19.2 The state space is a space of artistic theories

Let \(S\) be a sketch presenting a visual theory. Its generators may name objects, colors, textures, parts, spatial relations, viewpoints, composition rules, and rendering operations. Its equations and designated constructions state which composites agree and which visual structures must be preserved. A realization

interprets this presentation in a semantic and generative environment \(\mathcal E\). The realization may assign an image distribution rather than a single artifact to a semantic scene.

For a fixed sketch \(S\), write \(M_S\) for the space of admissible models, modulo the chosen observational equivalence. This quotient is important: two neural parameter vectors that induce the same relevant artistic behavior should not count as different visual theories. Allowing the sketch itself to change produces a larger space whose points are pairs

It is generally unsafe to call this entire space a smooth manifold. Different sketches have different generators and model dimensions, and their moduli may contain singular strata. A more faithful global object is a fibred or Grothendieck-style assembly of the local model spaces \(M_S\). The differential construction below is made on a regular stratum; finite sketch maps connect different strata.

This distinction separates two operations that a parameter-only account can collapse: