# RH-W-02: Weil Normalization Alignment and Notation Closure
## Riemann Hypothesis GAP Engineering Note v0.2

**Research Project:** RH GAP Atlas / AI Mathematical Engineering Relay  
**Parent Node:** `RH-W-02`  
**Current Node:** `RH-W-02-NORMALIZATION`  
**Prerequisite Nodes:** `RH-W-01`, `RH-W-02-RANGE`  
**Status:** `CLOSED_FOR_ENDPOINT_NULL_CORE / GLOBAL_BRIDGE_OPEN`  
**Date:** 2026-07-23  
**Nature:** Literature normalization, notation audit, and interface closure; not an RH proof

---

# 0. Conclusion of this Round

The explicit formula "trace-negativity" used by Bombieri/Clay and the Weil "covariance-positivity" used by Lagarias are not identical on general test functions, as they differ by the endpoint terms:

$$
\widehat f(0),\qquad \widehat f(1).
$$

However, for the core of double vanishing moments in this engineering project

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

and the correlation-type function

$$
C_{g,h}:=g*h^*,
$$

the endpoint terms vanish exactly, hence we have:

$$
\boxed{
E_{B0}(C_{g,h})
=
-\langle g,h\rangle_W
}
$$

where:

- $E_{B0}$ is the arithmetic side of the Bombieri/Clay explicit formula;
- $\langle\cdot,\cdot\rangle_W$ is Lagarias's Weil covariance form;
- $h^*$ is the Hermitian involution on the multiplicative group.

On the diagonal:

$$
\boxed{
Q_{B0}(g)
:=-E_{B0}(C_{g,g})
=
\langle g,g\rangle_W
=
W[C_{g,g}]
}
$$

Therefore:

$$
E_{B0}(C_{g,g})\le 0
\quad\Longleftrightarrow\quad
Q_{B0}(g)\ge 0.
$$

This is not a new RH equivalence criterion, but rather locks two existing formulations into the same engineering interface.

---

# 1. Fixing a Unique Mellin Convention

This project permanently fixes:

$$
\boxed{
\widehat f(s)
=
\int_0^\infty f(x)x^s\frac{dx}{x}
}
$$

It must not be memorized separately from another common convention

$$
\int_0^\infty f(x)x^{s-1}\,dx
$$

both are numerically identical, but programs and documentation must only retain the first notation to avoid adding or losing an $x$ during variable substitution.

The multiplicative convolution is fixed as:

$$
(a*b)(x)
=
\int_0^\infty
 a(y)b\!\left(\frac{x}{y}\right)\frac{dy}{y}.
$$

Then:

$$
\widehat{a*b}(s)
=
\widehat a(s)\widehat b(s).
$$

---

# 2. Fixing the Hermitian Involution

The geometric involution used by Lagarias is:

$$
\widetilde h(x)
=
\frac1x h\!\left(\frac1x\right),
$$

and:

$$
\widehat{\widetilde h}(s)
=
\widehat h(1-s).
$$

To establish a Hermitian form, this project fixes:

$$
\boxed{
 h^*(x)
 :=
 \widetilde{\overline h}(x)
 =
 \frac1x\overline{h\!\left(\frac1x\right)}
}
$$

Therefore:

$$
\boxed{
\widehat{h^*}(s)
=
\overline{\widehat h(1-\overline s)}
}.
$$

The complex conjugation here cannot be omitted. Omitting it yields a bilinear form, not a Hermitian quadratic form.

---

# 3. Clay Correlation Function and Lagarias Convolution are Exactly Identical

Definition:

$$
C_{g,h}(x)
:=(g*h^*)(x).
$$

Expanding directly:

$$
\begin{aligned}
(g*h^*)(x)
&=
\int_0^\infty
 g(y)
 \frac{y}{x}
 \overline{h\!\left(\frac{y}{x}\right)}
 \frac{dy}{y}
\\
&=
\frac1x
\int_0^\infty
 g(y)
 \overline{h\!\left(\frac{y}{x}\right)}dy.
\end{aligned}
$$

Letting $y=xu$, we obtain:

$$
\boxed{
C_{g,h}(x)
=
\int_0^\infty
 g(xu)\overline{h(u)}\,du
}.
$$

The diagonal is precisely the correlation function used by Bombieri/Clay:

$$
\boxed{
C_g(x)
=
\int_0^\infty
 g(xu)\overline{g(u)}\,du
}.
$$

Therefore, `correlation integral` and `multiplicative convolution + Hermitian involution` are not two candidate definitions, but two coordinates of the same object.

---

# 4. Mellin Correlation Identity

By the convolution theorem:

$$
\boxed{
\widehat{C_{g,h}}(s)
=
\widehat g(s)
\overline{\widehat h(1-\overline s)}
}.
$$

On the diagonal:

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

If $s$ lies on the critical line:

$$
\operatorname{Re}(s)=\frac12,
$$

then:

$$
1-\overline s=s,
$$

Therefore:

$$
\boxed{
\widehat{C_g}(s)=|\widehat g(s)|^2
}.
$$

Note: This is merely a squared modulus identity on the critical line; it does not by itself prove that the zeros lie on the critical line.

---

# 5. Three Functionals Must Be Named Separately

## 5.1 Zero / Covariance Side

Fixed:

$$
\boxed{
W[f]
:=
\sum_{\rho}^{\prime}\widehat f(\rho)
}
$$

where the prime indicates the symmetric truncation limit specified by the source.

## 5.2 Trace Side

Fixed:

$$
\boxed{
T[f]
:=
\widehat f(0)-W[f]+\widehat f(1)
}.
$$

## 5.3 Arithmetic Side

The right-hand side of the Bombieri/Clay explicit formula is uniformly named:

$$
\boxed{
E_{B0}[f]
}
$$

and is fixed by the explicit formula:

$$
\boxed{
E_{B0}[f]=T[f]
}.
$$

Thus, in the general case:

$$
\boxed{
E_{B0}[f]
=
\widehat f(0)-W[f]+\widehat f(1)
}.
$$

If any AI directly writes

$$
E_{B0}[f]=-W[f]
$$

without first proving that the two endpoint terms vanish, this step is automatically flagged as:

```text
INVALID_ENDPOINT_DROP
```

---

# 6. How Double Vanishing Moments Eliminate Endpoints

If:

$$
\widehat g(0)=\widehat g(1)=0,
$$

and:

$$
C_g=g*g^*,
$$

then:

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

and:

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

Therefore:

$$
\boxed{
E_{B0}[C_g]
=
-W[C_g]
}.
$$

More generally, if $g,h\in\mathcal C_{00}$, then:

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

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

Hence:

$$
\boxed{
E_{B0}[C_{g,h}]
=
-W[C_{g,h}]
}.
$$

---

# 7. Engineering Definition of the Weil Inner Product

Fixed:

$$
\boxed{
\langle g,h\rangle_W
:=
W[C_{g,h}]
=
W[g*h^*]
}.
$$

Therefore:

$$
\boxed{
E_{B0}[C_{g,h}]
=-\langle g,h\rangle_W
}
\qquad
(g,h\in\mathcal C_{00}).
$$

Diagonal:

$$
\boxed{
Q_{B0}(g)
:=-E_{B0}[C_g]
=\langle g,g\rangle_W.
}
$$

If RH holds, for every non-trivial zero we have:

$$
\rho=1-\overline\rho,
$$

Therefore:

$$
\begin{aligned}
Q_{B0}(g)
&=
\sum_\rho
\widehat g(\rho)
\overline{\widehat g(1-\overline\rho)}
\\
&=
\sum_\rho
|\widehat g(\rho)|^2
\ge 0.
\end{aligned}
$$

The reverse direction must still rely on the sufficiency of the test space being "adequate to separate all zeros"; this is not provided by the normalization identity itself.

---

# 8. Sign Conversion Table

| Formulation | Engineering Notation | RH Criterion Direction |
|---|---:|---:|
| Bombieri/Clay Arithmetic Side | $E_{B0}[C_g]$ | $\le 0$ |
| Lagarias Weil Covariance | $W[C_g]$ | $\ge 0$ |
| Engineering Unified Quadratic Form | $Q_{B0}(g):=-E_{B0}[C_g]$ | $\ge 0$ |
| Weil Inner Product Diagonal | $\langle g,g\rangle_W$ | $\ge 0$ |

On the double vanishing moment core:

$$
\boxed{
Q_{B0}(g)
=
W[C_g]
=
\langle g,g\rangle_W
=-E_{B0}[C_g].
}
$$

This boxed identity is the only valid interface for all subsequent programs, AI agents, and formalized versions.

---

# 9. Which Differences Cannot Be Eliminated by This Normalization

## 9.1 Endpoint Terms of General Functions

If $\widehat f(0)$ or $\widehat f(1)$ is non-zero, then:

$$
E_{B0}[f]
\neq
-W[f]
$$

holds in general. In this case, we must retain:

$$
\widehat f(0)+\widehat f(1).
$$

## 9.2 Singular Test Functions of the Li Class

Lagarias's Li test functions may have poles at $s=0$ and $s=1$, and the trace form might even be undefined; the covariance form, however, can be extended via cutoff/regularization.

Therefore:

$$
\mathcal C_{00}
\longrightarrow
\text{Li singular class}
$$

cannot be automatically completed by the identities of this round. New GAP added:

```text
RH-W-02-REGULARIZATION-BRIDGE
```

## 9.3 Global Sufficiency

Even though a unique quadratic form has been fixed on $\mathcal C_{00}$, it is not yet proven that:

$$
Q_{B0}(g)\ge0
\quad\forall g\in\mathcal C_{00}
$$

is sufficient to deduce RH. The following still needs to be completed:

- Density of the compactly supported core in some RH-equivalent test space; or
- Directly prove that the compactly supported core is already sufficient to separate any off-axis zeros.

Therefore:

```text
RH-W-02-GLOBAL-DENSITY = OPEN
RH-W-02-RH-SUFFICIENCY = BLOCKED
```

---

# 10. Normalization Rules That AI Agents Must Not Violate

Every candidate proof node must carry:

```yaml
mellin_convention: integral_0_inf_f_x_x^s_dx_over_x
convolution_measure: dy_over_y
star_involution: x_inverse_conjugate_f_of_x_inverse
zero_sum_ordering: source_symmetric_cutoff
endpoint_zero_required: true
arithmetic_sign: negative_semidefinite
covariance_sign: positive_semidefinite
unified_quadratic_form: Q_B0_equals_minus_E_equals_W
```

Automatic rejection conditions:

1. Forgetting the complex conjugation;
2. Mixing $dy$ and $dy/y$;
3. Deleting $\widehat f(0),\widehat f(1)$ without verifying the endpoints;
4. Judging $E_{B0}\le0$ and $W\ge0$ as a contradiction;
5. Treating the Li singular class as $C_c^\infty$;
6. Inferring global positivity from the positive value of a finite sum over zeros;
7. Inferring RH from normalization consistency.

---

# 11. GAP Status Update

| GAP | New Status | Description |
|---|---|---|
| `RH-W-02-NORMALIZATION` | `CLOSED_FOR_ENDPOINT_NULL_CORE` | trace-negativity and covariance-positivity have been aligned term by term |
| `RH-W-02-ENDPOINTS` | `CLOSED` | Endpoint terms are exactly eliminated by the double vanishing moment correlation form |
| `RH-W-02-STAR` | `CLOSED` | Hermitian involution, convolution, and correlation integral are fully aligned |
| `RH-W-02-SIGN` | `CLOSED` | The unique unified quadratic form $Q_{B0}=-E_{B0}=W$ is locked |
| `RH-W-02-ADELTA-NORM` | `REFERENCE_LOCKED` | Lagarias strip uniform norm definition is fixed as the candidate completion space |
| `RH-W-02-REGULARIZATION-BRIDGE` | `OPEN` | The core to Li singular covariance class still requires a cutoff interface |
| `RH-W-02-GLOBAL-DENSITY` | `OPEN` | Density of the core in the RH-equivalent global space is unproven |
| `RH-W-02-RH-SUFFICIENCY` | `BLOCKED` | Still blocked by global density/separability |

---

# 12. Next Node

The normalization issue is no longer blocking. The next most natural engineering node is:

$$
\boxed{
\texttt{RH-W-03-SEPARATION}
}
$$

Research Question:

> If an off-axis zero orbit exists, is the compactly supported smooth double vanishing moment core already sufficient to construct a negative witness?

Formally, attempt to directly prove or disprove:

$$
\neg RH
\Longrightarrow
\exists g\in\mathcal C_{00},
\quad
Q_{B0}(g)<0.
$$

If true, one can bypass the abstract `GLOBAL-DENSITY` and directly prove the RH sufficiency of the core; if false, it will also generate a precise representation obstruction.

---

# 13. Sources

1. E. Bombieri, *The Riemann Hypothesis*, in **The Millennium Prize Problems**, Clay Mathematics Institute / AMS, explicit formula and Weil negativity criterion, pp. 121–122.
2. J. C. Lagarias, *Li Coefficients for Automorphic L-Functions*, Appendix 9, trace form, covariance form, test space $\mathcal A_\delta$, and Weil scalar product; arXiv:math/0404394.

---

# 14. Boundary Declaration

This document only completes the engineering closure of formulations and notations. It does not:

- Prove $Q_{B0}(g)\ge0$;
- Prove that the compactly supported core is sufficient to characterize RH;
- Establish any new zero-location results;
- Elevate finite numerical checks to infinite propositions.