Prime-Pattern Rank Rays in the 6-Wheel I / Order-Normalized Rank Rays, Quantitative Prime-Index Stencils, and Critical Pattern Order

Abstract

Prime-Pattern Rank Rays in the 6-Wheel I develops a common rank-coordinate framework for fixed admissible prime patterns. Each existing pattern occurrence is embedded into a global prime-index axis, where its prescribed components form a bounded prime-index stencil. The paper proves an exact order-normalized factorization separating the pattern-specific event factor from classical coordinate corrections, derives a third-order correction expansion, and establishes transfer of positive finite cluster values. Under the Hardy–Littlewood asymptotic for a fixed pattern, the unnormalized rank slope scales as S(H)(log⁡x)2−νS(H)(logx)2−ν, identifying pattern order ν=2ν=2 as the critical case with a finite nonzero slope scale. A finite cross-pattern audit supports the exact identities but is used only as a diagnostic layer. No prime-pattern infinitude or unconditional Hardy–Littlewood theorem is claimed.