The Geometry Behind the Barrier: Fisher Geometry and Linear Mode Connectivity in Wide Neural Networks

After permutation alignment, independently trained minima of wide neural networks often exhibit Linear Mode Connectivity (LMC), yet the geometric mechanism governing the interpolation loss barrier remains unclear. We show that Fisher information geometry explains the barrier through two decoupled axes: Shape, governing manifold curvature, and Length, governing the Fisher energy along the parameter displacement. Along the Shape axis, we show that the local variation of the Fisher metric vanishes in operator norm with width, driving Christoffel symbols and sectional curvature to zero with dimension-free bounds. However, this flattening is a local property of the metric: it neither implies that Fisher geodesics coincide with linear interpolations nor suffices for barrier collapse. Along the Length axis, we establish a Fisher-barrier inequality showing that the barrier is bounded by the Fisher quadratic form and predictive uncertainty. Experiments across architectures and training regimes show that while metric flattening occurs in every setting, geodesic-linear alignment is regime-dependent, and barrier collapse is driven by Fisher length reduction, most strongly when predictive uncertainty is controlled.