Phase Textures and Isocurvature from Folded Access to a Dynamic Reheating Boundary

This paper extends the dynamic reheating-boundary framework to spatially modulated reheating histories and studies how spatially varying access to a moving Phase-IV window imprints axion initial data. The central result is that, in the branched inverse sector, the fold singularity of the reduced reheating map is promoted to a codimension-one interface in physical space. Spatial modulation then changes not only the magnitude of local reduced observables but the selected inverse branch itself. The paper first formulates a spatial lift of the reduced boundary observables by promoting the effective reheating data to local fields A(u,x)A(u,\mathbf{x})A(u,x) and ω(u,x)\omega(u,\mathbf{x})ω(u,x). This yields local onset, weighted residence, and signed phase-displacement fields, which act as local access variables for the moving phase-sensitive reheating regime. It then shows that, in the branched sector, rank-deficient loci of the reduced inverse map become branch interfaces in physical space. The minimal generic structure is a fold interface, while cusp-type configurations provide the minimal endpoint or merger geometry. The signed phase displacement Δθ\Delta\thetaΔθ is identified as the cleanest local branch diagnostic, and its jump across the interface provides the leading branch-sensitive source term. The paper further connects these local phase data to isocurvature structure. In smooth sectors, spatially modulated reheating produces continuous phase fields and correspondingly smooth isocurvature patterns. In branched sectors, by contrast, the isocurvature source becomes piecewise defined and inherits nonanalyticity and interface-supported contributions from the singular inverse geometry. In this sense, the resulting phase textures are not merely smooth perturbative ripples induced by an inhomogeneous background, but spatial realizations of inverse-branch selection along a dynamic reheating boundary. Finally, the paper argues that metric and Palatini gravity acquire a new interpretation in this spatial setting. Because the two formulations can induce different local effective reheating histories even under a common Jordan-frame seed, they can shift branch loci and thereby act as texture-selecting formulations, generating different phase-texture morphologies and isocurvature patterns. Overall, the work establishes a direct link between dynamic reheating geometry, inverse singularity structure, and spatial axion imprints. V2: Version 2 clarifies the spatial fold-interface structure and its physical interpretation.We make explicit the local definition of the fold-control field and introduce a consistent branch-selection rule, ensuring that the phase field remains continuous but nonanalytic across the interface. Figure annotations and captions are revised to match the theoretical structure, including a clear representation of the fold displacement and the characteristic normal-gradient scaling near the interface. Minor corrections improve notation consistency and overall presentation without changing the main results.