Pre-Cusp Indicators and Formulation-Selective Fold-Limit Geometry in Reduced Reheating Reconstruction

This work characterizes where and how the fold normal form of reduced reheating reconstruction begins to break down, and how this breakdown differs between the Metric and Palatini formulations. Building on the fold‑singularity framework and deformation theory developed in earlier papers, we introduce a differential‑geometric description of fold arcs in reduced observable space—tangent t (s), normal n (s), curvature κ (s), and normal‑rotation rate dγ/ds. We define three pre‑cusp indicators that signal the approach to higher‑codimension onset before an actual cusp (A₃) forms: Curvature excess of the fold arc Quadratic‑recovery degradation of the reduced map Leverage‑hierarchy reorganization, measuring flattening of the discriminant‑normal weights Using these indicators, we construct a formulation‑selective pre‑cusp atlas across the (α, β) parameter plane. The Palatini formulation shows elevated curvature and flatter leverage hierarchy in the low‑α sector, while the Metric formulation exhibits stronger quadratic‑recovery degradation near the efficiency‑dominated boundary. These patterns overlap with—but do not coincide with—the fragility and switching structures identified in earlier work, revealing a structured and formulation‑dependent approach to fold‑limit breakdown. This paper closes the fold‑based analysis of reduced reheating geometry by identifying where the fold normal form loses validity and which formulation is more susceptible. It also provides the geometric substrate for a unified dual‑limit treatment of collapse and horizon structure across reheating and dynamical‑decoupling spectroscopy. V2: clarified the operational interpretation of the pre-cusp indicators (κκ, QQQ, ΛΛ), refined the fold-normal reduction and singular-vector definitions, added robustness tests for percentile thresholds and near-fold bandwidths, unified visualization conventions for log⁡10∣Q∣₁₀|Q|log10∣Q∣, and improved the formulation-selective pre-cusp atlas and discussion of leverage-ordering reorganization.