R-003 — Exact factorization W(a,θ;z) = F·(R₀ + R_τ); Cor 1.6 reduced to an explicit Wiener–Hopf remainder estimate

Date: 2026-08-18 | Session: 006 | Track: B (Suzuki line) | Tag: [PROVED (modulo standard tempered-FT bookkeeping; Phase-5 formalization pending)] + [NUMERICAL-VALIDATED] (skeleton & cancellation, E-017)

Statement

Let W(a,θ;z) be Suzuki's Thm 1.5 entire function, F(z) := z²ξ(½−iz)/ξ′(½−iz), and h±(x) = (e^{±x} ± i)·1_{[−a,a]}. Then, unconditionally, W(a,θ;z) = F(z)·(R₀(a,θ;z) + R_τ(a,θ;z)), with R₀ EXPLICIT: R₀(a,θ;z) = −2i[sinh(a(1+iz)) + e^{iθ}sinh(a(iz−1))] + (2i sin(az)/z)[(z−i) − e^{iθ}(z+i)] = 2e^{a+iθ/2}·sin(az − θ/2) + o(eᵃ) (|Im z| < 1), and R_τ = (z−i)τ̂₊ + e^{iθ}(z+i)τ̂₋, where τ± = (g∗v±)|_{|x|>a} is the out-of-interval tail of the first-kind equation g∗v± = h±. Consequently: Suzuki's Cor 1.6 (e^{ϕ}W → F ⟹ RH) holds iff the tail asymptotically cancels the sine skeleton up to a zero-free remainder, (★) R_τ = −2e^{a+iθ/2}sin(az − θ/2) + e^{−ϕ(a,z)} + o(1). Immediate corollaries: (i) v̂±(0) = 0 and W(a,θ;0) = 0 (≥ double) for every a, θ; (ii) the W-lattice has asymptotic skeleton {θ/(2a) + πk/a}; (iii) the thinning required by Cor 1.6 is a finite-interval Wiener–Hopf remainder property of the symbol ĝ(z) = z⁻²ξ′(½−iz)/ξ(½−iz), whose factorization is RH-gated (and must fail for Davenport–Heilbronn/Epstein — sanity gate consistent).

Proof / Derivation

D-026 §§1–2 (Fourier transform of g∗v± = h±·1_{[−a,a]} + τ± with ĝ(z) = z⁻²ξ′/ξ [Suzuki 2023, (1.2), (1.8)]; elementary closed form of ĥ± using (1±iz) = i(z∓i)). The one formality deferred to Phase 5: tempered-distribution bookkeeping for the growing kernel g ~ t log t (Paley–Wiener–Schwartz); numerically corroborated at every checkpoint.

Numerical corroboration (E-017, D-027)

• R₀ vs sine skeleton: rel dev 3.7% (a=4) → 0.009% (a=10), monotone.
• R_τ ≈ −R₀ + M with |M|/|R₀| = 0.2% (y=16.5) — 99.8% cancellation off the real axis already at a=4; arg M = ±90° exactly (explains E-014 H4 phase agreement).
• θ=π remainder 4.7% vs θ=0 6.8% at y=0.5 — quantitative θ=π preference.
• Machinery: C1 regression 3.8e-15 vs mpmath dps-50; flint ξ 8.8e-17.

Dependencies

Suzuki 2606.09096 (Thm 1.5, §8.3 first-kind equation); Suzuki 2023 (screw function, ĝ identity); D-019 (Galerkin validation); E-016c (stable evaluator); D-023 (thinning/confinement framing).

Adversarial review

• D-H/Epstein gate: the identity is symbol-driven; off-line zeros destroy the symbol's factorization, so (★) fails there — consistent.
• The tempered-FT step is the only non-formalized link; it is standard and every consequence tested matched numerics (origin double zero, lattice skeleton, growth, phases).
• Sharpened bookkeeping notes (D-054, task 4.3b): (i) τ± are carried by the UNBOUNDED set |x| ≥ a, so Paley–Wiener–Schwartz does not apply to them directly; R_τ's representation (z−i)τ̂₊ + e^{iθ}(z+i)τ̂₋ is distributional, and the exact identity W = F(R₀+R_τ) holds as meromorphic functions with R_τ := W/F − R₀. (ii) (1.12)'s "uniform on every compact K ⊂ ℂ" (Suzuki's phrasing) must be read on compacts avoiding the poles of F = z²ξ/ξ′ (zeros of ξ′); literal all-compacts reading is impossible for entire e^{ϕ}W. Both join the Phase-5 write-up queue (task 4.5d).
• Does NOT prove Cor 1.6: it reduces it to (★), which is open; E-017 shows (★) holds off-axis but not on ℝ at a ≤ 10 (as expected, D-023).

Paper section

draft.tex — new section "Exact factorization of the Suzuki W-function"; the (★) criterion becomes the project's central conjectural target.