Fold Geometry of Reduced Observation Maps: Branch Interfaces and Local Normal Forms in Reheating Reconstruction

This upload contains the manuscript “Fold Geometry of Reduced Observation Maps: Branch Interfaces, Sensitivity Hierarchies, and Local Normal Forms” by Hiroyuki Shioiri. The paper develops a singularity-theoretic framework for reduced inverse geometry in observation maps of the form F:Θ→DF:\Theta \to DF:Θ→D, with emphasis on reduced reheating diagnostics. The central claim is that the transition between locally regular and locally branched inverse sectors is generically governed by fold singularities. In this setting, the singular set in parameter space projects to a codimension-one discriminant in observable space, which acts as a branch interface for the reduced inverse problem. The manuscript further interprets the local interface-displacement law μ(D)≈ni(Di−D∗i)\mu(D) \approx n_i (D^i - D_*^i)μ(D)≈ni(Di−D∗i) as the first-order defining equation of the fold discriminant near a regular point D∗D_*D∗. In this formulation, the coefficients nin_ini are the components of the discriminant normal covector, and the observed sensitivity hierarchy acquires a direct geometric meaning: it identifies which reduced observable channel most effectively displaces the branch interface. In the reheating application studied here, the reduced observables are taken to be the onset diagnostic uIVu_{\mathrm{IV}}uIV, the efficiency diagnostic EEE, and the cumulative phase diagnostic Δθ\Delta \thetaΔθ. Numerically, the paper combines three elements: singular-value cartography based on the smallest singular value of the Jacobian, observable-space projection of the near-singular set as a proxy for the discriminant, and local extraction of normal covectors along the interface. The results show coherent near-singular bands, codimension-one observable-space structure consistent with fold geometry, and a clear dominance hierarchy in which the phase-displacement channel provides the leading interface leverage over most of the computable domain. This deposit is intended to document the current manuscript version and provide a citable public record of the work. V2: Added branch-stability verification and crossing-index diagnostics for the event-selected observable extraction. The revision clarifies the image-relative fold interpretation, distinguishes operational near-fold subsets from fixed-threshold classifications, and includes robustness checks against finite-difference step size, ODE resolution, and solver variations. New appendices and figures document smooth-FD-valid filtering, crossing-index structure, and stability of the observed Δθ-dominated transverse-direction hierarchy.