  We significantly advance the research program initiated in "Moment-Based
Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability
Probabilities" (J. Phys. A, 45, 095305 [2012]). A function P(alpha),
incorporating a family of six hypergeometric functions, all with argument 27/64
=(3/4)^3, is obtained here. It reproduces a series, alpha = 1/2, 1, 3/2,
2,...,32$, of sixty-four Hilbert-Schmidt generic 2 x 2 generalized separability
probabilities, advanced on the basis of systematic high-accuracy
probability-distribution-reconstruction computations, employing 7,501
determinantal moments of partially transposed 4 x 4 density matrices. For
generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional)
two-qubit, P(1) = 8/33, and (27-dimensional) two-quat(ernionic)bit systems,
P(2)= 26/323. The function P(alpha) is generated--applying the Mathematica
command FindSequenceFunction--using solely either) the thirty-two integral or
half-integral probabilities, yet successfully predicts the unused thirty-two.
