Dyadic Shell Constant and Positivity of the Predecessor Gap / Dyadic Shell Constant 与 Predecessor Gap 的正性

Full bilingual text and framework available at self-as-an-end.net Paper XXV in the ZFCρ series attacks the remaining technical bottleneck on the (L3) route of the D(N) → 1 proof chain — mean positivity of R_2 (Paper 24 reduction). The paper introduces the dyadic shell constant c_dyad(k; N), shifting the problem from the global c* lower bound (open since Selfridge 1953) to a shell-conditioned dyadic interval average — a strictly weaker proposition. Core results: Dyadic Shell Constant c_dyad(k; N): defined as E[pred gap | Ω=k, D_m=0, m≤N] / ln 2. R_2 mean positivity is equivalent to c_dyad > τ_k(N)/ln 2 ≈ 3.604. Numerically c_dyad > 3.604 on all tested shells (margin 0.29–0.52 at N = 10^9). Proposition 12 (Dyadic Telescoping, corollary of Paper XII): ρ_E(2m−1) − ρ_E(m−1) = m − Σ_{n=m}^{2m−1} j(n), where j(n) is the jump function from Paper XII. Predecessor gap equals m minus the jump function sum over the dyadic interval. R_2 positivity reduces to: E[Σj(n) | Ω=k, D_m=0, m≤N] < E[m | same] − τ_k(N), where τ_k(N) ≈ 2.498 is the target gap threshold. Three routes compared: Route A (prove c* > 3.61 globally, rejected — 70-year open problem + smooth numbers bottleneck), Route B (prove shell-conditioned dyadic average, main attack), Route C (smooth numbers only, auxiliary). Key numerical findings (N = 10^9): E[pred gap] > τ_k on all representative shells (pred gap margin 0.20–0.36). Jump rate statistically uniform across dyadic intervals (variation < 0.002). Worst-case ρ_E/ln n integers are all smooth numbers (2^a · 3^b type). ρ_E(10^9)/ln(10^9) = 3.6191 (first checkpoint above 3.61), but smooth family best ratio only 3.567. This "pointwise extreme low, interval average high" separation is the core argument for Route B over Route A. Keywords ZFCρ, dyadic shell constant, predecessor gap, telescoping identity, jump function, smooth numbers, integer complexity, bridge positivity, Cesàro average Related identifiers Paper 11: 10.5281/zenodo.18975756 (Continues) Paper 12: 10.5281/zenodo.18977948 (Continues) Paper 17: 10.5281/zenodo.19016958 (Continues) Paper 21: 10.5281/zenodo.19037934 (Continues) Paper 22: 10.5281/zenodo.19039953 (Continues) Paper 23: 10.5281/zenodo.19041689 (Continues) Paper 24: 10.5281/zenodo.19044696 (Continues)