# RH-W-06: First Prime Activation and Arithmetic Support Chambers
## From "Human-Computable" to Scalable Discrete-Continuous Coupling Engineering

**Version:** v0.1  
**Date:** 2026-07-23  
**Research Project:** RH GAP Atlas / AI Mathematical Engineering Relay  
**Parent Node:** `RH-W-05-REAL-MATRIX-ENCLOSURE`  
**Current Node:** `RH-W-06-PRIME-ACTIVE-MATRIX`  
**Status:** `CLOSED_FOR_PRIME2_2D_AND_5D_SPLINE_CHAMBERS`  
**Nature:** Finite-dimensional interval computation of the real Riemann–Weil explicit formula; neither a proof nor a counterexample of the RH

---

# 0. Problem of the Current Round

The previous round deliberately excluded all prime terms to complete the first real Weil matrix interval enclosure. The computation itself could, of course, be done by a human; the engineering value lies not in replacing a single manual calculation, but in establishing:

$$
\text{Function specification}
\longrightarrow
\text{Support analysis}
\longrightarrow
\text{Activated prime-power set}
\longrightarrow
\text{Explicit formula decomposition}
\longrightarrow
\text{Rational interval matrix}
\longrightarrow
\text{Small exact verifier}.
$$

This round introduces prime terms into the matrix for the first time, deliberately controlling the support within:

$$
\log 2<R<\log 3.
$$

## 0.1 Relationship with Existing Research

This node does not claim to reinvent the finite-dimensionalization of Weil's criterion. Suzuki has studied the Weil quadratic form via the screw-function and fixed interval forms; the 2026 work on the finite Guinand–Weil dictionary also explicitly connected the truncated form, finite Galerkin matrices, and Archimedean tail bounds. The role of this engineering package is narrower:

1. Select a B-spline dictionary that can be processed by rational piecewise polynomials;
2. Formulate the $\log n$ thresholds into machine-readable support chambers;
3. Allow the single $n=2$ term to be isolated and observed;
4. Generate reproducible rational interval matrices and small verifiers;
5. Separately log verifier failures and mathematical property failures.

Therefore, the outcome of this round is a research engineering and certificate pipeline, asserting no new RH equivalence theorems.

Thus, in the von Mangoldt sum, only:

$$
\boxed{n=2}
$$

can be activated. This allows the effect of the first prime to be observed independently, without being mixed with $3,4,5,\ldots$.

---

# 1. Fixed Explicit Formula

For a suitable test function $f$ in additive coordinates, this project adopts:

$$
\begin{aligned}
W(f)
={}&
\int_{-\infty}^{\infty}
f(x)(e^{x/2}+e^{-x/2})\,dx
\\
&-
\sum_{n\ge1}
\frac{\Lambda(n)}{\sqrt n}
f(\log n)
-
\sum_{n\ge1}
\frac{\Lambda(n)}{\sqrt n}
f(-\log n)
\\
&-(\log4\pi+\gamma)f(0)
\\
&-
\int_0^\infty
\frac{e^{x/2}(f(x)+f(-x))-2f(0)}{e^x-e^{-x}}\,dx.
\end{aligned}
$$

For real functions $v_i,v_j$, let:

$$
\widetilde v_j(x)=v_j(-x),
$$

$$
f_{ij}=v_i*\widetilde v_j,
$$

$$
M_{ij}=W(f_{ij}).
$$

The prime block is:

$$
P_{ij}
=
-
\sum_{n\ge2}
\frac{\Lambda(n)}{\sqrt n}
\bigl(f_{ij}(\log n)+f_{ij}(-\log n)\bigr).
$$

If:

$$
\operatorname{supp}(f_{ij})\subseteq[-R_{ij},R_{ij}],
$$

then only $n$ satisfying:

$$
\log n\le R_{ij}
$$

can possibly contribute. This is the basic rule of the "arithmetic support chamber" in this round.

---

# 2. Arithmetic Support Chambers

An **arithmetic support chamber** for a set of basis parameters is defined as a parameter region where:

1. The support endpoints of each correlation function do not cross any $\log n$;
2. The activated set of von Mangoldt indices remains fixed;
3. Each $f_{ij}(\pm\log n)$ stays within a fixed spline polynomial segment.

Within the same chamber, the discrete combinatorial structure of the explicit formula does not change; the matrix elements vary only smoothly or piecewise-analytically with the continuous parameters. Only when crossing:

$$
R=\log n
$$

is a new prime-power sample added.

Therefore, the support radius is not an ordinary numerical parameter, but a switch for the discrete arithmetic structure:

$$
R<\log2
\Rightarrow
\text{prime-free},
$$

$$
\log2<R<\log3
\Rightarrow
\text{only }n=2,
$$

$$
\log3<R<\log4
\Rightarrow
n=2,3,
$$

and so on. Note that $\Lambda(n)$ is non-zero only on prime powers, so the actual activated set must be further filtered by the von Mangoldt function.

---

# 3. Two-Dimensional Isolated Coupling Experiment

Taking the centered cubic B-spline $\beta_3$, let:

$$
h=\frac1{10},
\qquad
t_1=-\frac14,
\qquad
t_2=\frac14,
$$

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

The support of each basis function is:

$$
\operatorname{supp}(v_1)
=
\left[-\frac9{20},-\frac1{20}\right],
$$

$$
\operatorname{supp}(v_2)
=
\left[\frac1{20},\frac9{20}\right].
$$

The two do not overlap, so:

$$
\langle v_1,v_2\rangle_{L^2}=0.
$$

The correlation function remains:

$$
f_{ij}(x)
=
\beta_7\!\left(
\frac{x-(t_i-t_j)}h
\right).
$$

The diagonal correlation support is:

$$
\operatorname{supp}(f_{11})
=
\operatorname{supp}(f_{22})
=
\left[-\frac25,\frac25\right]
\subset(-\log2,\log2),
$$

Therefore:

$$
P_{11}=P_{22}=0.
$$

The cross-correlation support is:

$$
\operatorname{supp}(f_{12})
=
\left[-\frac9{10},-\frac1{10}\right],
$$

$$
\operatorname{supp}(f_{21})
=
\left[\frac1{10},\frac9{10}\right].
$$

From:

$$
\log2<\frac9{10}<\log3
$$

it can be seen that the cross term only activates $n=2$. Specifically:

$$
f_{12}(\log2)=0,
\qquad
f_{12}(-\log2)>0,
$$

So:

$$
P_{12}
=
-
\frac{\log2}{\sqrt2}
f_{12}(-\log2)<0.
$$

The strict interval is:

$$
\boxed{
P_{12}
\in
[-0.014521033659419367232,\,
 -0.014521033659419367232]
}.
$$

The interval displays identical decimals only because the upper and lower bounds are extremely tight; distinct rational endpoints are still preserved in the certificate.

---

# 4. Two-Dimensional Matrix: The First Prime as Non-Local Coupling

When the prime block is removed, the cross term is:

$$
M_{12}^{(0)}
\in
[0.080368663737242510003,\,
 0.080368664575458659272].
$$

After adding $n=2$:

$$
M_{12}
=
M_{12}^{(0)}+P_{12}
$$

it becomes:

$$
\boxed{
M_{12}
\in
[0.065847630077823141037,\,
 0.065847630916039290305]
}.
$$

The diagonal terms are:

$$
M_{11}=M_{22}
\in
[0.17817969315586257295,\,
 0.17981738868199476089].
$$

Thus, the complete matrix is strictly positive definite in this two-dimensional space.

More importantly, the splitting of the parity modes. For:

$$
e_+=(1,1),
\qquad
e_-=(1,-1),
$$

the negative cross-coupling of $n=2$ causes:

$$
Q(e_+)=M_{11}+M_{12}
$$

to decrease, while:

$$
Q(e_-)=M_{11}-M_{12}
$$

increases.

Without $n=2$:

$$
Q^{(0)}(e_+)
\in
[0.25854835689310506908,\,
 0.26018605325745342016],
$$

After adding $n=2$:

$$
Q(e_+)
\in
[0.24402732323368570011,\,
 0.24566501959803405120].
$$

Without $n=2$:

$$
Q^{(0)}(e_-)
\in
[0.097811028580403913679,\,
 0.099448724944752264765],
$$

After adding $n=2$:

$$
Q(e_-)
\in
[0.11233206223982328265,\,
 0.11396975860417163373].
$$

This provides a clean local picture:

> The first prime is not added to the local energy of each wave packet itself, but acts as a non-local arithmetic coupling between two wave packets separated by approximately $\log2$.

This statement merely describes the matrix structure of the present finite basis and explicit formula; it is not a new RH theory.

---

# 5. Five-Dimensional Toeplitz Support Chamber

To step beyond the scale of manual calculation, take five translations:

$$
t_k
\in
\left\{
-\frac3{10},
-\frac3{20},
0,
\frac3{20},
\frac3{10}
\right\}.
$$

Still let:

$$
v_k(x)
=h^{-1/2}\beta_3\!\left(\frac{x-t_k}{h}\right),
\qquad h=\frac1{10}.
$$

By translation invariance, the Weil matrix depends only on $|i-j|$, so it is a symmetric Toeplitz matrix:

$$
M_{ij}=T_{|i-j|}.
$$

The correlation support radius for the $k$-th lag is:

$$
R_k
=
\frac25+\frac{3k}{20}.
$$

Namely:

$$
R_0=\frac25,
\quad
R_1=\frac{11}{20},
\quad
R_2=\frac7{10},
\quad
R_3=\frac{17}{20},
\quad
R_4=1.
$$

From:

$$
R_0,R_1<\log2<R_2,R_3,R_4<\log3
$$

we obtain the exact activation pattern:

$$
\boxed{
[\mathrm{off},\mathrm{off},\mathrm{on},\mathrm{on},\mathrm{on}]
}.
$$

So $n=2$ only couples bases separated by at least two grid units, and all $n\ge3$ are excluded by the support.

---

# 6. Five-Dimensional Real Matrix

The lag intervals obtained are:

$$
T_0
\in
[0.17830890976252,\,0.17961569170656],
$$

$$
T_1
\in
[-0.11350184334353,\,-0.11316488850491],
$$

$$
T_2
\in
[-0.0046696978281104,\,-0.0046656145320484],
$$

$$
T_3
\in
[0.064390925313298,\,0.064390925592652],
$$

$$
T_4
\in
[-0.025760245628262,\,-0.025760245348908].
$$

where the $n=2$ block is:

$$
P_0=P_1=0,
$$

$$
P_2
\approx
-6.901841001562\times10^{-13},
$$

$$
P_3
\approx
-0.0022566345837752,
$$

$$
P_4
\approx
-0.12737368742383.
$$

$P_2$ is extremely small because $R_2=0.7$ is only slightly larger than $\log2$, and the sample has just entered the edge of the spline support. This shows that "activation" itself also has a continuous intensity: when

$$
R\downarrow\log2
$$

the newly added discrete sample is born smoothly from the support boundary.

---

# 7. Why Direct Interval LDL Fails

Executing a naive interval $LDL^T$ directly on the complete interval matrix yields for the fourth pivot:

$$
[-0.0194265,\,0.0826456],
$$

making it impossible to determine its sign.

This is not because the matrix is truly indefinite, but because naive interval arithmetic suffers from dependency blow-up when the same variables are reused. This failure is preserved as an engineering record; a "verifier failure" must not be mischaracterized as a "mathematical property failure."

This round instead uses:

$$
\text{Exact midpoint spectral margin}
+
\text{Interval perturbation operator norm bound}.
$$

See the second document for the complete certificate.

---

# 8. Completed and Uncompleted in This Round

Completed:

1. Strict support chambers activating only $n=2$;
2. Prime-only cross coupling of two-dimensional orthogonal wave packets;
3. Five-dimensional Toeplitz automated matrix;
4. Rational interval evaluation for each prime sample;
5. Analytic exclusion of $n\ge3$;
6. Five-dimensional complete positive-definite certificate;
7. An exact witness of a prime-free negative direction being flipped to positive by $n=2$.

Uncompleted:

1. Did not find a real Weil negative direction;
2. Did not advance the truth or falsity of the RH;
3. Did not handle multi-prime chambers for $n=3,4,5,\ldots$;
4. Did not establish global parameter optimization or effective search complexity;
5. Have not yet ported to a fully formalized backend in Arb, Lean, or Coq.

The next engineering node is:

$$
\boxed{
\texttt{RH-W-07-MULTIPRIME-CHAMBER-COMPILER}
}
$$

The goal is to automatically compile:

$$
(h,t_1,\ldots,t_N)
$$

into:

- active prime-power set;
- support and knot chamber;
- sparse prime block;
- archimedean block;
- interval matrix;
- exact positivity / negative-witness certificate.

---

# References

1. E. Bombieri, *Problems of the Millennium: The Riemann Hypothesis*, in *The Millennium Prize Problems*, Clay Mathematics Institute / AMS, 2006.
2. M. Suzuki, *Weil's quadratic form via the screw function*, arXiv:2606.09096, 2026.
3. A. Groskin, *A finite Guinand–Weil dictionary and archimedean tail order for the truncated Weil quadratic form*, arXiv:2607.02828, 2026.