Towards a Complete Kerr Soft-Hair Theory: Integrability, Counterterm Equivalence,and Quantum State Counting

Abstract

The horizon symmetry construction of Haco, Hawking, Perry, and Strominger yield two Virasoro central charges 12J after a specified Wald-Zoupas correction. Turning that calculation into a black hole microstate theory requires separate control of nonlinear charge integrability, the allowed counter-term class, and quantum state multiplicities. We formulate exact local and global integrability tests, prove uniqueness of the counter-term coefficients within a one-parameter ansatz, and identify the quotient in which a physically meaningful uniqueness theorem must be stated. A coadjoint-orbit model supplies globally defined classical moment maps, while unitary highest-weight modules supply an explicit algebraic quantum realization for macroscopic positive angular momentum, neither construction is proved equivalent to the gravitational phase space. A character calculation shows why a single Virasoro representation, despite having the desired central charge, does not provide the required large-central-charge entropy. We derive a curvature improvement invisible to the constant mode, quantize a regulated area-boost corner polarization, and find a continuous kinematic spectrum with divergent Gibbs trace. Explicit local modular CFT spectra with identical central charges yield different finite-temperature entropies, demonstrating that the available symmetry and quantum consistency data do not select the microscopic theory. We give a conditional sufficiency theorem and a concrete sequence of calculations needed to establish a gravitational completion.