Twisted Convolution Conjecture

Establish that for every admissible tuple t = (d, r, Q), the set of shifts Z_t contains 0 and 1, and determine that if t = (d, r, Q) and t' = (d, r, Q') have the same discriminant, then Z_t = Z_{t'}.

Background

The Twisted Convolution Conjecture is introduced to ensure that the ghost fiducial constructed from Shintani–Faddeev cocycle values is idempotent, which the authors were unable to prove directly. It guarantees the existence of permissible "shifts" (including 0 and 1) and invariance of the shift set under change of form with the same discriminant. Combined with the Stark-type conjectures, it implies the existence of live fiducials and yields SICs.