A generalized Hartle–Hawking wave function

The Hartle-Hawking wave function is known to be the Fourier dual of Chern-Simons or Kodama state reduced mini-superspace, using an integration contour covering whole real line. But since a general solution Hamiltonian constraint (with given ordering), its should provide (i.e. beyond mini-superspace) Wheeler DeWitt equation representing in metric representation. We write down formal expression for such function, seen as generalization mini-superspace function. Its explicit evaluation (or simplification) depends only on symmetries problem, and we illustrate procedure with anisotropic Bianchi models Kantowski-Sachs model. A significant difference this approach that may leave torsion inside functions when set up ansatz connection, rather than setting it zero before quantization. This allows quantum fluctuations torsion, far reaching consequences.