Preface

This book develops the themes first introduced in Categories for AGI¹1. Mahadevan [ 2026c ] , revised July 25, 2026. ↩ by synthesizing them around a new core principle for machine learning. Rather than beginning with a scalar-valued loss function, it begins with the failure of a declared composition. Such a failure retains information that a single number discards: what was meant to compose, how it failed, and what kind of repair might restore it.

The framework developed here is lincs, short for Learning in Infinitesimal Non-Compositional Sketches. Its central claim is simple:

Compositionality is a desirable structural property of learning systems, whether natural or artificial. A deviation from it is therefore a potential signal for learning, but not yet a scalar error: it is a typed, generally non-scalar obstruction recording which declared composition failed, where it failed, and along which directions. Scalar observations may be chosen to optimize against that obstruction, but they neither exhaust its content nor determine an admissible repair.

Most learning systems begin with an objective and ask how to optimize it. lincs begins one level earlier by declaring what structure the system promises to preserve. A failure produces an obstruction that may later be observed through a norm, likelihood, test statistic, or decision functional. Those scalarizations can drive optimization, but remain accountable to the type, location, and semantics of the obstruction. Optimization is therefore placed inside an explicit structural contract rather than allowed to define it.