The Theory of Data: Governed Analytical Objects, Lawful Transformation, and Certification

This paper develops a foundational framework for governed analytical objects, lawful transformation, and certification. Version 4.0 is the terminology revision: content-preserving, vocabulary-changing. No definition, law, proposition, or open problem is altered in substance, and the proved fragment chain is unchanged. The revision retires the word atom, which named smallestness while the theory's object is a family with laws. The vocabulary is now: a datum is one given value at one typed anchor point; a series is a homogeneous typed binding of datums over one anchor — the shape, prior to citizenship; a measure is the governed family, its laws and its members; and a member of a measure is a governed series admitted under a complete contract. Membership is a status conferred by law: a family is constituted by its law. The relational vocabulary is unchanged — member closure, canonical member identity, member equivalence — and a full terminology-history section records the lineage from Versions 1.0–3.1. The framework: a measure such as revenue is not one column at one grain; a column realizes one member of the measure. Members carry complete contracts (type, universe, anchor, eligibility, observed support, observation process, sufficient state, boundaries, evidence, lineage). Transformations have three identity outcomes — member closure, measure synthesis, or non-closure — and closure is decided by law, not by successful evaluation: within the proved fragments of the accompanying Contract Calculus, a result can be evaluable, deterministic under every execution schedule, and still not the analytical object it claims to be. Canonical member identity with member equivalence and diagram commutation, bind consistency, capability-resolved boundaries, projection coherence between the broad framework and its proved kernel, and ask denotation by identity tuples complete the account.