# Zero-Side Leakage Budget: Scale Diagnostics from Local Negative Certificates to Global Zero Sums

**Version:** v0.1  
**Date:** 2026-07-24  
**Status:** Engineering diagnostic prototype; not an RH proof

## 1. The Problem

The previous version strictly proved via the same explicit test function:

\[
\sup_{w\in K}2\Re(G(w)^2)
\le-2.2416560599e-06,
\]

and that the arithmetic quadratic value is strictly positive.

However, the complete zero side is:

\[
Q_{\rm zero}
=
Q_{\rm target}
+
Q_{\rm axis}
+
Q_{\rm finite,off}
+
Q_{\rm tail}.
\]

A local negative value can only imply a negative complete zero sum if its absolute value dominates the other three terms.

## 2. The First Critical Line Zero Exceeds the Target Negative Margin

Numerically evaluating the first critical line zero using the same piecewise linear test function:

\[
\gamma_1\approx14.1347251417,
\]

yields:

\[
|G(\gamma_1)|^2
\approx 0.00535215750176.
\]

Meanwhile, the strict negative margin of the target rectangle is only:

\[
\eta_K=2.2416560599e-06.
\]

The ratio is:

\[
\frac{|G(\gamma_1)|^2}{\eta_K}
\approx 2387.591.
\]

That is, the scale of the positive contribution from the first on-axis zero is already about three orders of magnitude larger than the local negative margin.

## 3. The First Fifty On-Axis Zeros

The numerical cumulative mass of the first fifty zeros is:

\[
\sum_{n=1}^{50}|G(\gamma_n)|^2
\approx 0.0237237823405.
\]

This is:

\[
10583.150
\]

times the target negative margin.

These numerical zero values are used solely for diagnostics and do not serve as formal zero verification data.

## 4. High-Altitude Tail

Since \(\psi\) is a compactly supported piecewise linear function and \(\psi'\) has bounded total variation, we have:

\[
|G(t)|
\le
\frac{\operatorname{TV}(\psi')}{t^2}.
\]

For this function, we obtain:

\[
\operatorname{TV}(\psi')
\approx 877.527968242.
\]

Combined with an explicit zero-counting shell budget, the prototype yields a conservative tail budget for altitudes above 60:

\[
Q_{\rm tail}^{\rm abs}
\lesssim 8.66760062477.
\]

This tail bound is more conservative than the actual value and has not yet been upgraded to a full formal certificate.

## 5. Unknown Off-Axis Zeros in the Finite Window

For \(|\Im w|\le1/2\), we use:

\[
|G(w)|
\le
\int|\psi(t)|e^{|t|/2}dt
\approx 5.44903286959.
\]

Therefore, the worst-case absolute value for each unknown off-axis conjugate block can be roughly bounded by:

\[
2|G(w)|^2
\le 59.3839184277.
\]

Currently, the target negative margin can only tolerate:

\[
3.77485373019e-08
\]

such unknown orbits of this worst-case scale, which is far less than one.

## 6. Conclusion

This round is not a failure, but rather quantifies the next necessary condition:

\[
\boxed{
\text{The next optimization cannot solely maximize the negativity of the target rectangle;}
\text{it must simultaneously suppress the critical line spectral mass.}
}
\]

It is recommended to reformulate the objective function as:

\[
\min_c\left[
\sup_{w\in K}B_w(c)
+
\lambda
\int_{\mathbb R}|G_c(t)|^2\,d\mu_{\rm axis}^+(t)
+
\mu\,\mathcal T(c)
\right],
\]

where:

- The first term shapes the target negative block;
- The second term penalizes critical line leakage;
- The third term controls the high-altitude tail.

A stronger approach is to directly require:

\[
B_w(c)\le0
\]

to hold throughout the entire finite off-axis candidate window, rather than being negative only on a single small rectangle.

## 7. Legitimate Conclusion

Currently, it can be legitimately stated that:

\[
\boxed{
\text{The existing strict intersection function proves that local negativity is compatible with arithmetic positivity,}
\text{but its on-axis leakage is far greater than the local negative margin.}
}
\]

Therefore, the research node should shift from "whether the intersection exists" to:

\[
\boxed{
\text{Joint optimization with on-axis suppression and full-window sign constraints.}
}
\]