# RH-W-04: Finite-Dimensional Completeness and Rational Negative Certificates

**Version:** v0.1  
**Date:** 2026-07-23  
**Positioning:** Non-RH proof; establishing the finite-dimensional reachability of the Weil quadratic form and machine-verifiable negative certificates

---

## 0. Conclusions of the Current Round

This round splits `RH-W-04-GALERKIN-CERTIFICATE` into two mutually independent problems:

1. **Finite-Dimensional Completeness:** If a negative direction exists for the Weil quadratic form with a fixed support radius $a$, will some explicitly enumerable finite-dimensional space eventually capture it?
2. **Numerical Rigor:** After finding a floating-point negative direction, how can it be converted into a finite certificate that does not rely on floating-point eigenvectors?

This round provides an affirmative answer to the first question; for the second question, it reduces the certificate to a rational vector and a purely rational interval inequality.

What remains incomplete is: the rigorous interval enclosure for every element of the true $\mathrm{zeta}$ Weil matrix.

---

# 1. Fixed-Support Weil Form

Fix $a>0$, and let

$$
H_a=L^2(-a,a).
$$

Let $Q_a$ denote the closed, bounded-below extension of the Weil quadratic form on $(-a,a)$, with its form domain denoted by

$$
D(Q_a).
$$

Define the spectral bottom:

$$
\lambda(a)
=
\inf_{0\ne v\in D(Q_a)}
\frac{Q_a(v)}{\|v\|_{L^2}^2}.
$$

It is known that $C_c^\infty(-a,a)$ is the associated form core; more specifically, the space spanned by periodic exponential functions

$$
E_a
=
\operatorname{span}
\left\{
 e_n(x)=e^{i\pi n x/a}:n\in\mathbb Z
\right\}
$$

is a form core for $Q_a$, and each $e_n$ can be approximated in the form norm by compactly supported smooth functions after a boundary cutoff.

This allows us to transform the abstract density statement into an enumerable dictionary.

---

# 2. Cutoff–Fourier Complete Dictionary

Select a sequence of real-valued smooth cutoffs:

$$
\eta_k\in C_c^\infty(-a,a),
$$

satisfying:

$$
0\le \eta_k\le1,
$$

$$
\eta_k(x)=1
\quad
\text{for }|x|\le a-\varepsilon_k,
$$

where:

$$
\varepsilon_k\downarrow0.
$$

For example, one can take $\varepsilon_k=2^{-k}a$ and use a standard flat cutoff in the two boundary layers.

Define the real-valued dictionary:

$$
b_{0,k}(x)=\eta_k(x),
$$

$$
b^c_{n,k}(x)
=
\eta_k(x)\cos\left(\frac{\pi n x}{a}\right),
\qquad n\ge1,
$$

$$
b^s_{n,k}(x)
=
\eta_k(x)\sin\left(\frac{\pi n x}{a}\right),
\qquad n\ge1.
$$

Let:

$$
V_N(a)
=
\operatorname{span}
\left\{
 b_{0,k},b^c_{n,k},b^s_{n,k}:
 1\le k\le N,
 1\le n\le N
\right\}.
$$

Then:

$$
V_1(a)\subseteq V_2(a)\subseteq\cdots.
$$

## Proposition 2.1: Form-Core Completeness

$$
\boxed{
\overline{\bigcup_{N\ge1}V_N(a)}^{\|\cdot\|_{Q_a}}
=
D(Q_a)
}
$$

### Proof Skeleton

For each fixed $n$:

$$
\eta_k e_n
\longrightarrow e_n
$$

in the form norm of $Q_a$. The real and imaginary parts respectively yield:

$$
\eta_k\cos(\pi nx/a)
\longrightarrow
\cos(\pi nx/a),
$$

$$
\eta_k\sin(\pi nx/a)
\longrightarrow
\sin(\pi nx/a).
$$

Therefore, the form-norm closure of $\bigcup_N V_N(a)$ contains $E_a$. Since $E_a$ is a form core:

$$
\overline{E_a}^{\|\cdot\|_{Q_a}}
=
D(Q_a),
$$

thus the conclusion holds.

---

# 3. Rayleigh–Ritz Reachability

Define the finite-dimensional spectral bottom:

$$
\lambda_N(a)
=
\inf_{0\ne v\in V_N(a)}
\frac{Q_a(v)}{\|v\|_{L^2}^2}.
$$

By the nested property of the spaces:

$$
\lambda_1(a)
\ge
\lambda_2(a)
\ge\cdots
\ge
\lambda(a).
$$

## Theorem 3.1: Finite-Dimensional Convergence

$$
\boxed{
\lambda_N(a)\downarrow\lambda(a)
}
$$

### Proof

For any $\varepsilon>0$. By $\varepsilon$-minimality, we can choose $v\in D(Q_a)$, normalized such that $\|v\|_{L^2}=1$, satisfying:

$$
Q_a(v)<\lambda(a)+\frac{\varepsilon}{2}.
$$

By Proposition 2.1, there exists $v_N\in V_N(a)$ such that:

$$
\|v_N-v\|_{Q_a}\to0.
$$

Therefore:

$$
\|v_N\|_{L^2}\to1,
$$

and by the bilinearization of the closed form and form-norm convergence:

$$
Q_a(v_N)\to Q_a(v).
$$

For sufficiently large $N$:

$$
\frac{Q_a(v_N)}{\|v_N\|_{L^2}^2}
<
\lambda(a)+\varepsilon.
$$

Thus:

$$
\limsup_{N\to\infty}\lambda_N(a)
\le\lambda(a)+\varepsilon.
$$

The proof is completed by the arbitrariness of $\varepsilon$ and $\lambda_N(a)\ge\lambda(a)$.

## Corollary 3.2: Finite-Dimensional Reachability of Negative Directions

If:

$$
\lambda(a)<0,
$$

then there exists a finite $N$ such that:

$$
\boxed{
\lambda_N(a)<0
}.
$$

Therefore, if RH is false, it is known that a negative direction exists for some finite support window; by this corollary, a negative direction must appear in some finite cutoff–Fourier matrix.

This is an existence conclusion and does not provide an effective upper bound for $a$ or $N$.

---

# 4. No Need to Verify the Entire Eigenvalue Problem

Take any real basis of $V_N(a)$:

$$
\phi_1,\ldots,\phi_m.
$$

Define the Weil matrix:

$$
M_{ij}=Q_a(\phi_i,\phi_j),
$$

and the $L^2$ Gram matrix:

$$
G_{ij}=\langle\phi_i,\phi_j\rangle_{L^2}.
$$

Floating-point programs can use the generalized minimum eigenvector to search for candidate directions, but the formal certificate does not need to save or trust this floating-point eigenvector.

It only needs to output a non-zero rational vector:

$$
c\in\mathbb Q^m,
$$

and rigorously prove:

$$
\boxed{
c^TMc<0
},
$$

Let:

$$
v_c=\sum_{j=1}^m c_j\phi_j,
$$

then we have:

$$
Q_a(v_c)<0.
$$

The negative value itself already excludes $v_c=0$, so **the refutation semantics only require the negativity of the numerator**. Proving:

$$
c^TGc>0
$$

is not logically necessary, but it can serve as a basis degeneracy check and allows for the output of a rigorous Rayleigh quotient interval.

---

# 5. Component-wise Interval Negative Certificates

Assume each matrix element already has a rational interval:

$$
M_{ij}\in
[\underline M_{ij},\overline M_{ij}].
$$

For a fixed rational vector $c$, let:

$$
a_{ij}=c_ic_j.
$$

Define the monomial upper bound:

$$
U_{ij}(c)=
\begin{cases}
 a_{ij}\overline M_{ij},&a_{ij}\ge0,\\
 a_{ij}\underline M_{ij},&a_{ij}<0.
\end{cases}
$$

Then:

$$
\boxed{
 c^TMc
\le
U_M(c):=
\sum_{i,j}U_{ij}(c)
}.
$$

Thus:

$$
\boxed{
U_M(c)<0
\Longrightarrow
c^TMc<0
}.
$$

The entire verification only uses rational addition, multiplication, and order comparison.

Similarly, for $G$, we can obtain a lower bound:

$$
L_G(c)
\le c^TGc.
$$

If:

$$
U_M(c)<0,
\qquad
L_G(c)>0,
$$

then the Rayleigh quotient strictly satisfies:

$$
\frac{c^TMc}{c^TGc}
\le
\frac{U_M(c)}{L_G(c)}<0.
$$

---

# 6. Equivalent Form of Perturbation Budgets

If there is a central matrix $\widehat M$ and component-wise errors:

$$
|M_{ij}-\widehat M_{ij}|
\le\epsilon_{ij},
$$

then:

$$
\boxed{
 c^TMc
\le
c^T\widehat Mc
+
\sum_{i,j}|c_i||c_j|\epsilon_{ij}
}.
$$

This quantity is called the **certificate margin**:

$$
\mathfrak m(c)
:=
-
\left(
 c^T\widehat Mc
+
\sum_{i,j}|c_i||c_j|\epsilon_{ij}
\right).
$$

Only when:

$$
\mathfrak m(c)>0
$$

can it be upgraded to a formal negative certificate.

The floating-point minimum eigenvalue itself has no certificate status.

---

# 7. Firewall from Floating-Point Candidates to Rational Witnesses

The candidate searcher can be completely untrusted. The standard workflow is:

1. A floating-point or AI program generates a candidate eigenvector $v$;
2. Choose an integer scale $B$;
3. Take:

   $$
   p_i=\operatorname{round}(Bv_i)\in\mathbb Z;
   $$

4. Divide by the greatest common divisor of the integers to obtain $p$;
5. Pass $c=p$ to the purely rational verifier;
6. The verifier only reads the interval matrix and does not read floating-point residuals or model confidence.

If the negativity disappears after rationalization, increase $B$ or change the candidate; the original floating-point result must not be treated as a substitute certificate.

This structure is equivalent to:

$$
\text{untrusted search}
\longrightarrow
\text{proof-carrying witness}
\longrightarrow
\text{small exact verifier}.
$$

---

# 8. Minimum Trusted Boundary for a True RH Certificate

A certificate claiming to belong to the true zeta Weil form must at least fix:

1. Support radius $a$;
2. Exact formula and parameters of the cutoff;
3. Basis ordering and file hashes;
4. Fourier/Mellin normalization;
5. Version of the Weil form and arithmetic-side formulas;
6. Rational upper and lower bounds for each $M_{ij}$;
7. Sources of each error:
   - Endpoint terms;
   - Prime sums;
   - Archimedean terms;
   - Removable singularities;
   - Integral truncations;
   - Rounding errors;
8. Rational witness;
9. Attachments and hashes;
10. Verifier version.

The current round's verifier only guarantees: **given the interval endpoints, the purely rational negativity inference is correct.**

It has not yet proven that the interval endpoints actually enclose the true Weil matrix; that is the next layer's `MATRIX-ENCLOSURE` GAP.

---

# 9. One-Sided Semantics

Formally locked:

$$
\boxed{
\text{rigorous Weil interval matrix}
+
U_M(c)<0
\Longrightarrow
\neg RH
}.
$$

The converse does not hold:

$$
U_M(c)\ge0
$$

only means that this witness was not successful.

Even if a certain finite matrix is strictly positive definite, it cannot prove RH.

---

# 10. GAP Updates for the Current Round

- `RH-W-04-CUTOFF-FOURIER-CORE`: `CLOSED_BY_FORM_CORE_CONSTRUCTION`
- `RH-W-04-RAYLEIGH-RITZ-LIMIT`: `CLOSED`
- `RH-W-04-FINITE-DETECTION`: `CLOSED_NON_EFFECTIVE`
- `RH-W-04-REAL-BASIS`: `CLOSED`
- `RH-W-04-RATIONAL-WITNESS-REDUCTION`: `CLOSED`
- `RH-W-04-EXACT-VERIFIER`: `CLOSED_FOR_SUPPLIED_INTERVALS`
- `RH-W-04-MATRIX-ENCLOSURE`: `OPEN`
- `RH-W-04-ARCHIMEDEAN-ENCLOSURE`: `OPEN`
- `RH-W-04-TRUE-ZETA-CERTIFICATE`: `NOT_PRODUCED`
- `RH-W-04-POSITIVE-NONCERTIFICATE`: `LOCKED_WARNING`

---

# 11. Reference Dependencies

1. M. Suzuki, *Weil's quadratic form via the screw function*, arXiv:2606.09096, 2026. Used for its fixed-support closed form, form core, boundary cutoff approximation, and generalized eigenvalue framework.
2. A. Imakura, K. Morikuni, A. Takayasu, *Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh–Ritz procedure for generalized Hermitian eigenvalue problems*, JCAM 424 (2022). Serves as the background for complete verified eigenvalue algorithms; this project adopts a smaller single-witness purely rational certificate.

---

# 12. Conclusion

This round shifts the problem from:

> A finite-dimensional scan might forever miss the true negative direction.

to:

> The specified cutoff–Fourier dictionary is complete under the form norm; if a negative direction exists, finite dimensions will eventually reach it. The only remaining problem is how to rigorously enclose the matrix elements.

This narrows down `RH-W-04` from "whether finite-dimensional methods have logical gaps" to a well-defined verified numerics engineering problem.