation theory of prospective health introduced in Papers I–II of the Health, Formally Defined series. A declared health-query family induces a canonical query-visible quotient (q_A). Paper III proves that a representation is sufficient for that query family exactly when it refines the canonical quotient. In bundled form, the sufficient representations constitute the principal refinement upset generated by (Q_A). This identifies the minimal task-visible information required to answer a declared family of health questions. The same representation order governs exact deterministic transport. For a target query family (A_Q), the canonical quotient (qA_Q) is the least sufficient target: it is determinately transportable from a source representation exactly when the set of sufficient determinate target representations is nonempty. A finite witness inherited from Paper II shows how raw target capacity can remain indeterminate while its query-visible quotient is determinate from the same source representation. The paper also develops a state-level observation criterion. For a full health family (H), an observation is logically sufficient exactly when it refines the corresponding health quotient (q_H). The prospective-health normal form is connected explicitly to this result through the representation ((,q_A∘ C)). Concrete applications to controlled response, viable cost, route structure, finite probability, and failure-aware robustness distinguish richer response-grounded representations from minimal task-visible ones. In the probability and cost settings, concrete statistics are proved to realize their complete query-language quotients through explicit attained-range equivalences. The formal development is machine-checked in Lean. The core definition of health is unchanged from the preceding papers. The framework is scale-neutral and is intended to support future general and domain-specific instantiations together with further formal work in measurement science.