Swallowtail Onset in Reduced Reheating Geometry: Multi-Cusp Precursors and A4 Normal-Form Candidates

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.