D-031 — Task 1.7 (T4b scoping): higher-derivative node mechanism — iterated formally, CLOSED structurally

Date: 2026-08-18 | Session: 006 cont.6 | Task: 1.7 Status: complete (scope-only, per task definition). Verdict: CLOSE with rationale. Revival condition noted.

1. The coefficient family (from Zeta23/XiPrime/Coeff/Basic.lean)

lamLogConv j := (Λ·log) ∗ Λ^{∗j}; xiCoeff(Λ⋆, N) = −Λ(N) + Σ_{j<ΩN} Λ⋆^{−(j+1)}·lamLogConv j N. At a PRIME p (Ω=1, j=0 only; Λ^{∗0} = convolution identity): C(p; Λ⋆) = Λ(p)·(log p/Λ⋆ − 1) — a DEGREE-1 polynomial in log p. The node of D₁ comes from its root at log p = Re Λ⋆ = l/2 (prime scale p ~ √(T/2π), D-022).

2. Mechanism iteration (formal)

A ξ⁽ᵏ⁾ certificate family would pair −Λ with partners through a degree-k prime polynomial Π_{i=1..k}(log p/Λ⋆_i − 1) (L-series B/Π(Λ⋆_i − A) generalizing Coeff/LSeries's B/(Λ⋆ − A)), giving a diagonal law with k nodes: D^{(k)}(s) ≈ s·Π_i(2s/c_i − 1)². Leading-order pair integrals (flat window, computed): s(1−2s)² : 0.06667 (= 1/15; full series fills ×2.124 → 0.14162 ✓ matches E-013 exactly) s(1−2s)⁴ : 0.02857 → with the same fill: ≈ 0.0607 → flat certificate ≈ 0.939 s(1−3s)²(2−3s)² (split nodes): 0.09048 (worse — co-located node wins) So the mechanism ITERATES: a ξ″ certificate plausibly reaches ~0.94 flat-window (leading-order estimate [HEURISTIC]; real value needs the full family development + H1–H3 hyps per derivative).

3. Why it cannot prove RH (structural wall)

Speiser/Levinson–Montgomery: RH ⟺ all zeros of ξ⁽ᵏ⁾ lie on the critical line (Laguerre–Pólya closure; RH ⟹ all on line by Rolle). So a 100% ξ⁽ᵏ⁾-on-line certificate WOULD prove RH. BUT: the certificate sees only 2-moment data of the ξ⁽ᵏ⁾ zeros, and the G-001 substitution symmetry (depth→0 off-line pairs ≡ on-line doubles in all 2-moment data) applies to EVERY ξ⁽ᵏ⁾ world unchanged — the adversary-side ceiling of each such data class is strictly below 100% (exactly as the ξ-world's 0.682 and the ξ′-world's analogue). The route is capped below 100% by the same identifiability wall; the last mile is unreachable in this data class, at every derivative order.

4. Verdict: CLOSE T4b

• (i) Certificates certify ξ⁽ᵏ⁾ zeros only — no improvement to the ξ (2/3) or ξ′ (0.86864, window-saturated per E-020b) records.
• (ii) The 100% threshold needed for RH (§3) is unreachable in the 2-moment data class at every k (G-001 symmetry).
• (iii) Cost: full coefficient-family development per derivative (Reexpansion machinery, DiagLaw H3, window certification, Lean).
• (iv) Levinson–Montgomery heuristics: the on-line fraction of ξ⁽ᵏ⁾ zeros grows with k anyway — a ~0.94 ξ″ constant would not be a surprising new theorem. REVIVAL CONDITION: if a route opens that uses ξ⁽ᵏ⁾ on-line fractions as INPUT to a ξ-side argument (none known), or if the (★)/Suzuki line ever needs a stronger derivative-side certificate, T4b reopens.

5. Honesty notes

• The 0.939 figure is a leading-order × fill-ratio estimate, not a certificate; quoted only to size the forgone gain.
• The Lean structure (degree-1 prime polynomial per family) was read directly from Zeta23/XiPrime/Coeff/Basic.lean + LSeries.lean (D-031 §1) — the degree-k generalization is a formal sketch, not a proved construction.