The Choice of Time Direction and the Stage for Amplitudes: the Marking Theorem, Derived Z4 Phases, Inversion Holonomy, and the Canonical Counting Theorem (Dual Geometry of Wavelength and Frequency Spaces, Paper 14)

Paper 14 of the series Dual Geometry of Wavelength Space and Frequency Space. Two questions remained after Paper 13: why does a particular axis look like time, and where do amplitudes (complex numbers, phases) come from? Both are theorematized within the axiom inventory. Choosing a time direction is choosing a complex structure C_u inside the quaternions; the polar decomposition yields clock and phase simultaneously (Theorem 1), with the proven range-unlimited Proposition 1' joining the norm rung to odd Gaussian norms via the factor (1+i). The marking theorem (Theorem 2): exactly 16 lattice-compatible quaternion structures, a single transitive B4 orbit with 24-element stabilizers; the real axis runs over all four axes; the shared first-order canonical readings are exactly the norm (R) and the central character (Q). Transport phases are derived and always land on Z4 (Theorem 3, zero exceptions at s=9/11/13). The forced one bit of the channel-consistent aggregate is identified as the Z2 chain holonomy of canonical inversion, not the arrow (Theorem 4, with explicit scope: the diagnostic is the channel-consistent sum, adjudicated in Paper 16). The canonical counting theorem (Theorem 5): identical-particle symmetrization is forced by set structure, granularity-robust. Appendix A collects general transport-algebra lemmas (the single-axis branch-flip +/-i lemma over 103,168 flips with zero exceptions; single-sign support; charge-like parity; the mirror-flip lemma). This paper does not select an aggregation rule (measure) for path amplitudes. Reviewed in two rounds under a two-party independent verification protocol. Japanese and English full texts. No physical identification is made.