Regression Has an Anchor: Members, State, and Lawful Movement in the Most Performed Analysis in Statistics

Regression is the most performed analysis in statistics, and its standard presentation begins after a data foundation has been fixed: a response vector, a design matrix, and rows understood as observations. This paper restates regression under the Theory of Data, in which the inputs are typed objects — members of measures, co-anchored over one universe — and asks what the standard presentation forgets. Three findings organize the paper. First, a fitted model is a member, not a law: it is estimated at a home anchor, on a home universe, with an approximation contract, while the field's language describes a measure-level relation quantified over anchors and universes. Everything called generalization is movement of that member to places its contract does not cover, and the paper reads distribution shift, train/test splitting, transfer, and monitoring as informal maintenance of clauses that could be stated — connecting the account to the formal literatures that already write them, covariate shift and transportability. Second, model classes stratify by what a block of data must carry. Conditional on the design, every canonical generalized linear model has finite, additive sufficient state for the coefficients; linear regression separates from the others not by the existence of that state but by whether the fit can be finalized without retaining the design, a boundary that coincides exactly with a quadratic cumulant function. General nonlinear models carry no fixed-dimension state at all, which the Pitman–Koopman–Darmois theorem certifies under its regularity conditions. The stratification gives distributed fitting an exact reading in each stratum, including the precision-weighted combination that makes display averaging exact for the linear case and the shared-point protocol that makes it exact per round for generalized linear models.Third, predictors are not one kind of thing. They range from co-anchored members of other measures, through transported members and images of declared coordinate maps, to raw anchor coordinates entered as if they were values. Fertility, transportability, and several classical pathologies sort along this typology: the incidental-parameters problem and cold start are the same object met at estimation and at prediction time, the ecological fallacy and Simpson's reversal are non-closure across grains, and Yule's nonsense correlations are a shared anchor acting as a confounder. Random effects appear as the lawful form of fixed effects, with the conditions on their combinability stated. The framework proposes no new estimator and modifies no classical result. It gives typed locations to premises that every regression already makes, and it connects results the field has treated as separate anecdotes for a century.