Unbounded Mean Gain and the Closure of Assumption A' / 无界均值增益与假设 A' 的闭合

Full bilingual text and framework available at self-as-an-end.net Paper XXII in the ZFCρ series establishes a Chebyshev reduction of Assumption A' (p_∞(k) → 1) and decomposes the unbounded mean gain conjecture (E[G_spf | Ω = k] → ∞) into precise arithmetic inputs via the insertion recursion. Core results in three layers: Layer 1 (Proved): Chebyshev Bridge Theorem (Theorem 4) — if the liminf of shell means μ_k diverges and the variance is subquadratic (σ²_k = o(μ_k²)), then limsup P(G_spf ≤ 0 | Ω = k) → 0 as k → ∞. Combined with Assumption A's limit existence, this gives A'. Layer 2 (Exact identity + numerical confirmation): Insertion recursion μ_{k+1} = E_{I_k}[G_spf(m)] + E_{I_k}[K_p], verified to four decimal places across 12 shells. Step increment decomposes into cofactor representativeness gap (< 0.09) and bridge gain (E[K_p] > 0, always positive). Layer 3 (Conditional Theorem 6): If (L1) cofactor representativeness, (L3) bridge positivity, (D) net increment divergence, and (V) subquadratic variance hold, then Assumption A' holds. All conditions have overwhelming numerical support (N = 10^7). Key findings: σ²/μ² monotonically decreasing: 0.81 (k=5) → 0.05 (k=19) Var(G_spf|k) ≈ 1.24 uniform in k (R² = 0.018 for linear regression on k) Cov(A,B|k) ≈ −0.10 stable — anti-correlation engine stabilizes total variance E[K_p] ∈ [0.18, 0.49], always positive — each prime factor insertion increases G_spf Scaling test (N = 10^4 to 3×10^7): Var(ΔM|k) = O(ln ln N), but fixed-k Var(G_spf) converges — the O(ln ln N) growth is a cross-shell red herring Includes Appendix A on AI collaboration methodology: the "data before direction" principle, the research trinity (computation + AI array + human mind), review as force multiplier (ChatGPT 3-round review: moderate → minor → accept), and trust calibration per task. After Paper 22, D(N) → 1 proof chain status: B (known), A' (conditionally closed), A (open — deepest gap). Keywords ZFCρ, remainder theory, unbounded mean gain, Assumption A', Chebyshev bridge, Cantelli inequality, insertion recursion, bridge positivity, cofactor representativeness, SPF positivity, AI collaboration methodology Related identifiers Paper 15: 10.5281/zenodo.19007312 (IsPartOf) Paper 16: 10.5281/zenodo.19013602 (Continues) Paper 18: 10.5281/zenodo.19023418 (Continues) Paper 20: 10.5281/zenodo.19027893 (Continues) Paper 21: 10.5281/zenodo.19037934 (Continues)