This short technical note supplements the ZPS v1 dynamical framework. It corrects an inconsistency in an earlier draft—which conflated the argument of the complex history embedding zH = x + iZΔ with a phase κZΔ —and instead constructs a separate phase-lift ζ = ReiθZ , with θZ := κZΔ defined from history depth and R := R0eλZΔ from accumulated amplification. In this lift, history generates phase and amplification generates an exact logarithmic spiral R (θZ ) = R0e(λ/κ)θZ . The result is a representation theorem: R and θZ are both defined from ZΔ , so the spiral follows by construction, and empirical relevance requires an independent interpretation of R . This is offered as a supplement, not as a standalone theory.
Sana Kamiki (Sat,) studied this question.