Chapter 8 shows the imaginary unit is a logically independent scalar. Insight into the mechanisms of indeterminacy reduces to understanding how and where Quantum Mathematics drives the necessity for this number's presence in the theory. This chapter eliminates fundamental symmetry as a source; and shows that (complex) unitarity arises logically independently, due to complementarity.
Examination of the homogeneity of space shows that the Canonical Commutation Relation cannot be derived purely from homogeneity alone, but is a composite requiring extra information. That extra information is an imaginary unit, logically independent of the homogeneity symmetry, which arises from the demand that position and momentum spaces be complementary, as well as mutually consistent. The significance is that this single example contradicts the textbook doctine that symmetries underlying quantum systems are ontologically unitary, and instead, unitarity originates partly in the complementarity of mixed states.