# Rigorous Intersection Certificate v0.2: From Floating-Point Candidates to Continuous Regions and Arithmetic Positive Intervals

**英文名稱：** RH Validated Separation–Positivity Intersection Certificate v0.2  
**日期：** 2026-07-24  
**性質：** Riemann Hypothesis Research Project / Interval Validation / Explicit Formula Test Function  
**狀態：** Reproducible validated-numerics certificate; NOT a proof of RH

---

## I. What Was Fixed in This Version

Version 0.1 had found a single set of floating-point coefficients such that:

\[
2\operatorname{Re}(G(w)^2)<0
\]

holds on a dense rectangular grid, while simultaneously:

\[
c^TM_{\mathrm{arith}}c>0.
\]

However, this still left two loopholes:

1. Negative values on a grid do not imply that the continuous rectangular region is strictly negative everywhere;
2. A positive floating-point value cannot be directly regarded as a rigorous arithmetic lower bound.

Version 0.2 no longer uses black-box arrays; instead, it reconstructs the candidate as an explicit piecewise linear function, and then establishes interval certificates for both signs separately.

---

## II. Explicit Test Function

Let:

\[
h=0.01,
\qquad
 t_i=-3+ih,
\qquad
0\le i\le600,
\]

and define the hat function:

\[
\phi_h(t)=\max\left(1-\frac{|t|}{h},0\right).
\]

The test function is:

\[
\psi(t)=\sum_{i=0}^{600}y_i\phi_h(t-t_i).
\]

The base node values are stored in:

```text
 data/base_nodes.csv
```

The central node is not artificially rounded and forced to zero, but is redefined by:

\[
y_{300}
=b_{300}-\frac{\sum_i b_iq_i}{q_{300}}
\]

where:

\[
q_i=\int\phi_h(t-t_i)e^{-t/2}dt.
\]

Thus, in the mathematical model:

\[
G(i/2)=0.
\]

The actual interval residual is:

\[
G(i/2)
\in[-6.52,6.52]\times10^{-36}.
\]

---

## III. Fourier Transform No Longer Relies on Numerical Integration

The Fourier transform of the hat function is:

\[
\Phi_h(w)
=h\left(\frac{\sin(wh/2)}{wh/2}\right)^2.
\]

Therefore:

\[
G(w)=
\Phi_h(w)
\sum_{i=0}^{600}y_ie^{iwt_i}.
\]

This is a finite exponential polynomial multiplied by a known entire function. The program can directly compute using complex intervals:

\[
G(w_0),
\qquad
G'(w_0).
\]

---

## IV. Continuous Rectangular Negative Certificate

The target rectangle is:

\[
K=[8,8.5]+i[-0.2,-0.1].
\]

For each sub-rectangle, using the center \(w_0\) and radius \(r\):

\[
G(w)=G(w_0)+G'(w_0)(w-w_0)+R_2,
\]

\[
|R_2|
\le\frac12M_2r^2.
\]

From the integral representation, we obtain:

\[
M_2
\ge\sup_{w\in K}|G''(w)|,
\]

In this instance, we use:

\[
M_2\le11.4820525926.
\]

If all \(G(w)\) in a sub-rectangle fall within a complex disk of center \(c\) and radius \(\varepsilon\), then:

\[
\operatorname{Re}(G(w)^2)
\le
\operatorname{Re}(c^2)
+2|c|\varepsilon
+\varepsilon^2.
\]

Rectangles that cannot be immediately determined are further subdivided into four pieces.

Final results:

- Certified sub-rectangles: 480;
- Undecided sub-rectangles: 0;
- Maximum subdivision depth: 3;
- Global upper bound:

\[
\boxed{
\sup_{w\in K}
2\operatorname{Re}(G(w)^2)
\le-2.2416560599\times10^{-6}
}.
\]

This is now a statement over a continuous region, not a grid-based conjecture.

---

## V. Finite Cubic Spline Representation of Autocorrelation

Let:

\[
r_k=\sum_i y_iy_{i+k}.
\]

Then:

\[
C(x)=\int\psi(t)\psi(t-x)dt
=\sum_k r_kK_h(x-kh),
\]

where \(K_h\) is the cubic B-spline kernel supported on \([-2h,2h]\).

Therefore:

- No FFT is required;
- No linearly interpolated correlation matrix is required;
- The interval for each \(C(x)\) can be evaluated using at most four cubic polynomial terms.

---

## VI. Finite Place Terms

Since the support radius is 3, the convolution support scale is 6, activating only:

\[
m\log p<6
\]

prime powers, totaling 98.

Each sample point uses:

\[
m\,\operatorname{iv.log}(p)
\]

as an interval parameter, rather than treating the double-precision \(\log p\) as an exact number.

Yielding:

\[
Q_{\mathrm{fin}}
\in
[-0.099762166120387,
 -0.099762166120386].
\]

---

## VII. Archimedean Term

Adopting the compactly supported time-domain formula fixed in the previous version:

\[
Q_\infty
=-(\log(4\pi)+\gamma)C(0)
-\int_0^6
\frac{e^{x/2}C(x)-C(0)}{\sinh x}dx
-\log(\tanh3)C(0).
\]

### Near Zero

In \([0,0.01]\):

\[
C(x)=C(0)+a_2x^2+a_3x^3,
\]

so after rewriting the removable singularity, interval subdivision is performed directly.

### Remaining Regions

Using the interval midpoint values, and from:

\[
|C(x)|\le C(0),
\]

\[
|C'(x)|
\le\|\psi\|_2\|\psi'\|_2
\]

we establish an analytical derivative upper bound for each integration block, and then add the composite midpoint error.

Result:

\[
Q_\infty
\in
[0.133524840678940,
 0.161109862461679].
\]

---

## VIII. Intersection Conclusion

The total arithmetic value is:

\[
\boxed{
Q_{\mathrm{arith}}(\psi)
\in
[0.033762674558557,
 0.061347696341296]
}.
\]

Thus, the same explicit test function simultaneously satisfies:

\[
\sup_{w\in K}2\operatorname{Re}(G(w)^2)<0,
\]

and:

\[
Q_{\mathrm{arith}}(\psi)>0.
\]

This proves that in the current model:

\[
\boxed{
\mathcal G_{\mathrm{sep}}
\cap
\mathcal G_{\mathrm{arith}+}
\ne\varnothing
}
\]

there is at least one reproducible, interval-validated concrete instance.

---

## IX. Why This Is Still Not a Proof of RH

These two signs are not directly contradictory.

Hypothetical zeros within the off-axis rectangle would yield negative blocks, but the complete zero side also includes:

\[
Q_{\mathrm{axis}},
\qquad
Q_{\mathrm{finite\ off}},
\qquad
Q_{\mathrm{tail}}.
\]

They might compensate for the negative values of the target rectangle. Version 0.2 has not yet proven:

\[
Q_{\mathrm{rest}}^{\mathrm{zero}}
<
- Q_K^{\mathrm{zero}}.
\]

Therefore, the current achievement is:

> The separation requirement and the arithmetic positivity requirement are not only compatible in floating-point optimization, but there also exists a common test function whose continuous region and arithmetic value are both interval-validated.

The next truly difficult bottleneck has shifted to:

\[
\boxed{
\text{Control of non-target zero leakage and on-axis positive mass within a finite window}
}.
\]