# RH-W-12: Mixed-Order Dictionary and Cross-Cancellation Mode v0.1

- Node: `RH-W-12-MIXED-ORDER-DICTIONARY`
- Date: 2026-07-23
- Status: The first true $m=1/3$ mixed-order Weil dictionary is completed; 10-dimensional interval matrix and exact certificates have been established
- Conclusion boundary: Finite-dimensional positivity does not imply RH

---

## 1. Research Problem

`RH-W-11` proved that in the B-spline family, lower-order kernels are more sensitive to the prime boundary, but their Archimedean tail bounds are more difficult; higher-order kernels more easily form strict certificates, but they smooth the new arithmetic layers into higher-order small quantities.

This round no longer requires a single kernel to simultaneously undertake two conflicting tasks, but instead places two families of bases into the same test space:

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

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

By the convolution closure of centered cardinal B-splines:

$$
\beta_m*\beta_n=\beta_{m+n+1},
$$

The three types of correlation kernels in the matrix are:

$$
1\times1\longmapsto\beta_3,
$$

$$
1\times3\longmapsto\beta_5,
$$

$$
3\times3\longmapsto\beta_7.
$$

Therefore, the same Hermitian matrix simultaneously contains three activation orders of the prime layer:

$$
\varepsilon^3,\qquad \varepsilon^5,\qquad \varepsilon^7.
$$

---

## 2. The First Strict Chamber

This round fixes:

$$
h=\frac3{20},\qquad d=\frac9{40},\qquad N=5
$$

Each channel has five translations:

$$
t_j=\left(j-2\right)d,\qquad j=0,1,2,3,4.
$$

The total dimension is:

$$
2N=10.
$$

The complete matrix has a block Toeplitz structure:

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

The Gram matrix is similarly:

$$
G=
\begin{pmatrix}
G^{11}&G^{13}\\
G^{31}&G^{33}
\end{pmatrix}.
$$

---

## 3. Multi-Distance Arithmetic Sensing Map

Since the half-support widths of the three correlation kernels are respectively:

$$
2h,\qquad3h,\qquad4h,
$$

The same lag will not have the same prime-power field of view.

### $1\times1$ block: degree 3

| lag | Activated prime powers |
|---:|---|
| 0 | None |
| 1 | None |
| 2 | $2$ |
| 3 | $2$ |
| 4 | $2,3$ |

### $1\times3$ block: degree 5

| lag | Activated prime powers |
|---:|---|
| 0 | None |
| 1 | None |
| 2 | $2$ |
| 3 | $2,3$ |
| 4 | $2,3$ |

### $3\times3$ block: degree 7

| lag | Activated prime powers |
|---:|---|
| 0 | None |
| 1 | $2$ |
| 2 | $2$ |
| 3 | $2,3$ |
| 4 | $2,3,4$ |

The maximum correlation radius is:

$$
4d+4h=\frac32<\log5.
$$

Thus, all von Mangoldt terms for $n\ge5$ are strictly excluded, and the complete arithmetic set is exactly:

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

Here, $4=2^2$ must be treated as an independent prime-power layer; its coefficient is:

$$
\frac{\Lambda(4)}{\sqrt4}=\frac{\log2}{2},
$$

However, the sampling distance is $\log4$, which cannot be merged with $n=2$.

---

## 4. Three Archimedean Tail Bounds Working Simultaneously

For correlation degree $r\in\{3,5,7\}$, let:

$$
F(x)=f(x)+f(-x).
$$

Integration by parts retains all available even-order derivatives:

### degree 3

$$
I(a)=\frac{F(0)}a+\frac{F''(0)}{a^3}+R_4(a),
$$

$$
|R_4(a)|\le\frac{B_3}{a^4}.
$$

### degree 5

$$
I(a)=\frac{F(0)}a+\frac{F''(0)}{a^3}+\frac{F^{(4)}(0)}{a^5}+R_6(a).
$$

### degree 7

$$
I(a)=\frac{F(0)}a+\frac{F''(0)}{a^3}+\frac{F^{(4)}(0)}{a^5}+\frac{F^{(6)}(0)}{a^7}+R_8(a).
$$

This round uses respectively:

$$
K_3=900,\qquad K_5=320,\qquad K_7=180.
$$

The results reflect the regularity cost:

- The degree-$3$ lag interval width is approximately $5.08\times10^{-8}$;
- degree-$5$ is approximately $2.46\times10^{-10}$;
- degree-$7$ is approximately $3.77\times10^{-12}$.

The lower-order channel indeed requires a larger tail term budget.

---

## 5. Differences Between Single and Mixed Channels

Exploratory generalized eigenvalues are used only for explanation and do not participate in the proof path:

$$
\lambda_{\min}^{(1)}\approx6.0218\times10^{-2},
$$

$$
\lambda_{\min}^{(3)}\approx4.9910\times10^{-3},
$$

$$
\lambda_{\min}^{\mathrm{mixed}}\approx9.7385\times10^{-4}.
$$

The important phenomenon is not that the lower-order channel alone produces a lower spectral value. Quite the opposite:

$$
\lambda_{\min}^{(1)}>\lambda_{\min}^{(3)}>\lambda_{\min}^{\mathrm{mixed}}.
$$

That is to say, the new low mode comes from:

$$
\boxed{\text{the cross-coupling of } m=1 \text{ and } m=3}
$$

rather than any isolated kernel.

---

## 6. Exact Spectral Sandwich Certificates

### 6.1 Mixed Space Lower Bound

A purely rational verifier proves:

$$
C-\frac1{2000}G-\epsilon I\succ0,
$$

where the maximum column error is:

$$
\epsilon\approx1.53285\times10^{-7}.
$$

Therefore:

$$
\boxed{
\lambda_{\min}(M,G)>\frac1{2000}
}.
$$

### 6.2 Mixed Space Upper Bound Witness

For the integer vector:

$$
c=
(-1198150,-2003160,-2002921,-2003160,-1198150,
$$

$$
5477016,10000000,9638591,10000000,5477016)^T,
$$

An interval verifier proves:

$$
c^TMc<\frac1{1000}c^TGc.
$$

Thus:

$$
\boxed{
\frac1{2000}
<\lambda_{\min}^{\mathrm{mixed}}
<\frac1{1000}
}.
$$

### 6.3 Isolated Channel Lower Bounds

The same verifier additionally proves:

$$
\boxed{
\lambda_{\min}^{(1)}>\frac1{20}
},
$$

$$
\boxed{
\lambda_{\min}^{(3)}>\frac1{250}
}.
$$

Therefore, there is a strict separation:

$$
\lambda_{\min}^{\mathrm{mixed}}
<\frac1{1000}
<\frac1{250}
<\lambda_{\min}^{(3)},
$$

and:

$$
\lambda_{\min}^{(1)}>\frac1{20}.
$$

This is not a floating-point ordering, but an exact rational certificate.

---

## 7. Cross-Cancellation Mode

For the above rational witness, after normalizing by $c^TGc$:

$$
\frac{Q_{11}(c_1)}{c^TGc}
\in
[0.00663962797,0.00663965873],
$$

$$
\frac{2Q_{13}(c_1,c_3)}{c^TGc}
\in
[-0.01378995092,-0.01378994947],
$$

$$
\frac{Q_{33}(c_3)}{c^TGc}
\in
[0.00812415339,0.00812415345].
$$

Both self-blocks are strictly positive:

$$
Q_{11}>0,\qquad Q_{33}>0,
$$

but the cross-block is strictly negative:

$$
2Q_{13}<0.
$$

It cancels approximately:

$$
93.40\%
$$

of the self-energy, ultimately leaving:

$$
\frac{c^TMc}{c^TGc}
\in
[0.000973830446,0.000973862704].
$$

Therefore, this low mode can be named:

$$
\boxed{\text{cross-regularity cancellation mode}}
$$

or in Chinese:

$$
\boxed{\text{cross-regularity cancellation mode}}.
$$

---

## 8. The Effect of Prime-Powers on the Witness

On the same rational witness, the normalized contributions of each arithmetic layer are:

$$
P_2\approx-0.30391314387,
$$

$$
P_3\approx-0.02581098712,
$$

$$
P_4\approx-4.19481\times10^{-6}.
$$

These numbers cannot be interpreted individually as complete spectral values, because the endpoints, constants, and Archimedean background simultaneously provide a large amount of positive compensation; they merely indicate how the witness reads the different arithmetic distance layers.

---

## 9. Conclusion Boundary

What is proved in this round is:

1. The Gram matrix of the mixed-order B-spline dictionary is exactly positive definite;
2. The 10-dimensional interval matrix of the complete explicit formula is exactly positive definite;
3. The mixed spectral bottom is exactly sandwiched between $1/2000$ and $1/1000$;
4. The spectral bottoms of the two isolated channels are both strictly higher than the mixed spectral bottom;
5. The cross-block contribution on the rational witness is strictly negative and forms a near-cancellation.

What is not proved in this round is:

- Weil positivity in arbitrary dimensions or the complete limit;
- RH;
- That a mixed kernel is necessarily closer to any global extremum than a single kernel;
- That cross-regularity cancellation necessarily corresponds one-to-one with a specific structure of zeta zeros.

The correct conclusion is:

$$
\boxed{
\text{Mixed regularity is not channel splicing, but generates new spectral modes that do not exist in a single channel.}
}
$$