# RH-W-09: Adaptive Chamber Continuation
## From Fixed Grids to Local Tracking Near Prime-Power Activation Boundaries

**Version:** v0.1  
**Date:** 2026-07-23  
**Research Project:** RH GAP Atlas / AI Mathematical Engineering Relay  
**Parent Node:** `RH-W-08-CHAMBER-SEARCH-AND-REFINEMENT`  
**Current Node:** `RH-W-09-ADAPTIVE-CHAMBER-CONTINUATION`  
**Status:** `CLOSED_FOR_ONE_LOCAL_CONTINUATION_POLICY_AND_ONE_CERTIFIED_15D_CHAMBER`  
**Boundary:** This round only investigates a family of finite-dimensional translated cubic B-spline chambers; it does not prove the RH, nor does it find a counterexample to the RH.

---

# 1. Why Fixed Grids Are Still Insufficient

`RH-W-08` has completed:

$$
\text{Coarse search}
\longrightarrow
\text{Candidate sorting}
\longrightarrow
\text{Interval reconstruction}
\longrightarrow
\text{exact verifier}.
$$

However, a fixed grid can only answer:

> Among the pre-listed finite points, which one is the lowest?

It cannot answer:

1. Starting from the current lowest mode, which parameter direction will continue to descend?
2. Is the descent concentrated near a certain $\log p^k$ activation boundary?
3. Between dimension increase and geometric parameter movement, which is the primary source?
4. After the searcher finds a candidate at the $10^{-9}$ scale, can the certificate layer keep up?

Therefore, this round reformulates the chamber search into a local continuation problem.

---

# 2. Chamber Parameters and Generalized Spectral Bottom

Fixing the centered cubic B-spline $\beta_3$, let

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

and construct the Weil matrix $M(h,d,N)$ and the Gram matrix $G(h,d,N)$ in the $N$-dimensional space.

The exploration objective is

$$
\lambda_{\min}(h,d,N)
=
\min_{c\ne0}
\frac{c^TM(h,d,N)c}{c^TG(h,d,N)c}.
$$

The floating-point values here are only responsible for guidance; their status is always:

```text
NUMERICAL_CANDIDATE_ONLY
```

---

# 3. Local Continuation Policy

This round fixes $N=15$, starting near the candidate from the previous round:

$$
(h_0,d_0)=\left(\frac3{20},\frac9{40}\right).
$$

Each round evaluates nine local neighboring points:

$$
(h,d),
\quad
(h\pm\Delta h,d),
\quad
(h,d\pm\Delta d),
$$

$$
(h\pm\Delta h,d\pm\Delta d).
$$

If a neighboring point lowers the exploratory generalized spectral bottom, move to that point; otherwise, execute:

$$
\Delta h\leftarrow\frac{\Delta h}{2},
\qquad
\Delta d\leftarrow\frac{\Delta d}{2}.
$$

This is neither global optimization nor a gradient method. It is a replayable finite-state policy aimed at:

- Generating candidates;
- Preserving the complete path;
- Finding the nearest prime-power support boundary;
- Handing the final point over to the rigorous layer.

The complete path is saved in:

- `adaptive_continuation_path.csv`
- `adaptive_continuation_path.json`

---

# 4. Prime-Power Activation Boundary Tracker

The correlation function of the $k$-th Toeplitz lag is centered at $-kd$, with support

$$
[-kd-4h,-kd+4h].
$$

The boundary condition for the sample $-\log n$ entering the left endpoint is

$$
\boxed{
\log n=kd+4h
}.
$$

The entry and exit of the sample $+\log n$ are also determined by the corresponding support endpoint conditions.

Therefore, every prime power $n=p^m$ generates several boundary curves in the parameter plane. At each step, the searcher records:

$$
\min_{k,n,\pm}
\operatorname{dist}
\left(
\pm\log n,
\partial\operatorname{supp}(f_k)
\right).
$$

This makes "which prime threshold it is approaching" an explicit state, rather than an afterthought.

---

# 5. Continuation Results

The continuation endpoint is

$$
\boxed{
h=\frac{87}{400},
\qquad
d=\frac{117}{512},
\qquad N=15
}.
$$

That is,

$$
h=0.2175,
\qquad
d=0.228515625.
$$

The floating-point exploration layer yields

$$
\lambda_{\min}^{\mathrm{explore}}
\approx
1.3197955822649145\times10^{-9}.
$$

Compared to the initial fixed-grid candidate

$$
1.2020901345911073\times10^{-5},
$$

it decreases by a factor of approximately

$$
9.11\times10^3.
$$

This decrease is merely a candidate generation result and cannot be directly taken as a positivity or near-zero theorem.

---

# 6. Nearest Activation Boundary

The closest to the endpoint is:

$$
\boxed{
\text{lag}=1,
\qquad n=3,
\qquad -\log3\text{ sample}
}.
$$

The boundary equation is

$$
\log3=d+4h.
$$

Currently,

$$
d+4h
=
\frac{117}{512}+4\cdot\frac{87}{400}
=
1.098515625,
$$

while the rigorous interval proof shows

$$
\boxed{
\log3-(d+4h)
>
9.6663668109691\times10^{-5}
}.
$$

Thus, $-\log3$ is still outside the lag-$1$ correlation support, but very close to the entry boundary.

The significance of this is not that "the prime number three is particularly mysterious," but rather:

> The searcher automatically pushes the lowest mode toward a geometric boundary where the discrete arithmetic coupling is about to change.

Whether all low modes tend toward such boundaries remains an open research question.

---

# 7. What Was Not Achieved This Round

This round did not prove that:

$$
\lambda_{\min}(h,d,N)
$$

attains a local minimum at this boundary; nor did it prove that the spectral bottom necessarily rises or falls after crossing the boundary.

Furthermore, it did not prove:

$$
\inf_{h,d,N}\lambda_{\min}(h,d,N)=0.
$$

Currently, we have only obtained:

1. A replayable local continuation path;
2. A candidate at the $10^{-9}$ scale;
3. An explicit nearest prime-power boundary;
4. A completed rigorous certificate for this candidate, see the next document.

---

# 8. GAP Updates

```text
RH-W-09-LOCAL-CONTINUATION
    CLOSED_FOR_ONE_DETERMINISTIC_POLICY

RH-W-09-BOUNDARY-TRACKER
    CLOSED_FOR_TRANSLATED_BSPLINE_FAMILY

RH-W-09-GLOBAL-OPTIMUM
    OPEN

RH-W-09-BOUNDARY-CROSSING-LAW
    OPEN

RH-W-09-NEGATIVE-WITNESS
    NOT_FOUND
```

---

# 9. Next Research Direction

After this round, research should not merely continue to blindly descend numerically. It is necessary to split "before the boundary, on the boundary, and after the boundary" into three comparable states:

$$
\theta_-,\qquad\theta_0,\qquad\theta_+,
$$

where

$$
\theta_0:\ \log3=d+4h.
$$

Next, we will investigate:

1. The local order of the newly added matrix block for prime-$3$;
2. The coupling sign between the lowest generalized eigenvector and the newly added block;
3. The one-sided derivative of the spectral bottom with respect to the boundary normal parameter;
4. Whether there exists an analytically predictable avoided crossing or a true zero crossing.

This will become:

```text
RH-W-10-PRIME-BOUNDARY-LOCAL-MODE
```