# RH-W-07: Multi-Prime Support Chamber Compiler
## Advancing from a Single $n=2$ Coupling to a Prime-Power Sparse Arithmetic Graph

**Version:** v0.1  
**Date:** 2026-07-23  
**Research Project:** RH GAP Atlas / AI Mathematical Engineering Relay  
**Parent Node:** `RH-W-06-PRIME-ACTIVE-MATRIX`  
**Current Node:** `RH-W-07-MULTIPRIME-CHAMBER-COMPILER`  
**Status:** `CLOSED_FOR_ONE_9D_RATIONAL_SPLINE_FAMILY`  
**Boundary:** This document establishes finite-dimensional interval matrices and local positive definite certificates for the true Riemann–Weil explicit formula; it does not constitute a proof of RH, nor has an RH counterexample been found.

---

# 1. Why Not Just Look at the "Maximum Support Radius"

In the previous round, with only $n=2$, using the maximum correlation support radius was sufficient to exclude $n\ge3$. After moving to multiple primes, this approach is too coarse.

For translated bases $v_i,v_j$, let:

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

If:

$$
\operatorname{supp}(f_{ij})=[L_{ij},R_{ij}],
$$

then the discrete term for the prime power $n=p^k$ is active only when:

$$
\log n\in[L_{ij},R_{ij}]
$$

or:

$$
-\log n\in[L_{ij},R_{ij}].
$$

Therefore, what truly needs to be compiled is the **complete translated support window**, not a single radius. As the correlation window moves along the real axis, a certain $\log p^k$ will enter and subsequently leave.

This gives the prime-power activation graph a band-pass property, rather than a low-pass structure where "the larger the support, all old primes are permanently retained."

---

# 2. Fixed 9-Dimensional B-spline Dictionary

Take the centered cubic B-spline $\beta_3$, and fix:

$$
h=\frac1{10},
$$

along with nine translations, directly fixed by index:

$$
\boxed{t_j=\frac{j}{5},\qquad j=-4,-3,\ldots,4.}
$$

Define:

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

The correlation function is:

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

Its support is:

$$
\boxed{
[t_i-t_j-4h,\ t_i-t_j+4h].
}
$$

The maximum correlation support only reaches:

$$
|x|\le2<\log8.
$$

Thus, all $n\ge8$ are strictly excluded. Among $2\le n\le7$, only:

$$
\boxed{2,3,4,5,7}
$$

have non-zero von Mangoldt weights.

Note:

$$
\Lambda(2)=\Lambda(4)=\log2,
$$

However, $n=2$ and $n=4$ are sampled at the geometric positions of $\log2$ and $\log4$ respectively, so they are two distinct sparse coupling layers.

---

# 3. Compiled Prime-Power Activation Graph

Due to Toeplitz translation invariance, we only need to compute the lags $\ell=0,\ldots,8$.

| lag $\ell$ | $|t_i-t_j|$ | Maximum Absolute Value of Support | Truly Active Prime Powers |
|---:|---:|---:|---|
| 0 | $0$ | $2/5$ | None |
| 1 | $1/5$ | $3/5$ | None |
| 2 | $2/5$ | $4/5$ | $2$ |
| 3 | $3/5$ | $1$ | $2$ |
| 4 | $4/5$ | $6/5$ | $2,3$ |
| 5 | $1$ | $7/5$ | $2,3,4$ |
| 6 | $6/5$ | $8/5$ | $3,4$ |
| 7 | $7/5$ | $9/5$ | $3,4,5$ |
| 8 | $8/5$ | $2$ | $4,5,7$ |

The most important phenomenon is:

$$
\boxed{
\text{At }\ell=6\text{, }2\text{ has left the support window, but }3,4\text{ remain inside.}
}
$$

The furthest lag also does not see all of $2,3,4,5,7$, but only sees:

$$
\boxed{4,5,7.}
$$

Therefore, each prime power forms a set of matrix edges with a finite bandwidth and a finite life cycle.

---

# 4. Sparse Arithmetic Graph Decomposition

The complete 9-dimensional matrix can be written as:

$$
\boxed{
M=A_{\infty}+P_2+P_3+P_4+P_5+P_7,
}
$$

where:

- $A_{\infty}$ contains the endpoint, constant, and Archimedean blocks;
- $P_n$ is the sparse Toeplitz block for the prime power $n$;
- the non-zero lags of $P_n$ are entirely determined by whether $\pm\log n$ falls into the correlation support window.

This can be viewed as an arithmetic graph:

- the basis functions are nodes;
- each $p^k$ is an edge label;
- edges light up only when the relative displacement is approximately equal to $\log p^k$;
- the complete Weil geometry is a superposition of the continuous Archimedean background and discrete prime-power edges.

---

# 5. True Intervals for the Nine Toeplitz Lags

The computations in this round yield:

$$
\begin{aligned}
T_0&\in[0.178584904159,\ 0.178613529500],\\
T_1&\in[-0.100987756288,\ -0.100982154010],\\
T_2&\in[0.0493818162380,\ 0.0493818714463],\\
T_3&\in[-0.0257602730928,\ -0.0257602178844],\\
T_4&\in[0.0270514132119,\ 0.0270514684202],\\
T_5&\in[0.00228735758772,\ 0.00228741279606],\\
T_6&\in[0.0171981453966,\ 0.0171982006049],\\
T_7&\in[0.0216954720465,\ 0.0216955272549],\\
T_8&\in[-0.127273287125,\ -0.127273231916].
\end{aligned}
$$

The matrix is:

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

These intervals are generated directly from the complete explicit formula, without using zero truncation.

---

# 6. Second Engineering Correction of the Archimedean Tail Bound

The old tail bound enclosed:

$$
\sum_{k\ge K}
F(0)
\left(
\frac1{2k+1/2}-\frac1{2k+1}
\right)
$$

with a rough $O(K^{-1})$ majorant, which would drown out the near-critical minimum eigenvalue.

This round uses the closed form:

$$
\boxed{
\sum_{k=0}^{\infty}
\left(
\frac1{2k+1/2}-\frac1{2k+1}
\right)
=
\frac{\pi}{4}+\frac{\log2}{2}.
}
$$

Thus, the leading tail can be precisely enveloped by subtracting the finite rational part from the closed-form sum, leaving only the higher-derivative tail terms for the $p$-series majorant.

Furthermore, an adaptive cutoff is adopted:

- lags $0,1$ use $K=1000$;
- lags $2,\ldots,8$ use $K=100$ because the endpoint values are zero or the tail bounds are already extremely small.

This is not a change in the mathematical form, but rather concentrates the verification cost on the lags that truly control the overall error.

---

# 7. 9-Dimensional Exact Positive Definite Certificate

For each interval element, a fixed rational center matrix $C$ with a denominator of $10^{20}$ is selected, and the center offset is also included in the error matrix $E$:

$$
M=C+E.
$$

Yielding the maximum row-radius:

$$
\epsilon
=
\|E\|_{\infty}^{\mathrm{bound}}
\approx
2.0080572380713665\times10^{-5}.
$$

The verifier proves via purely rational $LDL^T$:

$$
\boxed{
C-\frac1{2000}I\succ0.
}
$$

Therefore:

$$
\lambda_{\min}(M)
>
\frac1{2000}-\epsilon.
$$

That is:

$$
\boxed{
\lambda_{\min}(M)
>
4.799194276192863\times10^{-4}.
}
$$

Thus, every true interval matrix contained in the certificate is positive definite:

$$
\boxed{M\succ0.}
$$

It must be reiterated:

$$
\boxed{
\text{Positive definiteness of this 9-dimensional subspace }\centernot\Longrightarrow RH.
}
$$

---

# 8. Prime-Power Ablation Experiment

To observe the effect of each discrete layer, we define an artificial cumulative matrix:

$$
M_S=A_{\infty}+\sum_{n\in S}P_n.
$$

Only the complete set:

$$
S=\{2,3,4,5,7\}
$$

is the true explicit formula; the other stages are merely mechanistic ablations, not new zeta objects.

In this round, four rational integer witnesses were found, strictly proving that when $2,3,4,5$ are added respectively, the same direction flips from negative to positive.

### Adding $2$

$$
c=(3,-3,-8,-6,0,6,8,3,-3),
$$

$$
c^TM_{\varnothing}c< -28.002,
$$

$$
c^TM_{\{2\}}c>26.543.
$$

### Adding $3$

$$
c=(12,2,-16,-16,-9,-16,-16,2,12),
$$

Flips from less than $-95.41$ to greater than $153.16$.

### Adding $4$

$$
c=(2,1,-2,-2,0,2,2,-1,-2),
$$

Flips from less than $-0.606$ to greater than $0.568$.

### Adding $5$

$$
c=(16,9,2,2,0,-2,-2,-9,-16),
$$

Flips from less than $-74.89$ to greater than $107.82$.

These results only demonstrate a strict sign flip in specified directions and are not equivalent to a complete stage-by-stage inertia proof.

For $7$, no strict sign flip was found in the fixed candidate set. This does not mean $P_7$ has no effect; it only means that this round did not prove a zero-crossing for any candidate direction.

---

# 9. What Was Actually Accomplished in This Round

Accomplished:

1. Upgraded from "support radius" to full shifted-window compilation;
2. Strictly enumerated the five von Mangoldt layers for $2,3,4,5,7$;
3. Established an activation graph where prime powers enter and leave;
4. Generated the true $9\times9$ multi-prime Weil interval matrix;
5. Improved the Archimedean leading tail;
6. Established an exact positive definite certificate on a fixed rational grid;
7. Proved with four rational witnesses that different prime-power layers can change the sign in specified directions.

Not accomplished:

1. A universal chamber compiler for arbitrary bases and arbitrary supports;
2. Unbounded support enumeration for all prime powers;
3. Automatic search for subspaces closest to zero or truly negative;
4. An RH proof or RH counterexample;
5. Lean/Coq formalized certificates.

---

# 10. Next Node

The next node is fixed as:

$$
\boxed{
\texttt{RH-W-08-CHAMBER-SEARCH-AND-REFINEMENT}
}
$$

Its task is not to blindly increase dimensionality, but to establish a two-tier system:

$$
\text{Fast floating-point/AI searcher}
\longrightarrow
\text{Candidate chambers and witnesses}
\longrightarrow
\text{Strict interval reconstructor}
\longrightarrow
\text{Purely rational verifier}.
$$

The searcher can be untrusted; the final certificate must be reproducible.

---

# References

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

The above sources provide background on the Weil form, fixed interval/finite-dimensional research, and tail bounds; the B-spline support chambers, interval implementations, and certificate decompositions in this engineering package are engineering instances within this research workflow.