RH-W-02: Weil Test Function Core, Range, and Topology

Proving that D(D+1)C_c^∞(0,∞) is exactly equal to the compactly supported smooth double-vanishing-moment core C₀₀ — the GBUMP generated core from the previous round is not an arbitrary small subfamily, but exactly covers all valid core test functions; simultaneously excluding the bare L² completion, and separately marking the density bridge to the RH equivalent space as the next true GAP.

CORE_CLOSED / GLOBAL_BRIDGE_OPEN — Self-stated nature from the files in the package: "Function space and closure engineering; not a proof of RH, does not establish new Weil positivity results", reproduced here as-is.

Connections

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

Belongs toResearch Starting Point v0.4 · Core Topology Closure

Previous PackageRH-W-01-D/E/F/G: Double Vanishing Moments, Correlation Closure, and Mellin–Fourier Interface

Next Package · v0.2RH-W-02: Weil Normalization Alignment and Sign Closure

Batch 012 / 20

"This round proves: AC_c^∞(0,∞)=C₀₀. Therefore, the analytic generating kernel established in RH-W-01 does not merely cover a portion of legitimate functions, but exactly covers all compactly supported, smooth test functions with double vanishing moments. However, this still does not equate to RH. The truly unfinished bridge is: C₀₀ → a complete Weil test space known to be equivalent to RH." — Excerpted from Section 0 "Conclusions of this Round" in the main document of this package.

GAP Ledger

From RH-W-02_subgaps_v0.1.csv within the package; the status and next steps are recorded by the package itself, not my retrospective judgments.

Reproduce

Range inversion, moment conditions, and numerical regression of the L² failure witness (only checks consistency between implementation and formula; the closure result is provided by the analytic proof in the main document):

python rh_w02_validate.py

Main Document

Exact range theorem (necessity/sufficiency/uniqueness), logarithmic coordinate inversion formula, atomic dictionary density, B0 functional continuity, valid closure transitivity, why bare L² is rejected, and the global density bridge yet to be crossed.

Verification Output

From VALIDATION.txt within the package, the raw output of the numerical regression, reproduced line by line.

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Files

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