Establish an inverse map from KMS crossing to the thermal Polyakov hierarchy

Establish a general inverse of the thermal KMS residual map that recovers the thermal Polyakov hierarchy from KMS crossing data, including the subtraction-sector ambiguity.

Background

The paper relates the thermal Polyakov equations to the conventional KMS crossing bootstrap through a residual analytic function expanded in classical double-twist blocks. Vanishing of the Polyakov rows implies both OPE matching and KMS covariance, but the converse is obstructed because KMS crossing has an invariant kernel, exemplified by the bosonic constant block.

The authors explain that obtaining the converse requires a dispersion prescription that reproduces the relevant discontinuities and pole terms, satisfies growth bounds, fixes homogeneous terms, and justifies spectral interchanges. They establish this only under restrictive reconstruction assumptions and leave a general inverse construction unresolved.