Morphogen-Controlled Topology-Preserving Sparse Growth

Integrate the module into a spatial CNN, graph neural network, or Transformer linear layer represented by a sparse bipartite graph. Maintain a binary mask $n_r$ for candidate edges and a scalar field $c_r$ on edges or nodes. Every $K$ optimizer steps, compute edge utility $s_r$ from the exponential moving average of $|w_r\,\partial L/\partial w_r|$, diffuse it with $c\leftarrow c+(\Delta t/\tau_c)(s-c+D_c\Delta_{graph}c)$, and estimate nonnegative directional responses using $\bar\partial_e^+c_r=\max(c_{r+e}-c_r,0)$ and $\bar\partial_e^-c_r=\max(c_r-c_{r+e},0)$. For each minibatch of proposals, sample additions with probability $1-\exp(-\Delta t\nu_r^+)$ and removals with probability $1-\exp(-\Delta t\nu_r^-)$. Before accepting a removal, temporarily delete the edge and run a cheap local connectivity or simple-point test; set $f_r=0$ if an output unit loses all inputs, an input loses all outputs, or the graph's monitored connected-component count changes. Keep the parameter budget fixed by pairing accepted additions and removals. The paper supplies the birth-death construction and topology-preserving local gating; the utility source, diffusion discretization, and graph topology test are engineering adaptations. First run a CIFAR-10 experiment with a 90%-sparse 3-layer CNN, comparing magnitude pruning, RigL-style rewiring, and this method at equal FLOPs. Sweep $\rho=\bar\nu^+/\bar\nu^-$. Measure graph radius of gyration versus active-edge count, validation accuracy, and connected components. The predicted signature is a compact-to-branched transition as $\rho D_c$ increases, while the topology-preserving regime keeps component count constant and yields $R_g\propto N^\alpha$ with $\alpha$ approaching approximately $0.67$ for sufficiently large clusters. A failure of this exponent, absence of a reproducible transition, or a large accuracy loss at fixed FLOPs falsifies the transfer.