D-051 — L3/L4 assembly: the symbol-class question for ĝ, and the correct frame

Date: 2026-08-19 | Session: 008 | Task: 2.9 remainder (L3/L4 of D-041) Status: complete (verdict + correct frame identified).

1. The question (from D-041 §4)

L3: show ĝ(z) = z⁻²ξ′(½−iz)/ξ(½−iz) belongs to a symbol class for which Wiener–Hopf factorization theory gives finite-interval asymptotics. L4: assemble the Krein 1958/62 / Gohberg–Feldman finite-interval resolvent convergence for that class.

2. Verdict: the classical classes do NOT apply directly

The Gohberg–Feldman finite-section/finite-interval theory is developed for symbols in function classes on the contour: Wiener algebra, piecewise-continuous, or (for the strongest asymptotics) symbols with a canonical factorization in those classes. Our ĝ is MEROMORPHIC WITH POLES ON ℝ (at z = ±γ under RH): it lies in none of these classes — this is exactly Suzuki's remark that "A_∞ is only a formal symbol". No smoothing trick fixes this: the poles ARE the content (they are the zeros). The classical GF route is therefore the wrong frame — documented as the answer to L3: no class inclusion of the naive kind exists.

3. The correct frame: Krein–Langer screw-function theory

The operator family is already inside the right theory: the screw function g (Krein–Langer, IEOT 78) — continuous hermitian functions with a finite number of negative squares in the continuation sense. Within that theory:

• the deficiency vectors v±(a) solve finite-interval convolution equations with the screw kernel (Suzuki §8);
• the continuation theory of hermitian-indefinite functions (Krein–Langer) plays the role of the WH factorization: the continuation of g's Fourier data off the interval IS the (★) remainder problem (D-026) in its native form;
• the relevant "factorization" is the one for the ASSOCIATED STRINGS/2×2 canonical systems (Krein–Langer), whose spectral measures are our Clark measures (E-022) — connecting to the spectral-purity criterion (D-001 sweep entry: Eichinger–Lukić– Woracek universality dichotomy). So L4's content exists but lives in the Krein–Langer continuation theorems, not in Gohberg–Feldman. The exact theorem to cite/prove: continuation asymptotics for the screw function of ζ ⟺ (★) — i.e. L4 collapses onto (★) itself, as expected from D-035's circularity analysis (W-3).

4. Status of the D1 chain after this

5. Note for the paper

Cite: Krein–Langer (screw functions / continuations), Gohberg–Feldman (contrast: classical classes), Suzuki [13][14] + 2606.09096, and the universality criterion (Eichinger–Lukić–Woracek 2409.18045) for the spectral-purity reading of the equilibrium.