V164_4 — A Finite‑Window Flux Reduction for the de Bruijn–Newman Closure Problem

Description: This paper refines the finite‑window flux program for the de Bruijn–Newman deformation. It does not claim an unconditional proof of the Riemann Hypothesis (RH). Instead, it provides a patched conditional reduction framework, combining both convex affine‑tail tests and localized signed probes to handle boundary‑layer obstructions.

🔹 Exact Finite‑Window Balance

• For a finite consecutive set of zeros and a smooth test function, the balance identity splits into:

  1. Interior repulsion term — nonnegative for convex tests.

  2. Exterior boundary flux term — the genuine obstacle.

• This algebra is exact and deterministic, requiring no limiting argument.

🔹 Endpoint Stability via Hurwitz

• Removes endpoint stability as an assumption.

• If Ht→H0 locally uniformly and Ht has only real zeros for all t>0, then Hurwitz’s theorem implies H0 also has only real zeros.

• This ensures RH once A≤0 is established.

🔹 Paired Cancellation vs Boundary Layer

• Paired Laguerre–Polya cancellation controls far exterior fields on fixed compact sets.

• However, globally convex tests cannot have compactly supported derivatives.

• Therefore, convex affine‑tail tests require a weighted boundary‑layer estimate near the moving edge of the window.

🔹 Localized Coercive Probes

• An alternative route uses localized signed probes with compactly supported derivatives.

• These avoid the moving boundary layer but sacrifice global convexity.

• A localized coercive defect domination principle is introduced, requiring cutoff‑error control.

🔹 Coercive Threshold Defect

• A positive threshold A>0 must produce a nonnegative defect.

• This defect must be dominated by residual boundary flux, far‑field error, weighted escape tails, and (in the localized route) cutoff errors.

• If those residuals vanish, the defect is impossible, forcing A≤0.

🔹 Conditional Closure Theorem

Theorem 9.1: If the following hold:

1. Finite‑window balance identity on regular intervals.

2. Exterior flux vanishing for localized probes, plus weighted boundary‑layer control for convex affine‑tail tests.

3. Far‑field errors, weighted no‑escape tails, and localized cutoff errors vanish.

4. Finite‑window statistics converge to infinite‑window limits.

5. Coercive threshold defect estimate holds (either convex or localized route).

Then A ≤ 0, hence RH follows by Hurwitz endpoint stability.

🔹 Conclusion

V164_5 patches the finite‑window flux program by:

• Allowing two complementary proof routes: convex affine‑tail tests (requiring boundary‑layer estimates) or localized signed probes (requiring cutoff‑error domination).

• Removing endpoint stability as an assumption via Hurwitz.

• Replacing informal threshold contradictions with a precise defect domination principle.

The remaining analytic tasks are:

1. Prove weighted boundary‑layer estimates for convex affine‑tail tests, or replace with localized coercive probes plus cutoff‑error control.

2. Prove coercive threshold defect estimates showing that positive thresholds create defects dominated by residual terms.

👉 Key message: V164_5 reframes RH through a patched finite‑window flux reduction lens, showing that persistent obstructions must manifest as boundary flux or threshold defects, and under explicit analytic hypotheses, these obstructions vanish, yielding A = 0.