1 Axioms of Structural Learning

LINCS and a Repair Calculus

Chapter 0 supplied the working language of sketches, factorization, quotients, descent, tangent structure, repair, and admission. This chapter now states the structural contract of LINCS before the remaining foundations develop its ingredients one at a time. Its organization takes methodological inspiration from synthetic differential geometry. Kock begins that subject with an axiom saying that every map from the infinitesimal object \(D\) to the line \(R\) has a unique affine form [ Kock , 2006 ] :

Equivalently, a canonical map \(R\times R\to R^D\) is invertible. The axiom supplies a universal decomposition into value and infinitesimal slope; the subsequent theory explores what this single structural promise makes possible.

LINCS begins from an analogous organizing move, although not from the same algebraic axiom. Its primitive object is the typed obstruction: a universal carrier through which every admissible observation of a declared compositional failure must factor. This makes a failure prior to any chosen norm, loss, or coordinate system. Scalar losses become observers of the obstruction rather than definitions of it.

An axiomatization is not yet a calculus. Axioms give the semantic structures in which reasoning takes place. A calculus gives rules for transforming one licensed judgment into another. The distinction is familiar from causal inference. A structural causal model supplies semantics for interventions; do-calculus supplies rules for replacing expressions involving actions and observations when graphical separation conditions hold [ Pearl , 2009b ] .

LINCS suggests a related, but more general, question:

Under what declared conditions may one change the observer, quotient, localization, tangent level, repair target, or theory while preserving the meaning of a structural-learning judgment?

This chapter first states one universal obstruction axiom and then identifies the additional structural laws required to transport its meaning through tangent lift, presentation, localization, and repair. It then proposes a repair calculus as a disciplined interface for structural intervention. The rules are sound only relative to their stated sketch-theoretic hypotheses. They are not do-calculus rules for identifying causal effects, and no completeness theorem is claimed.

¹1. The word calculus is used in the proof-theoretic sense: a small collection of transformations on typed judgments. It is not another name for gradient calculus. ↩