# RH-W-16: Three-Parameter Near-Zero Spectral Tube

**版本：** v0.1  
**日期：** 2026-07-24  
**定位：** RH Engineering Relay — Batch 01, Round 16  
**聲明：** This document only establishes a strict parameter box over a fixed finite-dimensional Weil dictionary; it does not prove or disprove the Riemann Hypothesis.

---

## Abstract

RH-W-15 established a two-dimensional near-zero positive spectral tube for $(d,\sigma)$ at the fixed scale

$$
h_0=\frac{1797}{10000}
$$

This round incorporates the kernel scale $h$ itself into the parameter domain, establishing the first three-dimensional rational box:

$$
\boxed{
\left|h-\frac{1797}{10000}\right|\le10^{-8},\qquad
\left|d-\frac{893}{5000}\right|\le10^{-7},\qquad
|\sigma|\le10^{-7}
}.
$$

For every point within this box, a purely rational certificate proves:

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

The certificate consists of eight rigorous corner matrices, trilinear convex interpolation, pure second-order Taylor remainders, exact $LDL^T$ and a fixed integer Rayleigh witness.

---

## 1. Mixed-Order Dictionary

Each channel uses five basis functions:

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

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

where

$$
t_j^{(1)}=(j-2)d-\frac\sigma2,
\qquad
t_j^{(3)}=(j-2)d+\frac\sigma2.
$$

The correlation blocks remain:

$$
1\times1\to\beta_3,
\qquad
1\times3\to\beta_5,
\qquad
3\times3\to\beta_7.
$$

The full matrix dimension is ten.

---

## 2. Why $h$ is a Genuinely New Direction

A global channel scaling $\alpha$ only causes a congruence change of basis and does not alter the generalized spectrum; however, $h$ simultaneously changes:

1. The physical support width of each B-spline;
2. The positions of the samples $\pm\log p^k$ in normalized coordinates;
3. The Gram blocks;
4. The endpoint integrals;
5. The prime-power sampled values;
6. The Archimedean jump positions and tail terms.

Therefore, $h$ is not a gauge parameter, but a direction that actually changes the test subspace.

---

## 3. Trilinear Corner Architecture

The parameter box has a total of eight corners:

$$
(h_\pm,d_\pm,\sigma_\pm).
$$

For each corner, a complete rigorous Weil interval matrix $M_{abc}$ and an exact Gram matrix $G_{abc}$ are independently constructed.

The trilinear interpolation for any point within the box is a convex combination of the eight corner matrices. Thus, if the nonlinear curvature remainders are absorbed into diagonal perturbations, the positive definiteness of the entire box can be deduced from the positive definiteness of the eight corners.

---

## 4. Second Derivatives in the Scale Direction

Let

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

Then

$$
\partial_h f=-\frac{\nu}{h}\beta_r'(\nu),
$$

and

$$
\partial_h^2f
=\frac{2\nu\beta_r'(\nu)+\nu^2\beta_r''(\nu)}{h^2}.
$$

Using

$$
|\nu|\le q,
\qquad
\|\beta_r'\|_\infty\le1,
\qquad
\|\beta_r''\|_\infty\le4,
\qquad
\|\beta_r'''\|_\infty\le8,
$$

we obtain

$$
|\partial_h^2f|
\le
\frac{2q+4q^2}{h_{\min}^2},
$$

$$
|\partial_x\partial_h^2f|
\le
\frac{2+16q+8q^2}{h_{\min}^3}.
$$

Substituting these two bounds into the full Weil explicit formula, including the out-of-support Archimedean tails corrected in RH-W-15, yields the second-order integer bounds for the scale:

| correlation degree $r$ | center second-order bound | $h$ second-order bound |
|---:|---:|---:|
| 3 | 2494 | 17279 |
| 5 | 3110 | 40860 |
| 7 | 3697 | 78886 |

The scale curvature is significantly larger than the pure translation curvature, so this round adopts a smaller

$$
\rho_h=10^{-8}.
$$

---

## 5. Three-Dimensional Taylor Remainders

For a matrix element $F(c,h)$, the change in center coordinates is

$$
\Delta c=(i-j)\Delta d+b\Delta\sigma,
$$

where $b=1$ for cross-channel and $b=0$ for self-channel.

The conservative remainder for the trilinear interpolation is taken as

$$
|R_F|
\le
\frac12L_{cc}
\left((i-j)^2\rho_d^2+b^2\rho_\sigma^2\right)
+
\frac12L_{hh}\rho_h^2.
$$

The Gram remainder uses the same structure and the analytical second-order bounds of the B-splines.

The maximum combined row remainder for the final full matrix is

$$
\boxed{
\epsilon_{3D}
=1.128736500411401062\times10^{-9}
}.
$$

The maximum row remainder for the Gram matrix is

$$
4.114011040652843\times10^{-11}.
$$

---

## 6. Prime-Power Chambers Remain Fixed

The maximum correlation support radius in the entire three-dimensional box is

$$
R_{\max}=1.43320054<\log5.
$$

Therefore, the globally active von Mangoldt indices remain exactly

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

The minimum strict distance from all $\pm\log n$ samples to the nearest spline knot is

$$
\boxed{
2.1252319440054689\times10^{-2}
}.
$$

Thus, there are no prime-powers entering or exiting the support within the box, nor are there any identity switches of the polynomial pieces.

---

## 7. Exact Lower Bound

For each corner, the center of the interval matrix is first quantized to a fixed rational grid, and then it is proved that

$$
C_{abc}-10^{-8}G_{abc}-(\epsilon_{abc}+\epsilon_{3D})I\succ0.
$$

The exact $LDL^T$ pivots for all eight corners are strictly positive.

By the convexity of the positive definite cone and trilinear interpolation, we obtain for the entire parameter box:

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

Therefore,

$$
\lambda_{\min}>10^{-8}.
$$

---

## 8. Exact Upper Bound

We continue to use the fixed integer witness from RH-W-13. Taking the convex hull of the quadratic forms and Gram values at the eight corners, and then adding the three-dimensional Taylor remainders, yields the Rayleigh quotient upper bound for the entire box:

$$
\frac{c^TM(h,d,\sigma)c}{c^TG(h,d,\sigma)c}
<5\times10^{-8}.
$$

The actual worst-case upper bound in the certificate is approximately

$$
4.023762155948623\times10^{-8}.
$$

---

## 9. Non-Rigorous Spectral Observations

The midpoint generalized eigenvalues of the eight corners fall approximately in:

$$
3.9958900011\times10^{-8}
\le\lambda_0^{\mathrm{mid}}
\le
3.9959197359\times10^{-8}.
$$

These numbers are for exploratory observation only and do not participate in the certificate.

They show that the actual drift caused by $h$ within a radius of $10^{-8}$ is still much smaller than the global remainder reserved in the certificate.

---

## 10. Conclusion and Scope

This round establishes the first three-dimensional continuous near-zero positive spectral parameter box:

$$
\boxed{
(h,d,\sigma)\in\mathcal B_3
\Longrightarrow
10^{-8}<\lambda_{\min}<5\times10^{-8}
}.
$$

It proves that the near-zero positive spectral structure exists not only at a single point or on a fixed-scale plane, but extends to at least a non-degenerate three-dimensional volume.

However, it remains a finite-dimensional result over a fixed ten-dimensional dictionary:

$$
\boxed{
\text{Three-dimensional finite parameter tube}\centernot\Longrightarrow\mathrm{RH}.
}
$$