The Theory of Data Applied

Familiar analytical failures—averaging averages, summing exact distinct counts across overlapping groups, unsafe fan-out, invalid calendar rollups, summing inventory across time, ambiguous aggregation identity, and denominator errors—are already well known. The contribution of the Theory of Data is not to rediscover these warnings, but to show that they arise from a small set of structural obligations that can be represented as governed information about analytical data itself. This paper applies Theory of Data Version 6.0 to seven classical cases. A universe governs which root points exist; anchors govern how those points are partitioned; a measure is a measure family at an anchor; family law determines lawful continuation; lineage records identity-bearing construction; edge contracts govern licensed movement; and sufficient state determines whether a lawful continuation remains possible from what was retained. Across the cases, a common dependency order emerges: B→F@B→B≻A→Γ(e)→sufficient state. The cases distinguish failures in which the requested derivation itself is not established from those in which the derivation is lawful but the available evidence or retained state is insufficient to produce the exact result. The paper's applied claim is structural: many familiar analytical failures can be located at the first unmet analytical obligation before successful execution is mistaken for analytical validity. The architecture for using such findings to govern user intent, risk, authorization, and execution is developed separately in Analytical Governance: From User Intent to Governed Analytical Execution.