Swallowtail Onset in Reduced Reheating Geometry: Swallowtail Precursor Index and A4 Susceptibility Atlas

This paper extends the singularity‑theory programme of reduced reheating reconstruction to the next level of the Arnol'd–Thom hierarchy. Papers 13–16 established the A₂ (fold) and A₃ (cusp) structure of the reduced observation map F: Θ→D, including fold normal forms, deformation modes, formulation‑selective fragility, and the pre‑cusp atlas. The present work asks the natural next question: where does the reduced map enter the A₄ / swallowtail regime? We introduce a swallowtail precursor index Υ (α, β) =log⁡10∣Q∣−min⁡ (log⁡10∣Q∣) ∣∇log⁡10∣Q∣∣+ε, which detects the simultaneous conditions required for A₄ onset: the quadratic‑recovery coefficient Q must be small (approaching cusp onset) and spatially flat (vanishing gradient), corresponding to a codimension‑2 degeneracy in the (α, β) plane. Applied to the 80×80 Metric and Palatini grids of Paper 16, two complementary candidate sets are identified: bottom‑5% Υ within the near‑fold zone composite A₄ atlas satisfying Q<Q25 and Υ<Υ25 The composite atlas reveals 165 high‑susceptibility points for the Metric formulation (9. 4% of the near‑fold zone) and 207 for Palatini (12. 2%). The spatial patterns differ: Metric candidates cluster in the high‑β sector, while Palatini candidates are more broadly distributed across the α axis. This pattern partially inherits—but does not replicate—the A₃ formulation selectivity of Paper 16, indicating that the transition from cusp to swallowtail reorganises the susceptibility landscape. These results provide the first precursor‑level atlas of A₄/swallowtail onset in reheating reconstruction. Together with Papers 13–16, they complete the precursor‑level account of the A₂→A₃→A₄ hierarchy. The connection to the Dual‑Limit Correspondence suggests that the DD spectroscopy side should exhibit an analogous A₄ structure (simultaneous coincidence of two fragility loci), a direction reserved for future work. All numerical data and figure‑generation scripts are included in the Zenodo deposit. V2: Swallowtail-precursor susceptibility atlas for reduced reheating reconstruction geometry. This release introduces an operational A4/swallowtail-precursor index Ψ∼ (log⁡∣Q∣−min⁡log⁡∣Q∣) (∥∇log⁡∣Q∣∥+ε), (|Q|- |Q|) \, (\| |Q|\|+), Ψ∼ (log∣Q∣−minlog∣Q∣) (∥∇log∣Q∣∥+ε), constructed from the pre-cusp indicator QQQ of Paper 16. Metric and Palatini reduced reheating maps are analysed on 80×80 near-fold grids to identify flat small-∣Q∣|Q|∣Q∣ regions suggestive of higher-order degeneracy beyond the A3 cusp regime. The deposit includes: paper PDF and LaTeX source figure-generation scripts verification / robustness scripts input NPZ datasets reproducibility statistics and candidate-atlas outputs Robustness tests include: Gaussian smoothing dependence additive vs product precursor index comparison interpolated refinement stability checks ε-sensitivity and physical-isotropy verification This work presents an operational precursor atlas rather than a full A4 normal-form proof.