ifc-0304

19.11 A paintbrush over the space of generative dynamics

The bracket experiments operate on update fields after a generative model has already been fixed. A stronger interpretation changes the dynamics by which an image is made. Let \(\mathcal S\) denote a local space of admissible stochastic dynamics, with a point

\[ \mathsf S=(f,g,s_\theta ) \]

specifying the drift, diffusion coefficient, and score field of a reverse-time image process. A conventional conditional sampler evolves according to

\[ dX_t= \bigl[f(X_t,t)-g(t)^2s_\theta (X_t,t,\gamma )\bigr]dt +g(t)\, d\overline W_t. \]

An elementary paintbrush inserts a registered state-level field \(\rho _C(c)\):

\begin{equation} dX_t= \bigl[f-g^2s_\theta +\rho _C(c)(X_t,t,\gamma )\bigr]dt +g_c(X_t,t,\gamma )\, d\overline W_t. \end{equation}

19.1

This equation already permits a reusable operation to act throughout the trajectory rather than as a terminal image filter. It is nevertheless the lower-level shadow of the desired construction. A family of parameterized SDEs is only a bundle of possible tools. It becomes Lie-algebroidal only after one specifies admissible sections, an anchor into \(T\mathcal S\), a bracket, and the Jacobi, Leibniz, and anchor-morphism laws (or their declared internal involution-algebroid analogues).

The proposed meta-dynamics therefore separates the evolution of the visual state from the evolution of its generative law:

\begin{align} \frac{d\mathsf S_\tau }{d\tau } & =\rho _C(c_\tau )(\mathsf S_\tau ), \\ dX_t & =b_{\mathsf S_\tau }(X_t,t,\gamma )dt +\sigma _{\mathsf S_\tau }(X_t,t,\gamma )d\overline W_t. \end{align}

The image-time variable \(t\) traces one act of generation, while the meta-time variable \(\tau \) traces a change in the available mode of generation. In this sense, DILATE seeks a new paintbrush rather than merely a new picture. The creative artifact is a reusable deformation rule for a family of samplers.

This formulation also makes the double direction explicit. Assimilation changes an image or sampler inside the current theory; proto-accommodation probes a change to the family of admissible generative laws. A proposed law becomes accommodation only after finite realization, transport, and admission. For assimilatory section \(a\) and paintbrush section \(c\), the corresponding mixed obstruction is

\[ \Omega _{A,C}(a,c)= \bigl[\rho _A(a),\rho _C(c)\bigr] \pmod{\operatorname {im}\rho _A+\operatorname {im}\rho _C}. \]

A nonzero raw bracket records order sensitivity. A persistent nonzero quotient class says something stronger: the current registered image and sampler edits do not close under their interaction. Only the latter can motivate a finite extension of the paintbrush language.

. Equation (19.2) is a target semantics, not a theorem that the implemented brush family already forms a global Lie algebroid. The registered experiments below estimate a local anchor, metric, and connection-like transport rule in a fixed observer chart. Bracket closure, global integrability, and equivalence under changes of observer remain mathematical obligations.