CPTP-Invariant Obstruction Theory Trace-Norm Geometry and Quantum Dual Witnesses

This preprint develops the CPTP-compatible quantum observer instance of finite obstruction calculus. For finite directed networks whose edge defects are Hermitian trace-zero operator differences, the paper studies obstruction diagnostics whose local gauge is the operational trace distance and whose global aggregation is replica-extensive across edges. The main result is conditional: CPTP monotonicity alone does not uniquely force the trace norm. Instead, trace distance is selected only after adding further operational assumptions such as unitary invariance, tensor stability, exact reduction to classical total variation on diagonal trace-zero operators, and binary distinguishability structure. Under this declared observer class, the global obstruction geometry becomes l1 (E; trace norm). The paper defines the corresponding quantum quotient obstruction Phiₜr as the minimum trace-distance residual modulo exact vertex repairs, and derives its dual witness formulation. The dual witnesses are divergence-free observable fields bounded in operator norm. With the convention ||X||ₜr = 1/2 ||X||₁, the dual bound is ||Mₑ||ₒp <= 1/2; using the unhalved trace norm gives the bound ||Mₑ||ₒp <= 1. The result is observer-relative and finite-dimensional. It does not claim a derivation of quantum mechanics, a universal quantum metric, or trace-norm uniqueness from CPTP monotonicity alone. It identifies the trace-distance obstruction geometry for one explicitly declared quantum distinguishability observer class.