D-035 — Task (★)b: attack design on the tail-cancellation — resolvent-convergence route and the three-layer wall

Date: 2026-08-18 | Session: 006 cont.8 | Task: (★) analytic attack Status: complete (design). Route chosen: strong resolvent convergence in the (indefinite) limit space; achievable intermediate deliverables ranked; wall mapped in three layers.

1. Restating (★) in its three equivalent forms (D-026, R-003)

(a) Remainder form: R_τ = −2e^{a+iθ/2}sin(az−θ/2) + e^{−ϕ(a,z)} + o(1) on compacts. (b) Function form: e^{ϕ(a,z)}M(a,θ;z) → 1 on compacts, M = W/F zero-free in the limit (Hurwitz ⟺ γ-locking + outflow of W's zeros, D-023/D-028). (c) Spectral form: D_{a,θ(a)} → D (Hilbert–Pólya operator) in strong resolvent sense with REAL spectrum in the limit (Suzuki §7.5 made unconditional).

2. The symbol factorization, concretely

ĝ(z) = z⁻²ξ′(½−iz)/ξ(½−iz) = z⁻²·(log ξ)′(½−iz): poles at z = i(½−ρ) — ALL REAL ⟺ RH. A Wiener–Hopf factorization ĝ = ĝ₊·ĝ₋ assigns poles/zeros to half-plane factors; the placement of the ζ-zero poles determines the factorization INDICES. RH ⟺ canonical factorization with all poles on the contour. This is the precise sense in which (★) is RH-gated: the finite-interval resolvent convergence theorems (Krein, Gohberg–Feldman) require the factorization, and the factorization is the zeros. [NOTE: this is not circular for a CONDITIONAL theorem — "if the factorization has form X, then (★) follows" is a legitimate implication; it IS circular as an RH proof.]

3. The resolvent-convergence route (the attack)

Step 1 (unconditional frame): ℋ(A∞) without RH is a KREIN space — Q_W indefinite, negative index = number of off-line pairs (Bombieri). The minimal operator D_a and its extensions D_{a,θ} live in ℋ(T_a) (Hilbert, since T_a > 0 for λ < λ_a). The limit operator D = the translation generator (Suzuki §7.5) exists in the Krein sense. Step 2 (the convergence theorem to prove): v±(a,·) → v±(∞,·) in the ℋ(T)-norm of the limit space ⟹ (Krein resolvent formalism) D_{a,θ(a)} → D in strong resolvent sense. THIS is the precise unconditional form of the original Q-S3a question — a concrete, attackable theorem [DELIVERABLE D1]. Step 3 (the paywall): strong resolvent convergence in a Krein space does NOT preserve real spectrum — pairs of eigenvalues can leave ℝ exactly where the metric is indefinite. Proving no pair leaves ℝ = proving (★) = Cor 1.6 ⟹ RH. The last step is the wall, not the route.

4. The three-layer wall

• W-1 (index): the negative index of Q_W (= #off-line pairs) is UNKNOWN — possibly infinite. The Krein frame needs finite index for the clean theory; this is an unprovable-today input.
• W-2 (persistence): real-spectrum persistence under Krein-space strong resolvent limits fails in general — no theorem of the needed form exists (D-034 §3). This layer IS RH.
• W-3 (circularity for proofs): the factorization of ĝ needs the zero locations; usable only for conditional statements.

5. Achievable deliverables (ranked)

• D1 (analysis, criterion-3-shaped): the unconditional Krein-space frame + the exact deficiency-convergence statement ⟺ strong resolvent convergence; proves "Cor 1.6 ⟺ the Krein-space operator D has real spectrum" — a genuine reduction sharpening R-003. Effort: paper-work; Krein–Langer + resolvent-formalism literature in hand.
• D2 (numerics, E-021/E-022 candidate): spectral-measure convergence: atomic measures of D_{a,π} (atoms at W-zeros, weights 1/‖v_z‖²_T from T_a v_z = e^{−izx} — computable with existing Galerkin machinery) vs the target atomic measure at γ's; measures the resolvent convergence rate directly. Novel observable.
• D3 (estimate): transition-scale lower bound: from E-019 (lattice stiffening to a=12) + the skeleton model, estimate the a-scale at which any (★)-consistent theory must begin the consolidation — quantifies D-023's "a ≫ 10".

6. Why (★) looks 99.8% done but is hard (the honest reading)

The off-axis cancellation is nearly tautological: M = W/F with W and F both entire of exponential type 2a, and the tail τ is FORCED to absorb R₀'s skeleton off the real axis by the entirety of W (the sine skeleton is the only candidate growth term). The remaining ~0.5% envelope ON ℝ carries the entire zero-consolidation problem (form (b)). So E-017/E-018's "closeness" is structural, not asymptotic — the measured trend (F5: slow INCREASE of the leftover with a) is the honest signal: nothing is converging yet.

7. Verdict

(★) = RH-equivalent difficulty (W-2 layer). The resolvent route converts Cor 1.6 into a sharp unconditional operator statement with a named paywall — pursue D1 as the next Suzuki-line paper-work package; D2 as the supporting numerics; do not attempt W-2 head-on.