SO(k)-Invariant Gauge Classification and the ℓ¹(E; ℓ²) Obstruction Geometry

This preprint classifies the obstruction geometry selected by SO(k)-invariant local gauges on finite graph cochains with vector-valued edge defects. Within the M-compatible gauge framework, SO(k) invariance and the standard norm axioms force each local gauge on R^k to be proportional to the Euclidean norm. When this local geometry is combined with replica-extensive, edge-decomposable aggregation, the resulting global diagnostic is the mixed norm ℓ¹(E;ℓ²), with edge weights retained unless edge-exchange symmetry or explicit normalization is imposed. The paper proves the associated primal-dual obstruction formula: the quotient obstruction Φ₂,₁ is the infimum of the ℓ¹(E;ℓ²) residual modulo exact repairs, and its dual witnesses are Euclidean-unit-bounded divergence-free vector circulations. It also presents a finite triangle example separating scalar-coordinate ℓ¹ observers from SO(k)-invariant vector observers: the same cycle class receives obstruction magnitude 2 under coordinate-separable ℓ¹ geometry and √2 under Euclidean vector geometry. The result is observer-relative and conditional. It does not claim universal Euclidean physics or universal ℓ¹ geometry. Rather, it identifies the precise two-stage selection mechanism: SO(k) symmetry determines the local ℓ² gauge, while replica-extensive edge aggregation determines the outer ℓ¹ sum.