Prove the viscosity repulsion bound for confined isolated bad Gram points

Prove that every isolated bad Gram point of the Hardy Z-function associated with a confined pair of zeros satisfies the viscosity lower bound μ(g_n)>4.

Background

The paper defines the viscosity of a Gram point by μ(g_n)=|Z'(g_n)/Z(g_n)| and studies isolated bad Gram points, particularly those whose neighboring zero branches remain confined to the local Gram strip. Numerical computations for the Hardy Z-function show that such points appear to satisfy a repulsion threshold near μ=4, whereas non-isolated bad points and Davenport–Heilbronn Gram points can have substantially smaller viscosity.

The threshold is interpreted dynamically: for a confined pair of zeros, sufficiently large viscosity represents transverse motion of the continued extremum that prevents the two real zero branches from colliding and leaving the real axis. Establishing μ(g_n)>4 rigorously would therefore convert the observed viscosity barrier into a mathematical repulsion law for confined isolated bad Gram points.

References

In the data considered there, no isolated bad Gram point associated with a confined pair was observed below this threshold, motivating the repulsion Conjecture.

— On the Sampling of Certain Dirichlet and Prime Sums at Gram Points and their Viscosity  (2610.21952 - Jerby, 15 Oct 2026) in Section 4, subsection “The distribution of μ(g_n^{DH}) for isolated bad DH Gram points”; the underlying numerical observation is introduced in Section 1, Remark “Confined zeros and Rosser's law.”

Understanding why such information survives in the simpler prime cloud, remains an open problem.

— On the Sampling of Certain Dirichlet and Prime Sums at Gram Points and their Viscosity  (2610.21952 - Jerby, 15 Oct 2026) in Section 5, “Summary and Concluding Remarks.”