The First Asymptotic Theory of ρ_E / ρ_E 的第一渐近理论

ZFCρ Series, Paper 11. This paper establishes the first global asymptotic law for ρ_E(n), the minimum cost among compact history terms with extensional value n in the term model of ρ-arithmetic.

The core tool is a structural bridge between ρ_E and integer complexity ||n|| (the minimum number of 1's needed to represent n using 1, +, ×). By constructing exact translations between compact ρ-terms and {1, +, ×}-expressions, we prove:

• Theorem 10: ρ_E(n) ≥ ||n||
• Theorem 11: ρ_E(n) ≤ 2||n|| − 2 (n ≥ 2)

Combined with the classical Selfridge bounds 3·log₃n ≤ ||n|| ≤ 3·log₂n, this immediately yields:

• Theorem 12: ρ_E(n) = Θ(ln n), with explicit constants (3/ln 3)·ln n ≤ ρ_E(n) ≤ (6/ln 2)·ln n + O(1)

Corollaries: δ(n) = n − O(log n), δ(n)/n → 1; deficiency d(n) = ρ_E(n) − ||n|| satisfies 0 ≤ d(n) ≤ ||n|| − 2 = O(ln n).

Numerically (n ≤ 10000), ρ_E/ln n oscillates in [2.73, 4.95]; existing data does not show convergence to a single constant. The exact values of liminf and limsup remain open (Conjecture F).

Chinese and English versions under one DOI.

Description (中文)

ZFCρ 系列,论文十一。本文建立了 ρ_E(n)——ρ-算术项模型中外延值为 n 的紧凑历史项的最小代价——的第一个全局渐近律。

核心工具是 ρ_E 与整数复杂度 ||n||(用 1 和 +, × 表示 n 所需的最少 1 的个数)之间的结构桥梁。通过在紧凑 ρ-项和 {1, +, ×}-表达式之间建立精确翻译,证明:

• 定理10:ρ_E(n) ≥ ||n||
• 定理11:ρ_E(n) ≤ 2||n|| − 2(n ≥ 2)

联合 Selfridge 经典界 3·log₃n ≤ ||n|| ≤ 3·log₂n,立即得到:

• 定理12:ρ_E(n) = Θ(ln n),显式常数 (3/ln 3)·ln n ≤ ρ_E(n) ≤ (6/ln 2)·ln n + O(1)

推论:δ(n) = n − O(log n),δ(n)/n → 1;deficiency d(n) = ρ_E(n) − ||n|| 满足 0 ≤ d(n) ≤ ||n|| − 2 = O(ln n)。

数值层面(n ≤ 10000):ρ_E/ln n 在 [2.73, 4.95] 之间波动,现有数据未显示向单一常数收敛。liminf 和 limsup 的精确值仍为开放问题(猜想F)。

中英文版本同一 DOI。

Keywords

ρ-arithmetic, integer complexity, asymptotic theory, minimum cost, Selfridge bounds, structural translation, deficiency, term model, ZFCρ

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References

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