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W Engineering Package v0.1 2026-07-23 CLOSED_FOR_ONE_LOCAL_CONTINUATION_POLICY_AND_ONE_CERTIFIED_15D_CHAMBER

RH-W-09: Adaptive Chamber Continuation

Rewrote the fixed-grid search into a local continuation problem — starting from the W-08 candidate, evaluating nine neighboring points per round, automatically pushing the 15-dimensional lowest generalized spectral bottom all the way down from 1.2×10⁻⁵ to 1.32×10⁻⁹ (a decrease of about 9110 times). The endpoint closely approaches the lag-1 prime activation boundary of log3, differing by only about 9.67×10⁻⁵.

RH_CLAIM = False — The claim status output by the verifier in the package itself, reproduced exactly as is. "15-dimensional positive margin ⇏ RH"; the floating-point continuation path can only output NUMERICAL_CANDIDATE_ONLY, without proving that the spectral bottom achieves a local minimum at this boundary.

Connections

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

Belongs toResearch Starting Point v1.2 · Adaptive Continuation and One-Billionth Margin

Previous PackageRH-W-08: Chamber Search and Rigorous Refinement

Next PackageRH-W-10: Prime Boundary Local Modes and 7th-Order Soft Activation

Batch 019 / 20

"The searcher automatically pushes the lowest mode toward a geometric boundary where the discrete arithmetic coupling is about to change... The significance of this is not that 'the prime number three is particularly mysterious', but that the searcher automatically pushes the lowest mode toward a geometric boundary where the discrete arithmetic coupling is about to change. Whether all low modes tend toward such boundaries remains an open research question." — Excerpt from Section 6 of Document 01 in this package.

GAP Ledger

From RH-W-09_subgaps_v0.1.csv in the package; the status and dependencies are recorded by the package itself, not my retrospective judgment.

Reproduce

Adaptive continuation (requires NumPy/SciPy) → rigorous interval certificate → purely rational exact verifier:

python adaptive_continuation.py
python build_continuation_certificate.py
python verify_continuation_certificate.py

01 · Adaptive Chamber Continuation

Main Document — Local continuation policy (nine-neighbor stepping + halving), prime-power activation boundary tracker, continuation endpoint and 9110-fold decrease, and the nearest activation boundary (lag=1, n=3).

02 · One-Billionth Positive Margin Certificate

15-dimensional real matrix decomposition, rational center and perturbation radius, exact LDL^T certificate, and why the 15-dimensional positive margin is still not an RH proof.

03 · Precision Upgrade and Trust Boundaries

Why 10⁻⁹ scale candidates require redesigning constants and tail bounds — Binet/digamma intervals for the Euler constant, two-stage continuation of the Archimedean power tail, and decoupling of computational cost and certificate precision.

Verification Output

From EXACT_VERIFY.txt within the package, reproduced line by line.

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Files

The complete continuation path (adaptive_continuation_path.json) and the 15-dimensional interval certificates are not listed individually; they are all included in the complete zip file.

Download Complete Package (88.1 KB)

sha256 61871e8c9c8fbdf9ca788a873140049c194386b816fde99dd718659f6ec0166b