# RH AI Mathematical Engineering Milestone v1.0

**Date:** 2026-07-23  
**Status:** Advanced from source cleanup and GAP registration to a real multi-prime-power finite-dimensional certificate pipeline.  
**Boundary:** No proof of RH, no counterexample to RH found.

## Established Engineering Chain

$$
\text{Old manuscript claim removal}
\rightarrow
\text{GAP Atlas}
\rightarrow
\text{Valid test function core}
\rightarrow
\text{Weil normalization}
\rightarrow
\text{Compact support separation}
\rightarrow
\text{Finite-dimensional reachability}
\rightarrow
\text{Real explicit formula interval matrix}
\rightarrow
\text{Prime-power support chamber compiler}.
$$

## Replayable Artifacts of v1.0

1. Real 9-dimensional Riemann–Weil interval matrix;
2. Five prime-power sparse blocks for $2,3,4,5,7$;
3. Lag-by-lag activation graph;
4. Purely rational $LDL^T$ positive definite certificate;
5. Four cumulative prime-power sign-flip witnesses;
6. Generator, JSON certificates, and independent verifier;
7. All conclusions carry scope and non-RH claim labels.

## Key Observations of v1.0

As the relevant support window moves along the logarithmic coordinate, prime powers enter and leave. The discrete arithmetic terms can thus be expressed as:

$$
P_{p^k}=\text{Sparse coupling layer at a specific logarithmic distance}.
$$

The complete finite-dimensional form is:

$$
M=A_{\infty}+\sum_{p^k}P_{p^k}.
$$

This provides a discrete-continuous decomposition that subsequent AI searchers can directly manipulate.

## Next Phase

$$
\boxed{\texttt{RH-W-08-CHAMBER-SEARCH-AND-REFINEMENT}}
$$

Establish a two-tier architecture consisting of an untrusted fast searcher and a trusted rigorous reconstructor, allowing the AI to propose chambers and witnesses in bulk, while only the exact verifier can change the GAP state.