Arithmetic origin and stability of the mod-30 sector-weight partition ratio

Summary. This note investigates the arithmetic origin of the empirical constant κ ≈ 0.8509, observed at the distinguished parameter τ* = 0.145 in prior numerical studies on prime proximity graphs. By performing a gap-ring sector decomposition of the reduced residues modulo primorials and assigning power-law weights w(s) = s^(−n), it obtains a closed-form sector-weighted partition ratio. Evaluated at mod 30, this ratio matches the empirical κ to within 3×10⁻⁴. A primorial ladder test across {6, 30, 210, 2310, 30030} and sector-shuffle null tests demonstrate that only mod 30 approaches the point (τ*, κ) with minimal deviation, and that the measured ratio is sensitive to the specific cyclic arrangement of sector sizes. Context. This paper forms a self-contained arithmetic chapter within the author's broader research program on prime proximity graphs and their arithmetic structures (see Related identifiers). It preserves full technical detail while remaining accessible independently, assuming only basic knowledge of reduced residues modulo primorials. This is a revised and substantially expanded version of the previous record, with a closed-form derivation, an extended primorial ladder test, and a new stability analysis. is new version of 10.5281/zenodo.19450506