Pressure-Controlled Neural IFS

1. Exact integration point: define $K$ neural maps $g_i(z)=c_i+\rho\,\mathrm{MLP}_i(z)$ on a two-dimensional latent state, or use the same maps as a multi-branch recurrent cell $h_{t+1}=g_{i_t}(h_t)$. Choose branch centers $c_i$ so their image regions are initially disjoint. Apply spectral normalization or explicit Jacobian scaling to keep each branch in the intended contraction or expansion regime. 2. Pseudocode: sample latent states $z$ and branch words of length $n$; initialize $A=I$ and $z_t=z$; for each branch, compute the Jacobian $Dg_{i_t}(z_t)$ with autodiff, update $A\leftarrow Dg_{i_t}(z_t)A$, and accumulate $\log\varphi^s(A)$. Estimate $\widehat P(s)$ with log-sum-exp over all words when $K^n$ is small, or with Monte Carlo importance sampling and beam search otherwise. Add the squared pressure penalty, a branch-image separation penalty, and a Jacobian-distortion penalty to the task loss. 3. Computed versus estimated: Jacobian-vector products and singular values are computed by autodiff; the infinite-time pressure, spatial supremum, and attractor dimension are estimated empirically. The paper supplies the pressure-to-dimension mechanism, while the implementation must verify separation, bounded distortion, and contraction or expansion numerically. 4. First cheap experiment: use four two-dimensional affine maps with trainable small MLP residuals, fit samples from a Sierpinski-like or anisotropic synthetic fractal, and compare ordinary training against pressure-controlled training. Sweep target dimensions $s_{\mathrm{target}}\in\{0.5,1.0,1.5\}$ and word lengths $n\in\{2,4,8,16\}$. The quantitative prediction is that the empirical pressure curve crosses zero near $s_{\mathrm{target}}$, and the measured box-counting dimension agrees with that crossing within about $0.1$ when separation holds. Removing the pressure penalty should cause systematic drift of the pressure zero and a corresponding dimension mismatch; deliberately overlapping branch images should break the prediction.