# RH GAP Status Update v0.3

**Date:** 2026-07-23  
**Update Node:** `RH-W-01`  
**New Status:** `IN_PROGRESS_PARTIAL_CLOSURE_GBUMP`

## Closed Finite Scope

Let:

$$
\mathcal G_{\mathrm{bump}}
=
\left\{D(D+1)h:h\in C_c^\infty(0,\infty)\right\},
\qquad
D=x\frac{d}{dx}.
$$

For this family, we have established:

$$
\widetilde g(s)=s(s-1)\widetilde h(s),
$$

Therefore:

$$
\widetilde g(0)=\widetilde g(1)=0.
$$

And proved that:

$$
f_g(x)=\int_0^\infty g(xy)\overline{g(y)}dy
\in C_c^\infty(0,\infty)\subset\mathcal W.
$$

Simultaneously, the Mellin–Fourier transform has been fixed:

$$
\phi(u)=e^{u/2}g(e^u),
$$

$$
e^{v/2}f_g(e^v)
=\int_{\mathbb R}\phi(u+v)\overline{\phi(u)}du.
$$

## Still Unclosed

- `RH-W-02`: Selection of the completion space and topology;
- `RH-W-03`: Compression of the generating family for negative witnesses;
- `RH-W-04`: Arithmetic decomposition and controllable remainder terms on the generating family;
- `RH-W-05`: Positivity closure and propagation;
- Lean/Isabelle formalization;
- Any Weil positivity or RH conclusions.

## Engineering Significance

This iteration transforms the artificial hurdle of "finding a function that simultaneously satisfies two moment conditions"—which every agent previously had to resolve independently—into a reusable generating operator. It is the first RH GAP node to be partially closed with accompanying programmatic regression tests.