D-022 — Thread T4: the ξ′-analogue — why is the method stronger on ξ′?

Date: 2026-08-17 | Session: 004 | Fork 4 of 4 (§8.4a) Status: complete — mechanism identified and numerically reproduced (E-013); portability verdict: NOT portable to ξ. Thread T4 CLOSED; T4b opened.

Question

The paper's method certifies 2/3 (simple+on-line) for ξ but 0.85838/0.86864 for ξ′ (Remark 7.1). D-013 item 4 + G-001 salvage asked: WHY — does the extra derivative break the s₂/p substitution symmetry, and is the structure portable back to ξ?

1. The s₂/p symmetry does NOT break (correction to G-001 salvage)

The G-001 substitution argument is measurement-side: bandwidth-one 2-moment data see zero locations/multiplicities only, and an on-line double ≡ an off-line pair at depth δ→0 in every such datum — regardless of WHICH entire function the zeros belong to. Applied to ξ′-zero configurations the argument is unchanged. The strength gain therefore cannot come from the adversary side; it must come from the data side.

2. The actual mechanism (from zeta-23-lean Zeta23/XiPrime)

The ξ′ certificate uses the coefficient family b_N = C(N; L_T), C(N;Λ⋆) = −Λ(N) + Σ_{j<Ω N} Λ⋆^{−(j+1)} lamLogConv_j(N), L_T = l/2 + iπ/4, l = log(T/2π). Leading structure: C ≈ −Λ + (Λ·log)/L_T, with Re(1/L_T) ~ 2/l. On the diagonal (primes p ~ e^{sl}, so Λ(p) = log p = sl): C(p) ≈ −sl + (2/l)(sl)² = sl(2s − 1) ⟹ the diagonal second-moment law is a PERFECT SQUARE with a node: D₁(s) = s − 4s² + 4s³ − … ≈ s(1 − 2s)² (higher terms fill the node to D₁(½)/½ ≈ 0.089). Interpretation: the derivative contributes the (Λ·log) partner whose diagonal ANTI-CANCELS the −Λ diagonal at prime scale p ~ e^{l/2} = √(T/2π). The pair kernel of the R-functional (E-001: R = [∫ψ² + ∬K(|u−v|)ψψ]/(∫ψ)² with K = s for ξ) becomes K = D₁.

3. Quantitative confirmation (E-013, all checks pass)

R with kernel D₁ (series, c_k = 2·4^{k+1}k!/(2k+2)!):

• flat window: 2 − R = 0.85838496 vs certified 0.85838 (5e-6 above)
• quartic window: 2 − R = 0.86864152 vs certified 0.86864 (2e-6 above)
• bookkeeping identity: pair integral 1/3 → 0.14162 equals the certificate gain 0.85838 − 2/3 exactly (V3c).
• sanity: same code with kernel s gives R = 4/3 (V4). So Remark 7.1's constants are reproduced from first principles, and the entire gain over 2/3 is the pair-integral reduction caused by the node.

4. Portability verdict: NOT portable to ξ

The cancellation requires the (Λ·log) partner, which IS the derivative (log-derivative differentiation). Injecting a log-weight into the ξ certificate = differentiating the kernel = computing a ξ′ moment — one certifies ξ′ zeros, not ξ zeros. There is no ξ-side coefficient family with a tunable anti-correlating partner (b_N = Λ(N) is forced). Hence: the ξ′ gain is real but function-specific; Track A's ξ-walls (W1–W6, D-013) are unaffected.

5. Corollaries and open threads

• ξ′-world is nearly saturated: 0.86864 simple+on-line vs Wu's 0.86957 on-line-only [Wu15] — headroom ~ 1e-3. Not a promising push target.
• T4b (OPEN): higher derivatives ξ⁽ᵏ⁾. The node mechanism should iterate: C^{(k)} involves (−Λ + c₁(Λ·log)/l + c₂(Λ·log²)/l² + …) — more cancellation partners, plausibly more nodes in the diagonal law, stronger constants. Costs: zeros of ξ⁽ᵏ⁾ are less classical (no Levinson–Montgomery pairing control for k ≥ 2?), and any ξ⁽ᵏ⁾ certificate certifies ξ⁽ᵏ⁾ zeros only. Worth one scoping pass if a Track A result past the paper is ever needed for its own sake.
• Also open: the ξ′ bandwidth-one CEILING (LawN256 analogue for kernel D₁) — is 0.86864 near the ξ′-world optimum? Cheap finite-program computation, mirror of T3.

6. Adversarial review

• D-H/Epstein gate: E-013 is a reproduction of a proved result, not a new certificate — gate N/A. Note the D₁ node relies on the Λ-vs-Λ·log diagonal correlation (prime-specific arithmetic input); for Davenport–Heilbronn-type analogues the same machinery would need its own diagonal law — no shortcut to RH claimed.
• The "perfect square" is the leading three terms; higher-order terms fill the node to ≈ 0.089 (measured). Mechanism claims are stated at leading order; constants themselves are exact series computations.