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

Manoj Saravanan, Rohit Kumar Salla

Keywords: weight-space symmetries, quotient geometry, branched quotient geometry, gauge fixing, permutation symmetries, exchangeable blocks, atlas-based learning, canonicalization, invariant learning, sectional category

Abstract: 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.

Email Sharing: We authorize the sharing of all author emails with Program Chairs.

Data Release: We authorize the release of our submission and author names to the public in the event of acceptance.

Submission Number: 1