Algebraic Structure and Conditional Distribution of Jump Gain
Paper 12 established the local geometry of the δ-staircase: jumps are purely multiplicative (Theorem 13), all growth of δ comes from jump accumulation (Theorem 14), density/jump/plateau locked by algebraic identities (Theorem 15). Conjecture H asserted D(N) → d ≈ 0.57. This paper studies the jump gain G(n) — the signed advantage of the best multiplicative path over the successor path. Four exact results:
Theorem 16 (Gain decomposition): G_n(a,b) = G_IC(n;a,b) + G_d(n;a,b) where G_IC lives in integer complexity and G_d is the deficiency contribution. Numerically (n ≤ 50,000), 86.9% of jumps occur with G_IC ≤ 0: the classical term opposes the jump, deficiency drives it.
Theorem 17 (Operation count): ρ_E(n) = s(n) + 2m(n), where s counts successor nodes and m counts multiplication nodes.
Theorem 18 (Deficiency splitting): if the canonical optimal expression roots at mul(a,b), then d(n) = d(a) + d(b) + 2 + Δ_IC where Δ_IC = ‖a‖+‖b‖−‖n‖ ≥ 0.
Theorem 19 (Tie criterion): G(n) = 0 iff Δs + 2Δm = 0 — a lattice equation producing the 22.6% zero-point atom. Conditional on Ω(n), gain G follows a zero-inflated lattice normal: a discrete atom at G=0 from algebraic locking, plus a shifted discrete-normal background with mean ≈ −1.93 + 2.21 ln Ω, σ ≈ 1. Conjecture H revised to H': D(N) → 1.
1. Introduction: Jump Gain as Lattice Phenomenon
1.1 Definition of Jump Gain
For n ≥ 2 composite, Theorem 13 states n jumps iff M_n < S_n where M_n = min_{ab=n, a,b≥2}(ρ_E(a)+ρ_E(b)+2) and S_n = ρ_E(n−1)+1. Define jump gain:
for each factorization ab=n with a,b≥2, and G(n) = max_{ab=n} G_n(a,b). Then:
- n jumps ⟺ G(n) > 0
- n ties (zero-point) ⟺ G(n) = 0
- n does not jump ⟺ G(n) < 0
1.2 Motivation: Why Study Gain?
G(n) is the signed advantage of the multiplicative path over the successor path. Understanding its sign is understanding when shortcuts (multiplication) beat incrementalism (succession). The numerical discovery that 86.9% of jumps have G_IC ≤ 0 inverts classical intuition: the classical complexity measure opposes the jump, yet jumps happen. Why? Deficiency dominates.
2. Canonical Choices and Notation
2.1 Canonical Optimal Expression
Among all compact terms h ∈ Hist*(n) achieving ρ(h) = ρ_E(n), select the one with: (1) fewest multiplication nodes m(h); (2) if tied, lexicographically smallest pre-order serialization. This makes s(n), m(n), path(n) well-defined.
The canonical factor pair (a*, b*) maximizes G_n(a,b); when several pairs tie, take lexicographically smallest with a* ≤ b*.
2.2 Key Notation
G_n(a,b): gain for specific factorization ab=n
G(n): optimal gain = max G_n over all ab=n
(a*,b*): canonical factor pair achieving G(n)
s(n): successor node count in canonical optimal expression
m(n): multiplication node count in canonical optimal expression
d(n): deficiency = ρ_E(n) − ‖n‖
Ω(n): number of prime factors with multiplicity
3. Theorem 16: Gain Decomposition
3.1 Statement
Theorem 16: For every ab = n with a, b ≥ 2:
where
(integer complexity term, lives in classical world), and
(deficiency contribution).
3.2 Proof Sketch
Substitute ρ_E(m) = ‖m‖ + d(m) into the definition of G_n(a,b):
3.3 Numerical Result 1: Deficiency Dominates
Among composites n ≤ 50,000 that jump (G(n) > 0):
- Average G_IC = −0.585 (opposes jump)
- Average G_d = +2.345 (drives jump)
- In 86.9% of all jumps, G_IC ≤ 0: the classical measure opposes the jump
Interpretation: A jump is triggered by escaping a predecessor whose deficiency is too high — not by finding a cheap factorization. The successor path inherits d(n−1); the multiplicative path distributes cost across d(a)+d(b). When the former exceeds the latter, the splitting gain triggers a jump. The classical integer complexity measure fights against jumps; deficiency overturns the verdict.
4. Structure of Deficiency
4.1 Theorem 17: Operation Count
Theorem 17: Under default parameters (c_S, c_⊕, c_⊗) = (1, 1, 2), the canonical optimal expression satisfies:
Proof: By the add-dominance theorem (Paper 6 §4.2), no add node appears in the optimal expression. Each succ node contributes c_S = 1 and each mul node contributes c_⊗ = 2.
4.2 Corollary and Numerical Result 2
Corollary: d(n) = 2m(n) + (s(n) − ‖n‖) with s(n) − ‖n‖ ≥ 0.
Numerically (n ≤ 10,000), the term 2m(n) accounts for 88% of d(n), confirming that deficiency is overwhelmingly driven by multiplication node overhead.
4.3 Theorem 18: Deficiency Splitting
Theorem 18: If the canonical optimal expression for n roots at mul(a,b), then:
where Δ_IC = ‖a‖+‖b‖−‖n‖ ≥ 0 measures the sub-additivity surplus of integer complexity.
Proof: ρ_E(n) = ρ_E(a)+ρ_E(b)+2, then
Non-negativity of Δ_IC: a×b is a valid {1,+,×}-expression for n, so ‖n‖ ≤ ‖a‖+‖b‖.
4.4 Numerical Result 3: Δ_IC Distribution
For n ≤ 10,000:
- Δ_IC = 0 in 61% of cases
- Δ_IC = 1 in 25%
- Δ_IC = 2 in 9%
4.5 Recursion for m(n)
m(n) = m(n−1) when canonical path is succ; m(n) = m(a)+m(b)+1 when path is mul(a,b).
5. Theorem 19: Tie Criterion and Zero-Point Atom
5.1 Definition and Main Result
Define Δs = s(n−1)−s(a*)−s(b*)−1 and Δm = m(n−1)−m(a*)−m(b*). Then:
Theorem 19: G(n) = Δs + 2Δm. In particular, G(n) = 0 iff Δs = −2Δm — a lattice equation.
Proof: By Theorem 17:
5.2 Tie Patterns (n ≤ 10^6)
| (Δs, Δm) | Fraction | Interpretation |
|---|---|---|
| (−2, 1) | 54.1% | Factor pair saves one mul, costs two succ |
| (0, 0) | 29.7% | Identical (s, m) decomposition |
| (−4, 2) | 13.1% | Factor pair saves two mul, costs four succ |
| (2, −1) | 2.0% | Reverse compensation |
| (−6, 3) | 1.1% | Saves three mul, costs six succ |
5.3 Distribution of Jump Gain (n ≤ 10^6)
| G Value | Percentage |
|---|---|
| −1 | 9.8% |
| 0 | 22.6% |
| 1 | 29.8% |
| 2 | 22.2% |
| 3 | 8.4% |
Two highest peaks at G = 0 and G = 1. Nearly a quarter of all composites at the exact jump/no-jump boundary — structural concentration on the integer lattice, not a measure-zero artifact.
6. Conditional Distribution of G Given Ω
6.1 Jump Rates by Prime Factor Count (n ≤ 10^6)
| Ω | p(Ω) jump | q(Ω) tie | P(G<0) |
|---|---|---|---|
| 2 | 24.8% | 31.9% | 43.3% |
| 3 | 52.9% | 32.7% | 14.4% |
| 4 | 75.0% | 20.5% | 4.5% |
| 5 | 87.3% | 11.2% | 1.5% |
| 6 | 93.5% | 6.0% | 0.6% |
| 7 | 96.5% | 3.3% | 0.3% |
| ≥10 | ≥99.7% | ≤0.3% | ≤0.0% |
6.2 Conditional Moments of G | Ω = k
| Ω | ⟨G⟩ | σ(G) | Mode |
|---|---|---|---|
| 2 | −0.37 | 1.32 | 0 |
| 3 | 0.56 | 1.05 | 1 |
| 4 | 1.13 | 1.00 | 1 |
| 5 | 1.58 | 1.01 | 2 |
| 6 | 1.98 | 1.04 | 2 |
| 7 | 2.32 | 1.09 | 2 |
6.3 Key Observations
Three features: (a) σ ≈ 1.0 approximately constant across Ω. (b) Mean grows logarithmically: μ(Ω) ≈ −1.93 + 2.21 ln Ω (R² = 0.998). (c) Distribution shifts rightward with Ω while maintaining shape.
7. Zero-Inflated Lattice Normal Model
7.1 Normal Approximation
Discrete normal approximation: p(Ω) ≈ Φ((μ(Ω)−0.5)/σ). Comparison with late window [800k, 10^6):
| Ω | Φ prediction | Actual p | Error |
|---|---|---|---|
| 2 | 0.246 | 0.240 | +0.006 |
| 3 | 0.508 | 0.514 | −0.006 |
| 4 | 0.723 | 0.736 | −0.013 |
| 5 | 0.851 | 0.865 | −0.014 |
| 7 | 0.949 | 0.963 | −0.014 |
Fits to within 1–2%, systematic underestimation ≈ 1.4%.
7.2 Zero-Inflated Lattice Normal
G | Ω = k is a superposition of: (1) a discrete atom at G=0 from algebraic locking condition Δs+2Δm=0; (2) a background fluctuation field approximately following shifted discrete normal N(μ(k), σ²).
7.3 N-Drift Analysis
Jump probability p(Ω, N) decreases with N:
| Ω | [2, 200k) | [800k, 10^6) |
|---|---|---|
| 2 | 0.264 | 0.240 |
| 3 | 0.559 | 0.514 |
| 5 | 0.892 | 0.865 |
Decline decelerates; fit p(Ω,N) ≈ p_∞(Ω) + b(Ω)/ln N gives R² > 0.95, p_∞(Ω) > 0 for all Ω.
8. Revision of Conjecture H to H'
8.1 Observed Trend
D(N) monotonically increasing. Values:
- D(10^3) = 0.551
- D(10^4) = 0.561
- D(10^5) = 0.570
- D(10^6) = 0.574
8.2 Conjecture H'
Conjecture H': D(N) → 1 as N → ∞.
8.3 Supporting Evidence
(a) Erdős-Kac implies P(Ω(n) ≤ K | n ≤ N) → 0 for every fixed K; as N grows, the typical value of Ω(n) grows like ln ln N, driving p(Ω) → 1.
(b) Numerical fits yield p_∞(Ω) > 0 with p_∞(Ω) → 1 as Ω → ∞.
8.4 Obstacles to Proof
(a) Existence of limits p_∞(Ω) not established. (b) Proof requires uniform control of p(k,N) within Erdős-Kac drift window |k − ln ln N| ≤ A√(ln ln N). Estimated rate: 1 − D(N) ∼ C(ln N)^{−α} with α ≈ 0.5–0.7.
9. Discussion: Connections and Predictions
9.1 Non-Trivial Predictions
(1) p_∞(3) ≈ 0.50 (50% crossing at Ω = 3). (2) D(10^100) ≈ 0.81 if H' holds. (3) σ(G | Ω) ≈ 1.0 constant across Ω and N — no theoretical explanation known; may reflect discrete fluctuation-dissipation relation.
9.2 Classical Connections
Ω(n) → Erdős-Kac theorem (1940); deficiency d(n) bridges ρ_E and ‖n‖; Selfridge (1953) asymptotics of ‖n‖/ln n remain open.
9.3 Two Surrogates
f(n) = Σ_{p^a ‖ n} ρ_E(p^a): Genuinely additive, Pearson correlation 0.947 with ρ_E, suitable for large-scale density theory but fails to preserve zero-point atom (only 15.6% of ties have G_f=0).
m(n): Accounts for 88% of d(n), has quasi-additive recursion, preserves discrete structure.
10. Conclusion
Theorems 16–19 reduce the jump mechanism from statistical phenomenon to discrete lattice structure. Jumps are driven not by classical efficiency but by deficiency splitting: 86.9% of jumps occur against the verdict of integer complexity. The 22.6% zero-point atom arises from exact algebraic constraint Δs+2Δm=0. Conditional on Ω, G follows zero-inflated lattice normal with logarithmically growing mean and constant σ. Conjecture H revised to H': D(N) → 1, rate controlled by interplay between Erdős-Kac drift and decline of conditional jump rates — an asymptotic racing problem.
11. References
- Han Qin, "Staircase Geometry of δ and Jump-Gap Theory," ZFCρ Series Paper XII, 2026. DOI: 10.5281/zenodo.18977948
- Han Qin, "The First Asymptotic Theory of ρ_E," ZFCρ Series Paper XI, 2026. DOI: 10.5281/zenodo.18975756
- Erdős, P., and Kac, M. "The Gaussian law of errors in the theory of additive functions." Proceedings of the National Academy of Sciences 25, no. 4 (1939): 206–207.
- Selfridge, J. L. "Generalized Mersenne numbers." Mathematics of Computation 8, no. 48 (1953): 147–152.
← ZFCρ Paper XII: Staircase Geometry of δ and Jump-Gap Theory
ZFCρ Series · Mathematical Foundations · Back to Papers
Paper 12确立了δ-阶梯的局部几何：跳跃是纯乘法（定理13），δ的所有增长来自跳跃累积（定理14），密度/跳跃/平台由代数恒等式锁定（定理15）。猜想H声称D(N) → d ≈ 0.57。本文研究跳跃增益G(n)——最佳乘法路径相对于后继路径的带符号优势。四个精确结果：
定理16（增益分解）：G_n(a,b) = G_IC(n;a,b) + G_d(n;a,b)，其中G_IC生活在整数复杂度中，G_d是缺陷贡献。数值上（n ≤ 50,000），86.9%跳跃发生在G_IC ≤ 0：经典项反对跳跃，缺陷驱动之。
定理17（操作计数）：ρ_E(n) = s(n) + 2m(n)，其中s计数后继节点，m计数乘法节点。
定理18（缺陷分裂）：若规范最优表达式根于mul(a,b)，则d(n) = d(a) + d(b) + 2 + Δ_IC其中Δ_IC = ‖a‖+‖b‖−‖n‖ ≥ 0。
定理19（并列判据）：G(n) = 0当且仅当Δs + 2Δm = 0——产生22.6%零点原子的格点方程。条件于Ω(n)，增益G遵循零膨胀格点正态：代数锁定产生G=0的离散原子，加上均值≈ −1.93 + 2.21 ln Ω、σ ≈ 1的移位离散正态背景。猜想H修正为H'：D(N) → 1。
1. 引言：跳跃增益作为格点现象
1.1 跳跃增益定义
对n ≥ 2合数，定理13陈述n跳跃当且仅当M_n < S_n其中M_n = min_{ab=n, a,b≥2}(ρ_E(a)+ρ_E(b)+2)且S_n = ρ_E(n−1)+1。定义跳跃增益：
对每个因子分解ab=n（a,b≥2），以及G(n) = max_{ab=n} G_n(a,b)。则：
- n跳跃 ⟺ G(n) > 0
- n并列（零点）⟺ G(n) = 0
- n不跳跃 ⟺ G(n) < 0
1.2 动机：为何研究增益？
G(n)是乘法路径相对于后继路径的带符号优势。理解其符号即理解何时捷径（乘法）胜出渐进（后继）。数值发现86.9%跳跃有G_IC ≤ 0反转了经典直觉：经典复杂度测度反对跳跃，然而跳跃发生。为何？缺陷主导。
2. 规范选择与记号
2.1 规范最优表达式
在所有达成ρ(h) = ρ_E(n)的紧凑项h ∈ Hist*(n)中，选择有：(1)最少乘法节点m(h)；(2)若并列，字典序最小的前序序列化。这使s(n)、m(n)、path(n)良定。
规范因子对(a*, b*)最大化G_n(a,b)；若多对并列，取字典序最小且a* ≤ b*的。
2.2 关键记号
G_n(a,b)：特定因子分解ab=n的增益
G(n)：最优增益 = max G_n在所有ab=n上
(a*,b*)：达成G(n)的规范因子对
s(n)：规范最优表达式中后继节点计数
m(n)：规范最优表达式中乘法节点计数
d(n)：缺陷 = ρ_E(n) − ‖n‖
Ω(n)：计重素因子数
3. 定理16：增益分解
3.1 陈述
定理16：对每个ab = n（a, b ≥ 2）：
其中
（整数复杂度项，生活在经典世界），且
（缺陷贡献）。
3.2 证明草图
将ρ_E(m) = ‖m‖ + d(m)代入G_n(a,b)定义：
3.3 数值结果1：缺陷主导
在跳跃的合数n ≤ 50,000中（G(n) > 0）：
- 平均G_IC = −0.585（反对跳跃）
- 平均G_d = +2.345（驱动跳跃）
- 在86.9%所有跳跃中，G_IC ≤ 0：经典测度反对跳跃
解释：跳跃由逃离缺陷过高的前驱触发——不是由找到便宜因子分解。后继路径继承d(n−1)；乘法路径分散成本于d(a)+d(b)。当前者超过后者，分裂增益触发跳跃。经典整数复杂度测度对抗跳跃；缺陷推翻判决。
4. 缺陷的结构
4.1 定理17：操作计数
定理17：在默认参数(c_S, c_⊕, c_⊗) = (1, 1, 2)下，规范最优表达式满足：
证明：由加法主导定理（Paper 6 §4.2），最优表达式中不出现加法节点。每个后继节点贡献c_S = 1，每个乘法节点贡献c_⊗ = 2。
4.2 推论与数值结果2
推论：d(n) = 2m(n) + (s(n) − ‖n‖)且s(n) − ‖n‖ ≥ 0。
数值上（n ≤ 10,000），项2m(n)占d(n)的88%，确认缺陷绝大由乘法节点开销驱动。
4.3 定理18：缺陷分裂
定理18：若n的规范最优表达式根于mul(a,b)，则：
其中Δ_IC = ‖a‖+‖b‖−‖n‖ ≥ 0测量整数复杂度的次加性盈余。
证明：ρ_E(n) = ρ_E(a)+ρ_E(b)+2，则
Δ_IC的非负性：a×b是n的有效{1,+,×}-表达式，故‖n‖ ≤ ‖a‖+‖b‖。
4.4 数值结果3：Δ_IC分布
对n ≤ 10,000：
- Δ_IC = 0占61%
- Δ_IC = 1占25%
- Δ_IC = 2占9%
4.5 m(n)递推
当规范路径是后继时m(n) = m(n−1)；当路径是mul(a,b)时m(n) = m(a)+m(b)+1。
5. 定理19：并列判据与零点原子
5.1 定义与主要结果
定义Δs = s(n−1)−s(a*)−s(b*)−1和Δm = m(n−1)−m(a*)−m(b*)。则：
定理19：G(n) = Δs + 2Δm。特别地，G(n) = 0当且仅当Δs = −2Δm——一个格点方程。
证明：由定理17：
5.2 并列模式（n ≤ 10^6）
| (Δs, Δm) | 比例 | 解释 |
|---|---|---|
| (−2, 1) | 54.1% | 因子对节省一个乘法，成本两个后继 |
| (0, 0) | 29.7% | 相同(s, m)分解 |
| (−4, 2) | 13.1% | 因子对节省两个乘法，成本四个后继 |
| (2, −1) | 2.0% | 反向补偿 |
| (−6, 3) | 1.1% | 节省三乘法，成本六后继 |
5.3 跳跃增益分布（n ≤ 10^6）
| G值 | 百分比 |
|---|---|
| −1 | 9.8% |
| 0 | 22.6% |
| 1 | 29.8% |
| 2 | 22.2% |
| 3 | 8.4% |
G = 0和G = 1处最高两峰。近四分之一所有合数在精确跳跃/不跳跃边界——整数格上的结构浓聚，非测度零人工品。
6. G给定Ω的条件分布
6.1 按素因子数的跳跃率（n ≤ 10^6）
| Ω | p(Ω)跳跃 | q(Ω)并列 | P(G<0) |
|---|---|---|---|
| 2 | 24.8% | 31.9% | 43.3% |
| 3 | 52.9% | 32.7% | 14.4% |
| 4 | 75.0% | 20.5% | 4.5% |
| 5 | 87.3% | 11.2% | 1.5% |
| 6 | 93.5% | 6.0% | 0.6% |
| 7 | 96.5% | 3.3% | 0.3% |
| ≥10 | ≥99.7% | ≤0.3% | ≤0.0% |
6.2 G | Ω = k的条件矩
| Ω | ⟨G⟩ | σ(G) | 众数 |
|---|---|---|---|
| 2 | −0.37 | 1.32 | 0 |
| 3 | 0.56 | 1.05 | 1 |
| 4 | 1.13 | 1.00 | 1 |
| 5 | 1.58 | 1.01 | 2 |
| 6 | 1.98 | 1.04 | 2 |
| 7 | 2.32 | 1.09 | 2 |
6.3 关键观察
三个特征：(a) σ ≈ 1.0在Ω上近似常数。(b) 均值对数增长：μ(Ω) ≈ −1.93 + 2.21 ln Ω（R² = 0.998）。(c) 分布随Ω向右移动同时保持形状。
7. 零膨胀格点正态模型
7.1 正态近似
离散正态近似：p(Ω) ≈ Φ((μ(Ω)−0.5)/σ)。与晚期窗口[800k, 10^6)比较：
| Ω | Φ预测 | 实际p | 误差 |
|---|---|---|---|
| 2 | 0.246 | 0.240 | +0.006 |
| 3 | 0.508 | 0.514 | −0.006 |
| 4 | 0.723 | 0.736 | −0.013 |
| 5 | 0.851 | 0.865 | −0.014 |
| 7 | 0.949 | 0.963 | −0.014 |
拟合在1–2%内，系统低估≈ 1.4%。
7.2 零膨胀格点正态
G | Ω = k是如下的叠加：(1)代数锁定条件Δs+2Δm=0产生的G=0处离散原子；(2)背景波动场近似遵循移位离散正态N(μ(k), σ²)。
7.3 N-漂移分析
跳跃概率p(Ω, N)随N递减：
| Ω | [2, 200k) | [800k, 10^6) |
|---|---|---|
| 2 | 0.264 | 0.240 |
| 3 | 0.559 | 0.514 |
| 5 | 0.892 | 0.865 |
下降减缓；拟合p(Ω,N) ≈ p_∞(Ω) + b(Ω)/ln N给出R² > 0.95，对所有Ω有p_∞(Ω) > 0。
8. 猜想H修正为H'
8.1 观察趋势
D(N)单调递增。值为：
- D(10^3) = 0.551
- D(10^4) = 0.561
- D(10^5) = 0.570
- D(10^6) = 0.574
8.2 猜想H'
猜想H'：当N → ∞时，D(N) → 1。
8.3 支持证据
(a) Erdős-Kac蕴含对每个固定K，P(Ω(n) ≤ K | n ≤ N) → 0；当N增长，Ω(n)的典型值像ln ln N增长，驱动p(Ω) → 1。
(b) 数值拟合给出p_∞(Ω) > 0且当Ω → ∞时p_∞(Ω) → 1。
8.4 证明的障碍
(a) 极限p_∞(Ω)的存在未建立。(b) 证明需要在Erdős-Kac漂移窗口|k − ln ln N| ≤ A√(ln ln N)内对p(k,N)的一致控制。估算率：1 − D(N) ∼ C(ln N)^{−α}其中α ≈ 0.5–0.7。
9. 讨论：联系与预测
9.1 非平凡预测
(1) p_∞(3) ≈ 0.50（Ω = 3处50%穿越）。(2) 若H'成立，D(10^100) ≈ 0.81。(3) σ(G | Ω) ≈ 1.0在Ω和N上常数——无已知理论解释；可能反映离散波动-耗散关系。
9.2 经典联系
Ω(n) → Erdős-Kac定理（1940）；缺陷d(n)桥接ρ_E和‖n‖；Selfridge（1953）关于‖n‖/ln n的渐近仍开放。
9.3 两个代理
f(n) = Σ_{p^a ‖ n} ρ_E(p^a)：纯加性，与ρ_E的Pearson相关0.947，适用于大规模密度理论但失败于保留零点原子（仅15.6%并列有G_f=0）。
m(n)：占d(n)的88%，有准加性递推，保留离散结构。
10. 结论
定理16–19将跳跃机制从统计现象简化为离散格点结构。跳跃不由经典效率驱动而由缺陷分裂：86.9%跳跃反对整数复杂度的判决发生。22.6%零点原子源自精确代数约束Δs+2Δm=0。条件于Ω，G遵循零膨胀格点正态且均值对数增长、σ常数。猜想H修正为H'：D(N) → 1，率由Erdős-Kac漂移与条件跳跃率衰退之间的相互作用控制——一个渐近竞赛问题。
11. 参考文献
- Han Qin, "Staircase Geometry of δ and Jump-Gap Theory," ZFCρ Series Paper XII, 2026. DOI: 10.5281/zenodo.18977948
- Han Qin, "The First Asymptotic Theory of ρ_E," ZFCρ Series Paper XI, 2026. DOI: 10.5281/zenodo.18975756
- Erdős, P., and Kac, M. "The Gaussian law of errors in the theory of additive functions." Proceedings of the National Academy of Sciences 25, no. 4 (1939): 206–207.
- Selfridge, J. L. "Generalized Mersenne numbers." Mathematics of Computation 8, no. 48 (1953): 147–152.