Geometry as Logic: Inducing Linear Manifolds in RNNs via Vector Gravity Training

Neural networks often rely on probabilistic fitting rather than intrinsic logical structure when performing symbolic reasoning tasks such as multi-digit arithmetic. This paper proposes a novel Vector Gravity Training (VGT) framework. By introducing a second-order norm-based gravity loss in the embedding space, we force the model to map discrete symbols onto a continuous geometric manifold. Experimental results show that, under identical conditions (256-dimensional hidden layer, 40k training steps), VGT significantly enhances the computational robustness of GRU models, improving three-digit addition accuracy from 72% to 99%. Visualization analysis further reveals that VGT successfully reshapes chaotic hidden states into linearly ordered geometric trajectories, demonstrating that logical robustness stems from geometric order, not mere parameter scaling.