The general self-consistent spin structures employed at the "zero-entropy" level of the topological expansion in the dual topological unitarization approach to particle theory are examined in detail. Special attention is given to the problem of the phase factors to be associated with the zero-entropy terms. We consider the discontinuity equation satisfied by the zero-entropy scalar function and find that the discontinuity has a sign that varies as ${\mathrm{}(-1\mathrm{})\mathrm{}}^L$ where $L$ is the number of quark loops. This result is shown to be a consequence of the requirement that the sum of zero-entropy terms be Hermitian analytic. The effects of charge conjugation, parity, and time reversal on the zero-entropy $S$-matrix elements are derived. The two- and four-component spin formalisms are also compared. Normalization conventions for amplitudes and phase space are given explicitly.

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