In Chapter 2, Quantum Mathematics, I used the quantum free particle to demonstrate how mixed states are unavoidably unitary, while pure states need not be. The job this chapter does is demonstrate how an uncaused, accidental self-referential mechanism facilitates a logical step-transition, from those non-unitary pure states to the unitary mixed states. And how, in this self-referential system, complementarity is an inherent consequence of the self-consistency which is necessary. Effectively, the procedure I work through is a formal logical approach to the derivation of the Fourier Integral Theorem.
The quantum free particle is used to demonstrate the crucial logical significance of the quantum mechanical imaginary unit; and how a step-transition from pure states to mixed is represented by formulae involving function spaces that are non-trivial evaluations of themselves; facilitated through an uncaused, unprevented, accidental self-referential mechanism; with a logically independent imaginary unit being the resulting fixed point. Effectively, the procedure worked through is a formal logical approach to the derivation of the Fourier Integral Theorem.