Complex Universal Phase Structure of Reheating-Imprinted Axion Initial Data: Spiral Saturation, Dual Diagnostics, and the Revival of Phase IV

This repository contains the source files for a paper on the complex completion of the universal reheating phase diagram for axion initial data. The work extends the scalar saturation framework to the full complex effective-data space, shows that saturation corresponds to spiral approach to a fixed point, and demonstrates that the previously saturated Phase IV remains phase-diagnostic even when amplitude sensitivity is suppressed. The paper formulates a refined universal phase structure and introduces a phase-based inverse problem sensitive to the ratio m_phi / H_reh. V2: This version resolves notation-level inconsistencies in the complex reheating-imprint framework. The axion mass mam_ama and inflaton mass mϕm_\phimϕ are now clearly separated throughout, including the definitions of the complex effective-data variable ZeffZ_{\rm eff}Zeff, the phase parameter ω\omegaω, and the reheating-rate estimate in Section 6.3. The discussion of spiral winding has been made mathematically precise by distinguishing accumulated phase ΩΔNreh\Omega \Delta N_{\rm reh}ΩΔNreh from the winding number ΩΔNreh/(2π)\Omega \Delta N_{\rm reh}/(2\pi)ΩΔNreh/(2π). The complex prefactor and scalar prefactor are also notationally disentangled, improving readability and eliminating ambiguity in the relation between the complex and scalar envelope descriptions. In addition, the wording around Phase IV has been refined so that the scalar envelope classification is preserved while the extended complex framework reveals an additional phase-diagnostic axis. V3: Refined the complex-response framework and Figure 1 for consistency with the asymptotic phase-shift structure. The presentation now cleanly distinguishes envelope decay from relational phase information, clarifies the coarse-grained scope of the ω\omegaω interpretation, and strengthens the statement that Phase IV retains a residual phase-encoded structure without altering the original envelope-level taxonomy.