Entropy Curvature Bends Neural Network Parameter Space An Information Field Equation for Neural Networks

We derive an exact decomposition of the stem:["K=1"] information time metric stem:[G_{ij} = \nabla_i\nabla_j(dt_{\mathrm{info}}^2)] into four source terms and experimentally validate a dominant-source approximation that yields a field equation for neural network parameter spaces. In the ViT-Tiny regime studied here, the entropy curvature term dominates (87–114% of stem:[\|G\|]), giving [stem] ++++ G_{ij} \;\approx\; -2\,dt_{\mathrm{info}}^2\;\Gamma_{ij}^{\mathrm{info}}, \qquad \Gamma_{ij}^{\mathrm{info}} = \frac{\nabla_i\nabla_j H}{H} - \frac{2\,\nabla_i H\,\nabla_j H}{H^2}, ++++ where stem:[\Gamma^{\mathrm{info}}] is the _information curvature tensor_. The exact decomposition achieves stem:[\mathrm{corr}=1.000] (12/12 sub-blocks); the approximate equation achieves stem:[\mathrm{corr}=0.988 \pm 0.006] across 3 seeds on ViT-Tiny. On ViT-Tiny, the field equation yields three empirical consequences. First, stem:[\|\Gamma^{\mathrm{info}}\|] decreases monotonically with network depth (1.23 at block 0 to 0.06 at block 11, stem:[\sim 20:1] ratio), providing a geometric explanation for why deep layers are safer to fine-tune. Second, stem:[\|\Gamma^{\mathrm{info}}\|] stably predicts per-layer forgetting (Spearman stem:[\rho = 0.822 \pm 0.054]) and learning capacity (stem:[\rho = -0.807 \pm 0.046]). Third, geometry-guided per-layer learning rates outperform uniform lr on all 3 seeds for 4 of 8 tested schedules (best: stem:[\Delta = +5.8 \pm 4.2\%]; most stable: stem:[\Delta = +2.7 \pm 1.1\%]). We establish that the _near-null subspace_ of stem:[\Gamma^{\mathrm{info}}] is a reproducible geometric structure on GPT-2 (124M): across 5 seeds and 42 parameter sub-blocks per seed (210 audits total), a cross-batch alignment audit achieves stem:[\bar{A} = 0.953] (98.6% pass rate), and null-space dimension tracks parameter-type curvature class (4.8× spread, layer-norm to large MLP weights). From this geometric result the field equation yields a new prediction: gradient updates confined to the audit-verified subspace should avoid high-curvature directions and therefore reduce catastrophic forgetting. A companion study {cite:li2026auditgated} tests this prediction and confirms it (20% forgetting reduction, stem:[p < 0.05], without degrading new-task learning), establishing stem:[\Gamma^{\mathrm{info}}] as the geometric foundation for an operable CL mechanism. Cross-architecture transfer confirms the field equation on GPT-2: the exact decomposition holds (stem:[\mathrm{corr} = 1.000], 24/24 sub-blocks) and the approximation remains accurate in the interior entropy-dominated regime (stem:[\mathrm{corr} > 0.975], blocks 2–9). In edge blocks where feature curvature dominates, the field equation degrades and the predictive sign of stem:[\|\Gamma\|] reverses relative to ViT-Tiny—a boundary-regime limitation addressed in §6.