V158_4 — Microscopic Shielding Obstructions in the de Bruijn–Newman Flow and Their Relation to Weil–Connes Positivity

Description: This paper develops a conditional local obstruction analysis for non‑real zeros in the de Bruijn–Newman flow, connecting the dynamics of conjugate pairs to the Weil–Connes positivity framework. It emphasizes explicit assumptions—zero‑field regularization and the existence of stopped simple non‑real branches—since these are nontrivial for the actual Riemann E‑function.

🔹 Conjugate Pair Attraction

• For a conjugate pair z=x±iy, vertical dynamics satisfy

• The −1/y term is attractive, pulling the pair toward the real axis.

• This contrasts with the repulsive intuition for nearby real zeros.

🔹 Occupation Bounds

• On intervals where shielding is uniformly absent (yℜ(Ez)≤1−c):

  • Time spent within separation ∣z−zˉ∣<R is bounded by O(R2).

  • Logarithmic pair‑energy occupation is also bounded.

• These sharp estimates show collapse occurs quickly without shielding.

🔹 Energy Balance Identity

• On general stopped intervals, monotonicity of y2 may fail.

• The paper introduces an exact energy balance formula:

• Long survival forces a large integrated positive residual field, meaning shielding must be strong.

🔹 Pointwise Shielding Alternatives

Strong shielding at a fixed time implies one of three explicit alternatives:

1. Close upper‑zero clustering → four‑zero real‑symmetric cluster, producing a corrected jet small‑ball event.

2. Far‑field/background shielding → residual contributions from distant zeros or analytic background.

3. Distributed upper‑zero shielding → density in dyadic annuli, producing large derivative‑normalized Jensen excess.

🔹 Corrected Jet Small‑Balls

• Proper normalization ensures derivative scales are respected.

• Clusters of zeros yield corrected jet small‑ball obstructions, especially in four‑zero symmetric configurations.

🔹 Conditional No‑Long‑Shielding Reduction

• If residual‑field blowup, far‑field shielding, close clusters, and Jensen‑doubling excess have negligible occupation, then a near‑real conjugate pair cannot survive longer than its natural y2 scale.

• This is not an unconditional theorem but a conditional obstruction reduction.

🔹 Relation to Weil–Connes Positivity

• The Weil–Connes framework interprets RH as a positivity condition in the explicit formula.

• This paper does not prove positivity or RH.

• Instead, it shows how any failure of RH must manifest dynamically:

  • Long‑lived near‑real non‑real pairs → residual shielding → explicit obstruction (cluster or Jensen excess).

• Thus, microscopic shielding obstructions are structurally compatible with the Weil–Connes picture.

🔹 Conclusion

V158_4 isolates the microscopic shielding obstructions that govern near‑real non‑real zeros in the de Bruijn–Newman flow.

• Near‑real pairs collapse in time O(y2) unless shielded.

• Persistent shielding forces explicit obstruction channels: far‑field/background, clusters, or Jensen excess.

This provides a precise, reviewer‑checkable mechanism linking local dynamics with the global Weil–Connes positivity program.