Unified Collapse Geometry of Reduced Observation Maps: Dual-Limit Structure Across Reheating Reconstruction and Dynamical-Decoupling Spectroscopy

This paper establishes a unified reduced‑geometry framework that connects two previously independent inverse‑problem programs: multi‑ratio dynamical‑decoupling (DD) spectroscopy and reheating reconstruction in early‑universe cosmology. Although these systems arise in entirely different physical contexts, both are governed by the same weakest‑direction hierarchy of a reduced observation map F: Θ→D. The central result is the Dual‑Limit Correspondence, which shows that the internal stratification of failure modes—weak singular direction, recoverable near‑null, observable horizon, supporting‑structure failure (pre‑cusp stress / lever degeneracy), and exact rank collapse—is controlled in both domains by the angular diversity of the rows of the leading‑order Jacobian J (1). The fragility locus α\* in DD and the fold‑normal covector ni in reheating are identified as structurally equivalent distinguished directions along which the approach to the dominant limit is accelerated. Likewise, the lever‑spread principle of DD spectroscopy and the normal‑covector spacing that governs branch‑interface displacement in reheating are shown to be two realizations of a single geometric design rule: maximize the angular diversity of J (1) within the accessible parameter range. By placing the results of Papers Q4–Q7 and Papers 13–16 into a common language, this work unifies the design‑side and fragility‑side conclusions of both series, revealing a shared mathematical structure underlying information limits across quantum‑sensing spectral reconstruction and early‑universe reheating inference. V2: Refined the unified Jacobian-level correspondence between reheating reconstruction and DD spectroscopy. Clarified the observable-subspace Fisher structure, introduced the scalarized pre-cusp coefficient QnQₙQn, strengthened the Level-3 structural analogy interpretation, and improved the dual-limit correspondence figures and asymptotic explanations.