Algebraic Structure and Conditional Distribution of Jump Gain / 跳跃增益的代数结构与条件分布

ZFCρ Series, Paper 13. This paper investigates the algebraic structure and conditional distribution of the jump gain G(n), the signed advantage of the best multiplicative path over the successor path.

Four exact results characterize the jump mechanism:

• Theorem 16 (gain decomposition): G decomposes into a classical integer-complexity term G_IC and a deficiency term G_d. Numerically, 86.9% of jumps are driven by G_d rather than G_IC.
• Theorem 17 (operation decomposition): ρ_E(n) = s(n) + 2m(n) under default parameters, where s counts successor nodes and m counts multiplication nodes.
• Theorem 18 (deficiency splitting): d(n) = d(a) + d(b) + 2 + Δ_IC, where Δ_IC = ||a|| + ||b|| − ||n|| ≥ 0 measures the sub-additivity surplus of integer complexity.
• Theorem 19 (tie characterization): G(n) = 0 if and only if Δs + 2Δm = 0, a lattice equation that explains the 22.6% 0-point atom in the G distribution.

The conditional distribution of G on fixed-Ω layers is described by a zero-inflated lattice normal model: a discrete atom at G = 0 (governed by algebraic locking) superposed on a background fluctuation field (approximately shifted discrete normal with constant σ ≈ 1 and logarithmically growing mean). Conjecture H of Paper 12 is revised to Conjecture H': D(N) → 1 at an extremely slow rate, contingent on an asymptotic racing problem between Erdős–Kac drift and ρ_E concentration.

Chinese and English versions under one DOI.

Description (中文)

ZFCρ 系列,论文十三。本文研究跳跃增益 G(n) 的代数结构与条件分布。

四个精确结果刻画跳跃机制:定理16(G 分解为经典复杂度项 + deficiency 项,86.9% 的跳跃由 deficiency 驱动);定理17(ρ_E = s + 2m 操作分解);定理18(deficiency 分裂恒等式 d(n) = d(a)+d(b)+2+Δ_IC);定理19(tie 等价于格点方程 Δs+2Δm=0,解释了 22.6% 的 0 点原子)。

G 在 fixed-Ω 层上的条件分布由零膨胀格点正态模型描述。Paper 12 的猜想H修正为猜想H'(D(N) → 1 极慢),但严格证明需要 Erdős-Kac 漂移窗口的统一控制。

中英文版本同一 DOI。

Keywords

jump gain, gain decomposition, deficiency, integer complexity, tie characterization, lattice equation, zero-inflated lattice normal, Erdős-Kac, asymptotic racing, ρ-arithmetic, ZFCρ

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References

1. Han Qin, Papers 1–12 of the ZFCρ series (DOIs listed above).
2. Selfridge, J. L. (1953). Integer representations (unpublished; see Guy, R. K., "Some suspiciously simple sequences," Amer. Math. Monthly 93 (1986), 186–190).
3. Erdős, P. and Kac, M. (1940). "The Gaussian law of errors in the theory of additive number theoretic functions." Amer. J. Math. 62, 738–742.
4. Fouvry, É. and Tenenbaum, G. (1996). "Entiers sans grand facteur premier en progressions arithmétiques." Proc. London Math. Soc. 63, 449–494.
5. Goudout, É. (2019). "Loi d'Erdős-Kac pour les entiers sans grand facteur premier." Acta Arith. 189, 1–38.