# RH AI Research Starting Point v1.9: Three-Parameter Near-Zero Spectral Tube

**Date:** 2026-07-24  
**Research Node:** RH-W-16-THREE-PARAMETER-TUBE  
**Batch 01 Progress:** 16/20

## New in this Version

Adding the kernel scale $h$, which genuinely alters the dictionary, to the $(d,\sigma)$ two-dimensional tube from RH-W-15, establishing:

$$
\boxed{
\left|h-\frac{1797}{10000}\right|\le10^{-8},\qquad
\left|d-\frac{893}{5000}\right|\le10^{-7},\qquad
|\sigma|\le10^{-7}
}.
$$

For every point inside the box, the purely rational verifier proves:

$$
\boxed{
10^{-8}<\lambda_{\min}(M(h,d,\sigma),G(h,d,\sigma))<5\times10^{-8}.
}
$$

## Methodology

This version uses:

$$
\boxed{
\text{Octagonal real Weil interval matrix}
+
\text{Trilinear convex interpolation}
+
\text{Center/scale pure second-order remainder}
}.
$$

The scale second-order Weil integer bounds are:

$$
\boxed{17279,\,40860,\,78886}
$$

Corresponding to correlation degrees $3, 5, 7$, respectively.

The complete three-dimensional merged row remainder is:

$$
1.1287365004114011\times10^{-9}.
$$

## Chamber

The entire three-dimensional box maintains:

- active prime powers: $\{2,3,4\}$;
- $R_{\max}=1.43320054<\log5$;
- minimum sample-to-knot margin of approximately $0.02125231944$;
- spline piece and activation graph remain unchanged.

## Reproducibility

v1.9/RH-W-16 fully incorporates the shared interval, B-spline, and jump-tail modules. Once the engineering package is extracted, the verifier can be executed independently, no longer relying on unpackaged external Python files.

## Scope of Claims

This is merely a finite-dimensional continuous near-zero positive spectral volume on a fixed ten-dimensional mixed-order dictionary; it neither proves nor disproves the RH.

## Next Node

$$
\boxed{
\texttt{RH-W-17-CHAMBER-AWARE-SUBDIVISION}
}
$$

The next round will allow large parameter boxes to approach or cross spline knot/prime-power boundaries, automatically subdividing them into fixed-chamber sub-boxes and establishing adjacency graphs and box-by-box certificates.