Staircase Geometry of δ and Jump-Gap Theory / δ 的阶梯几何与跳跃间距理论

Description (English)

ZFCρ Series, Paper 12. This paper concludes the M5 phase (internal structure theory) of the series by studying the local geometry of δ(n) = n − ρ_E(n).

δ is a monotone nondecreasing integer-valued staircase. Three exact results characterize its fine structure:

• Theorem 13 (jump criterion): A jump occurs at n if and only if the multiplicative path strictly beats the successor path (M_n < S_n). Jumps are a purely multiplicative phenomenon; they occur only at composites.
• Theorem 14 (telescoping identity): δ(N) = Σ_{n ∈ J, n ≤ N} j(n). All growth of δ comes precisely from the accumulation of jumps.
• Theorem 15 (density–jump–plateau reciprocity): If the jump density D(N) → d, then the mean jump size → 1/d and the mean plateau length → 1/d. The three quantities are locked together by algebraic identities.

Key numerical findings (n ≤ 2×10⁶): jump density stabilizes in 0.56–0.57; maximum plateau length is 5 up to 10⁶, with the first length-6 plateau at n = 1,072,218; 83% of jumps have size 1 or 2. The maximum plateau length is unbounded but grows extremely slowly (Conjecture G').

Papers 9–12 complete the M5 phase: fiber counting (Paper 9) → spectral statistics (Paper 10) → global asymptotic law (Paper 11) → local fine structure (Paper 12).

Chinese and English versions under one DOI.

Description (中文)

ZFCρ 系列,论文十二。本文是 M5 阶段(内部结构理论)的收口论文,研究 δ(n) = n − ρ_E(n) 的局部几何。

δ 是单调不减的整数值阶梯函数。三个精确结果刻画其精细结构:

• 定理13(跳跃判据):跳跃等价于乘法路径严格胜出后继路径(M_n < S_n)。跳跃是纯粹的乘法现象,只在合数处发生。
• 定理14(Telescoping identity):δ(N) = Σ_{n ∈ J, n ≤ N} j(n)。δ 的全部增长精确来自跳跃的累积。
• 定理15(密度-跳跃-平台互倒):若跳跃密度 D(N) → d,则平均跳跃大小 → 1/d,平均平台长度 → 1/d。三个量被代数恒等式锁在一起。

关键数值发现(n ≤ 2×10⁶):跳跃密度在 0.56–0.57 范围内稳定;平台最大长度到 10⁶ 为 5,首个长度 6 在 n = 1,072,218;83% 的跳跃大小为 1 或 2。最大平台长度无界但增长极慢(猜想G')。

Papers 9–12 完成 M5 阶段:纤维计数(Paper 9)→ 谱统计(Paper 10)→ 全局渐近律(Paper 11)→ 局部精细结构(Paper 12)。

中英文版本同一 DOI。

Keywords

ρ-arithmetic, delta staircase, jump criterion, telescoping identity, plateau length, jump density, multiplicative shortcuts, prime exclusion, local geometry, ZFCρ

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References

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10. Han Qin, "The Spectral Counting Polynomial and Fiber ρ-Statistics," 2026. DOI: 10.5281/zenodo.18973559
11. Han Qin, "The First Asymptotic Theory of ρ_E," 2026. DOI: 10.5281/zenodo.18975756
12. Cramér, H. "On the order of magnitude of the difference between consecutive prime numbers." Acta Arithmetica 2 (1936): 23–46.