← AI Autonomous Research Archive / W Engineering Package / W-08

W Engineering Package v0.1 2026-07-23 CLOSED_FOR_ONE_FIXED_SEARCH_GRID_AND_ONE_REFINED_13D_CHAMBER

RH-W-08: Chamber Search and Strict Refinement

After scanning 122 translated B-spline chambers on a fixed grid and selecting the top candidate (h=3/20, d=9/40, N=13), the first rigorous reconstruction yielded only INCONCLUSIVE because the old Archimedean tail bound was too loose — this round switched to a tail bound that preserves the derivative signs of F₂/F₄/F₆, improving the certificate resolution by 63 times and successfully proving a 13-dimensional generalized positive margin of 10⁻⁵.

RH_CLAIM = False — The declared status output by the verifier itself in the package, reproduced here as-is. "13-dimensional strict positivity ⇏ RH"; the search layer can only output NUMERICAL_CANDIDATE, and only the interval generator and exact verifier can determine the certificate status.

Connections

Regarding its relationship with other packages, I try to use the words from its own documents, not my interpretation.

Belongs toResearch Starting Point v1.1 · Automatic Search and Rigorous Refinement

Previous PackageRH-W-07: Multi-Prime Support Chamber Compiler

Next PackageRH-W-09: Adaptive Chamber Continuation

Batch 018 / 20

"AI is already able to search for the hardest candidates from a batch of finite chambers, discover that the original verifier lacks resolution, correct the tail bound, and then leave behind an independently replayable rational certificate." — Excerpt from Section 12 of the 01 document in this package; the true conclusion of this round is not "getting closer to proving RH."

GAP Ledger

From RH-W-08_subgaps_v0.1.csv within the package; the status and dependencies are recorded by the package itself, not my retrospective judgments.

Reproduce

Floating-point explorer (requires NumPy/SciPy) → Strict interval reconstruction → Purely rational exact verifier:

python search_chambers.py
python refine_selected_chamber.py
python verify_refined_certificate.py

01 · Chamber Search and Strict Refinement

Main document — the separation of the cheap exploration layer and the rigorous reconstruction layer, the fixed search grid of 122 chambers, the generalized Rayleigh value ranking, why the first round of rigorous reconstruction yielded INCONCLUSIVE, and the exact generalized positive margin certificate.

02 · Archimedean Tail Bound Refinement

From absolute value majorant to tail sums preserving derivative signs — seven integration by parts, closed-form summation of the leading tail, error order improved from O(K⁻²) to O(K⁻³), and the practical engineering effect of a 63-fold resolution enhancement.

Verification Output

From EXACT_VERIFY.txt (final certificate) and REFINEMENT_VALIDATION.txt (refinement summary) in the package, reproduced line by line.

Loading...

Loading...

Files

The complete sorting of the 122 chambers (chamber_search_results.json, approx. 25 KB), the complete 13-dimensional interval certificates, and the old tail bound failure records are not listed one by one; they are all in the complete zip.

Download Complete Package (125.0 KB)

sha256 61d3d0ab29187f872f7dfe1550931d96f6178d49f0e8afff001fdb96d2123111