The Three-Point Law, Self-Correction, and the Narrowing of the Variance Gap / 三点律、自纠正与方差缺口的收窄

Full bilingual text and framework available at self-as-an-end.net We report the M9 exploration results that reduce the B-bound to a precise algebraic condition and characterize the microstructure of the DP recursion. The factorization deficiency B(n) concentrates on three values {-2, -1, 0} at high k, with P(B = -1) ≈ 0.76 and approximately symmetric wings. The data suggest a three-point law near weights (1/8, 3/4, 1/8), which would imply Var(B) → 1/4. A conditional reduction theorem is proved: if the shell-averaged second moment of the prime-power correction ε_{P^-} is finite, then Var(B) = O_k(1) uniformly in N. The dominant contribution to Var(B) comes from the v_{P^-} ≥ 4 regime (63% of the Ω = 8 shell, 81% of variance), overturning the early-series emphasis on squarefree behavior. DP self-correction is confirmed: after large jumps (j ≥ 3), the next jump size drops by 95% to 0.113. G has shell-order autocorrelation -0.037 at lag 1 and ≈ 0 at lag ≥ 2. The predecessor-target correlation is 0.9676, giving 97% cancellation of O(ln N)-scale variances. The A-tail is consistent with exponential-type decay on the tested range. The proof landscape is updated: B-bound is reduced to ε_p second-moment control (conditional theorem); Local Lipschitz remains the most leveraged open target, now supported by self-correction and tail structure. Appendix documents four-AI parallel exploration and thermodynamic interface role. Bilingual edition: English original with Chinese translation. Keywords integer complexity, ρ-arithmetic, ZFCρ, three-point law, B-bound, self-correction, DP recursion, exponential tail, autocorrelation, variance cancellation, prime-power correction License Creative Commons Attribution 4.0 International (CC BY 4.0) Language English Related Identifiers (all "continues") 10.5281/zenodo.19024385 (Paper 18) 10.5281/zenodo.19016958 (Paper 17) 10.5281/zenodo.19013602 (Paper 16) 10.5281/zenodo.19007312 (Paper 15) 10.5281/zenodo.18991986 (Paper 13) Additional Notes Part of the ZFCρ series (Papers 1–19). This is Paper 19, Milestone 9: Three-point law, DP self-correction, and B-bound reduction. Bilingual edition: English original with Chinese translation. Contains one conditional theorem (B-bound Reduction), seven numerical discoveries, and corrections to Papers 16 and 18. Four-AI parallel exploration methodology documented in Appendix A.