RH-W-10: Prime Boundary Local Modes and Seventh-Order Soft Start

Precisely derived the soft-start law of prime-3 at the log3=d+4h support boundary — new matrix elements soft-start with (μ₊)⁷, with regularity exactly C⁶ but not C⁷. This overturns the intuition that "the matrix jumps sharply as soon as a prime enters," and proves that the direct magnitude upon entry (about 10⁻²⁸) is far smaller than the current near-zero spectral bottom of 10⁻⁹. Therefore, the searcher approaching the boundary is a genuine geometric phenomenon, not caused by this entry event.

RH_CLAIM = False — The claim status output by the verifier within the package itself, reproduced exactly as is. Both complete fifteen-dimensional chambers before and after the boundary (θ₋, θ₊) are proven by pure rationals to have a positive margin >10⁻⁹; local scanning did not observe an avoided crossing, but this is a numerical diagnostic, not a full-parameter interval theorem.

Connections

Regarding its relationship with other packages, I try to use the words from its own documents as much as possible, not my interpretation. The attribution listed at the beginning of the main document of this package is worth reproducing exactly as is: "Original research concept: Neo.K; Mathematical engineering, derivation, and implementation: Aletheia (GPT-5.6 Thinking)" — this is the first appearance of a named division of labor in this W-series of documents, different from the previous nine packages. Quoting it exactly does not imply my endorsement or denial of its authenticity.

Belongs toResearch Starting Point v1.3 · Prime Boundary 7th-Order Soft Activation

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"One might initially intuit that as soon as a prime power enters, the matrix will immediately exhibit sharp spectral changes. This round proves that this intuition does not hold for the current B-spline dictionary... Geometric support crossing the threshold ⇏ low-order non-smooth spectral jumps. The smoothness of the threshold is determined by the vanishing order of the test function at the support boundary." — Excerpt from Section 7 of Document 01 in this package.

GAP Ledger

From RH-W-10_subgaps_v0.1.csv in the package; the status and dependencies are all recorded by the package itself, not my retrospective judgments.

Reproduce

Local exploration → prime-3 soft-start certificate → purely rational verification before and after the boundary:

python local_boundary_exploration.py
python build_prime3_activation_certificate.py
python verify_prime3_activation.py
python verify_theta_minus.py
python build_theta_plus_certificate.py
python verify_theta_plus.py

01 · Prime Boundary Local Mode and Seventh-Order Soft Start

Main Document — Local boundary parameter μ, prime-3 exact soft-start law and regularity proof, action signs on arbitrary finite-dimensional directions, direct change magnitude of the current lowest mode, three states before, during, and after the boundary (θ₋/θ₀/θ₊), and whether an avoided crossing exists.

02 · Certificates and Local Scan Audit

Three mutually isolated evidence paths (symbolic resolution / rigorous interval / floating-point exploration), local prime-3 interval after the boundary, why the full matrix still needs reconstruction, and software trust boundaries.

Verification Output

From ALL_VERIFY.txt within the package (merged output of three segments: prime-3 soft-start certificate + θ₋ + θ₊), reproduced line by line.

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Files

The complete 15-dimensional interval certificates before and after the boundary (theta_minus_15x15_interval.json, theta_plus_15x15_interval.json) exceed the size threshold for individual listing and are all included in the full zip.

Download Full Package (168.2 KB)

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