Formulation-Selective Fragility in Reduced Reheating Geometry

This work compares the Metric and Palatini formulations at the level of the full reduced Jacobian in reheating reconstruction and identifies which geometric properties of the fold discriminant are formulation‑selective and which are formulation‑robust. Using the reduced observation map FX: Θ→D, X∈M, P, we analyze three geometric layers of formulation dependence: Accessibility, measured by the singular‑value contrast Δσ (α, β) =σmin⁡ (JM) −σmin⁡ (JP), which identifies where one formulation is locally more fragile than the other. Leverage, measured by the difference of discriminant‑normal weights δwi=wi (P) −wi (M), which determines whether the dominant observational direction of the inverse problem changes under formulation switch. Switching–fragility coupling, measured by the intersection between the formulation‑switching frontier and the formulation‑specific near‑fold bands, identifying regions where formulation change and inverse fragility become geometrically entangled. At the reference slice cP=1. 3, the parameter plane is split almost evenly between Metric‑advantage and Palatini‑advantage regions (45. 4% vs. 54. 6%). The dominant leverage channel flips on 11. 6% of valid grid points, and the mean aggregate normal‑weight difference is ⟨∑i∣δwi∣⟩=0. 274. A localized coupling zone is identified where the switching frontier passes through the fold‑compatible fragility layer, showing that formulation dependence is amplified precisely where the inverse problem is most delicate. These results elevate the Metric/Palatini comparison from proxy‑level distinctions to a full Jacobian‑level comparison of reduced fold geometry. In this sense, formulation choice reshapes not only the reheating dynamics but the inverse geometry by which those dynamics can be reconstructed. V2: clarified the accessibility-sign convention (Δσ=σmin⁡ (JM) −σmin⁡ (JP) =_ (JM) -_ (JP) Δσ=σmin (JM) −σmin (JP) ), corrected the formulation-fragility interpretation, refined the operational switching-layer and near-fold definitions, improved valid-mask conditioning and robustness diagnostics, and revised the figure set for clearer reduced-geometry visualization and reproducibility.