# RH-W-03: Compact Support Separation, Existence of Negative Witnesses, and Dual-Core Architecture
## Riemann Hypothesis GAP Engineering Note v0.1

**Research Project:** RH GAP Atlas / AI Mathematical Engineering Relay  
**Parent Node:** `RH-W-03-SEPARATION`  
**Prerequisite Nodes:** `RH-W-01`, `RH-W-02`  
**Status:** `EXISTENTIAL_SEPARATION_CLOSED / CONSTRUCTIVE_SEARCH_OPEN`  
**Date:** 2026-07-23  
**Nature:** Engineering deconstruction of existing equivalent criteria; not a proof of the RH

---

# 0. Conclusion of this Round

The core determination obtained in this round is:

$$
\boxed{
\neg RH
\Longrightarrow
\exists\,g\in C_c^\infty(0,\infty)
\quad Q_{\mathrm{full}}(g)<0
}
$$

Moreover, this negative witness has a finite logarithmic support radius: there exists some $a>0$ such that the corresponding additive coordinate test function

$$
\psi(u):=e^{u/2}g(e^u)
$$

satisfies

$$
\operatorname{supp}(\psi)\subset(-a,a).
$$

Therefore, "whether off-axis zeros can be seen by compactly supported smooth test functions" is not a new unresolved GAP; this is an existence conclusion already provided by the Weil–Bombieri–Yoshida type criteria.

What remains truly incomplete is:

1. No actual off-axis zeros are known, thus their negative witnesses cannot be directly generated;
2. The existence theorem must be transformed into a finite-dimensional, computable, and strictly verifiable negative eigenvalue certificate;
3. Whether the previous dual-vanishing-moment subspace $\mathcal C_{00}$ independently retains full separation capability still requires an independent proof.

---

# 1. Why the Need to Split into Two Cores

Previous engineering only used:

$$
\mathcal C_{00}
=
\left\{
 g\in C_c^\infty(0,\infty):
 \widehat g(0)=\widehat g(1)=0
\right\}.
$$

Its advantage is that the two endpoint terms of the explicit formula vanish exactly, but it is a codimension-two subspace of $C_c^\infty(0,\infty)$ and should not be treated as the complete Weil test space without proof.

This round officially adopts a dual-core architecture.

## 1.1 Full Separation Core

$$
\boxed{
\mathcal C_{\mathrm{full}}
:=C_c^\infty(0,\infty)
}
$$

Purposes:

- To carry the Weil criterion refined by Bombieri;
- To guarantee that if the RH is false, a compactly supported negative witness exists;
- To serve as the logical parent space for finite-support spectral scanning.

## 1.2 Endpoint-Zero Simplified Core

$$
\boxed{
\mathcal C_{00}
:=\ker \widehat{(\cdot)}(0)
\cap
\ker \widehat{(\cdot)}(1)
}
$$

And it was previously proven that:

$$
\mathcal C_{00}
=
D(D+1)C_c^\infty(0,\infty),
\qquad
D=x\frac{d}{dx}.
$$

Purposes:

- To eliminate endpoint terms;
- To make the arithmetic side and the Weil covariance differ only by a negative sign;
- To facilitate local computations by AI, CAS, and formalization systems.

However, it cannot currently be claimed that:

$$
Q(g)\ge0\quad\forall g\in\mathcal C_{00}
\Longrightarrow RH.
$$

This is a new precise sub-GAP: `RH-W-03-C00-SEPARATION`.

---

# 2. Complete Endpoint Correction Formula

Fix the Mellin transform:

$$
\widehat g(s)
=
\int_0^\infty g(x)x^s\frac{dx}{x}.
$$

Fix the Hermitian correlation form:

$$
C_g(x)
=
\int_0^\infty g(xy)\overline{g(y)}\,dy.
$$

Then:

$$
\widehat{C_g}(s)
=
\widehat g(s)
\overline{\widehat g(1-\overline s)}.
$$

The Weil explicit formula adopts:

$$
\widehat f(0)
-
\sum_\rho^{\prime}\widehat f(\rho)
+
\widehat f(1)
=
E_{B0}(f),
$$

where $E_{B0}$ is the arithmetic side composed of the prime terms, the $x=1$ term, and the Archimedean terms.

Let:

$$
Q_{\mathrm{full}}(g)
:=
\sum_\rho^{\prime}
\widehat g(\rho)
\overline{\widehat g(1-\overline\rho)}.
$$

Substituting $f=C_g$ yields the complete arithmetic interface:

$$
\boxed{
Q_{\mathrm{full}}(g)
=
\widehat{C_g}(0)
+
\widehat{C_g}(1)
-
E_{B0}(C_g)
}
$$

Also, since:

$$
\widehat{C_g}(0)
=
\widehat g(0)\overline{\widehat g(1)},
$$

$$
\widehat{C_g}(1)
=
\widehat g(1)\overline{\widehat g(0)},
$$

Therefore:

$$
\boxed{
Q_{\mathrm{full}}(g)
=
2\operatorname{Re}
\left(
\widehat g(0)\overline{\widehat g(1)}
\right)
-
E_{B0}(C_g)
}.
$$

Only when $g\in\mathcal C_{00}$ does this degenerate to:

$$
Q_{\mathrm{full}}(g)=-E_{B0}(C_g).
$$

Thus, the endpoint-zero condition is not a necessary condition for the Weil criterion itself, but rather a **normalization choice**.

---

# 3. Additive Coordinates and the Weil Quadratic Form

Let:

$$
\psi(u)=e^{u/2}g(e^u),
\qquad
u=\log x.
$$

Then:

$$
g(x)=x^{-1/2}\psi(\log x).
$$

If we define the Fourier–Laplace transform:

$$
\mathcal F\psi(z)
:=
\int_{-\infty}^{\infty}
\psi(u)e^{izu}\,du,
$$

Then:

$$
\widehat g\!\left(\frac12+iz\right)
=
\mathcal F\psi(z).
$$

Writing the non-trivial zeros as:

$$
\rho=\frac12+i\gamma,
$$

where $\gamma$ are all real when the RH is true. The quadratic form becomes:

$$
\boxed{
Q_W(\psi)
=
\sum_{\gamma\in\Gamma}^{\prime}
 m_\gamma
\mathcal F\psi(\gamma)
\overline{\mathcal F\psi(\overline\gamma)}
}.
$$

If the RH holds, $\gamma=\overline\gamma$, hence:

$$
Q_W(\psi)
=
\sum_{\gamma\in\Gamma}
 m_\gamma
\left|\mathcal F\psi(\gamma)\right|^2
\ge0.
$$

If there exists a non-real $\gamma_0$, Paley–Wiener type interpolation can select a compactly supported smooth $\psi$ such that its Fourier–Laplace transform has opposite phases at $\gamma_0$ and $\overline{\gamma_0}$, while suppressing the contributions of other zeros, thereby obtaining:

$$
Q_W(\psi)<0.
$$

Suzuki's proof of the sufficiency of the Weil criterion explicitly uses this separation mechanism of "specifying one non-real zero and suppressing the remaining zeros".

---

# 4. Engineering Version of the Compact Support Separation Theorem

For $a>0$, define:

$$
\mathcal V_a
:=C_c^\infty(-a,a).
$$

And define the lowest Rayleigh quotient:

$$
\boxed{
\lambda(a)
:=
\inf_{0\neq\psi\in\mathcal V_a}
\frac{Q_W(\psi)}{\|\psi\|_{L^2}^2}
}.
$$

Then:

$$
\boxed{
RH
\Longleftrightarrow
\lambda(a)\ge0
\quad\forall a>0
}.
$$

Its contrapositive is:

$$
\boxed{
\neg RH
\Longrightarrow
\exists a>0:
\lambda(a)<0
}.
$$

Here, $a$ is a finite support scale, not an infinite limit.

## 4.1 Monotonicity

If $0<a_1<a_2$, then:

$$
\mathcal V_{a_1}\subset\mathcal V_{a_2},
$$

hence:

$$
\boxed{
\lambda(a_2)\le\lambda(a_1)
}.
$$

Therefore, $\lambda(a)$ is non-increasing as the support radius increases.

## 4.2 Support Phase Transition Point

Recent work by Suzuki proves that $\lambda(a)$ is continuous with respect to $a$, and recovers Yoshida's conclusion: if the RH is false, there exists some finite $a$ such that $\lambda(a)<0$; on the other hand, for sufficiently small $a$, $\lambda(a)>0$.

Therefore, under the assumption of $\neg RH$, one can define:

$$
a_*
:=
\inf\{a>0:\lambda(a)\le0\},
$$

and obtain:

$$
\lambda(a_*)=0.
$$

This $a_*$ can be viewed as the "minimum logarithmic window that can first accommodate a negative witness". It is not a known numerical value of the RH, nor can it be calculated without off-axis zeros; however, it provides clear engineering scanning parameters.

---

# 5. Existence is Closed, but Constructivity Remains Open

This round must strictly distinguish between three levels.

## 5.1 Logical Existence

$$
\neg RH
\Longrightarrow
\exists a,\psi:
Q_W(\psi)<0.
$$

**Status: Closed by known theory.**

## 5.2 Witness Construction After Knowing Off-Axis Zeros

Given an off-axis zero $\gamma_0$ with error bounds, construct $\psi$ using entire function interpolation and control the tail sum.

**Status: Theoretical mechanism exists, but has not yet been engineered into an executable strict certificate generator.**

## 5.3 Blind Search Without Knowing Off-Axis Zeros

Build finite matrices solely from the prime side, and scan for the appearance of strictly negative eigenvalues.

**Status: Next phase.**

If a negative eigenvalue proven by interval arithmetic is found, it directly refutes the RH; if the finite matrices are all non-negative, it only proves that there are no negative witnesses on that finite subspace, and cannot prove the RH.

---

# 6. Updates to Old GAPs

## `RH-W-02-GLOBAL-DENSITY`

The original question was: whether the compactly supported core needs to first be proven dense in the strip-analytic completion space before implying the RH.

This round's determination:

- For the full core $\mathcal C_{\mathrm{full}}$, **this bridge is not needed**; the Bombieri–Yoshida compact support criterion directly provides sufficiency.
- For the endpoint-zero core $\mathcal C_{00}$, whether it is independently sufficient remains unresolved.

Therefore, the status is changed to:

`BYPASSED_FOR_FULL_CORE / OPEN_FOR_C00`.

## `RH-W-02-RH-SUFFICIENCY`

Split into:

$$
\texttt{FULL-COMPACT-SUFFICIENCY}=\texttt{CLOSED},
$$

$$
\texttt{C00-SUFFICIENCY}=\texttt{OPEN}.
$$

---

# 7. Next Node

The next unit of work is:

$$
\boxed{
\texttt{RH-W-04-GALERKIN-CERTIFICATE}
}
$$

The goal is to select a set of computable basis $\{\psi_j\}_{j=1}^N\subset\mathcal V_a$, and establish:

$$
M_{ij}(a):=Q_W(\psi_i,\psi_j),
$$

$$
G_{ij}(a):=\langle\psi_i,\psi_j\rangle_{L^2},
$$

and compute the generalized minimum eigenvalue:

$$
\lambda_N(a)
=
\min_{c\neq0}
\frac{c^*M(a)c}{c^*G(a)c}.
$$

From the finite subspace inclusion relation:

$$
\lambda_N(a)\ge\lambda(a).
$$

Therefore:

$$
\boxed{
\lambda_N(a)<0
\Longrightarrow
\lambda(a)<0
\Longrightarrow
\neg RH
}
$$

as long as all matrix elements and negative eigenvalues have strict error certificates.

The reverse direction does not hold:

$$
\lambda_N(a)\ge0
\centernot\Longrightarrow RH.
$$

---

# 8. References

1. Enrico Bombieri, *The Riemann Hypothesis*, in **The Millennium Prize Problems**, Clay Mathematics Institute, explicit formula and Weil negativity criterion.  
   https://www.claymath.org/library/monographs/MPPc.pdf
2. Masatoshi Suzuki, *Aspects of the screw function corresponding to the Riemann zeta function*, arXiv:2206.03682; compactly supported Weil positivity criterion.  
   https://arxiv.org/abs/2206.03682
3. Masatoshi Suzuki, *On the Hilbert space derived from the Weil distribution*, arXiv:2301.00421; explicit off-axis-zero separation construction.  
   https://arxiv.org/abs/2301.00421
4. Masatoshi Suzuki, *Weil's quadratic form via the screw function*, arXiv:2606.09096; finite-support quadratic form, lowest eigenvalue, continuity in the support parameter.  
   https://arxiv.org/abs/2606.09096

---

# 9. Epistemic Status

This document does not:

- Find off-axis zeros;
- Find actual negative witnesses;
- Prove any new RH equivalent conditions;
- Prove that the endpoint-zero core $\mathcal C_{00}$ is independently sufficient;
- Prove that any finite-dimensional non-negative result can be extended to infinite dimensions.

The actual achievement of this round is:

> Moving "whether a negative witness exists" out of the pending research GAPs, and precisely shifting the research focus to "the construction and complete convergence of finite-dimensional verifiable negative certificates".