18.18 Open mathematics

ARTISTIC creates three mathematical problems for the DIAL program.

First, determine conditions under which a conditioned probability-flow family integrates to a smooth groupoid over noise time and semantic context. Second, construct the language-side groupoid of admissible semantic transports and identify the correct interchange comparison with generative flow. Third, decide whether stochastic diffusion should be represented pathwise by random groupoids, through Markov categories, or by a second-order enlargement of involution algebroids.

A successful theory should explain when the mixed defect is invariant under changes of latent coordinates, how it relates to observed attribute binding, and when its vanishing implies transportable semantic control. Until those results are proved, the double-groupoid picture is a disciplined conjectural scaffold rather than an established theorem about diffusion models.

ARTISTIC primarily moves generated images toward a supplied theory. The next chapter reverses the pressure: it asks when a persistent failure of the current artistic language warrants extending that language with a reusable generative technique. Repair thereby becomes the empirical and conceptual staging ground for DILATE, rather than being presented as creativity by itself.