R-002 — Mechanism of the ξ′-certificate strength + reproduction of Remark 7.1 constants

Date: 2026-08-17 | Session: 004 | Track: A (thread T4) | Tag: [NUMERICAL-VALIDATED] (mechanism: [PROVED] at leading order by inspection of the proved coefficient family)

Statement

The paper's ξ′ certificate (Remark 7.1: 0.85838 flat / 0.86864 quartic, simple+on-line, unconditional) exceeds the ξ certificate (2/3) because the ξ′ coefficient family C(N;L_T) = −Λ(N) + (Λ·log)(N)/L_T + O(l⁻²), Re(1/L_T) ~ 2/l, gives the diagonal second-moment law a perfect-square node: D₁(s) = s − 4s² + 4s³ − … ≈ s(1−2s)² — the −Λ and (2/l)(Λ·log) diagonals cancel at prime scale p ~ √(T/2π). The R-functional's pair integral (flat window) drops from 1/3 (ξ) to 0.14162 (ξ′), exactly the certificate gain. The s₂/p substitution symmetry (G-001) does NOT break — the gain is data-side, not adversary-side.

Proof / Derivation

D-022 (mechanism from Zeta23/XiPrime Lean sources); E-013 (all checks pass): R with kernel D₁ reproduces 0.85838496 / 0.86864152 vs the certified decimals 0.85838 / 0.86864 (5e-6 / 2e-6 above, consistent with certified truncation); bookkeeping identity pair-reduction = gain (V3c); kernel-s sanity R = 4/3 (V4); node D₁(½)/½ ≈ 0.089 (V3b).

Dependencies

E-001 (R-functional), G-001 (substitution symmetry), D-006 §4 (Remark 7.1 constants), zeta-23-lean Zeta23/XiPrime (D₁, coefficient family).

Adversarial review

• Portability to ξ: NONE — the (Λ·log) partner is the derivative itself; log-weighting the ξ certificate = computing a ξ′ moment (certifies ξ′ zeros only). Track A ξ-walls unaffected (D-013).
• ξ′-world nearly saturated (0.86864 vs Wu's 0.86957 on-line-only).
• Open threads spawned: T4b (higher derivatives ξ⁽ᵏ⁾ — mechanism should iterate, more nodes), ξ′-world bandwidth-one ceiling (LawN256 analogue).

Paper section

Obstruction-map / mechanism appendix: explains the ξ vs ξ′ gap and rules out log-weight imports into the ξ certificate.