19.3 Two local artistic directions

Fix a regular model stratum \(M=M_S\). DILATE places two interpreted algebroids over it:

The side \(A\) contains assimilatory skills: operations that refine a work or generator while preserving the current visual theory. Examples include registered color transformations, compositional adjustments, brush or denoising operations, and corrections that remain expressible in \(S\).

The side \(B\) contains proto-accommodative probes: local variations that test possible reorganizations of the visual language. They may perturb token bindings, viewpoint conventions, compositional laws, geometric invariants, or the interface between a semantic declaration and its renderer. A section of \(B\) is not yet a new technique. It is an infinitesimal question asked of the current theory.

Their anchors

map abstract artistic operations to observable local changes in the current model space. Sections \(a\in \Gamma (A)\) and \(b\in \Gamma (B)\) therefore induce vector fields \(\rho _A(a)\) and \(\rho _B(b)\) on the same stratum. The side interpretations are part of the target card; two arbitrary neural gradients do not become assimilation and accommodation merely by being named \(A\) and \(B\).

The proposed double involution structure adds compatibility, core, and interchange data to these sides. At the classical smooth level, the picture is a candidate double Lie-algebroid geometry; internally, its relationship with involution algebroids in a tangent category remains one of the DIAL bridge problems described earlier in the book [ Mackenzie , 2000b , Burke and MacAdam , 2019 ] .