V148 — What Logarithmic Low-Strip Gram Control Can and Cannot Imply

Description:
This paper clarifies the exact zero-exclusion meaning of logarithmic low-strip Gram control. It shows that such control excludes zeros only inside a logarithmic cone, not throughout the whole critical strip. Therefore, by itself, it does not imply the Riemann Hypothesis.

🔹 Weighted Vertical Gram Energy

The paper studies a weighted vertical energy involving the logarithmic derivative of the zeta function.

If a zero lies on the integration line at nonzero height, the logarithmic derivative has a pole and the energy diverges.

🔹 Finite-Height Zero Exclusion

Finite Gram control on a vertical line excludes zeros only up to the height covered by the integral.

Thus, finite-height boundedness gives a local zero-exclusion result, not a global RH statement.

🔹 Logarithmic Cone Exclusion

For logarithmic low strips, the excluded region is a logarithmic cone to the right of the critical line.

This is meaningful, but it does not cover all possible off-critical zeros.

🔹 Why This Is Not RH

The Riemann Hypothesis requires excluding all zeros with real part greater than one half.

A logarithmic cone leaves open possible high-frequency zeros outside the controlled region.

🔹 Full-Height Gram Criterion

If full-height Gram energy were finite on every line to the right of the critical line, then RH would follow.

This is a sufficient criterion, but the paper does not prove such full-height finiteness.

🔹 Finite-Height Covering Criterion

A finite-height condition can imply RH only if the chosen height function covers all possible off-critical zero heights.

Logarithmic low strips do not provide such a covering by themselves.

🔹 Missing High-Frequency Mechanism

To turn logarithmic low-strip control into RH, one needs an additional high-frequency zero-exclusion mechanism.

No such mechanism is proved in this paper.

🔹 Relation to Rational-Kernel Estimates

Rational-kernel Gram estimates do not automatically imply boundedness of the actual analytic Gram energy.

One must still justify the passage from the zeta logarithmic derivative to the packet model, including regular terms, cross terms, and infinite-packet limits.

🔹 Conclusion
V148 gives a diagnostic boundary for the current Gram-form approach: logarithmic low-strip Gram control implies logarithmic-cone zero exclusion, but not RH. The remaining route requires full-height control, a true finite-height covering theorem, or a high-frequency propagation mechanism.