D-034 — Task (★)a: literature map for the Wiener–Hopf / resolvent-convergence attack on (★)

Date: 2026-08-18 | Session: 006 cont.8 | Task: (★) analytic attack Status: complete. Sources recorded per §4A.3.

1. Direct hits: none — the (★) framing is new territory

• arXiv sweep "Wiener–Hopf + Riemann": 26 hits, ALL Riemann–Hilbert / Riemann-surface / gravitational — NO paper connects Wiener–Hopf factorization to RH, ζ-zeros, or explicit formulas. The WH reading of Cor 1.6 (D-026) appears to be genuinely uncharted as a framing.
• Screw-function corpus (all Suzuki): 2206.03682 (JLMS 2023 — screw function of ζ, RH-equivalences, ĝ identity [13]); 2209.04658 (screw line under RH, ℋ(A∞) ≅ de Branges space [14]); 2606.09096 (the Cor 1.6 paper — our R-003 source); 2409.00888 (M-functions/ Goldbach); 2308.11860 (survey, no new results).

2. The machinery that EXISTS and fits the wall

• Krein–Langer [paper ref 8] (IEOT 78, 2014): screw-function theory — continuation of hermitian positive definite functions; g is a screw function ⟺ RH (Suzuki [13] Thm 1.2). The representing measure of the screw function is the spectral object behind ĝ.
• Finite-interval convolution asymptotics (Krein 1958/62; Gohberg–Feldman, "Convolution Equations and Projection Methods", 1974): the finite-interval resolvent of g∗v = h on [−a,a] converges as a → ∞ to the half-line Wiener–Hopf resolvent PROVIDED the symbol admits the appropriate factorization. This is the classical engine behind the v±(a) → v±(∞) question (original Q-S3a) — and the factorization of ĝ(z) = z⁻²ξ′(½−iz)/ξ(½−iz) is RH-gated (poles of ĝ at z = i(½−ρ) — all real ⟺ RH; D-035 §2).
• Strong resolvent convergence of self-adjoint extensions (classical Krein resolvent formalism; e.g. Gorbachuk–Gorbachuk): norm-convergence of deficiency elements ⟹ strong resolvent convergence of the extensions D_{a,θ(a)} → D. This is the exact theorem-class the v±-convergence question belongs to.
• de Branges spaces / indefinite (de Branges–Krein) spaces (de Branges 1968; Krein–Langer): ℋ(A∞) ≅ de Branges space under RH [14]; without RH, Q_W is indefinite — negative index = off-line pairs (Bombieri [Bom00]) — the right frame is a Krein-space de Branges space.
• Connes 1999 trace formula (and Connes–Consani–Moscovici 2511.22755 — zeta spectral triples, the sister v̂a → ξ conjecture (1.2)): the scaling-action/Weil-positivity framing of the same positivity; NOT the WH framing but the same gate.

3. What does NOT exist (the gap)

• No WH-factorization analysis of ĝ = z⁻²ξ′/ξ anywhere.
• No unconditional (Krein-space) version of Suzuki's [14] isomorphism.
• No convergence theorem for D_{a,θ} as a → ∞ in ANY metric (Suzuki's §7.5 expectation is stated under RH only, as a heuristic).
• No known example of a strong-resolvent limit of real-spectrum operators being used to FORCE real spectrum in the limit (the phenomenon needed for (★)) — in Krein spaces, spectrum can leave ℝ in pairs; controlling that = the wall (D-035 §3).