The Theory of Data

A value becomes analytical data when it is bound to a governed analytical point. The Theory of Data describes the objects and laws needed to identify such data independently of its physical representation. A universe establishes which root points exist. An anchor partitions those points into analytical locations. A measure family supplies a governed analytical identity, and a measure is that family at one current anchor, written \(F@A\). Analytical lineage records how family identity is constituted. Version 7.1 retains Version 7.0's constructive family formation and sufficient-state family relations, and clarifies their continuation, evidence, and reuse obligations. A family law specifies its asserted target, constitutive inputs, admitted anchors and movements, participation, semantic value domains, sufficient-state constructions, and exceptional cases. Sufficient state remains a relationship among ordinary measure families: SUM and COUNT can construct MEAN; a set-valued family can construct exact distinct count. Each admitted basis must correctly construct the independently specified target on its own domain. Sufficiency does not imply necessity, equal material availability, or interchangeability for every later operation. A family law may depend on a governed order of analytical points while remaining invariant to evidence enumeration. The admitted multidimensional construction specifies complete point orders within constituent dimensions and precedence among those dimensions. These induce an order on the existing analytical point space, without creating absent coordinate combinations. FIRST and LAST use ordinary witness-valued families whose finite continuation satisfies the same commutative-state coherence theorem as other self-sufficient families. A further proposition establishes the exact compression from a retained point–value family to the extremum witness. The paper separates contextual formation from continuation, mathematical coherence from current evidence adequacy, and a stored answer from retained reusable state. A coherent target may be established even when a particular sufficient-state basis or staged materialization is unavailable. Lossy state does not certify its own input-consistency premises. The foundation includes a compact law catalog and an index to the inherited statistical extensions, preserved in an accompanying reference supplement under their declared contracts. It states the boundaries with semantic types, analytical requests, assurance, physical realization, and statistical inference. Its claim is conditional and focused: given governed definitions and adequate evidence, analytical identity, derivability, consistency, and reuse can be made explicit and independently adjudicable.