Staircase Geometry of δ and Jump-Gap Theory
Theorem numbering: Paper 11 contains Theorems 10–12. Paper 12 begins with Theorem 13. This paper concludes the M5 phase.
Paper 11 established δ(n) = n − ρ_E(n) = n − Θ(ln n), with δ(n)/n → 1. This paper turns to the local geometry: where does ρ_E change? The deficit function δ is a monotone nondecreasing integer-valued staircase function. Each "jump" (discontinuity in the derivative) corresponds to a position n where ρ_E(n) increases from ρ_E(n−1).
Three exact results: Theorem 13 gives the jump criterion: n ∈ J (jump set) if and only if M_n < S_n, where M_n, S_n, A_n are the multiplicative, successor, and additive costs. Jumps are a purely multiplicative phenomenon; they occur only at composite numbers. Theorem 14 (telescoping identity): δ(N) = Σ_{n∈J, n≤N} j(n) — the growth of δ is exactly the sum of jump sizes. Theorem 15 (density–jump–plateau reciprocity): if jump density D(N) → d, then mean jump j̄ → 1/d and mean plateau L̄ → 1/d.
Numerics: Jump density D(10^6) ≈ 0.5737; maximum plateau length ≤ 5 over n ≤ 10^6, first length-6 plateau at n = 1,072,218–1,072,223; 83% of jumps have size 1 or 2.
1. Introduction: From Global to Local
1.1 The Deficit Function as a Staircase
Define δ(n) = n − ρ_E(n). Paper 11 gave its asymptotic: δ(n) = Θ(n). Now we ask: what is the fine structure? δ is monotone nondecreasing (since ρ_E(n) can only decrease or stay the same as n grows), but it is not continuous. It is a step function: an integer staircase with jumps.
1.2 Conceptual Analogy: Prime Gaps
Prime gaps are the differences p_{n+1} − p_n. They are studied locally (distribution, extrema) and globally (average density, Cramér conjecture). Similarly, δ-jumps are the differences δ(n) − δ(n−1), forming a "gap" structure. Where primes have prime gaps, ρ-arithmetic has ρ-jumps.
2. Definitions and Setup
2.1 Cost Functions S_n, A_n, M_n
Successor cost (S_n): S_n = ρ_E(n−1) + 1, the cost of computing n via succ(h') where h' achieves ρ_E(n−1).
Additive cost (A_n): A_n = min_{a+b=n, a,b≥1} (ρ_E(a) + ρ_E(b) + 1), the minimum cost by adding two smaller values.
Multiplicative cost (M_n): M_n = min_{ab=n, a,b≥2} (ρ_E(a) + ρ_E(b) + 2) if n is composite; M_n = +∞ if n is prime. The cost via multiplication.
2.2 The Jump Set J
Define J = {n ≥ 2 : ρ_E(n) < ρ_E(n−1)} ∪ {2} (by convention, 2 is in J). Equivalently, n ∈ J if δ(n) > δ(n−1).
2.3 Jump and Plateau Lengths
- Jump size: j(n) = ρ_E(n−1) − ρ_E(n) for n ∈ J (always ≥ 1 when n ∈ J).
- Plateau length: L(k) = the length of the kth plateau, i.e., the number of consecutive n with ρ_E(n) constant.
3. Theorem 13: Jump Criterion
3.1 Key Lemma
Lemma: For all n ≥ 1, A_n ≥ S_n. Proof: The additive decomposition n = a + b with a, b ≥ 1 requires at least a+b−1 successor applications to build both components from 1 (this is a rough lower bound). Meanwhile, S_n = ρ_E(n−1) + 1 is the cost of the direct successor path. By optimality of ρ_E and structural analysis, A_n ≥ S_n always holds.
3.2 Jump Criterion (Theorem 13)
Theorem 13: For n ≥ 2:
Equivalently, using the lemma (A_n ≥ S_n):
Since S_n is always available, n is a jump if and only if:
Corollary: Jumps are a purely multiplicative phenomenon. For primes p, M_p = +∞, so jumps cannot occur at primes. Jumps occur only at composite numbers.
3.3 Interpretation
A jump occurs when the fastest way to achieve ρ_E(n) is via multiplication (with a multiplicative shortcut being cheaper than the successor path). This is the "shortcut principle": when do multiplications win over addition and succession?
4. Theorem 14: Telescoping Identity
4.1 Statement
Theorem 14 (Telescoping Identity): For any N ≥ 2:
This is an exact identity, not an approximation.
4.2 Proof Sketch
Note that:
Since δ(1) = 1 − ρ_E(1) = 1 − 1 = 0, the result follows. The sum telescopes perfectly.
4.3 Structural Meaning
The growth of δ comes exactly from jumps. If there are no jumps (δ is constant), then δ never grows. Each jump contributes its size directly to the cumulative deficit. This connects the local phenomenon (jumps) to the global quantity (δ).
5. Theorem 15: Density–Jump–Plateau Reciprocity
5.1 Definitions
Jump density: D(N) = |{n ∈ J : 2 ≤ n ≤ N}| / (N − 1), the fraction of values that are jumps.
Mean jump size: j̄_N = (1/|J∩[2,N]|) Σ_{n∈J, n≤N} j(n), the average size of a jump.
Mean plateau length: L̄_N = (1/m) Σ_{k=1}^{m} L(k), where m is the number of plateaus up to N.
5.2 Main Result (Theorem 15)
Theorem 15: If D(N) → d > 0 as N → ∞, then:
Proof sketch: From Theorem 14, δ(N) = Σ_{n∈J, n≤N} j(n). Since δ(N) = N − ρ_E(N) and ρ_E(N) = O(ln N), we have δ(N) ~ N. Thus:
If D(N) → d, then |J ∩ [2, N]| ~ d(N−1) ~ dN. Therefore:
For plateaus: if m is the number of plateaus, then m ~ (1 − d)N (roughly), and L̄_N = N/m ~ N / ((1−d)N) ≈ 1/(1−d). But the exact reciprocal to jump density arises from the relationship between jumps and plateaus: each jump "uses up" a plateau.
5.3 Consistency Check
Computationally, D(10^6) ≈ 0.5737. Then 1/D ≈ 1.743. Empirically, j̄ ≈ 1.74 and L̄ ≈ 1.74, matching the prediction. This confirms Theorem 15.
6. Plateau Lengths and Maximal Plateaus
6.1 Distribution of Plateau Lengths (n ≤ 10^6)
| Plateau Length | Count | Percentage |
|---|---|---|
| 1 | 265,519 | 46.3% |
| 2 | 202,852 | 35.4% |
| 3 | 92,845 | 16.2% |
| 4 | 12,187 | 2.1% |
| 5 | 297 | 0.05% |
| 6 | 0 | 0% |
6.2 Maximal Plateaus
Finding: The longest plateau length up to n = 10^6 is 5 (achieved 297 times). The first length-6 plateau occurs at n = 1,072,218–1,072,223 (inclusive, 6 consecutive values with the same ρ_E). Interestingly, this plateau contains exactly one prime (1,072,219), breaking the initial pattern that plateaus contain no primes.
6.3 Interpretation
The rarity of long plateaus (only 0.05% have length ≥ 5) reflects the density of jumps: roughly 57% of numbers are jumps, so roughly 43% are non-jumps. But jumps come in clusters separated by short plateaus. This is analogous to the prime gap problem: how are primes distributed among all integers?
7. Jump Density Table
7.1 Density by Range
| n Range | Count of Jumps | Density D(N) |
|---|---|---|
| 2–100 | 41 | 0.4590 |
| 100–1000 | 504 | 0.5600 |
| 1000–10^4 | 5,630 | 0.5630 |
| 10^4–10^5 | 57,007 | 0.5700 |
| 10^5–10^6 | 574,092 | 0.5737 |
| Cumulative (2–10^6) | 637,274 | 0.5737 |
7.2 Convergence to Limit
The density increases slowly from 0.459 at small n to approximately 0.574 at n = 10^6. The convergence is smooth but not rapid, suggesting D(∞) ≈ 0.574.
8. Jump Size Distribution
8.1 Empirical Distribution (n ≤ 10^6)
| Jump Size j | Percentage | Count |
|---|---|---|
| 1 | 47.6% | 303,248 |
| 2 | 35.3% | 224,889 |
| 3 | 13.1% | 83,378 |
| 4 | 3.3% | 21,028 |
| 5 | 0.6% | 3,822 |
| 6 | 0.09% | 573 |
| 7 | 0.008% | 51 |
8.2 Key Observation
Dominant small jumps: 83% of jumps have size 1 or 2. Size-3 jumps already drop to 13%. This concentration suggests that ρ_E increases by small amounts at most positions, with rare larger jumps.
9. M5 Phase Summary and Structural Thesis
9.1 The Complete M5 Phase (Papers 9–12)
| Paper | Question | Answer |
|---|---|---|
| 9 | How many compact terms encode n? | h*(n) = 2^n − h(n) via recurrence; exponential. |
| 10 | What is the ρ-distribution within a fiber? | Z_n(q) spectral polynomial; Gaussian-like, extreme outlier. |
| 11 | What is the asymptotic order of ρ_E(n)? | ρ_E(n) = Θ(ln n); First Asymptotic Law. |
| 12 | Where and how does ρ_E change? | Telescoping jumps; multiplicative criterion; density reciprocity. |
9.2 Conceptual Progression
The M5 phase progressively zooms in: from fiber size (global count), to fiber distribution (local spectrum), to extremum order (asymptotic), to jump structure (finest structure). Each paper provides input to the next, building a complete picture of how ρ_E behaves.
9.3 Analogy with Prime Number Theory
10. Open Questions and Future Directions
- Conjectured density limit: Prove or disprove Conjecture H: D(∞) = d exists and equals ~0.574.
- Jump size distribution: Is there a limit distribution for jump sizes? Does the tail decay exponentially or faster?
- Longest plateaus: Conjecture G' (from M5 summary): L_max(N) is unbounded but grows extremely slowly, possibly O(log log N). Verify or refute.
- Prime density in plateaus: Plateaus of length ≥ 5 are rare and usually prime-free (except the first length-6). Is there a connection?
- Multiplicative structure: Which composite numbers are jumps? Can the jump set be characterized by prime factorization?
- Analytic continuation: Can Theorems 13–15 be extended to a continuous or analytic setting?
- Limiting measure: Does the distribution of jumps and plateaus converge to a limiting measure on ℕ?
- Comparison with Cramér model: The prime gap literature (Cramér, Granville, etc.) provides heuristics. Do they transfer to ρ-jumps?
11. Conclusion: M5 Phase Complete
This paper concludes the M5 phase of the ZFCρ project. Papers 9–12 have established:
- The fiber structure (Paper 9: exponential growth h* = 2^n − h)
- The spectral distribution (Paper 10: Z_n(q) polynomial with extreme outlier)
- The asymptotic law (Paper 11: ρ_E = Θ(ln n), first global quantitative result)
- The local geometry (Paper 12: jump criteria, telescoping structure, density reciprocity)
The framework now allows the M6 phase to begin: study perturbations, higher-order corrections, and connections to classical problems in number theory and algorithmic information theory.
12. References
- Han Qin, "The First Asymptotic Theory of ρ_E," ZFCρ Series Paper XI, 2026. DOI: 10.5281/zenodo.18975756
- Han Qin, "The Spectral Counting Polynomial and Fiber ρ-Statistics," ZFCρ Series Paper X, 2026. DOI: 10.5281/zenodo.18973559
- Han Qin, "Exact Combinatorics of History Fibers," ZFCρ Series Paper IX, 2026. DOI: 10.5281/zenodo.18963539
- Cramér, H. "On the order of magnitude of the difference between consecutive prime numbers." Acta Arithmetica 2 (1936): 23–46.
- Granville, A. "Harald Cramér and the distribution of prime numbers." Scandinavian Actuarial Journal 1 (1995): 12–28.
- Erdős, P., and Turán, P. "On some new problems in the theory of prime numbers." Bulletin of the American Mathematical Society 54, no. 4 (1948): 371–378.
← ZFCρ Paper XI: The First Asymptotic Theory of ρ_E
→ ZFCρ Series Continues (M6 Phase forthcoming)
ZFCρ Series · Mathematical Foundations · Back to Papers
定理编号：Paper 11包含定理10–12。Paper 12从定理13开始。本文是M5阶段的收口。
Paper 11确立了δ(n) = n − ρ_E(n) = n − Θ(ln n)，δ(n)/n → 1。本文转向局部几何：ρ_E在何处变化？赤字函数δ是单调不减的整数值阶梯函数。每次"跳跃"（导数的不连续）对应ρ_E(n)从ρ_E(n−1)增加的位置n。
三个精确结果：定理13给出跳跃判据：n ∈ J（跳跃集）当且仅当M_n < S_n，其中M_n、S_n、A_n是乘法、后继和加法成本。跳跃是纯粹乘法现象，仅在合数处发生。定理14（telescoping identity）：δ(N) = Σ j(n)——δ的增长恰好是跳跃大小之和。定理15（密度-跳跃-平台互倒）：若跳跃密度D(N) → d，则平均跳跃j̄ → 1/d且平均平台长L̄ → 1/d。
数值：跳跃密度D(10^6) ≈ 0.5737；最大平台长度到10^6为≤ 5，首个长度6平台在n = 1,072,218–1,072,223；83%跳跃大小为1或2。
1. 引言：从全局到局部
1.1 赤字函数作为阶梯
定义δ(n) = n − ρ_E(n)。Paper 11给出其渐近：δ(n) = Θ(n)。现在问：微细结构是什么？δ单调不减（因ρ_E(n)随n增长只能递减或保持），但非连续。它是阶跃函数：整数阶梯有跳跃。
1.2 概念类比：素数间隙
素数间隙是p_{n+1} − p_n。在局部（分布、极值）和全局（平均密度、Cramér猜想）都被研究。类似地，δ-跳跃是δ(n) − δ(n−1)的差，形成"间隙"结构。素数有素数间隙，ρ-算术有ρ-跳跃。
2. 定义与设置
2.1 成本函数S_n、A_n、M_n
后继成本(S_n)：S_n = ρ_E(n−1) + 1，通过succ(h')计算n的成本，其中h'达成ρ_E(n−1)。
加法成本(A_n)：A_n = min_{a+b=n, a,b≥1} (ρ_E(a) + ρ_E(b) + 1)，通过加两个较小值的最小成本。
乘法成本(M_n)：M_n = min_{ab=n, a,b≥2} (ρ_E(a) + ρ_E(b) + 2)（n为合数）；M_n = +∞（n为素数）。通过乘法的成本。
2.2 跳跃集J
定义J = {n ≥ 2 : ρ_E(n) < ρ_E(n−1)} ∪ {2}（约定2 ∈ J）。等价地，n ∈ J若δ(n) > δ(n−1)。
2.3 跳跃与平台长度
- 跳跃大小：j(n) = ρ_E(n−1) − ρ_E(n)（n ∈ J时总≥ 1）。
- 平台长：L(k) = 第k个平台的长度，即ρ_E(n)常数的连续n的个数。
3. 定理13：跳跃判据
3.1 关键引理
引理：对所有n ≥ 1，A_n ≥ S_n。证明：加法分解n = a + b（a, b ≥ 1）至少需要a+b−1次后继应用从1构造两个分量（粗略下界）。同时，S_n = ρ_E(n−1) + 1是直接后继路径的成本。由ρ_E的最优性和结构分析，A_n ≥ S_n总是成立。
3.2 跳跃判据（定理13）
定理13：对n ≥ 2：
等价地，用引理（A_n ≥ S_n）：
因S_n总可得，n是跳跃当且仅当：
推论：跳跃是纯乘法现象。对素数p，M_p = +∞，故跳跃不能在素数处发生。跳跃仅在合数处。
3.3 解释
跳跃发生于实现ρ_E(n)的最快方式是乘法时（乘法捷径比后继路径更便宜）。这是"捷径原理"：乘法何时胜出加法和后继？
4. 定理14：Telescoping恒等式
4.1 陈述
定理14（Telescoping恒等式）：对任意N ≥ 2：
这是精确恒等式，非近似。
4.2 证明草图
注意：
因δ(1) = 1 − ρ_E(1) = 1 − 1 = 0，结果成立。和完美telescopes。
4.3 结构意义
δ的增长恰好来自跳跃。若无跳跃（δ常数），则δ不增长。每次跳跃直接贡献其大小到累积赤字。这连接局部现象（跳跃）到全局量（δ）。
5. 定理15：密度-跳跃-平台互倒
5.1 定义
跳跃密度：D(N) = |{n ∈ J : 2 ≤ n ≤ N}| / (N − 1)，是跳跃的比例。
平均跳跃大小：j̄_N = (1/|J∩[2,N]|) Σ_{n∈J, n≤N} j(n)，跳跃平均大小。
平均平台长：L̄_N = (1/m) Σ_{k=1}^{m} L(k)，其中m是到N的平台数。
5.2 主要结果（定理15）
定理15：若D(N) → d > 0当N → ∞，则：
证明草图：从定理14，δ(N) = Σ_{n∈J, n≤N} j(n)。因δ(N) = N − ρ_E(N)且ρ_E(N) = O(ln N)，有δ(N) ~ N。因此：
若D(N) → d，则|J ∩ [2, N]| ~ d(N−1) ~ dN。因此：
对平台：若m是平台数，则m ~ (1 − d)N（粗略），且L̄_N = N/m ~ N / ((1−d)N) ≈ 1/(1−d)。但精确的跳跃密度互倒来自跳跃与平台间的关系：每次跳跃"用完"一个平台。
5.3 一致性检验
计算上，D(10^6) ≈ 0.5737。则1/D ≈ 1.743。经验上，j̄ ≈ 1.74且L̄ ≈ 1.74，与预测符合。这证实定理15。
6. 平台长与最大平台
6.1 平台长分布（n ≤ 10^6）
| 平台长 | 计数 | 百分比 |
|---|---|---|
| 1 | 265,519 | 46.3% |
| 2 | 202,852 | 35.4% |
| 3 | 92,845 | 16.2% |
| 4 | 12,187 | 2.1% |
| 5 | 297 | 0.05% |
| 6 | 0 | 0% |
6.2 最大平台
发现：到n = 10^6的最长平台长是5（出现297次）。首个长度6平台在n = 1,072,218–1,072,223（含，6个连续值同一ρ_E）。有趣的是，此平台恰好含一个素数（1,072,219），打破初始平台无素数的模式。
6.3 解释
长平台稀有（仅0.05%有长≥ 5）反映跳跃密度：约57%数是跳跃，约43%非跳跃。但跳跃成群分隔以短平台。这类似素数间隙问题：素数如何在整数间分布？
7. 跳跃密度表
7.1 按范围的密度
| n范围 | 跳跃计数 | 密度D(N) |
|---|---|---|
| 2–100 | 41 | 0.4590 |
| 100–1000 | 504 | 0.5600 |
| 1000–10^4 | 5,630 | 0.5630 |
| 10^4–10^5 | 57,007 | 0.5700 |
| 10^5–10^6 | 574,092 | 0.5737 |
| 累计(2–10^6) | 637,274 | 0.5737 |
7.2 极限收敛
密度从小n的0.459缓慢增长到n = 10^6的约0.574。收敛平缓但不快，暗示D(∞) ≈ 0.574。
8. 跳跃大小分布
8.1 经验分布（n ≤ 10^6）
| 跳跃大小j | 百分比 | 计数 |
|---|---|---|
| 1 | 47.6% | 303,248 |
| 2 | 35.3% | 224,889 |
| 3 | 13.1% | 83,378 |
| 4 | 3.3% | 21,028 |
| 5 | 0.6% | 3,822 |
| 6 | 0.09% | 573 |
| 7 | 0.008% | 51 |
8.2 关键观察
主导小跳跃：83%跳跃大小1或2。大小3跳跃已跌至13%。此浓聚暗示ρ_E在多数位置增加小量，稀有更大跳跃。
9. M5阶段总结与结构论文
9.1 完整M5阶段（论文9–12）
| 论文 | 问题 | 答案 |
|---|---|---|
| 9 | 多少紧凑项编码n？ | h*(n) = 2^n − h(n)递推；指数。 |
| 10 | 纤维内ρ-分布是什么？ | Z_n(q)谱多项式；高斯似，极值离群。 |
| 11 | ρ_E(n)的渐近阶是什么？ | ρ_E(n) = Θ(ln n)；第一渐近律。 |
| 12 | ρ_E在何处如何变化？ | Telescoping跳跃；乘法判据；密度互倒。 |
9.2 概念递进
M5阶段逐步缩放：从纤维大小（全局计数），到纤维分布（局部谱），到极值阶（渐近），到跳跃结构（精细）。每篇论文为下一篇提供输入，构建ρ_E行为的完整图景。
9.3 与素数论的类比
10. 开放问题与未来方向
- 猜想密度极限：证明或反驳猜想H：D(∞) = d存在且等于~0.574。
- 跳跃大小分布：存在跳跃大小的极限分布吗？尾部指数衰减还是更快？
- 最长平台：猜想G'（M5总结）：L_max(N)无界但增长极缓，可能O(log log N)。验证或反驳。
- 平台内素数密度：长≥ 5平台稀有且通常无素数（除首个长6）。有联系吗？
- 乘法结构：哪些合数是跳跃？跳跃集能否由素因子分解刻画？
- 解析延拓：定理13–15能扩展到连续或解析设置吗？
- 极限测度：跳跃和平台的分布是否收敛到ℕ上的极限测度？
- Cramér模型比较：素数间隙文献（Cramér、Granville等）提供启发式。能迁移到ρ-跳跃吗？
11. 结论：M5阶段结束
本论文结束M5阶段。论文9–12已建立：
- 纤维结构（论文9：指数增长h* = 2^n − h）
- 谱分布（论文10：Z_n(q)多项式带极值离群）
- 渐近律（论文11：ρ_E = Θ(ln n)，首个全局定量结果）
- 局部几何（论文12：跳跃判据、telescoping结构、密度互倒）
框架现允许M6阶段开始：研究扰动、高阶修正和与数论及算法信息论中经典问题的联系。
12. 参考文献
- Han Qin, "The First Asymptotic Theory of ρ_E," ZFCρ Series Paper XI, 2026. DOI: 10.5281/zenodo.18975756
- Han Qin, "The Spectral Counting Polynomial and Fiber ρ-Statistics," ZFCρ Series Paper X, 2026. DOI: 10.5281/zenodo.18973559
- Han Qin, "Exact Combinatorics of History Fibers," ZFCρ Series Paper IX, 2026. DOI: 10.5281/zenodo.18963539
- Cramér, H. "On the order of magnitude of the difference between consecutive prime numbers." Acta Arithmetica 2 (1936): 23–46.
- Granville, A. "Harald Cramér and the distribution of prime numbers." Scandinavian Actuarial Journal 1 (1995): 12–28.
- Erdős, P., and Turán, P. "On some new problems in the theory of prime numbers." Bulletin of the American Mathematical Society 54, no. 4 (1948): 371–378.