The First Asymptotic Theory of ρ_E

Qin, Han

doi:10.5281/zenodo.18975756

English

中文

Abstract

Theorem numbering: Paper 10 contains Theorems 6–9. Paper 11 begins with Theorem 10.

Paper 10 established that ρ_E(n) lies 5–14 standard deviations below the fiber mean, marking it as an extreme-value outlier. This paper determines the asymptotic order of ρ_E(n). The core tool is a structural bridge between ρ_E and integer complexity ‖n‖. By constructing exact translations between compact ρ-terms and {1,+,×}-expressions, we prove ‖n‖ ≤ ρ_E(n) ≤ 2‖n‖ − 2 (Theorems 10, 11).

Main result: Combined with the classical Selfridge bounds on ‖n‖, we obtain (3/ln 3)ln n ≤ ρ_E(n) ≤ (6/ln 2)ln n + O(1), establishing the first asymptotic law: ρ_E(n) = Θ(ln n) (Theorem 12). Corollary: δ(n) = n − ρ_E(n) = n − Θ(ln n), hence δ(n)/n → 1.

Numerical findings: Over n ≤ 10000, ρ_E/ln n oscillates in [2.73, 4.95]. The oscillation reflects multiplicative structure; exact values of liminf and limsup remain open (Conjecture F).

1. Introduction and Motivation

1.1 From Extreme Value to Asymptotics

Paper 10 showed that ρ_E(n) is an extreme outlier: its z-score grows unboundedly (|z_n| → ∞), meaning ρ_E becomes ever more exceptional relative to its fiber. But we still need the quantitative order: is ρ_E(n) = O(1), O(ln n), O(√n), O(n)? This paper answers: ρ_E(n) = Θ(ln n).

1.2 Structural Position: A PNT Analogue

The role of Theorem 12 within the ZFCρ series is analogous to the Prime Number Theorem within number theory. PNT gives π(n) ~ n/ln n as the first quantitative global law for prime distribution. Theorem 12 gives ρ_E(n) = Θ(ln n) as the first quantitative global law for the extremum of ρ-arithmetic.

2. Integer Complexity and ρ-Terms

2.1 Definition of ‖n‖

Definition: The integer complexity ‖n‖ is the minimum number of 1's needed in a {1, +, ×}-expression evaluating to n. Formally:

‖n‖ := min{ s(e) : e is a {1,+,×}-expression, eval(e) = n }

where s(e) counts 1's in the expression tree.

2.2 Selfridge Bounds

The classical Selfridge bounds (1953) give:

3 log₃(n) ≤ ‖n‖ ≤ 3 log₂(n)

Equivalently (base-3 to natural log):

(3/ln 3) ln n ≤ ‖n‖ ≤ 3 ln n / ln 2 = (3/ln 2) ln n

2.3 Compact ρ-Terms Encode Minimally

A key observation: compact ρ-terms h ∈ Hist*(n) are precisely the minimal-ones representations of n in {1, +, ×}-arithmetic (possibly with some structural flexibility in associativity). Thus, the cost function ρ(h) is closely related to ‖n‖.

3. Bridge Between ρ_E and ‖n‖

3.1 Lower Bound (Theorem 10)

Theorem 10 (Lower Bound): ρ_E(n) ≥ ‖n‖ for all n ≥ 1.

Proof: Let h ∈ Hist*(n) be any compact ρ-term for n. Consider the {1,+,×}-expression induced by h (reading off the tree structure). The number of 1's in this expression is s(h) = ρ(h), since each leaf corresponds to a 1 in ρ-arithmetic and each internal add or mul node contributes 0 ones. Thus ‖n‖ ≤ s(h) = ρ(h). Taking the minimum over all h ∈ Hist*(n) gives ‖n‖ ≤ min ρ_E(n), hence ρ_E(n) ≥ ‖n‖. ✓

3.2 Upper Bound (Theorem 11)

Theorem 11 (Upper Bound): ρ_E(n) ≤ 2‖n‖ − 2 for all n ≥ 2.

Proof sketch: Conversely, given a minimal {1,+,×}-expression for n (achieving ‖n‖ ones), construct a compact ρ-term h from it by reversing the correspondence. The number of add/mul nodes in the expression is ‖n‖ − 1 (since an expression with k ones has k−1 binary operations). In ρ-arithmetic, each succ contributes 1, and each add/mul contributes 1 or 2. The detailed accounting yields ρ(h) ≤ ‖n‖ + (‖n‖ − 1) = 2‖n‖ − 1, and a careful analysis reduces this to 2‖n‖ − 2 for n ≥ 2. (Full structural induction in main text.)

3.3 Combined Estimate

Theorems 10 and 11 give:

‖n‖ ≤ ρ_E(n) ≤ 2‖n‖ − 2

Applying Selfridge bounds:

(3/ln 3) ln n ≤ ρ_E(n) ≤ 2 · (3/ln 2) ln n = (6/ln 2) ln n

4. Theorem 12: The First Asymptotic Law

4.1 Main Statement

Theorem 12 (First Asymptotic Law for ρ_E):

ρ_E(n) = Θ(ln n)

Precisely:

(3/ln 3) ln n ≤ ρ_E(n) ≤ (6/ln 2) ln n + O(1)

Equivalently, 2.73 ln n (approx) ≤ ρ_E(n) ≤ 2.73 ln n · 2/ln 3 ≈ 4.33 ln n (up to lower order terms).

4.2 Interpretation

The extremum cost ρ_E(n) grows logarithmically, not linearly. Compared to the mean cost E[ρ] ≈ 1.72n (from Paper 10), the extremum is vastly smaller. This is the signature of extreme-value behavior: the ground state ρ_E is a rare outlier that costs only O(ln n), while typical compact terms cost linear in n.

4.3 Numerical Validation

The bounds (3/ln 3) ≈ 2.7306 and (6/ln 2) ≈ 8.6644 give rough range [2.73 ln n, 8.66 ln n]. Over n ≤ 10000, the actual ρ_E/ln n oscillates in the narrower [2.73, 4.95], indicating the lower bound is tight but the upper bound has room.

5. Corollary: The Deficit Function

5.1 Definition

Define the deficit function δ(n) = n − ρ_E(n), i.e., the "gap" between the value and the extremum cost.

5.2 Asymptotic Behavior

Corollary (from Theorem 12): δ(n) = n − ρ_E(n) = n − Θ(ln n) = Θ(n), hence δ(n)/n → 1.

This means: as n grows, almost all of the value n is "not involved" in computing ρ_E(n). The extremum cost scales only logarithmically, leaving a linear surplus.

6. Numerical Analysis of ρ_E/ln n

6.1 Data Table: Extremum Cost Normalized by ln n

n Range	Min(ρ_E/ln n)	Max(ρ_E/ln n)	Oscillation Width
2–100	2.73	4.66	1.93
100–1000	3.91	4.86	0.95
1000–5000	4.04	4.95	0.91
5000–10000	4.10	4.82	0.72

6.2 Interpretation

The ratio ρ_E/ln n is not constant but oscillates. This oscillation reflects the multiplicative structure of n: powers of 2 and 3 achieve different ρ_E/ln n ratios due to their special factorization properties. As n increases, the oscillation amplitude narrows (from ~1.9 at small n to ~0.7 at n~10000), suggesting possible convergence to a limiting distribution.

7. Special Cases: Powers of Primes

7.1 Powers of 3

Observation: For n = 3^k, the optimal representation is essentially k iterations of doubling (or multiplication chains). Numerically, ρ_E(3^k) ≤ 5k − 2, with equality for small k. Thus:

ρ_E(3^k) / ln(3^k) = (5k−2) / (k ln 3) → 5/ln 3 ≈ 4.551 as k → ∞

7.2 Powers of 2

Observation: For n = 2^k, the chain of doublings yields ρ_E(2^k) ≤ 3k (approx), hence:

ρ_E(2^k) / ln(2^k) → 3/ln 2 ≈ 4.328 as k → ∞

This ratio is close to the lower Selfridge bound.

7.3 Implications

From 3/ln 2 ≈ 4.328 and 5/ln 3 ≈ 4.551, we see the oscillation is driven by factorization. The infimum and supremum of all ρ_E/ln n across all n are not yet proven, but bounded computations suggest:

Conjecture F: liminf_n ρ_E(n)/ln n = c_min and limsup_n ρ_E(n)/ln n = c_max exist, with 3/ln 2 ≤ c_min ≤ 4.328 and 2.73 ≤ c_min ≤ 5/ln 3.

8. Connection to Paper 12

8.1 Global Order as Input to Local Theory

Paper 11 provides the global quantitative order: ρ_E = Θ(ln n) and δ/n → 1. Paper 12 takes this as given and investigates the local structure: where exactly do jumps occur, how do they accumulate, and what is their density? The global theorem enables the local analysis by providing the asymptotic backdrop.

8.2 Structural Analogy

Just as the Prime Number Theorem gives a global count (π(n) ~ n/ln n) which then enables study of local phenomena (prime gaps, distribution), Theorem 12 establishes global ρ_E order, enabling local study in Paper 12.

9. Discussion: Why ρ_E = Θ(ln n)?

9.1 Intuition from Complexity Theory

In complexity theory, the "addition chain" length for computing n (i.e., building n from 1 by repeated doublings and additions) grows logarithmically. The ρ-arithmetic is a generalization of this: ρ_E(n) is the minimal cost in a rich instruction set, and richness keeps it logarithmic.

9.2 Comparison to Other Measures

Successor-only (ρ ≡ n): Cost is linear; no compression possible.
Bit-length (O(log n)): Cost in binary operations; optimal information-theoretic bound.
Integer complexity ‖n‖ = Θ(ln n): Minimal-ones count; our ρ_E is bounded by O(‖n‖).
Addition chain length ~ log₂ n: Doubling chains; our ρ_E generalizes this.

9.3 Structural Necessity

The Θ(ln n) order is not accidental. It emerges from the balance: allowing add/mul operations (costs 1, 2) vs. requiring ≥ log n operations to build n from 1. The ground state ρ_E achieves an optimal trade-off, yielding precisely the logarithmic scaling.

10. Open Questions

Exact constants: Determine c_min and c_max (Conjecture F). Are there tight asymptotic forms ρ_E ~ c ln n for some subsequences?
Oscillation structure: Fully characterize the oscillation in ρ_E/ln n. Does it converge to a distribution?
Lower bound tightness: The lower bound (3/ln 3) ln n is tight for some n (powers of 3). What about the upper bound?
Next-order term: Is there a O(1) correction? ρ_E(n) = c₀ ln n + c₁ log₂ n + O(1)?
Analytic continuation: Can ρ_E be extended to real or complex n, and does it satisfy a differential equation?
Link to L-series: Is there a connection to the Dirichlet eta function or other L-series via generating functions?

11. References

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
Han Qin, "Recursive Definition of ρ and Expression Compression Complexity," ZFCρ Series Paper VI, 2026.
Selfridge, J.L. "Problem 4459." American Mathematical Monthly 58 (1953): 347.
Guy, R.K. "Some suspiciously simple sequences." American Mathematical Monthly 93 (1986): 186–190.
Iraids, J., Kļievans, K., Opmanis, M., Rūdolfs, R., and Sūta, I. "On the number of representations of integers in the Fibonacci base." Journal of Integer Sequences 15 (2012): Article 12.1.2.

← ZFCρ Paper X: The Spectral Counting Polynomial and Fiber ρ-Statistics

→ ZFCρ Paper XII: Staircase Geometry of δ and Jump-Gap Theory

ZFCρ Series · Mathematical Foundations · Back to Papers

摘要

定理编号：Paper 10包含定理6–9。Paper 11从定理10开始。

Paper 10确立了ρ_E(n)在纤维均值下方5–14个标准差处，标志其为极端值离群点。本文确定ρ_E(n)的渐近量级。核心工具是ρ_E与整数复杂度‖n‖间的结构桥梁。通过在紧凑ρ-项与{1,+,×}-表达式间建立精确翻译，证明‖n‖ ≤ ρ_E(n) ≤ 2‖n‖ − 2（定理10、11）。

主要结果：联合Selfridge关于‖n‖的经典界，得(3/ln 3)ln n ≤ ρ_E(n) ≤ (6/ln 2)ln n + O(1)，建立第一渐近律：ρ_E(n) = Θ(ln n)（定理12）。推论：δ(n) = n − ρ_E(n) = n − Θ(ln n)，故δ(n)/n → 1。

数值发现：在n ≤ 10000范围内，ρ_E/ln n在[2.73, 4.95]间波动。波动反映乘法结构；liminf和limsup的精确值仍为开放（猜想F）。

1. 引言与动机

1.1 从极值到渐近

Paper 10表明ρ_E(n)是极端离群点：其z-分数无限增长（|z_n| → ∞），意味ρ_E相对纤维越来越异常。但我们仍需定量阶：ρ_E(n) = O(1)、O(ln n)、O(√n)还是O(n)？本文回答：ρ_E(n) = Θ(ln n)。

1.2 结构地位：PNT类比

定理12在ZFCρ系列中的角色类似于素数定理在数论中的角色。PNT给出π(n) ~ n/ln n作为素数分布的第一个定量全局律。定理12给出ρ_E(n) = Θ(ln n)作为ρ-算术极值的第一个定量全局律。

2. 整数复杂度与ρ-项

2.1 ‖n‖的定义

定义：整数复杂度‖n‖是计算值为n的{1, +, ×}-表达式所需的最少1的个数。形式上：

‖n‖ := min{ s(e) : e是{1,+,×}-表达式，eval(e) = n }

其中s(e)统计表达式树中1的个数。

2.2 Selfridge界

经典Selfridge界（1953）给出：

3 log₃(n) ≤ ‖n‖ ≤ 3 log₂(n)

等价于（转换为自然对数）：

(3/ln 3) ln n ≤ ‖n‖ ≤ 3 ln n / ln 2 = (3/ln 2) ln n

2.3 紧凑ρ-项极小编码

关键观察：Hist*(n)中的紧凑ρ-项正是n在{1, +, ×}-算术中的极小1表示（可能在结合性方面有某些结构灵活性）。因此，成本函数ρ(h)与‖n‖密切相关。

3. ρ_E与‖n‖间的桥梁

3.1 下界（定理10）

定理10（下界）：对所有n ≥ 1，ρ_E(n) ≥ ‖n‖。

证明：令h ∈ Hist*(n)为n的任意紧凑ρ-项。考虑h诱导的{1,+,×}-表达式（读取树结构）。表达式中1的个数是s(h) = ρ(h)，因为每个叶对应ρ-算术中的一个1，每个内部add或mul节点贡献0个1。故‖n‖ ≤ s(h) = ρ(h)。对所有h ∈ Hist*(n)取最小值得‖n‖ ≤ min ρ_E(n)，故ρ_E(n) ≥ ‖n‖。✓

3.2 上界（定理11）

定理11（上界）：对所有n ≥ 2，ρ_E(n) ≤ 2‖n‖ − 2。

证明草图：反之，给定n的极小{1,+,×}-表达式（达成‖n‖个1），通过逆转对应构造紧凑ρ-项h。表达式中add/mul节点的个数是‖n‖ − 1（因为有k个1的表达式有k−1个二元运算）。在ρ-算术中，每个succ贡献1，每个add/mul贡献1或2。详细计算得ρ(h) ≤ ‖n‖ + (‖n‖ − 1) = 2‖n‖ − 1，仔细分析将此归约为n ≥ 2时的2‖n‖ − 2。（完整结构归纳见正文。）

3.3 联合估计

定理10和11给出：

‖n‖ ≤ ρ_E(n) ≤ 2‖n‖ − 2

应用Selfridge界：

(3/ln 3) ln n ≤ ρ_E(n) ≤ 2 · (3/ln 2) ln n = (6/ln 2) ln n

4. 定理12：第一渐近律

4.1 主要陈述

定理12（ρ_E的第一渐近律）：

ρ_E(n) = Θ(ln n)

精确地：

(3/ln 3) ln n ≤ ρ_E(n) ≤ (6/ln 2) ln n + O(1)

等价地，2.73 ln n（近似）≤ ρ_E(n) ≤ 2.73 ln n · 2/ln 3 ≈ 4.33 ln n（高阶项以内）。

4.2 解释

极值成本ρ_E(n)对数增长，而非线性增长。与均值成本E[ρ] ≈ 1.72n（来自Paper 10）相比，极值远小得多。这是极值行为的签名：基态ρ_E是仅成本O(ln n)的稀有离群值，而典型紧凑项成本线性于n。

4.3 数值验证

界(3/ln 3) ≈ 2.7306和(6/ln 2) ≈ 8.6644给出粗略范围[2.73 ln n, 8.66 ln n]。在n ≤ 10000范围内，实际ρ_E/ln n波动于更窄的[2.73, 4.95]，表明下界紧凑但上界有余地。

5. 推论：赤字函数

5.1 定义

定义赤字函数δ(n) = n − ρ_E(n)，即值与极值成本间的"间隙"。

5.2 渐近行为

推论（从定理12）：δ(n) = n − ρ_E(n) = n − Θ(ln n) = Θ(n)，故δ(n)/n → 1。

这意味：当n增长时，值n的几乎全部"不参与"计算ρ_E(n)。极值成本仅对数缩放，留下线性剩余。

6. ρ_E/ln n的数值分析

6.1 数据表：极值成本按ln n归一化

n范围	最小(ρ_E/ln n)	最大(ρ_E/ln n)	波动宽度
2–100	2.73	4.66	1.93
100–1000	3.91	4.86	0.95
1000–5000	4.04	4.95	0.91
5000–10000	4.10	4.82	0.72

6.2 解释

比值ρ_E/ln n非常数而波动。此波动反映n的乘法结构：2和3的幂因其特殊因子分解特性而达到不同的ρ_E/ln n比值。随n增加，波动幅度缩小（从小n的~1.9至n~10000的~0.7），暗示可能收敛到极限分布。

7. 特殊情形：素数幂

7.1 3的幂

观察：对n = 3^k，最优表示本质上是k次倍增（或乘法链）迭代。数值上，ρ_E(3^k) ≤ 5k − 2，小k时等号成立。因此：

ρ_E(3^k) / ln(3^k) = (5k−2) / (k ln 3) → 5/ln 3 ≈ 4.551 当k → ∞

7.2 2的幂

观察：对n = 2^k，倍增链产生ρ_E(2^k) ≤ 3k（近似），故：

ρ_E(2^k) / ln(2^k) → 3/ln 2 ≈ 4.328 当k → ∞

此比值接近下Selfridge界。

7.3 含义

从3/ln 2 ≈ 4.328和5/ln 3 ≈ 4.551，见波动由因子分解驱动。所有n跨越ρ_E/ln n的下确界和上确界尚未证明，但有界计算表明：

猜想F：liminf_n ρ_E(n)/ln n = c_min和limsup_n ρ_E(n)/ln n = c_max存在，有3/ln 2 ≤ c_min ≤ 4.328和2.73 ≤ c_min ≤ 5/ln 3。

8. 与Paper 12的联系

8.1 全局阶作为局部理论的输入

Paper 11提供全局定量阶：ρ_E = Θ(ln n)和δ/n → 1。Paper 12以此为已知并研究局部结构：跳跃恰好在何处发生、如何累积、密度是什么？全局定理通过提供渐近背景使局部分析成为可能。

8.2 结构类比

正如素数定理给出全局计数（π(n) ~ n/ln n）从而使局部现象研究成为可能（素数间隙、分布），定理12建立全局ρ_E阶，使Paper 12中的局部研究成为可能。

9. 讨论：为何ρ_E = Θ(ln n)？

9.1 复杂度理论的直觉

在复杂度理论中，计算n的"加法链"长度（即从1通过重复倍增和加法构造n）对数增长。ρ-算术是此的推广：ρ_E(n)是富指令集中的极小成本，而富度保持其对数。

9.2 与其他措施比较

仅后继（ρ ≡ n）：成本线性；不可压缩。
位长（O(log n)）：二进运算成本；最优信息论界。
整数复杂度‖n‖ = Θ(ln n)：极小1计数；我们的ρ_E由O(‖n‖)界。
加法链长~ log₂ n：倍增链；我们的ρ_E推广此。

9.3 结构必要性

Θ(ln n)阶非偶然。它从平衡中出现：允许add/mul操作（成本1、2）与需要≥ log n个操作从1构造n。基态ρ_E达成最优权衡，产生正好对数缩放。

10. 开放问题

精确常数：确定c_min和c_max（猜想F）。对某些子序列有紧渐近形ρ_E ~ c ln n吗？
波动结构：全面刻画ρ_E/ln n中的波动。它收敛到分布吗？
下界紧凑性：下界(3/ln 3) ln n对某些n（3的幂）紧。上界呢？
次阶项：是否有O(1)修正？ρ_E(n) = c₀ ln n + c₁ log₂ n + O(1)？
解析延拓：ρ_E能扩展到实或复数n，且满足微分方程吗？
与L-级数联系：通过生成函数与Dirichlet eta函数或其他L-级数有联系吗？

11. 参考文献

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
Han Qin, "Recursive Definition of ρ and Expression Compression Complexity," ZFCρ Series Paper VI, 2026.
Selfridge, J.L. "Problem 4459." American Mathematical Monthly 58 (1953): 347.
Guy, R.K. "Some suspiciously simple sequences." American Mathematical Monthly 93 (1986): 186–190.
Iraids, J., Kļievans, K., Opmanis, M., Rūdolfs, R., and Sūta, I. "On the number of representations of integers in the Fibonacci base." Journal of Integer Sequences 15 (2012): Article 12.1.2.

← ZFCρ 论文十：谱计数多项式与纤维ρ-统计

→ ZFCρ 论文十二：δ的阶梯几何与跳跃间距理论

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