# RH-W-14: Strict Two-Dimensional Parameter Tube

**Version:** v0.1  
**Date:** 2026-07-23  
**Status:** Finite-dimensional continuous parameter certificate; neither proves nor disproves the Riemann Hypothesis

---

## Abstract

RH-W-13 established a near-zero positive spectral inclusion for the ten-dimensional mixed-order B-spline Weil dictionary at a single parameter point:

$$
10^{-8}<\lambda_{\min}(M,G)<5\times10^{-8}.
$$

This round expands the single-point result into a continuous parameter set for the first time. Fixing the scale

$$
h=\frac{1797}{10000},
$$

let the centers of the two channels be

$$
t_j^{(1)}=(j-2)d-\frac{\sigma}{2},
\qquad
t_j^{(3)}=(j-2)d+\frac{\sigma}{2},
\qquad j=0,\ldots,4.
$$

Within the rational rectangle

$$
\mathcal T=
\left\{
(d,\sigma):
\left|d-\frac{893}{5000}\right|\le4\times10^{-12},
\quad
|\sigma|\le4\times10^{-12}
\right\}
$$

, the purely rational verifier proves:

$$
\boxed{
10^{-8}
<
\lambda_{\min}\bigl(M(d,\sigma),G(d,\sigma)\bigr)
<
5\times10^{-8}
\qquad\forall(d,\sigma)\in\mathcal T.
}
$$

This is the first non-degenerate, continuous near-zero parameter tube certificate in this series that genuinely alters the test subspace.

---

## 1. Why $\alpha$ is Not Counted as a Parameter Dimension

If the entire $m=3$ channel is simply multiplied by a constant $\alpha\ne0$, then

$$
(M,G)\mapsto(T_\alpha^TMT_\alpha,T_\alpha^TGT_\alpha)
$$

is merely an invertible congruence transformation, leaving the generalized spectrum invariant. Therefore, $\alpha$ is a coordinate gauge or preconditioning parameter, not a new mathematical direction.

This round only computes parameters that genuinely alter the subspace:

- $d$: the basis spacing within the same channel;
- $\sigma$: the relative translation between the two regularity channels.

Thus, $\mathcal T$ is a genuine two-dimensional parameter tube.

---

## 2. Mixed-Order Dictionary

Two families of bases are used:

$$
v_j^{(1)}(x)=h^{-1/2}\beta_1\!\left(\frac{x-t_j^{(1)}}h\right),
$$

$$
v_j^{(3)}(x)=h^{-1/2}\beta_3\!\left(\frac{x-t_j^{(3)}}h\right).
$$

The correlation kernels are respectively:

$$
1\times1\longrightarrow\beta_3,
$$

$$
1\times3\longrightarrow\beta_5,
$$

$$
3\times3\longrightarrow\beta_7.
$$

The full matrix is

$$
M(d,\sigma)=
\begin{pmatrix}
M^{11}&M^{13}\\
M^{31}&M^{33}
\end{pmatrix},
$$

The Gram matrix is similarly denoted as $G(d,\sigma)$.

---

## 3. Global Derivative Bounds for B-splines

Let $\beta_r$ be the centered cardinal B-spline. By its convolutional representation and Young's inequality:

$$
0\le\beta_r\le1.
$$

The derivative has a finite difference representation:

$$
\beta_r'(u)
=
\beta_{r-1}\!\left(u+\frac12\right)
-
\beta_{r-1}\!\left(u-\frac12\right),
$$

therefore

$$
\|\beta_r'\|_\infty\le1.
$$

Differentiating once more:

$$
\beta_r''(u)
=
\beta_{r-2}(u+1)-2\beta_{r-2}(u)+\beta_{r-2}(u-1),
$$

thus we conservatively have

$$
\|\beta_r''\|_\infty\le4.
$$

These three bounds apply simultaneously to $r=3,5,7$.

---

## 4. Center Lipschitz Bounds for Weil Matrix Elements

Let

$$
f_c(x)=\beta_r\!\left(\frac{x-c}{h}\right),
\qquad
q=\frac{r+1}{2}.
$$

If the support within the entire parameter tube lies in $[-R_r,R_r]$, then the center derivatives of the four parts of the Weil explicit formula can be bounded respectively as:

### 4.1 Endpoint Integral

$$
\left|\frac{d}{dc}E(f_c)\right|
\le4q e^{R_r/2}.
$$

### 4.2 Constant Term

Let

$$
C_0=\log(4\pi)+\gamma,
$$

then

$$
\left|\frac{d}{dc}[-C_0f_c(0)]\right|
\le\frac{C_0}{h}.
$$

### 4.3 Prime-Power Term

For the set of von Mangoldt indices $\mathcal P_r$ that may enter the support:

$$
\left|\frac{d}{dc}P(f_c)\right|
\le
\frac2h
\sum_{n\in\mathcal P_r}
\frac{\Lambda(n)}{\sqrt n}.
$$

### 4.4 Archimedean Term

Let

$$
F_c(x)=f_c(x)+f_c(-x),
\qquad
g_c(x)=\partial_cF_c(x).
$$

After preserving the cancellation at $x=0$, using

$$
|g_c(x)|\le\frac2h,
\qquad
|g_c'(x)|\le\frac8{h^2},
$$

and

$$
e^{x/2}-1\le\frac{x}{2}e^{R_r/2},
\qquad
2\sinh x\ge2x,
$$

we obtain

$$
\left|\frac{d}{dc}A(f_c)\right|
\le
R_r\left(
\frac{e^{R_r/2}}{2h}
+
\frac4{h^2}
\right).
$$

### 4.5 Combined Bound

Therefore,

$$
\boxed{
L_r=
4q e^{R_r/2}
+
\frac{C_0}{h}
+
\frac2h\sum_{n\in\mathcal P_r}\frac{\Lambda(n)}{\sqrt n}
+
R_r\left(\frac{e^{R_r/2}}{2h}+\frac4{h^2}\right)
}
$$

satisfies

$$
|W(f_{c_1})-W(f_{c_2})|
\le L_r|c_1-c_2|.
$$

Purely rational reconstruction yields the integer upper bounds:

$$
\boxed{L_3\le175,\qquad L_5\le215,\qquad L_7\le253.}
$$

The Gram elements satisfy

$$
\boxed{
|G(c_1)-G(c_2)|
\le\frac1h|c_1-c_2|.
}
$$

---

## 5. Propagation of Parameter Variations to Each Matrix Element

For the $i$-th and $j$-th bases of the $a$-th and $b$-th channels, the variation in the center difference satisfies:

$$
|\Delta c_{ij}^{ab}|
\le
|i-j|\rho_d
+
\mathbf1_{a\ne b}\rho_\sigma,
$$

where

$$
\rho_d=\rho_\sigma=4\times10^{-12}.
$$

Thus, each element satisfies:

$$
|\Delta M_{ij}^{ab}|
\le
L_{r(a,b)}
\left(
|i-j|\rho_d+
\mathbf1_{a\ne b}\rho_\sigma
\right),
$$

$$
|\Delta G_{ij}^{ab}|
\le
\frac1h
\left(
|i-j|\rho_d+
\mathbf1_{a\ne b}\rho_\sigma
\right).
$$

Combining the original interval errors of the single-point matrix from RH-W-13, this round yields:

$$
\boxed{
\epsilon_{M-10^{-8}G}
\le
2.3021210580889285\times10^{-8}
}
$$

and

$$
\boxed{
\epsilon_G
\le
5.564830272676684\times10^{-10}.
}
$$

---

## 6. Lower Bound for the Entire Parameter Tube

Let $C$ be the fixed rational center of the single-point Weil interval matrix, and $G_0$ be the center Gram matrix.

The verifier performs a purely rational $LDL^T$ decomposition on

$$
C-10^{-8}G_0-
\epsilon_{M-10^{-8}G}I
$$

yielding ten strictly positive pivots.

By the column-sum bound for symmetric perturbations:

$$
\|E\|_2\le\|E\|_\infty,
$$

we obtain for all $(d,\sigma)\in\mathcal T$:

$$
M(d,\sigma)-10^{-8}G(d,\sigma)\succ0.
$$

Therefore:

$$
\boxed{
\lambda_{\min}(M(d,\sigma),G(d,\sigma))>10^{-8}.
}
$$

The verifier also independently proves:

$$
G(d,\sigma)\succ0
\qquad\forall(d,\sigma)\in\mathcal T.
$$

---

## 7. Upper Bound for the Entire Parameter Tube

Adopting the integer witness from RH-W-13:

$$
\begin{aligned}
c={}&(
68190193,
137154794,
187700175,
137154794,
68190193,\\
&-3577963013,
-7569824004,
-10000000000,
-7569824004,
-3577963013
)^T.
\end{aligned}
$$

Using the same set of Lipschitz bounds, we directly enclose the entire tube's

$$
c^TM(d,\sigma)c
$$

and

$$
c^TG(d,\sigma)c.
$$

Purely rational comparison proves:

$$
\boxed{
\frac{c^TM(d,\sigma)c}{c^TG(d,\sigma)c}
<5\times10^{-8}
\qquad\forall(d,\sigma)\in\mathcal T.
}
$$

Therefore, the lowest generalized eigenvalue also satisfies the same upper bound.

---

## 8. Prime-Power Chamber Remains Invariant Throughout the Tube

The maximum support radius of the entire dictionary is:

$$
R_{\max}
=
4(d_0+\rho_d)+4h
=
1.433200000016.
$$

Strictly, we have:

$$
R_{\max}<\log5.
$$

Thus all $n\ge5$ are excluded, and the globally activated von Mangoldt indices remain:

$$
\boxed{2,3,4.}
$$

Meanwhile, the minimum strict distance from all $\pm\log n$ samples to any spline knot is:

$$
\boxed{
2.125281942005469\times10^{-2}.
}
$$

This is far greater than the parameter tube width, therefore:

- No prime-power sample crosses the support boundary;
- No sample crosses any internal spline knot;
- The polynomial piece identity of each matrix element remains invariant;
- The prime-power activation graph remains invariant throughout the tube.

---

## 9. Main Conclusion

This round established:

$$
\boxed{
\forall(d,\sigma)\in\mathcal T,
\quad
10^{-8}
<
\lambda_{\min}(M(d,\sigma),G(d,\sigma))
<
5\times10^{-8}.
}
$$

This implies that the near-zero value from RH-W-13 is not an isolated single point, nor a numerical coincidence valid only at a single rational parameter; at least within a two-dimensional rectangle of strictly non-zero area, it forms a continuous near-zero positive spectral band.

However, this remains a result within a fixed ten-dimensional subspace:

$$
\boxed{
\text{Finite-dimensional continuous near-zero spectral band}
\centernot\Longrightarrow
RH.
}
$$

---

## 10. Scope of Claims

This round does not:

- Prove the RH;
- Disprove the RH;
- Find a negative Weil witness;
- Prove that the parameter tube can be extended to infinite dimensions;
- Prove a direct one-to-one correspondence between the current near-zero band and zeta zeros.

What it accomplishes is a reproducible finite-dimensional continuous perturbation certificate.