We consider retarded non-Hermitian systems of the form ∂t u (t) = L0 u (t) + M u (t − τ), τ > 0, whose Markovian realization lives on the history space Xτ = C (−τ, 0; H). The instantaneous spatial overlap of left and right eigenvectors does not quantify spectral degeneracy of the retarded semigroup. Under an analytic Fredholm framework for the delayed characteristic pencil, we show that the correct biorthogonal denominator is the Hale bilinear pairing between right and left history eigenmodes, and prove the exact identity hτ (Ψλ, Φλ) = ℓ^♭_λ∂λΔ (λ) rλ, with Δ (λ) = λI − L0 − M e^ (−λτ), for simple isolated characteristic roots of the analytic Fredholm pencil. This Hale–Keldysh identity identifies retarded spectral degeneracy with the derivative geometry of the delayed characteristic pencil. At a defective root the denominator vanishes by derivative self-orthogonality. For a generic EP3 unfolding we prove that hτ (Ψμ, Φμ) ∼ κτ μ^ (2/3), and therefore the Hale–Petermann factor satisfies KH (μ) ∼ Cτ |μ|^ (−4/3). Retarded feedback preserves the universal EP3 exponent while renormalizing the prefactor through the reduced unfolding coefficient of the delayed pencil. The norm of the rank-one spectral projector, given by the Keldysh residue formula for the retarded generator resolvent, diverges as |μ|^ (−2/3). We also present an exactly solvable delayed EP3 normal form—the Lambert-W EP3 tower—in which the coefficient Cτ ∝ |1 + τλEP|^ (−2) is computed in closed form on each retarded sheet of the exact scalar-reduced normal form.
CHAO MA (Wed,) studied this question.