# RH AI Research Starting Point v1.2
## Adaptive Chamber Continuation and $10^{-9}$ Finite-Dimensional Generalized Positive Margin

**Date:** 2026-07-23  
**Author Tags:** Original research direction by Neo.K; Research reconstruction and engineering implementation by Aletheia (GPT-5.6 Thinking)  
**Nature:** Non-proof research engineering package

---

# 1. Continuation in This Version

v1.1 completed:

$$
\text{Fixed grid search}
\rightarrow
\text{13-dimensional candidate}
\rightarrow
10^{-5}\text{ strict positive margin}.
$$

v1.2 further completed:

$$
\text{Local adaptive continuation}
\rightarrow
\text{prime-power boundary tracking}
\rightarrow
\text{15-dimensional near-critical candidate}
\rightarrow
10^{-9}\text{ strict positive margin}.
$$

---

# 2. Fixed Candidate

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

The exploratory generalized spectral bottom is approximately:

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

The candidate's distance to the lag-$1$ $n=3$ activation boundary:

$$
\log3=d+4h
$$

is only about:

$$
9.67\times10^{-5}.
$$

---

# 3. Rigorous Results

Completely enumerating all $24$ prime powers within the support, constructing the 15-dimensional true Weil interval matrix and exact Gram matrix.

A pure rational verifier proves:

$$
\boxed{
Q(c)>10^{-9}c^TGc
\qquad
\forall c\ne0
}.
$$

This only proves positivity in the fixed 15-dimensional subspace and does not imply RH.

---

# 4. Precision Engineering

New in this version:

1. Signed remainder intervals for the Euler constant of Binet/digamma;
2. Rational two-stage tail continuation for $S_3,S_5,S_7,S_8$;
3. Prime-power activation boundary tracking;
4. Preservation of adaptive coordinate continuation paths;
5. $10^{-32}$ grid rational centers;
6. Exact $LDL^T$ generalized positive margin verification.

The maximum row error of the matrix is approximately:

$$
2.01\times10^{-12}.
$$

---

# 5. Next Node

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

Instead of continuing to blindly push down the spectral value, we fix:

$$
\log3=d+4h
$$

and examine the three states near it: pre-boundary, on-boundary, and post-boundary, to study the one-sided effect of the newly entering prime-$3$ matrix block on the lowest mode.