# RH AI Research Starting Point v2.0: Chamber-Aware Subdivision

**Date:** 2026-07-24  
**Research Node:** RH-W-17-CHAMBER-AWARE-SUBDIVISION  
**Batch 01 Progress:** 17/20

## New in this Version

Fixing

$$
h=\frac{1797}{10000},\qquad \sigma=0,
$$

crossing along the $d$ direction

$$
\boxed{4d=\log2}.
$$

This event causes the furthest lagging $n=2$ sample to simultaneously cross the central spline knot of correlation degrees $3,5,7$.

The master interval:

$$
d\in[0.17328669,0.17328690]
$$

is subdivided into a left chamber, an event slab, and a right chamber, and for all three closed cells it is proven that:

$$
\boxed{\lambda_{\min}(M(d),G(d))>10^{-8}}.
$$

## Structural Distinction

The active prime powers across the entire interval remain as

$$
\{2,3,4\},
$$

thus this round is not a prime activation event, but a polynomial piece event:

$$
\boxed{
\text{activation graph remains unchanged}
\quad\text{but}\quad
\text{spline-piece identity changes}.
}
$$

## Methodology

$$
\text{event compiler}
\rightarrow
\text{rational event slab}
\rightarrow
\text{cellwise endpoint matrices}
\rightarrow
\text{$C^2$ interpolation remainder}
\rightarrow
\text{exact LDL}^T.
$$

The event value $\log2/4$ is bounded by a strict rational interval, leaving no deleted boundary points in the parameter domain.

## Added Materials

- Chamber-aware subdivision main document;
- Addendum on multi-regularity central knot events;
- Event surface catalog;
- Chamber adjacency graph;
- Three-cell exact certificate;
- 80-digit mpmath cross-check;
- Self-contained builder and verifier.

## Disclaimer

This version only proves local finite-dimensional positivity on a fixed ten-dimensional mixed-order dictionary; it does not prove or disprove the RH.

## Next Node

$$
\boxed{\texttt{RH-W-18-CERTIFICATE-BACKEND-CONSOLIDATION}}
$$

The next round will consolidate the currently scattered interval matrix, Gram, LDL, witness, prime-power, jump-tail, tube, and chamber certificates, forming a common certificate schema and a single verifier entry point for Batch 01.