Higher-Codimension Geometry in Reduced DD Spectroscopy: A4 Precursor Asymmetry with Reheating Reconstruction

This paper presents the first A₄‑level precursor atlas on the dynamical‑decoupling (DD) side of the Dual‑Limit Correspondence. The companion reheating paper (Paper SW) showed that A₄/swallowtail precursor regions—diagnosed through the flatness of the quadratic‑recovery coefficient Q—are broadly distributed across 9–12% of the near‑fold zone. Here we ask whether an analogous structure exists in reduced DD spectroscopy. For 4‑frequency protocols ω1, ω2, ω3, ω4, we compute the floor‑resilience coefficient C (α) =σmin⁡ (J (1) ) and define the DD swallowtail precursor index YDD=log⁡10Cmin⁡−min⁡ωlog⁡10Cmin⁡∣∂αlog⁡10C∣min⁡+ε. A 50×50 protocol scan over ω∈1, 100 reveals a structural asymmetry: interior minima of C (α) —the DD‑side diagnostic for A₄ precursor geometry—occur in only 4. 1% of configurations, all confined to a narrow near‑degenerate strip ω2/ω1∈1. 1, 1. 5. Extended scans show additional interior minima only under decade‑spanning endpoint separation (ωmax⁡/ωmin⁡=104). This asymmetry is structural, not numerical. Because the lever functions Ak (α) =ωkα are log‑linear, the Jacobian rows share a constrained angular‑diversity profile that generically forces the minimum of C (α) to the boundary of the physical range. The reheating‑side coefficient Q (α, β), derived from the second derivative of the full observation map, has no such algebraic restriction and therefore admits broadly distributed A₄ precursor regions. The result is a qualified A₄‑level extension of the Dual‑Limit Correspondence: the formal structure of the precursor index is shared between the two domains, but the distribution of precursor regions is not. This identifies the first structural limitation of the correspondence and constrains any future unified theory of reduced observation maps.