# Axis-Suppressed and Global-Window Leakage-Aware Optimizer
## Finite Critical-Line Annihilation, Collapse of the Arithmetic-Positive Cone, and Off-Axis Control Window Exchange Method

**English Title:** *Axis-Suppressed Global-Window Optimizer: Finite Critical-Line Annihilation, Collapse of the Arithmetic-Positive Cone, and Off-Axis Exchange Constraints*  
**Version:** v0.1  
**Date:** 2026-07-24  
**Nature:** RH Engineering Research / Floating-Point Finite-Dimensional Diagnostic Prototype  
**Status:** Not an RH proof; Not an interval certificate

---

## Abstract

The previous round quantified that the negative margin of existing local negative certificates is far smaller than the positive mass generated by the zeros on the critical line. Therefore, this round no longer solely maximizes the negativity of a single target rectangle, but establishes a joint optimization prototype that simultaneously features:

1. Finite critical-line sampling annihilation;
2. Arithmetic quadratic value lower bound;
3. Target rectangle negativity;
4. Finite off-axis control window exchange constraints;
5. Derivative energy and residual on-axis mass diagnostics.

The test space consists of 24 real-even paired-bump basis functions supported on \([-3,3]\). In addition to:

\[
G(0)=0,\qquad G(i/2)=0
\]

we further introduce:

\[
G(\gamma_j)=0,\qquad 1\le j\le q.
\]

The annihilation here refers to linear equalities within the finite basis and floating-point quadrature model, not a formalized zero certificate.

The most important result of this round is not finding a global negative window, but confirming a three-way trade-off:

\[
\boxed{
\text{Increase in on-axis annihilation}
\Longrightarrow
\text{Decrease in arithmetic-positive dimensions}
\Longrightarrow
\text{Reduction in phase degrees of freedom for target and global window}.
}
\]

In this basis, the number of arithmetic-positive eigen-directions drops from 12 at \(q=0\) to 1 at \(q=12\); it completely vanishes when \(q=15\).

---

# 1. Why the Control Window Cannot Directly Touch the Critical Line

If \(\psi\) is real and even, then for real \(x\):

\[
G(x)\in\mathbb R.
\]

Thus:

\[
B(x)=2\operatorname{Re}(G(x)^2)=2G(x)^2\ge0.
\]

If a non-zero entire function \(G\) satisfies:

\[
B(x+iy)\le0,
\]

in an off-axis region, and the closure of this region contains an open interval on the real axis, letting \(y\to0\) yields:

\[
G(x)=0
\]

holding over that interval. By the identity theorem for entire functions:

\[
G\equiv0.
\]

Therefore, a non-trivial test function cannot maintain non-positive orbital blocks in an entire neighborhood that extends all the way to the real axis. We must preserve:

\[
|\operatorname{Im}w|\ge\beta_{\min}>0.
\]

A complete approach can only study the cost as \(\beta_{\min}\downarrow0\), rather than covering the axis all at once with a single non-zero function.

---

# 2. Basis Modification

In the previous global cosine-window basis, the arithmetic-positive directions rapidly disappeared after adding a small number of on-axis annihilation conditions. This round switches to local paired-bumps:

\[
\phi_j(t)
=
b\left(\frac{t-a_j}{h}\right)
+
b\left(\frac{t+a_j}{h}\right),
\]

where:

\[
b(u)=
\begin{cases}
(1-u^2)^3,& |u|<1,\\
0,& |u|\ge1.
\end{cases}
\]

This basis is automatically real-even, compactly supported, and possesses more local degrees of freedom.

---

# 3. Finite Critical-Line Annihilation

Let:

\[
\Gamma_q=\{\gamma_1,\ldots,\gamma_q\}.
\]

Add:

\[
G(\gamma_j)=0,\qquad \gamma_j\in\Gamma_q.
\]

The direct modulus-squared contributions of the finite on-axis prefix are thus annihilated by the model. However, each condition reduces the dimension of the constrained space and alters the inertia index of the arithmetic matrix.

---

# 4. Collapse of Arithmetic-Positive Directions

Scan results:

| \(q\) | Constrained Dimension | Arithmetic-Positive Directions | Max Arithmetic Eigenvalue |
|---:|---:|---:|---:|
| 0 | 22 | 12 | 5.89381875 |
| 4 | 18 | 8 | 0.42111899 |
| 8 | 14 | 4 | 0.00290303 |
| 10 | 12 | 2 | 0.00203168 |
| 12 | 10 | 1 | 0.00119356 |
| 15 | 7 | 0 | -0.00014260 |

Thus, within this basis, support, and discretization:

\[
q=15
\Longrightarrow
\text{No arithmetic-positive directions exist in the constrained subspace}.
\]

This is not a general impossibility theorem, but it proves that "annihilating more on-axis zeros" is not a free operation.

---

# 5. Target Rectangle

The target rectangle is:

\[
K=[20,20.5]+i[-0.2,-0.1].
\]

For each \(q\), solve the finite minimax problem:

\[
\min_c\max_{w\in K_{\rm grid}}B_w(c),
\]

subject to:

\[
c^\top C_0c=1,
\]

\[
c^\top M_{\rm arith}c\ge\delta_q,
\]

and all annihilation conditions.

At \(q=0\), the target negative margin is approximately \(1.95\times10^{-5}\). After adding two to four on-axis annihilation conditions, the negative margin drops to about \(6\times10^{-7}\). Under more conditions, the common negative cone gradually approaches degeneracy.

---

# 6. Global-Window Exchange Method

The control window is set to:

\[
W=[10,60]+i[-0.45,-0.05].
\]

First, solve on sparse control points:

\[
\min_c\max_{w\in W_{\rm active}}B_w(c),
\]

while requiring the target rectangle to remain negative and the arithmetic quadratic value to remain positive. Then, find the worst-case points on a denser grid, add them to the active constraint set, and resolve.

This is a floating-point exchange method prototype for a semi-infinite quadratically constrained problem.

---

# 7. Selected Candidate at \(q=12\)

The maximum linear condition residual of the candidate in the finite model is:

\[
1.7333621857e-16.
\]

Normalization:

\[
c^\top C_0c
\approx 1.
\]

Arithmetic value:

\[
Q_{\rm arith}
\approx 5.00000000001e-05>0.
\]

Dense target grid:

\[
\max_{w\in K}B(w)
\approx -2.64607989612e-08<0.
\]

Among the first fifty on-axis samples, after excluding the first twelve annihilated points:

\[
\sum_{n=13}^{50}|G(\gamma_n)|^2
\approx 0.000154365729672.
\]

This quantity is still approximately:

\[
5833.75
\]

times the target negative margin.

---

# 8. Failure of Global-Window Non-Positivization

On the high-density finite control window grid:

\[
\max_{w\in W\setminus K}B(w)
\approx 0.267543612562>0.
\]

The maximum positive peak in the control window is approximately:

\[
1.011e+07
\]

times the target negative margin.

Therefore, in the current finite model, it is possible to simultaneously achieve:

\[
\text{Finite on-axis annihilation}
+
\text{Target negativity}
+
\text{Arithmetic positivity},
\]

but after adding:

\[
B(w)\le0
\qquad
\forall w\in W
\]

no feasible solution has yet been found.

---

# 9. Geometric Interpretation

What is truly being studied now is the intersection of four sets:

\[
\mathcal C_{\rm target-},
\qquad
\mathcal C_{\rm axis0},
\qquad
\mathcal C_{\rm arith+},
\qquad
\mathcal C_{\rm window-}.
\]

Finite-dimensional results indicate:

- An intersection exists among the first three;
- No intersection is found after adding the fourth;
- As \(q\) increases, the dimension of \(\mathcal C_{\rm arith+}\) collapses rapidly.

---

# 10. Next Milestone

Moving forward, we should not merely increase penalty weights, but rather change the certificate structure.

## 10.1 Support and Dimension Scanning

Investigate:

\[
R=3,4,5,6
\]

and higher basis dimensions to confirm whether the collapse of the positive cone dimension is merely a low-dimensional phenomenon.

## 10.2 Banded Certificates

Partition the off-axis band into shallow, medium, and deep regions, assigning different test functions to each, rather than requiring a single function to control the entire critical band.

## 10.3 Certificate Covering Families

Search for finite or countable families of test functions:

\[
\{G_\alpha\}_{\alpha\in A},
\]

such that every possible off-axis orbit is negatively captured by at least one member, while the arithmetic side can provide an additive positivity certificate for the entire family.

---

# 11. Valid Conclusions

This round validly confirms:

\[
\boxed{
\text{A finite number of critical-line samples can be annihilated in the same finite model,}
\text{while still preserving target negativity and arithmetic positivity.}
}
\]

It also confirms:

\[
\boxed{
\text{Annihilation conditions rapidly compress the arithmetic-positive directions;}
\text{global off-axis window non-positivization for a single function fails in the current basis.}
}
\]

Therefore, the next phase should shift from a "single global test function" to:

\[
\boxed{
\text{A banded, multi-certificate, covering-based zero-barrier system.}
}
\]