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W Engineering Package v0.1 2026-07-23 CLOSED_FOR_PRIME2_2D_AND_5D_SPLINE_CHAMBERS

RH-W-06: First Prime Activation and Arithmetic Support Chamber

By strictly controlling the relevant support between log2 and log3, n=2 becomes the only active von Mangoldt index — the same rational integer witness c=(1,1,0,-1,-1) is strictly negative after manually removing the prime terms, and strictly positive after adding back the true n=2 terms, demonstrating that prime terms can genuinely change the inertia of the finite-dimensional Weil form and are not merely decorative.

RH_CLAIM = False — The declared status output by the verifier itself in the package, reproduced here as-is. "5-dimensional positivity ⇏ RH"; the naive interval LDL decomposition failed at the fourth pivot, which has been recorded as a verifier method failure, not a mathematical property failure.

Connections

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

Belongs toResearch Starting Point v0.9 · First Prime Activation

Previous PackageRH-W-05: The First Real Weil Matrix Rational Interval

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

Batch 016 / 20

"prime-free negative (artificially truncated formula) + prime-2 positive correction (real discrete arithmetic term) = full positive (real finite-dimensional Weil value)... The first prime is not added to the local energy of each wave packet itself, but acts as a non-local arithmetic coupling between two wave packets separated by approximately log2." — Excerpted from Documents 01 and 02 in this package.

GAP Ledger

From RH-W-06_subgaps_v0.1.csv within the package; the status and dependencies are recorded by the package itself, not my retrospective judgment.

Reproduce

2D and 5D matrix construction → Purely rational exact verifier → Independent high-precision cross-check:

python build_prime_active_matrix.py
python build_prime2_matrix_5x5.py
python verify_prime_active_certificates.py
python crosscheck_mpmath.py

01 · First Prime Activation and Arithmetic Support Chamber

Main Document — Definition of arithmetic support chambers, two-dimensional isolated coupling experiment (n=2 as non-local coupling), five-dimensional Toeplitz support chamber and activation mode [off,off,on,on,on], and why direct interval LDL fails.

02 · 5D Prime Stabilization Certificate

Midpoint-radius decomposition, exact midpoint spectral margin (purely rational LDL^T), interval family positive definiteness under Weyl perturbation bounds, sign flipping of the same witness, and trust boundaries.

Verification Output

From VALIDATION.txt (2D certificate) and VALIDATION_5x5.txt (5D raw interval construction — natural LDL obtains a zero-crossing interval at the fourth pivot, marked as INCONCLUSIVE rather than failure) within the package, reproduced line by line.

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