No Global Gauge in Neural Weight Space: Branched Quotient Geometry and Atlas-Optimal Learning

Neural parameter spaces with exchangeable blocks are quotient spaces under mixed continuous and discrete symmetries, but existing analyses are largely confined to regular strata where the quotient is smooth. For a positively homogeneous two-layer base model, we identify the regular quotient exactly as an unordered configuration space. For a signature-complete exchangeable-block family, we then show that every simple pairwise collision has local normal form $(u,v)\mapsto (u,[v])$; on every transverse two-dimensional slice this is the branched double cover $z\mapsto z^2$. Hence linked punctured neighborhoods admit no global continuous gauge, and any canonicalization written in branched invariant coordinates has unavoidable $\epsilon^{-1/2}$ conditioning blow-up. We further prove that the ambient relative sectional category $a(K;q)$ is the exact chart threshold for canonicalization-based invariant learning, section-valued learning on associated bundles, and symmetry-respecting memoryless optimizer proposals. Controlled experiments recover the predicted monodromy, the exact conditioning law with leading constants, and the sharp phase transition at chart count $2$. The resulting message is intrinsic: weight-space computation near singular strata should be atlas-based rather than globally gauge-fixed.