Singular Models and Moving Varieties: The Algebraic Geometry of Coordination-Dependent Data

Abstract

Two established fields apply algebraic geometry to data. Singular learning theory characterizes the generalization behavior of statistical models whose Fisher information degenerates, with deep neural networks the motivating case, through invariants obtained by resolution of singularities. Algebraic statistics treats statistical models as algebraic varieties and performs estimation as geometry on those varieties. A third literature, performative prediction and its relatives, formalizes distributions that respond to the act of prediction.

This paper identifies a coordinate that connects all three and that none of them carries: coordination-dependence, the degree to which a measured quantity's data-generating structure is constituted by the ongoing coordination of the observers around it, quantified in prior work as the fraction b of variance attributable to the coordination-dependent component. Two claims are staked. First, a deposited proposition of The Geometry of Constraint Knowledge, that conservation laws in training data induce Fisher rank deficiency in trained models, places that program's empirical predictions at the entry point of singular learning theory, and a conjecture follows with a specifiable differential test: training data structured as attributed, timestamped, facet-decomposed disclosure shapes the singularity structure of the trained model measurably differently from unstructured text about the same domain, with restricted variants of the local learning coefficient the natural instrument and background architectural degeneracy the identification problem the design must difference out. Second, the fixed-variety assumption of algebraic statistics fails for high-b quantities in a way performative prediction describes dynamically but does not geometrize: the variety moves under the observation directed at it because the observers partly constitute the structure being estimated. A candidate formalization is stated, a family of varieties fibered over a coordination-state manifold whose primary structure is the closed elliptope of similarity matrices, in its Bures-Wasserstein geometry, with the similarity structure already load-bearing in the corpus's formal core, under which the fixed-variety case is recovered as the constant family and the fraction b acquires a candidate geometric referent.

The contribution is the coordinate and its connections. The underlying geometry belongs to the fields that built it, and the paper states the boundary between what is established, what is struck in the corpus, and what is open.

Keywords: singular learning theory, algebraic statistics, performative prediction, Fisher information, local learning coefficient, coordination-dependence, training data, moving variety, OCOT (Origin Chain of Thought)

A note on vocabulary and sources:

This is a priority deposit drawing in part on a corpus not yet published. Published papers are cited in standard form with their deposits; unpublished works are not named, and the cover statement carries their attribution. Three terms of art recur. To strike a claim is to commit it to a dated, attributed record before the outcome is known. A facet-decomposed disclosure is a record in the corpus's six-part format (identity, input, method, output, scaling, transmit). The fraction b is defined counterfactually over variance components and is never computed by any runtime; a value traveling with a record is b-hat, an estimate struck by an identified originator with the method stated. OCOT, Origin Chain of Thought, names the field the corpus develops.