# Separation-Positivity Intersection Solver v0.1 Technical Notes

## 1. Purpose

The previous two engineering packages found respectively:

1. Coefficients that form negative orbital blocks on an off-axis rectangle;
2. An arithmetic matrix that remains numerically positive under another set of bases and constraints.

However, it has not yet been proven that both use the same coefficient vector. This package unifies the two for the first time as:

\[
\exists c\ne0:
\quad
\max_{w\in K}c^TM_{\mathrm{orb}}(w)c<0,
\quad
c^TM_{\mathrm{arith}}c\ge\delta.
\]

## 2. Unified Coordinates

All computations share:

- Real even paired-bump basis;
- Support interval `[-R,R]`;
- The same Fourier convention;
- The same endpoint constraints;
- The same central zero experimental constraints;
- Unit normalization of the zero-delay correlation matrix `C0`.

Therefore, the output `coefficients` simultaneously determine the regional negative direction and the arithmetic quadratic form.

## 3. Joint Solution

Let `W` be the joint mapping of the constraint null space and the `C0` whitening. All candidates are represented by `c=Wy` with `||y||=1`.

Solve on the fitting grid:

\[
\min_{y,t}t
\]

subject to:

\[
y^TM_k y\le t,\qquad\forall k,
\]

\[
y^TAy\ge\delta,
\]

\[
\|y\|_2=1.
\]

This is a non-convex QCQP; v0.1 uses multi-start SLSQP, thus it can only provide candidate and counterexample exploration, and cannot guarantee a global optimum.

## 4. Normalization Cross-Validation

For the selected coefficients, the program computes simultaneously in two ways:

\[
Q_\infty,
\quad
Q_{\mathrm{fin}},
\quad
Q_{\mathrm{total}}.
\]

The first is the matrix quadratic form; the second synthesizes `psi(t)` first, then directly computes its autocorrelation and the explicit formula time-domain kernel. The difference between the two is output to:

`outputs/normalization_audit.json`

This is used to check whether the basis scale, correlation direction, and finite position coefficients are consistent.

## 5. Valid Conclusion Boundaries

`intersection_found_on_grid=true` only indicates:

- The same floating-point coefficient vector;
- The blocks are all negative on a dense finite grid;
- The arithmetic quadratic form is above the specified floating-point safety margin;
- The linear constraint residuals are very small.

It does not indicate:

- The entire continuous rectangle is strictly negative;
- The arithmetic matrix has obtained an interval PSD certificate;
- Unknown zero leakage has been controlled;
- RH has been proven.

## 6. Next Version

v0.2 should add:

1. Complex interval rectangle subdivision;
2. Interval correlation matrices;
3. Rationalized coefficients;
4. Interval `LDL^T` or strict lower bounds for candidate vectors;
5. Two-dimensional phase diagrams for rectangle height, distance from the axis, and support scale;
6. Common non-positive shaping of the entire off-axis region within a finite window.


## 7. v0.1 Actual Results

For the synthetic spectral rectangle

\[
8.0\le\operatorname{Re}w\le8.5,
\qquad
-0.2\le\operatorname{Im}w\le-0.1
\]

when scanning \(R=1.5,2.0,2.5,3.0,3.5,4.0\), all six scales found a finite grid intersection on the same coefficient vector:

\[
\max_{w\in K_{\rm check}}2\operatorname{Re}(G_c(w)^2)<0
\]

and:

\[
c^TM_{\rm arithmetic}c>\delta>0.
\]

The maximum block on the dense grid improved from \(-1.08\times10^{-8}\) to \(-5.92\times10^{-5}\). The selected \(R=3.0\) candidate has:

\[
Q_\infty\approx0.1494534760,
\qquad
Q_{\rm fin}\approx-0.1002785731,
\]

\[
Q_{\rm total}\approx0.04917490294.
\]

The total difference between the matrix computation and the recalculation directly from the synthetic test function is approximately:

\[
4.86\times10^{-16}.
\]

This indicates that the two prototypes have completed normalized integration at the floating-point level; however, the continuous rectangle interval certificate and the arithmetic interval PSD are still not completed.

## 8. Grid Sensitivity

On the selected \\(R=3.0\\), after changing the time-domain integration grid to 1601, 2401, and 3201 points, the maximum block of the dense rectangle is maintained around:

\\[
-2.30782\\times10^{-5}
\\]

while the arithmetic value lies in:

\\[
0.04917\sim0.04951.
\\]

All three times maintained intersection feasibility. This only constitutes evidence of floating-point convergence, and is still not an interval proof.