ifc-0305

19.12 Observer-induced metric and discrete transport

A fixed coefficient vector is not a geometrically meaningful way to transport an artistic intention across heterogeneous latent states. The same numerical edit can have very different visible effects at different noise levels, prompts, and discretizations. The PB2.2 experiment therefore represents an intent in observer coordinates and lifts it separately at each state.

For a candidate skeleton, let

\[ \{ b_{\mathrm{focal}},b_{\mathrm{transition}}, b_{\mathrm{peripheral}}\} \]

be a local basis of typed fields. Let \(E_i\) be the high-frequency energy measured by an independent observer in region \(i\). Finite differences define the local response matrix

\[ J_{\mathsf S}(i,j)= \left.\frac{\partial }{\partial \epsilon _j} \log E_i(X+\epsilon _j b_j)\right|_{\epsilon _j=0}. \]

For the artistic contract “preserve focal detail, soften the transition by ten percent, and soften the periphery by twenty percent,” the frozen response coordinate is

\[ u=(0,\log 0.90,\log 0.80). \]

With observer weights \(W=\operatorname {diag}(3,1.5,2)\) and ridge \(\lambda =10^{-2}\), the regularized pullback form and weighted-response lift are

\begin{align*} G_{\mathsf S}& =J_{\mathsf S}^{\mathsf T}WJ_{\mathsf S}+\lambda I,\\ \alpha _{\mathsf S} & =G_{\mathsf S}^{-1}J_{\mathsf S}^{\mathsf T}Wu,\\ F_{\mathsf S}& =\sum _j\alpha _{\mathsf S,j}b_{\mathsf S,j}. \end{align*}

Equivalently, \(\alpha _{\mathsf S}\) minimizes

\[ \lVert J_{\mathsf S}\alpha -u\rVert _W^2 +\lambda \lVert \alpha \rVert _2^2. \]

The operational transport rule preserves the same observer-level intention, not the same coefficient vector. The resulting \(F_{\mathsf S}\) changes with the local response geometry.

Finally, temporal weights are normalized,

\[ \overline w_k=\frac{w_k}{\sum _jw_j}, \qquad X_{k+1}=D_k(X_k)+\Lambda \overline w_kF_{\mathsf S_k}, \]

so that \(\Lambda \) denotes total integrated action rather than an edit applied anew at every denoising step. This normalization is necessary for comparing samplers with different step counts. The metric and state-adaptive transport rule are then responsible for matching the desired regional response as the state changes. Calling this rule a discrete connection records its intended role. A mathematical connection would additionally require coherent transformation under changes of observer chart and compositional parallel transport; neither property is established by PB2.2.

. DILATE transports artistic intent through observer geometry. It does not assume that a paintbrush is a globally fixed vector in neural parameter space.