# RH-W-08: Chamber Search and Rigorous Refinement
## From Floating-Point Explorer to 13-Dimensional Generalized Positive Margin Certificate

**Version:** v0.1  
**Date:** 2026-07-23  
**Research Project:** RH GAP Atlas / AI Mathematical Engineering Relay  
**Parent Node:** `RH-W-07-MULTIPRIME-CHAMBER-COMPILER`  
**Current Node:** `RH-W-08-CHAMBER-SEARCH-AND-REFINEMENT`  
**Status:** `CLOSED_FOR_ONE_FIXED_SEARCH_GRID_AND_ONE_REFINED_13D_CHAMBER`  
**Boundary:** This document only proves a strict positive margin on a fixed 13-dimensional subspace; it does not constitute a proof of RH, nor does it find a counterexample to RH.

---

# 1. The True Object of Engineering in This Round

The previous round successfully constructed interval matrices for a pre-specified multi-prime supported chamber. However, if every set:

$$
(h,d,N)
$$

is manually selected by humans, and the decision of which set is worth rigorous computation is also made manually, the research still fails to form a scalable relay system.

Therefore, this round splits the workflow into two distinct, non-conflated layers:

$$
\boxed{
\text{Cheap Exploration Layer}
\rightarrow
\text{Candidate Ranking}
\rightarrow
\text{Rigorous Reconstruction Layer}
\rightarrow
\text{Pure Rational Verifier}
}
$$

The explorer may use floating-point numbers, numerical integration, eigenvalue algorithms, and heuristic ranking. It can only output:

```text
NUMERICAL_CANDIDATE
```

Only the interval generator and the exact verifier can output:

```text
CERTIFIED_POSITIVE_GENERALIZED_MARGIN
```

or:

```text
CERTIFIED_NEGATIVE_WITNESS
```

This round did not obtain the latter.

---

# 2. The Searched B-spline Chamber Family

Fixing the centered cubic B-spline $\beta_3$, we define:

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

where the translations adopt an equidistant configuration:

$$
t_j=jd.
$$

The correlation function of two basis elements is:

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

Its support is:

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

Thus, the parameters $(h,d,N)$ simultaneously control:

1. The width of a single wave packet;
2. Wave packet overlap and the condition number of the Gram matrix;
3. The maximum visible logarithmic distance;
4. Which $\log p^k$ enter the correlation support;
5. The lowest generalized Rayleigh quotient of the finite-dimensional Weil matrix.

---

# 3. Exploration Grid

This round fixes a finite grid:

$$
h\in
\left\{
\frac3{50},\frac2{25},\frac1{10},\frac3{25},\frac3{20}
\right\},
$$

$$
\frac dh\in
\left\{
\frac32,2,\frac52,3,\frac72,4
\right\},
$$

$$
N\in\{5,7,9,11,13\}.
$$

and excludes combinations where the maximum correlation radius exceeds:

$$
\frac{19}{5}
$$

The final actual scan covers:

$$
\boxed{122\text{ chambers}}
$$

The complete ranking is saved in:

- `chamber_search_results.csv`
- `chamber_search_results.json`
- `chamber_search_summary.json`

This grid is not an exhaustive search, but merely the first fixed, reproducible exploration batch.

---

# 4. The Exploration Layer Computes Generalized Eigenvalues

Under different translation spacings, the B-spline basis elements may overlap; therefore, one cannot simply compute the standard eigenvalues of the Weil matrix $M$.

The Gram matrix must be computed simultaneously:

$$
G_{ij}
=
\langle v_i,v_j\rangle_{L^2}.
$$

The exploration ranking uses:

$$
\boxed{
\lambda_{\min}^{\mathrm{explore}}
=
\min_{c\ne0}
\frac{c^TMc}{c^TGc}
}
$$

instead of:

$$
\lambda_{\min}(M).
$$

This avoids mistaking basis overlap or condition number issues for near-zero directions inherent to the Weil geometry itself.

---

# 5. Top Candidate

The top-ranked parameters from the coarse search are:

$$
\boxed{
h=\frac3{20},\qquad
d=\frac9{40},\qquad
N=13
}
$$

Its maximum correlation radius is:

$$
12d+4h
=
\frac{33}{10}.
$$

The exploration layer yields:

$$
\lambda_{\min}^{\mathrm{explore}}
\approx
1.20209013459\times10^{-5}.
$$

This numerical value is used solely for ranking and cannot be used for a proof of positivity.

---

# 6. Complete Prime-Power Enumeration

Because:

$$
\frac{33}{10}<\log28,
$$

all $n\ge28$ are excluded by the support geometry.

Between $2\le n\le27$, the indices with non-zero von Mangoldt weights are exactly:

$$
\boxed{
2,3,4,5,7,8,9,11,13,16,17,19,23,25,27
}.
$$

Among them:

- $8=2^3$;
- $9=3^2$;
- $16=2^4$;
- $25=5^2$;
- $27=3^3$.

Each prime power is sampled at its own geometric position:

$$
\log p^k
$$

, and cannot simply be merged by their underlying primes.

The activation set for the furthest lag is:

$$
\{9,11,13,16,17,19,23,25,27\}.
$$

This demonstrates once again that the prime-power graph features entries and exits, rather than simple accumulation.

---

# 7. Why the First Round of Rigorous Reconstruction Failed

Initially, the Archimedean tail bound from the previous version was still used:

$$
|F_2|S_3+|F_4|S_5+B_6S_7,
$$

which treats all higher-order tail terms as having unknown signs and uses an absolute value majorant.

Even with the first $1400$ terms computed exactly, the interval row-radius of the 13-dimensional matrix was still approximately:

$$
3.72847\times10^{-6}.
$$

This interval could not prove the matrix to be positive definite, so the status correctly remained as:

```text
INCONCLUSIVE
```

This does not mean the candidate was proven indefinite, nor does it indicate a counterexample to RH; it simply means **the resolution of the certificate was lower than the spectral scale of the candidate**.

The failure record is saved in:

`refinement_attempt_old_tail_inconclusive.json`

---

# 8. Results After Tail Bound Refinement

This round preserves the signs of $F_2, F_4, F_6$, and establishes upper and lower bounds respectively for:

$$
S_3(K),\qquad S_5(K),\qquad S_7(K)
$$

, placing only the seventh-derivative remainder term into the symmetric error.

Therefore, it only requires:

- lag $0,1,2$: $K=500$;
- other lags: $K=180$;

to compress the full 13-dimensional interval row-radius to:

$$
\boxed{
\epsilon
\approx
5.91302180744\times10^{-8}
}.
$$

Compared to the old method, this is a reduction of approximately:

$$
63.06\text{ times}.
$$

The exploratory generalized eigenvalue at the center of the rigorous interval is:

$$
1.20203314305\times10^{-5},
$$

which is consistent with the coarse search ranking value up to the significant scale.

---

# 9. Exact Generalized Positive Margin Certificate

Let $[M]$ be the complete 13-dimensional interval matrix, and $G$ be the exact rational Gram matrix.

On the fixed rational grid:

$$
10^{-26}\mathbb Z
$$

we select a center matrix $C$, and let $\epsilon$ be the maximum row-sum bound of all interval deviations.

The pure rational verifier proves:

$$
\boxed{
C-\frac1{100000}G-\epsilon I\succ0.
}
$$

For any true matrix contained within the interval matrix:

$$
M'=C+E,
$$

we have:

$$
|c^TEc|
\le
\epsilon\|c\|_2^2.
$$

Thus, for all non-zero $c\in\mathbb R^{13}$:

$$
\boxed{
c^TM'c
>
\frac1{100000}c^TGc.
}
$$

That is:

$$
\boxed{
\lambda_{\min}(M',G)>10^{-5}.
}
$$

The certificate status is:

```text
CERTIFIED_POSITIVE_GENERALIZED_MARGIN
```

However, its scope is limited to this fixed 13-dimensional subspace, so:

$$
\boxed{
\text{13-dimensional strict positivity}
\centernot\Longrightarrow RH.
}
$$

---

# 10. What Was Closed in This Round

This round did not rely on "guessing the answer" through search, but rather closed the following engineering chain:

$$
\boxed{
\text{Fixed Grid}
\rightarrow
122\text{ Candidates}
\rightarrow
\text{Generalized Spectral Ranking}
\rightarrow
\text{Top Candidate}
\rightarrow
\text{Complete Prime-Power Reconstruction}
\rightarrow
\text{Automatic Tail Bound Upgrade}
\rightarrow
\text{Exact Certificate}
}
$$

The roles of the search layer and the proof layer have also been strictly separated:

- AI / floating-point algorithms are responsible for finding regions worth the computational cost;
- The interval generator is responsible for translating candidates into rational envelopes;
- The exact verifier is responsible for determining the certificate status;
- A positive finite-dimensional result can never be elevated to a proof of RH.

---

# 11. Unfinished GAPs

This round did not resolve:

1. The parameter space outside the search grid;
2. Non-equidistant translations;
3. Different spline degrees;
4. Adaptively adding basis elements;
5. The effective convergence rate as the support radius tends to infinity;
6. True negative witnesses;
7. Lean / Coq formalized backends.

In particular, one cannot infer from this round that:

$$
\lambda_{\min}(a)>0
$$

holds for all support scales.

---

# 12. Next Node

The next round is fixed as:

$$
\boxed{
\texttt{RH-W-09-ADAPTIVE-CHAMBER-CONTINUATION}
}
$$

The goal is to start from the candidate of this round and perform local continuation along:

$$
(h,d,N)
$$

as follows:

- Automatically estimate which parameter direction causes the lowest generalized eigenvalue to decrease;
- Subdivide chambers near the prime-power activation boundaries;
- Track the shape and condition number of the lowest mode;
- Trigger rigorous reconstruction only when the candidate falls within the certificate resolution;
- If the rigorous interval crosses zero, mark it as `UNRESOLVED_NEAR_ZERO`, and do not arbitrarily assign a sign.

The conclusion of this round is not "closer to proving RH," but rather:

> **AI is now able to search for the hardest candidate among a finite batch of chambers, discover the insufficient resolution of the original verifier, correct the tail bounds, and then leave behind an independently reproducible rational certificate.**