Chiral Singular Sturm–Liouville Operators, Spectral Obstruction, and Evans-Zero Asymptotics

We prove three results on a chiral, non-self-adjoint Sturm–Liouville family H^ (m) ₌≠₀ on a singular endpoint interval. Result (i) is stated for the sign-positive chiral class: any coupling g with g′>0 and endpoint degeneration Δ (δ) =1−g (δ) ²→0, under the separated-real-part hypothesis on the endpoint Frobenius exponents Re r₁>1, Re r₂0, the linear-specialization corollary gives the forbidden-wedge estimate |Im λ| ≥ 2|m|k explicitly. Results (ii) and (iii) are stated for the linear family g (δ) =kδ, k>0 on I= (−1/k, 1/k). (ii) An exact connection matrix Mₘ (λ) = (Mᵢj^ (m) (λ) ) is defined from endpoint Frobenius bases and satisfies chiral involution identities. The entrywise real-axis nonvanishing package beyond M₂1^ (m) ≠0 is stated conditionally under explicit non-transmutation assumptions; the canonical Evans function Eₘ (λ): =M₂1^ (m) (λ) is cutoff-independent and is sharply distinguished from the cutoff-normalized transport ratios Rₘ^η, Rhatₘ^η used as numerical diagnostics. (iii) For m∈Z>₀ the leading Evans zeros are located unconditionally in branch-safe Rouché disks via the two-stage Rouché transfer of Appendix F (Gates 1–6). The result is unconditional with no uniform-matching assumption: it follows from the explicit Cramer-rule assembly of M₂1^ (m), a quantitative error budget |Eₘ|≤K/|Λ| on the Rouché strip, and a branch-safe Rouché radius r₌, ₊ that keeps every Rouché disk strictly inside the lower half Λ-plane. The m<0 case follows from the chiral conjugation symmetry of Gate 1. These results concern only the operator and Evans framework defined in this article. No identification is made between the Evans zeros or spectra below and zeros of the Riemann zeta function or any external arithmetic spectral object. Subleading depth coefficients, contour-counting data, and transport-ratio fits are separated as diagnostic appendices and are not used as proof substitutes. The separate entrywise non-transmutation assumption NT₌, ₊ is preserved but is not used for the Evans-zero asymptotic in (iii).