Inverse Problem for the Dynamic Reheating Boundary: Reconstruction, Deformation, and Hysteresis

This paper formulates the inverse problem associated with the dynamic reheating boundary. Building on the moving Phase-IV framework developed in the previous paper, it asks to what extent the underlying reheating history can be reconstructed from reduced dynamic boundary data. The central observation is that the forward map naturally factorizes into two stages: dynamical evolution from the effective history (AX(u),ωX(u))(\mathcal{A}_X(u), \omega_X(u))(AX(u),ωX(u)) to the complex trajectory ZX(u)\mathcal{Z}_X(u)ZX(u), and projection from that trajectory to reduced boundary observables DX(R∗)\mathbf{D}_X(R_*)DX(R∗). The essential difficulty of the inverse problem is shown to arise from loss of injectivity in the projection stage rather than from dynamical indeterminacy.

The paper develops a three-level inverse hierarchy. In Level I, corresponding to a regular four-parameter sector, the reduced boundary data define a closed inverse map and the reheating history is uniquely reconstructible within the model class. In Level II, the reduced map becomes near-singular, and reconstruction survives only up to a broadened inverse-equivalence class. In Level III, the reduced map branches, so that distinct reheating histories yield identical reduced data and the reconstruction becomes hysteretic. The onset of branching is described using a minimal fold-type normal form, while higher-codimension extensions are left for future work.

Conceptually, the dynamic reheating boundary is recast not only as a formulation discriminator but also as a reconstruction interface of limited fidelity. It is exactly invertible in regular sectors, deformed in near-singular sectors, and multivalued in branched sectors. This identifies the boundary between invertible and non-invertible reheating geometry and provides the natural bridge to spatially modulated phase textures and isocurvature structure in subsequent work.

V2:

This version improves the mathematical consistency of the inverse-reconstruction framework.

Main updates:

• Level-I reconstruction clarified with proper treatment of parameter dimensionality and multi-threshold data;
• determinant-based statements replaced by rank / singular-value language;
• trajectory solution and initial-time convention corrected;
• Heaviside-window expressions and δΘ\delta\ThetaδΘ terms explicitly treated as formal;
• fold behavior presented as a minimal model rather than a derived result.

These changes make the inverse hierarchy (unique, deformed, hysteretic) internally consistent while preserving the main conclusions.