Universal Phase Structure of Reheating-Imprinted Axion Initial Data

We develop a universal phase-diagram framework for reheating-imprinted axion and axion-like initial data. A broad class of analytic reheating maps factorizes as θ_eff − θ_i = A F(u), with a microphysical prefactor A and a universal saturation function F(u) = 1 − e^(−u) depending only on the coarse-grained reheating history u = κΔN. Two formulation-independent critical hypersurfaces, u = O(1) and |A F(u)| = θ_c, generate a four-phase taxonomy that separates history-sensitive from history-insensitive and subcritical from supercritical regimes. The same structure also controls contour geometry on fixed-w slices, allowing relevance and geometry to be treated as complementary projections of a single reheating map. Within a conservative common-portal baseline, the Palatini-versus-metric difference is expressed as a prior bias on the reheating plane, quantified by a Phase-II accessibility functional. The paper also presents an illustrative benchmark, a toy microphysical closure, and a boundary-layer analysis of the stiff limit w → 1. Version 2 update: this revision corrects the phase labels in Fig. 1, refines the interpretation of the diagnostic Phase II region, and weakens several statements that were previously too strong. In particular, the Palatini-versus-metric comparison is now stated explicitly in terms of illustrative reheating priors rather than as a formulation-independent general claim. The paper develops a universal phase-diagram framework for reheating-imprinted axion initial data based on the factorized map θeff−θi=A F(u),F(u)=1−e−u,u=κΔN.\theta_{\rm eff}-\theta_i = A\,F(u), \qquad F(u)=1-e^{-u}, \qquad u=\kappa\Delta N.θeff−θi=AF(u),F(u)=1−e−u,u=κΔN. Two critical hypersurfaces, u=ucu=u_cu=uc and ∣AF(u)∣=θc|AF(u)|=\theta_c∣AF(u)∣=θc, organize a four-phase taxonomy, while the order parameter SN=ueu−1S_N=\frac{u}{e^u-1}SN=eu−1u measures local sensitivity to reheating duration. This version supersedes the previous upload.