Thermodynamic Diffusion Inference with Minimal Digital Conditioning

Diffusion-model inference and overdamped Langevin dynamics are formally identical. A physical substrate that encodes the score function therefore equilibrates to the correct output by thermodynamics alone, requiring no digital arithmetic during inference and potentially achieving a $10{,}000\times$ reduction in energy relative to a GPU. Two fundamental barriers have until now prevented this equivalence from being realized at production scale: non-local skip connections, which locally coupled analog substrates cannot represent, and input conditioning, in which the coupling constants carry roughly $2{,}600\times$ too little signal to anchor the system to a specific input. We resolve both obstacles. \emph{Hierarchical bilinear coupling} encodes U-Net skip connections as rank-$k$ inter-module interactions derived directly from the singular structure of the encoder and decoder Gram matrices, requiring only $O(Dk)$ physical connections instead of $O(D^2)$. A \emph{minimal digital interface} -- a 4-dimensional bottleneck encoder together with a 16-unit transfer network, totalling \textbf{2,560 parameters} -- overcomes the conditioning barrier. When evaluated on activations drawn from a trained denoising U-Net, the complete system attains a decoder cosine similarity of \textbf{0.9906} against an oracle upper bound of 1.0000, while preserving theoretical net energy savings of approximately $10^7\times$ over GPU inference. These results constitute the first demonstration of trained-weight, production-scale thermodynamic diffusion inference.