This paper defines a secondary diagnostic functional, Ξ, for an ℓ1-primary finite obstruction framework. The construction begins with the quotient obstruction magnitude Φ1, inherited from the finite cochain complex C0(P)→C1(P), and introduces a conditional ℓ1-to-density lift using a mass-preserving, sign-preserving, phase-free square-root embedding. The paper then selects canonical primal and dual representatives through deterministic tie-breaking and defines Ξ as the trace-norm distance between the density of the selected primal residual h1∗ and the density of the optimal dual witness ϕ∗. The central distinction is that Φ1 remains the irreducible quotient obstruction magnitude, while Ξ is only a secondary primal-dual representative-distortion measure. Ξ is used to break ties when Φ1 and higher-priority diagnostics are unchanged; it is not an independent acceptance criterion and cannot override the primary obstruction hierarchy. The paper also clarifies the structural correspondence between the ℓ1 quotient-duality pattern and trace-distance duality in quantum information, while explicitly marking operator-level trace-distance identity, POVM correspondence, and Born-rule derivation as open or conditional rather than proven.
JEREMY H. CARROLL (Fri,) studied this question.