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 Rᵏ 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.