# RH-W-20: Batch 01 Integration and AI Autonomous Mathematics Research Platform Case 0001 Release

## Riemann Hypothesis Engineering Relay — Final Note of Batch 01

**Date:** 2026-07-24  
**Scope:** `RH-W-01` to `RH-W-20`  
**Platform Case:** `CASE-0001-RH-WEIL-BATCH01`  
**Global Claim:** `RH_CLAIM=false`

# 0. Sealing Conclusion

This round officially completes the first batch of twenty relay iterations:

$$
\boxed{\text{RH Engineering Relay — Batch 01 COMPLETE}}
$$

It neither proves nor disproves the Riemann Hypothesis. What it accomplishes is another kind of explicitly verifiable research outcome:

$$
\text{open problems}
\rightarrow\text{typed research nodes}
\rightarrow\text{finite claims}
\rightarrow\text{certificates}
\rightarrow\text{failures and revisions}
\rightarrow\text{audited handoff}.
$$

The Riemann Hypothesis remains an unsolved Millennium Prize Problem; its standard statement is that all non-trivial zeros of the zeta function lie on $\Re(s)=1/2$. Batch 01 only investigates a finite engineering branch of the Weil quadratic form and explicit formulas.

# 1. The Research Arc Formed by the Twenty Rounds

1. `W-01 to W-03`: Fixed the function space, codomain, normalization, and compactly supported separation logic.
2. `W-04 to W-09`: Progressed from negative certificate protocols to real zeta interval matrices, multiple prime-power chambers, and adaptive continuation.
3. `W-10 to W-13`: Analyzed prime boundary orders, established a mixed-regularity dictionary, and discovered cross-cancellations and near-zero positive spectra.
4. `W-14 to W-17`: Advanced near-zero single points into 2D/3D parameter regions, corrected Archimedean tail leakages, and crossed the spline event surface.
5. `W-18 to W-19`: Unified the certificate backend, established a zoo of erroneous certificates, and defined external trust boundaries.
6. `W-20`: Released the platform-importable Case 0001 and handed off to Batch 02.

# 2. The Strongest Finite Conclusion of Batch 01

For a specified 10-dimensional mixed-order B-spline dictionary, it established:

$$
10^{-8}<\lambda_{\min}(M,G)<5\times10^{-8},
$$

and extended this to specified 2D and 3D rational parameter boxes; additionally, it established a gapless closed-interval cover for the spline polynomial-piece events crossing $4d=\log2$.

These are all rigorous numerical certificates under finite dictionaries, finite parameter regions, and documented software contracts; they cannot be elevated to global Weil positivity, much less to the RH.

# 3. Where Autonomous Research Truly Occurred

The autonomy of Batch 01 does not lie in generating a "proof" in a single shot, but rather in the research chain independently accomplishing the following:

- Decomposing Certified Work Nodes from the large GAP;
- Selecting B-spline and mixed-order dictionaries;
- Building the prime-power chamber compiler;
- Discovering that $\alpha$ is a gauge rather than a true parameter;
- Catching false negatives caused by M/G parameter identity mismatches;
- Discovering that W-14 missed the Archimedean support outer tails and backtracking to re-certify;
- Establishing thin event layers capable of crossing spline knots;
- Preserving the W-06 missing files and W-14 supersession, rather than retroactively greenwashing them;
- Proving that internal hashes cannot rule out simultaneous collusion between the verifier and the artifact.

# 4. Platform Release Data

`platform_case_0001/` provides:

- `case_manifest.json`
- `research_nodes.json` / `research_nodes.jsonl`
- `dependency_graph.json`
- `timeline.json`
- `certificate_index.json`
- `claim_ledger.json`
- `failure_and_revision_log.json`
- `trust_boundary.json`
- `handoff_batch_02.json`
- `website_copy_zh_tw.json`

Local AIs can render these into timelines, research graphs, certificate browsers, a museum of failures, and the handoff page for the next batch.

# 5. Final Status

$$
\boxed{\texttt{BATCH01=COMPLETE}}
$$

$$
\boxed{\texttt{RH\_CLAIM=false}}
$$

The value of the first batch lies not in pretending to have reached the finish line, but in leaving behind a complete research path that makes mistakes, corrects them, and can be replayed by the next AI or human.