V146_1 — The Far-Far Error Term in the Natural Zeta-Coefficient Gram Form

Description:
This paper analyzes the far-far error term in the natural zeta-coefficient Gram form within the full-line rational kernel model. It proves that the absolute far-far error contribution is controlled in logarithmic low strips using only standard unit-interval zero counting and a gap from the origin.

🔹 Full-Line Rational Gram Kernel

The paper works with the sesquilinear Hermitian Gram kernel generated by Cauchy packets and a weighted full-line integral.

The leading far-far product term cancels in the natural symmetric zeta-coefficient model.

🔹 Far-Far Error Term

After close-pair control, the remaining kernel-level object is the far-far error term.

This paper isolates that error and reduces it to a weighted pair-spacing problem.

🔹 Weighted Pair-Spacing Energy

The far-far error is dominated by a pair-spacing energy over zero ordinates.

This reduction uses only elementary bounds and does not require cancellation in the error term.

🔹 Shell Decomposition

The ordinate set is decomposed into absolute-value shells.

Nearby shell pairs are uniformly controlled.

Distant shell pairs are controlled by separation between shell indices.

🔹 Logarithmic Low-Strip Control

The main estimate shows that the far-far error has at most mild logarithmic growth in general.

Inside logarithmic low strips, this growth becomes uniformly bounded.

🔹 Consequences for the Natural Zeta-Coefficient Form

Together with close-pair control and symmetric cancellation of the leading product term, the principal full-line rational-kernel blocks are controlled in logarithmic low strips.

The remaining issues lie beyond this model: finite-strip corrections, infinite zero-packet limits, regular and cross terms, and the final RH criterion.

🔹 Limitations

This paper does not prove the Riemann Hypothesis.

It does not prove a high-frequency mean-square estimate for the logarithmic derivative of the zeta function.

It does not establish a criterion connecting boundedness of the full-line rational Gram form with RH.

🔹 Conclusion
V146_1 shows that the far-far error term is controlled in the full-line rational Gram kernel model on logarithmic low strips. The result removes one more kernel-level obstruction, leaving the remaining difficulties outside the full-line rational framework.