When Obstruction Is Not Scalar - Vector and Quantum Observer Geometries

Scalar obstruction is not universal. In finite obstruction calculus, a residual class may be fixed by the declared carrier and repair map while its measured magnitude depends on the observer gauge used to measure local defects. This paper develops two non-scalar observer geometries for finite cochain obstruction systems. The first is a Euclidean vector gauge, L1(E;L2), where residuals are additive over edges but measured by Euclidean magnitude inside each vector-valued edge stalk. A three-edge cycle shows the basic split: the same residual class has magnitude 2 under coordinate-channel sensing, but magnitude sqrt(2) under Euclidean vector-displacement sensing. The second is a quantum trace-distance gauge, L1(E; trace norm), where operator-valued edge defects are measured by one-half trace norm. The paper gives finite-dimensional dual-witness formulations, closure bounds, and CPTP monotonicity results for the chosen trace-distance observer gauge. It explicitly does not claim that CPTP monotonicity selects trace distance; trace distance is treated as an admissible gauge unless an additional operational distinguishability axiom is supplied. The central point is observer-relative measurement discipline: the quotient class is structural, but the reported scalar magnitude is fixed by the declared observer geometry. The paper is finite-dimensional and finite-graph throughout. It makes no physical, dynamical, Born-rule, or quantum-substrate claim.