Vanishing of the diffuse phase-cancellation deficit

Determine whether, for a finite real signed measure on projective space whose reciprocal field is absolutely integrable almost surely and whose tangent projections are conditionally standard Cauchy on almost every oriented two-plane, the phase-cancellation deficit must vanish; equivalently, determine whether a nonzero diffuse negative part can remain dominated by the positive part in almost every phase pushforward.

Background

The paper studies signed reciprocal measures on projective space and their induced tangent fields. Under reciprocal integrability and a conditional standard-Cauchy law on almost every oriented two-plane, each planar phase pushforward is forced to be a positive probability measure. However, the authors establish only that the phase pushforwards can hide some negative mass through cancellation when the phase map is not injective.

The phase deficit is defined as the difference between the total variation of the signed measure and the essential supremum of the total variation of its planar phase pushforwards. The paper proves that this deficit equals twice the total negative mass and gives sufficient conditions—such as phase norming or suitable separation of opposite-sign carriers—under which it vanishes. It remains unresolved whether reciprocal integrability together with compatibility of the conditional laws across all planes is sufficient to force vanishing in general, particularly for diffuse negative parts.