Why κ ≈ 0.851: Chebyshev Closure and the 7-fold Lattice Stability of Spectral Partition Ratio

This note records an empirical invariant observed in τ-parameterized proximity graphs built on prime-driven embeddings: the normalized partition ratio κ remains stable near κ ≈ 0. 851 across large Pmax ranges (reported in the companion computations). We propose a conceptual explanation based on a Chebyshev closure identity (Σ₍=₁^m cos (nπ/m) = −1, integers m≥1), and suggest that the specific value of κ is located within a 7-fold lattice (m=7), since 1/7 = 0. 142857 lies closest to the empirically observed connectivity/locking transition window (~0. 14–0. 15), whose center τᵣef is near τ* ≈ 0. 145. We also report a reproducible "∧-oscillation" signal in κ across Pmax scales with non-overlapping bootstrap confidence intervals. The Gauss constant relation is included as an empirical numerical correspondence only; no first-principles derivation is claimed. This is an idea-disclosure note (not peer-reviewed). prime numbers; proximity graphs; spectral graph theory; universality; Chebyshev polynomials; lattice hypothesis; phase transition; bootstrap confidence interval; number theory; empirical invariantprime numbers; proximity graphs; spectral graph theory; universality; Chebyshev polynomials; lattice hypothesis; phase transition; bootstrap confidence interval; number theory; empirical invariant10. 5281/zenodo. 18638137 (Yang 2026a — Isotropy Sweep) 10. 5281/zenodo. 18661212 (Yang 2026b — Percolation/Null tests) 10. 5281/zenodo. 18627601 (Yang 2026c — Rotational Invariance)