D-018 — Q-3.4c scoping: Suzuki's finite-interval operators (what/why/how)

Date: 2026-08-17 | Session: 002 | Task: 3.4c scoping | Status: scoped; implementation planned Source: arXiv:2606.09096v1 (HTML full text, 2026-08-17); logged D-001 Part III.

1. The construction (extracted, precise)

Screw function of ζ (eq. 1.3), explicit: g(t) = −4(e^{t/2}+e^{−t/2}−2) + Σ_{n≤e^{|t|}} (Λ(n)/√n)(|t|−log n) − (|t|/2)(ψ(1/4) − log π) − ¼[Φ(1,2,¼) − e^{−|t|/2}Φ(e^{−2|t|},2,¼)] (ψ = digamma, Φ = Hurwitz–Lerch; prime sum = 0 for |t| < ½log 2.) Near origin (eq. 2.2): g(t) = ½|t|log|t| + A|t| + (prime sum) + r(t), r even C², r = O(t²); A = ½(log(2π) − ψ(2)) = ½(log(2π) + γ_E − 1) ≈ 0.707546. g is a Krein–Langer screw function IFF RH.

Operators on [−a,a]: (Gu)(x) = ∫g(x−y)u(y)dy on L²₀(−a,a) (mean-zero), G_a = P_a G P_a (P_a subtracts mean); D = i d/dx Dirichlet; B_a = DG_aD on H₀¹ — symmetric, not self-adjoint. Thm 1.1 [PROVED]: the Weil-form operator A_a (Q_W^a(v) = ⟨A_a v,v⟩) is the Friedrichs extension of B_a. Distributional action: convolution with −g″ (eq. 2.10). Thm 1.5 [PROVED, unconditional]: eigenvalues of each self-adjoint extension 𝒟̄_{a,θ} are exactly the zeros of an explicit entire function W(a,θ;z) (eq. 1.11), and all those zeros are REAL. Cor 1.6 [CONDITIONAL bridge]:* IF lim_{a→∞} e^{φ(a,z)}W(a,θ;z) = z²·ξ(½−iz)/ξ′(½−iz) uniformly on compacts, THEN RH follows. Mechanism: zeros of z²ξ(½−iz)/ξ′(½−iz) at real z ⟺ zeros of ξ on the line; Hurwitz: real-zero entire functions converge to real-zero limit.

2. What this is (project interpretation)

A Hilbert–Pólya ASSEMBLY LINE: unconditional self-adjoint operators with provably real spectra on finite intervals, whose conjectural limit's spectral function vanishes exactly at z = γ_ρ. The entire RH content is compressed into ONE convergence statement (Cor 1.6's hypothesis). This is the strongest concrete Track B entry point we have seen — superior to Berry–Keating (which has no self-adjointness theorem).

3. Implementability assessment (for our sandbox)

All of g is mpmath-computable (digamma ✔, lerchphi ✔, prime sums ✔). The quadratic form is Galerkin-friendly WITHOUT derivatives of g: Q(u,v) = ⟨G_a u′, v′⟩ — piecewise-linear hat functions (derivatives are differences) on a mesh of [−a,a]; matrix entries are double integrals of g(x−y) over element pairs — computable by quadrature (g continuous, the |t|log|t| singularity is integrable). This is the Kim et al. (2607.24830) finite-element route; a basic version is one focused sub-stage. Validation targets: bottom eigenvalues of A_a vs first zero ordinates (E-002); Kim et al. report an Archimedean spectral law to 30 digits — our independent cross-check.

4. Research questions opened (the real value)

• Q-S1 (cross-pollination with the paper): can the unconditional second-moment technology (Thm 5.7 machinery) say anything about the CONVERGENCE in Cor 1.6? The gap between finite-a and limit is where RH hides; our framework controls exactly such explicit-formula compressions. UNEXPLORED intersection.
• Q-S2 (canonical systems): W(a,θ;z) as a de Branges canonical- system object — Phase 2.4 link; the Krein–Langer screw theory is exactly the canonical-systems language (domain sweep hit, §4A.5).
• Q-S3 (the θ-family): Thm 1.5's self-adjoint extensions 𝒟̄_{a,θ} — which θ gives the zeta-aligned limit? Is there a distinguished θ?

5. Plan (next sub-stages, when user approves)

1. E-009a: implement g(t) in mpmath; validate against eq. 2.2 expansion numerically (self-test); tabulate g on a mesh (cached CSV).
2. E-009b: Galerkin discretization of Q (hat functions, ~200–500 elements); eigensolve; compare bottom spectrum vs γ_1..γ_10 at a few a values; document convergence behavior.
3. D-019: analysis — what the spectrum does, whether Q-S1 has a numerically visible signature. Effort: 2 sub-stages. Risk: moderate (quadrature care near t=0).