Gravity: A Physics-Inspired O(N) Framework with O(1) Streaming State Across Dimensions

We introduce Gravity, a neural network architecture that replaces quadratic self-attention with physics-derived field equations on lattice structures of any dimensionality. Each input element is mapped to K field parameters, compressed to a scalar density via a learnable nonlinear bottleneck, and propagated through multi-scale exponential kernel recurrence in O(N) time. Attention scores are computed only within a fixed-size local window, eliminating any N × N matrix. The same core mechanism—density bottleneck, multi-scale field propagation, local window interaction, and physics feature extraction—extends from 1D sequences to 2D grids, 3D volumes, and spatio-temporal video tensors by varying only the number of directional parallel scans. This design trades content-addressable retrieval for bounded-state inference. We characterize the tradeoff through data scaling from 100M to 1B tokens: the cross-entropy loss gap between Gravity and Transformer grows from 3.9% to 9.6% but decelerates sharply (+4.6pp then +1.1pp), consistent with convergence toward a stabilizing cost near ~10%. For target operating regimes, the practical gains are substantial: 92× memory reduction at 32K context, 7.5× throughput at 16K, native multi-dimensional processing across four data regimes (sequence, image, volume, spatio-temporal video), and bounded streaming state (O(1) per layer, 672 bytes total state across 1M characters with CV=0.14%). Source code, reference implementations, and reproduction notebooks are available at https://github.com/chiaweil/gravity-nn Provisional patent application filed March 2026 (U.S. Provisional Patent Application). Commercial licensing inquiries: chiangjw90@gmail.com