# RH-W-01: Fixing the Test Function Space for the Weil Approach
## Riemann Hypothesis GAP Engineering Note v0.1

**Research Project:** RH GAP Atlas / AI Mathematical Engineering Relay  
**Node:** `RH-W-01`  
**Status:** `IN_PROGRESS`  
**Nature:** Definition and scope locking document; not a proof of RH, nor does it propose new positivity theorems  
**Date:** 2026-07-23

---

## 0. Engineering Objectives

There exist multiple equivalent normalizations for Weil's explicit formula and Weil's criterion. If different AIs, papers, or programs silently swap Mellin/Fourier conventions, signs, conjugations, convolutions, summation orders over zeros, or test function spaces, subsequent derivations might end up studying entirely different quadratic forms, even if every line appears reasonable.

The purpose of this node is not to prove positivity, but to first establish a **baseline interface that cannot be silently altered**:

$$
\text{Input Function}
\longrightarrow
\text{Mellin Transform}
\longrightarrow
\text{Zero Side and Arithmetic Side}
\longrightarrow
\text{Weil Quadratic Form / Sign Criterion}.
$$

The completion standard is not "writing down familiar formulas," but ensuring that all subsequent agents can answer:

1. Which space does the function belong to?
2. In what sense does every sum and integral converge?
3. Which conventions for the Mellin transform, convolution, and involution are adopted?
4. How are the signs for positivity/negativity derived from these conventions?
5. If switching to another set of conventions, what is the transition map?

---

# 1. Baseline Version B0: Multiplicative Semigroup Formulation by Bombieri / Clay

The first phase of this engineering project selects Bombieri's RH chapter in the Clay Mathematics Institute's *Millennium Prize Problems* as the **B0 baseline**. This choice does not claim it to be the only or the most computationally suitable version, but rather because its function class, Mellin transform, choice of zero summation, and Weil criterion are explicitly given in one place.

## 1.1 Test Function Class

Let $\mathcal W$ be the class of complex-valued functions $f$ on the positive half-axis $\mathbb R_+=(0,\infty)$ satisfying:

1. $f$ is continuous, and continuously differentiable except at finitely many points;
2. At the exceptional points, $f$ and $f'$ have at most discontinuities of the first kind;
3. The function value at exceptional points is taken as the average of the left and right limits;
4. There exists $\delta>0$ such that

$$
f(x)=O(x^\delta),\qquad x\to0^+,
$$

and

$$
f(x)=O(x^{-1-\delta}),\qquad x\to+\infty.
$$

Here, $\delta$ is part of the function data; different functions may have different valid $\delta$.

## 1.2 Mellin Transform

Adopt:

$$
\widetilde f(s)
:=
\int_0^\infty f(x)x^s\frac{dx}{x}.
$$

Under the above conditions, $\widetilde f$ is analytic at least in the strip

$$
-\delta<\operatorname{Re}(s)<1+\delta.
$$

**Engineering Lock:** Within this package, the notation $\widetilde f$ strictly denotes this Mellin convention, and must not be changed to shifted versions other than $x^{s-1}dx$ without updating the version number. Note that both are formally identical since $x^s\,dx/x=x^{s-1}dx$; however, the programmatic interface still declares this formula as the sole standard.

## 1.3 von Mangoldt Function

$$
\Lambda(n)=
\begin{cases}
\log p,&n=p^a,\quad p\text{ is prime},\ a\geq1,\\
0,&\text{otherwise}.
\end{cases}
$$

## 1.4 Explicit Formula B0

For $f\in\mathcal W$, take the sum over non-trivial zeros as

$$
\lim_{T\to+\infty}
\sum_{|\operatorname{Im}(\rho)|<T}\widetilde f(\rho),
$$

Then the baseline explicit formula is:

$$
\widetilde f(0)
-
\sum_\rho\widetilde f(\rho)
+
\widetilde f(1)
=
\sum_{n=1}^\infty\Lambda(n)
\left(
 f(n)+\frac1n f\!\left(\frac1n\right)
\right)
+(\log4\pi+\gamma)f(1)
$$

$$
\qquad+
\int_1^\infty
\left(
 f(x)+\frac1x f\!\left(\frac1x\right)-\frac2x f(1)
\right)
\frac{dx}{x-x^{-1}}.
$$

To prevent subsequent agents from arbitrarily moving terms across the equation and changing signs, we define:

$$
E_{\mathrm{zero}}(f)
:=
\widetilde f(0)-\sum_\rho\widetilde f(\rho)+\widetilde f(1),
$$

$$
E_{\mathrm{arith}}(f)
:=
\sum_{n=1}^\infty\Lambda(n)
\left(
 f(n)+\frac1n f\!\left(\frac1n\right)
\right)
+(\log4\pi+\gamma)f(1)
$$

$$
\qquad+
\int_1^\infty
\left(
 f(x)+\frac1x f\!\left(\frac1x\right)-\frac2x f(1)
\right)
\frac{dx}{x-x^{-1}}.
$$

Thus, the valid equation for B0 is fixed as:

$$
E_{\mathrm{zero}}(f)=E_{\mathrm{arith}}(f).
$$

---

# 2. Special Functions Required for the Weil Criterion

## 2.1 Baseline Multiplicative Correlation Form

Take $g\in\mathcal W$, define

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

and require two vanishing moment conditions:

$$
\int_0^\infty g(x)\frac{dx}{x}=0,
\qquad
\int_0^\infty g(x)\,dx=0.
$$

Under the B0 convention, the Weil criterion can be stated as: RH is equivalent to the right-hand side of the explicit formula being non-positive for $f_g$, for all such $g$.

Therefore, this engineering project defines:

$$
Q_{B0}(g):=-E_{\mathrm{arith}}(f_g).
$$

The criterion is then written as:

$$
RH
\Longleftrightarrow
\forall g\in\mathcal G_{B0},\quad Q_{B0}(g)\geq0,
$$

where

$$
\mathcal G_{B0}
:=
\left\{
 g\in\mathcal W:
 \int_0^\infty g(x)\frac{dx}{x}=0,
 \ \int_0^\infty g(x)dx=0,
 \ f_g\in\mathcal W
\right\}.
$$

The last condition $f_g\in\mathcal W$ must not be taken for granted in engineering; it must be proven family by family or guaranteed by stronger input space conditions.

## 2.2 Sign Error-Prevention Rules

Literature may refer to the same criterion as Weil positivity or Weil negativity. The differences usually arise from:

- Defining either the zero side or the arithmetic side as the quadratic form;
- Moving all terms to the other side of the equation;
- Adding a minus sign in front of the quadratic form;
- Adopting different convolutions / involutions.

Therefore, simply writing "Weil positivity" is prohibited. Every subsequent document must explicitly write:

$$
Q(g):=\text{Full formula},
$$

and then state:

$$
Q(g)\geq0
\quad\text{or}\quad
Q(g)\leq0.
$$

---

# 3. `RH-W-01` Sub-GAP Decomposition

## `RH-W-01-A`: Proof of Function Class Scope

**Obligation:** For every rule of $\mathcal W$, prove that all terms in the explicit formula are well-defined.  
**Failure Witness:** Finding an $f\in\mathcal W$ such that any term diverges or the specified limit of the zero sum does not exist.  
**Status:** `REFERENCE_ESTABLISHED / LOCAL_RECHECK_REQUIRED`

From an engineering perspective, we do not claim to re-prove the overall theorem of Bombieri/Weil, but any newly generated family must still be re-checked to ensure it strictly falls within $\mathcal W$.

## `RH-W-01-B`: Locking the Zero Summation Rule

**Obligation:** Strictly adopt

$$
\lim_{T\to\infty}\sum_{|\operatorname{Im}\rho|<T}.
$$

If switching to summation by modulus, symmetric pairing, or Hadamard regularization, equivalence must be established.  
**Failure Witness:** Two orderings yield different limits, or one of them fails to converge.  
**Status:** `OPEN_FOR_ALTERNATIVE_NORMALIZATIONS`

## `RH-W-01-C`: Integrability of the Archimedean Integral at $x=1$

The integral kernel formally has a singular denominator at $x=1$:

$$
x-x^{-1}\to0.
$$

**Obligation:** For candidate function families, prove that the subtracted terms in the numerator

$$
f(x)+x^{-1}f(x^{-1})-2x^{-1}f(1)
$$

provide sufficient cancellation to make the integral valid around $x=1$.  
**Failure Witness:** Local expansion leaves a non-integrable first-order singularity.  
**Status:** `OPEN_PER_GENERATOR_FAMILY`

## `RH-W-01-D`: Closure of Multiplicative Correlation Functions

**Obligation:** Given a candidate $g$, prove that

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

exists and belongs to $\mathcal W$.  
**Failure Witness:** $f_g$ has insufficient decay at $0$ or $\infty$, lacks regularity, or the integral does not converge.  
**Status:** `OPEN`

## `RH-W-01-E`: Interface for the Two Vanishing Moment Conditions

By the B0 Mellin convention:

$$
\widetilde g(0)=\int_0^\infty g(x)\frac{dx}{x},
\qquad
\widetilde g(1)=\int_0^\infty g(x)dx.
$$

Thus, the two moment conditions are:

$$
\widetilde g(0)=\widetilde g(1)=0.
$$

**Obligation:** All generator constructions must explicitly preserve these two conditions; numerical approximations of $0$ cannot be substituted for exact equalities after the fact.  
**Failure Witness:** Any generator only approximately vanishes, or projection corrections destroy other conditions.  
**Status:** `OPEN_FOR_CONSTRUCTIVE_PARAMETERIZATION`

## `RH-W-01-F`: Consistency of Signs and Conjugation

**Obligation:** Verify that programmatic, natural language, and formalized versions all use the same:

- Definition of $f_g$;
- Position of complex conjugation;
- $Q_{B0}=-E_{\mathrm{arith}}$;
- Criterion $Q_{B0}\geq0$.

**Failure Witness:** For the same $g$, two modules output opposite signs but both claim to "pass."  
**Status:** `OPEN_UNTIL_CROSS_IMPLEMENTATION_TEST`

## `RH-W-01-G`: Alternative Representation Converter

**Obligation:** Establish explicit conversions between B0 and the Fourier-type formula under the additive variable

$$
x=e^u
$$

including the measure:

$$
\frac{dx}{x}=du,
$$

and function rescaling.  
**Failure Witness:** The conversion loses $e^{u/2}$-type weights, shifted Mellin parameters, or conjugation.  
**Status:** `OPEN`

## `RH-W-01-H`: Machine-Decidable Legality Preconditions

**Obligation:** Convert "regularity, decay, vanishing moments, closure, integrability" into metadata that can be checked by CAS, numerical programs, or Lean/Isabelle frontends.  
**Failure Witness:** An agent judges legality solely by the function's appearance, without providing bounds or proof objects.  
**Status:** `OPEN`

---

# 4. Fixed and Unfixed Items in This Round

## 4.1 Fixed Items

- Multiplicative formulation on the positive half-axis;
- B0 rules for the test function class $\mathcal W$;
- Mellin convention;
- Truncation method for the sum over non-trivial zeros;
- Naming of the two sides of the explicit formula;
- Correlation form of $f_g$;
- Two vanishing moments;
- Sign of the quadratic form $Q_{B0}$ for this engineering project.

## 4.2 Unfixed Items

- The generating family $\mathcal G$ most suitable for AI search;
- The topology used for subsequent closures on $\mathcal W$;
- Whether $Q_{B0}$ is continuous or closable in that topology;
- Whether negative witnesses can be compressed into a computable generating family;
- Complete translation between B0 and Schwartz / Paley–Wiener / additive Fourier versions;
- Whether formal function space libraries are sufficient to directly support all conditions.

These will respectively enter `RH-W-02` through `RH-W-05`, and must not be smuggled into completion within `RH-W-01`.

---

# 5. AI Task Interface

When any agent submits a candidate $g$, it must attach:

```yaml
candidate_id: string
formula: exact symbolic expression
domain: (0, infinity)
regularity_proof: reference_or_derivation
bound_at_zero: "|g(x)| <= C x^a"
bound_at_infinity: "|g(x)| <= C x^{-1-a}"
vanishing_mellin_0: exact_proof
vanishing_mellin_1: exact_proof
correlation_exists: proof_or_bound
correlation_in_W: proof_or_bound
sign_convention: Q_B0 = -E_arith
numerical_checks: optional_only
unproved_items: explicit_list
```

If any required field is missing, the candidate must not enter positivity testing and can only be marked as:

`INADMISSIBLE_PENDING_DOMAIN_CHECK`.

---

# 6. Engineering Verdict

`RH-W-01` is not yet fully closed, but it has transitioned from a vague sentence:

> "Take an appropriate test function space."

into eight relayable sub-GAPs. The value of the first version is not to solve RH, but to prevent subsequent AIs from unknowingly changing the problem.

Current engineering status:

$$
\boxed{
\texttt{RH-W-01: IN\_PROGRESS}
}
$$

The next recommended unit of work is not the entirety of `RH-W-02`, but rather:

$$
\boxed{
\texttt{RH-W-01-D/E}:
\text{Construct a parameterizable generating family that exactly satisfies the two vanishing moments and is closed under }f_g\text{.}
}
$$

---

# References

1. Enrico Bombieri, “The Riemann Hypothesis,” in *The Millennium Prize Problems*, Clay Mathematics Institute / Cambridge University Press, section 5, pp. 121–122.  
   https://www.claymath.org/library/monographs/MPPc.pdf
2. Jean-François Burnol, “The Explicit Formula in Simple Terms,” arXiv:math/9810169.  
   https://arxiv.org/abs/math/9810169