Recursive Definition of ρ and Expression Compression Complexity
Paper 5 provided structural induction — ρ-arithmetic's proof tools. This paper supplies the measurement tools.
Paper 3's conservation principle is transformed into a recursive definition of ρ on history terms (C1–C4). The compact history-term space Hist*(n) is introduced, and the first concrete measure ρ_w (α=1, β=2) is provided.
Core result: the recurrence for ρ_E(n) = min_{h ∈ Hist*(n)} ρ_w(h) is derived, simultaneously considering successor, addition, and multiplication paths. Defining δ(n) = n - ρ_E(n), it is proved that δ is monotonically non-decreasing and positive for all n ≥ 9.
δ(n) measures not primality but expression compression complexity — how much structural compression n can inherit from cheaper sub-expressions (especially multiplicative substructure). ρ_E has direct structural correspondence to the Integer Complexity Problem in theoretical computer science.
1. Introduction
1.1 From Proof Tools to Measurement Tools
Paper 5 provided structural induction. This paper brings Paper 3's conservation principle inside ρ-arithmetic, making ρ a bottom-up computable function on history terms.
1.2 Correction of Initial Intuition
During computation of ρ_E(n), an initial intuition was "primes have no multiplication shortcut, so ρ_E(p) = p." This intuition is wrong. Although primes' Hist*(p) contains no mul-root terms (Paper 4 §7.2), multiplication can appear at lower levels — e.g., succ(mul(h_2^std, h_5^std)) is a compact term of value 11 with cost 2+5+2+1=10<11. Primes can inherit compression from nearby composites' multiplicative structure.
The correct understanding: ρ_E(n) measures n's minimum expression cost in ρ-arithmetic's term grammar, without distinguishing primes from composites. δ(n) = n - ρ_E(n) is an expression compression complexity measure, not a primality criterion.
2. Compact History-Term Space Hist*(n) [Construction]
2.1 Trivial Operations
Add-zero: add(0, h) or add(h, 0), for any h.
Multiply-by-one: mul(succ(0), h) or mul(h, succ(0)), for any h.
Zero-containing multiplication: mul(0, h) or mul(h, 0), for any h (including h = 0).
2.2 Compact Terms
A history term h is compact iff no subterm of h is a trivial operation.
2.3 Hist*(n)
Lemma: Hist*(n) is non-empty and finite.
Non-emptiness: h_n^std ∈ Hist*(n).
Finiteness: Compactness requires every add child to have val > 0 and every mul child to have val > 1. Each child's val is strictly less than its parent's, so tree depth is bounded by n. Finite depth and finite branching guarantee finiteness.
3. Recursive Definition of ρ [Recursive Definition]
3.1 C1: ρ(0) = 0
3.2 C2: ρ(succ(h)) = ρ(h) + 1
3.3 C3: ρ(add(h₁, h₂)) = ρ(h₁) + ρ(h₂) + α
3.4 C4: ρ(mul(h₁, h₂)) = ρ(h₁) + ρ(h₂) + β
Default: α = 1, β = 2, β ≥ α > 0.
3.5 Conservation
Conservation is automatic: output ρ = sum of input ρ-values + operation cost.
4. Definition and Recurrence of ρ_E [Theorem]
4.1 Definition
By §2.3 lemma, the minimum exists.
4.2 Core Theorem: Recurrence for ρ_E
Proposition. For n > 0 (α = 1, β = 2),
Initial condition: ρ_E(0) = 0.
Note: The addition path (min_{a+b=n, a,b>0}(ρ_E(a)+ρ_E(b)+1)) is dominated by the successor path when α = 1, so it can be omitted. This is proved by the following lemma.
Lemma (Addition Domination). For all n ≥ 2, for all a, b > 0 with a + b = n: ρ_E(a) + ρ_E(b) ≥ ρ_E(n-1).
Proof. Two cases.
Case 1: b = 1 (i.e., a = n-1). ρ_E(n-1) + ρ_E(1) = ρ_E(n-1) + 1 > ρ_E(n-1). Immediate.
Case 2: b > 1. By subadditivity (add is a valid compact path, a > 0 and b-1 > 0, a + (b-1) = n-1):
By the successor branch: ρ_E(b) ≤ ρ_E(b-1) + 1, i.e., ρ_E(b-1) ≥ ρ_E(b) - 1. Substituting:
∎
Corollary. Addition path cost ρ_E(a) + ρ_E(b) + 1 ≥ ρ_E(n-1) + 1 = successor path cost. Addition never beats successor. The recurrence simplifies to:
Addition is dominated by successor — a strict consequence of α = 1.
4.3 Table of ρ_E for Small n
| n | ρ_E(n) | Optimal path |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | succ |
| 2 | 2 | succ |
| 3 | 3 | succ |
| 4 | 4 | succ (mul(2,2): 6>4) |
| 5 | 5 | succ |
| 6 | 6 | succ (mul(2,3): 7>6) |
| 7 | 7 | succ |
| 8 | 8 | succ (mul(2,4): 8=8, tie) |
| 9 | 8 | mul(3,3): 3+3+2=8 |
| 10 | 9 | succ from 9; or mul(2,5): 2+5+2=9 |
| 11 | 10 | succ from 10 |
| 12 | 9 | mul(3,4): 3+4+2=9 |
| 13 | 10 | succ from 12 |
| 14 | 11 | succ from 13 |
| 15 | 10 | mul(3,5): 3+5+2=10 |
| 16 | 10 | mul(4,4): 4+4+2=10 |
5. Properties of δ(n) [Theorem]
5.1 Definition
5.2 Monotonicity
Proposition. δ is monotonically non-decreasing: δ(n+1) ≥ δ(n) for all n ≥ 0.
Proof. From the successor branch: ρ_E(n+1) ≤ ρ_E(n) + 1. Therefore δ(n+1) = (n+1) - ρ_E(n+1) ≥ (n+1) - (ρ_E(n)+1) = δ(n). ∎
5.3 Table of δ for Small n
| n | Prime/Comp | ρ_E(n) | δ(n) |
|---|---|---|---|
| 0–8 | — | n | 0 |
| 9 | comp | 8 | 1 |
| 10 | comp | 9 | 1 |
| 11 | prime | 10 | 1 |
| 12 | comp | 9 | 3 |
| 13 | prime | 10 | 3 |
| 14 | comp | 11 | 3 |
| 15 | comp | 10 | 5 |
| 16 | comp | 10 | 6 |
5.4 δ Is Not a Primality Criterion
δ does not distinguish primes from composites. δ(11)=1 (prime), δ(10)=1 (composite). δ(13)=3 (prime), δ(12)=3 (composite).
This is because multiplicative subterms can appear at any depth in a compact term. A prime p cannot be directly written as non-trivial multiplication, but it can be written as succ^k(mul(...)) — inheriting multiplicative compression from nearby composites.
5.5 What δ Measures
δ(n) measures: how much expression cost n saves relative to the pure successor chain, under the given term grammar and weights. This is an expression compression complexity — it reflects how much structural compression n can inherit from multiplicative substructure.
δ may jump at composites (when new factorizations become available), then propagate compression gains to subsequent numbers via the successor path. This is why δ is non-decreasing — once some n gains compression, n+1 inherits at least that gain via succ.
6. ρ_E and the Integer Complexity Problem [Discussion]
6.1 Structural Correspondence
ρ_E(n)'s recurrence has direct structural correspondence to the Integer Complexity Problem. Integer complexity ‖n‖ is defined as the minimum number of 1's needed to represent n using 1, addition, and multiplication. Known: 3log₃ n ≤ ‖n‖ ≤ 3log₂ n (Selfridge). ρ_E(n) is a weighted variant with different primitives (0 + succ instead of 1 + addition).
6.2 Import of Known Results
If ρ_E(n) can be precisely translated into a known integer-complexity variant, existing bounds and structural results can be directly imported into ρ-arithmetic.
6.3 Computational Complexity
Computing ρ_E(n) requires factoring n to evaluate the multiplication branch. PA treats all natural numbers as flat; ρ-arithmetic reveals that the minimum operative cost of obtaining n is a function of n's factor structure.
7. Projection Properties [Definition]
7.1 P1–P3 (from Paper 5)
P1: val surjective. P2: val is Σ-homomorphism. P3: ≡_E = ker(val).
7.2 Projection Identity
where ρ_val(h) = ρ(h) - ρ_E(val(h)) ≥ 0.
Status: Projection identity (algebraic consequence of ρ_E definition), not an independent axiom. ρ_val(h) measures h's excess expression cost relative to the cheapest compact term of the same value.
8. Discussion
8.1 Why the Initial Intuition Was Wrong
"Primes have no multiplication shortcut" is wrong because it considers only root nodes. Paper 4 §7.2's prime characterization (Hist*_mul(p) = ∅) constrains only root nodes, not internal subterms. ρ_E's recurrence allows inheriting compression from internal multiplicative substructure via successor and addition.
8.2 Jump Points of δ
δ jumps at composites where new factorizations make multiplication paths cheaper. Highly composite numbers may be where δ grows fastest. The precise relation of δ's growth to d(n), Ω(n), etc. is a core M3 question.
8.3 Preliminary Discovery: Primes Are Fixed Points of δ
Theorem. If n is prime, then δ(n) = δ(n-1).
Proof. n is prime means n has no non-trivial factor pair a, b > 1 with ab = n. Therefore the multiplication branch of the ρ_E recurrence is empty. The addition branch is dominated by the successor branch (§4.2 note). The only competing path is the successor branch: ρ_E(n) = ρ_E(n-1) + 1. Therefore δ(n) = n - ρ_E(n) = n - ρ_E(n-1) - 1 = (n-1) - ρ_E(n-1) = δ(n-1). ∎
Equivalent formulation. δ(n) > δ(n-1) implies n is composite. That is: δ-jumps occur only at composites. Primes are "plateaus" in the δ sequence — they do not increase compression efficiency, only inheriting from the previous number.
Significance. This is a positive dynamical characterization of primes: primes are fixed points of the δ function. This property is not directly expressible in PA's base language {0, S, +, ×} — it requires the concepts of ρ_E and δ, which are native objects of ρ-arithmetic.
8.4 The Converse Does Not Hold: Some Composites Are Also Fixed Points
δ(n) = δ(n-1) does not imply n is prime. Composites can also be δ-fixed-points — when none of their multiplication paths is cheaper than the successor path.
Example: n = 4 = 2 × 2. ρ_E(2) + ρ_E(2) + 2 = 6 > 4 = ρ_E(3) + 1. Multiplication is more expensive. δ(4) = δ(3) = 0.
A more notable example: n = 21 = 3 × 7 (product of two primes). ρ_E(3) + ρ_E(7) + 2 = 3 + 7 + 2 = 12. ρ_E(20) + 1 = 11 + 1 = 12. Tie. δ(21) = δ(20).
Observation: Products of two primes (semiprimes) are more likely to be δ-fixed-points. Reason: primes' ρ_E has never gained compression from a multiplication root (primes have no non-trivial mul-root terms); their compression comes entirely from successor-inheritance from nearby composites. Therefore, when two primes p, q serve as factors, pq's multiplication path cost ρ_E(p) + ρ_E(q) + 2 is relatively large, making it more likely to not beat the successor path.
Note: one cannot say "ρ_E(p) = p" — large primes can have ρ_E < p (via successor-inheritance from nearby composites, e.g., ρ_E(11) = 10 < 11). Semiprimes become fixed points not because their factors "have no compression," but because their factors' compression comes from successor-inheritance rather than multiplication roots — the efficiency gain from combining inherited compression is insufficient.
By contrast, numbers whose factors are themselves composite (e.g., 12 = 3 × 4) more readily gain compression on the multiplication path.
8.5 Paper 7 Preview: Complete Characterization of δ-Fixed-Points
The δ-fixed-point set {n : δ(n) = δ(n-1)} contains all primes and also certain composites (especially those with "insufficiently rich" factor structure, such as semiprimes).
Paper 7 will investigate:
- Precise characterization of δ-fixed-points: which composites are fixed points? Are they exactly those whose "multiplication path cost ≥ successor path cost"?
- Density of the fixed-point set: what proportion of fixed points are prime? Asymptotically, are "almost all" fixed points prime?
- Distribution of δ-jump-points (δ(n) > δ(n-1)) and their precise relation to factor structure
- Possible connection between the δ-fixed-point characterization and the twin prime problem
These questions connect δ's dynamical analysis directly to classical number theory, forming the first test case of the M3 stage.
8.6 No New Primality Criterion — But a New Dynamical Property of Primes
ρ_w does not distinguish primes from composites — δ can be positive for both. But "primes are always δ-fixed-points" is a new theorem, characterizing a positive dynamical property of primes in δ's language. It is not a replacement for primality criteria, but ρ-arithmetic's first non-trivial answer to "why are primes special?": primes are those numbers that do not increase compression efficiency.
9. Open Problems
- Closed form for ρ_E(n). Does one exist? Precise relation to ‖n‖?
- Growth rate of δ(n). Since ‖n‖ ≤ 3log₂ n, we expect δ(n) ≥ n - O(log n) — nearly all successor cost can be compressed away for large n.
- Jump-point distribution of δ. At which n does δ strictly increase? Relation to highly composite numbers?
- Nested multiplication efficiency. Does nested mul(mul(...), ...) ever beat single-layer mul?
- Exact counting of |Hist*(n)|.
- Stronger normal forms. Compression-preorder irreducibility?
- Scale-dependent measures. If β depends on input size, how does δ behave?
References
- Han Qin, "On the Remainder of Choice," 2026. DOI: 10.5281/zenodo.18914682
- Han Qin, "The Quantitative Identity of the Remainder," 2026. DOI: 10.5281/zenodo.18927658
- Han Qin, "ρ-Conservation," 2026. DOI: 10.5281/zenodo.18929819
- Han Qin, "A Draft Term Model for ρ-Arithmetic," 2026. DOI: 10.5281/zenodo.18930810
- Han Qin, "Generation Axioms and Structural Induction," ZFCρ Series Paper 5, 2026.
- Peano, G. Arithmetices principia, nova methodo exposita. Turin: Bocca, 1889.
- Guy, R. K. "Some suspiciously simple sequences." Amer. Math. Monthly 93 (1986): 186–190.
← ZFCρ Paper V: Generation Axioms and Structural Induction
ZFCρ Series · Mathematical Foundations · Back to Papers
Paper 5给出了结构归纳——ρ-算术的证明工具。本文补上度量工具。
将Paper 3的守恒原理转化为ρ在历史项上的递归定义（C1–C4）。引入紧凑历史项空间 Hist*(n)，给出第一个具体度量 ρ_w（α=1, β=2）。
核心结果：推导 ρ_E(n) = min_{h ∈ Hist*(n)} ρ_w(h) 满足的递推公式，该公式同时考虑后继、加法和乘法三条路径的代价竞争。定义 δ(n) = n - ρ_E(n)，证明 δ 是单调不减的，且从 n = 9 起恒正。
δ(n) 度量的不是素性，而是表达式压缩复杂度——n 能从更便宜的子表达式（特别是乘法子结构）中继承多少结构压缩。ρ_E 与理论计算机科学中的整数复杂度问题（Integer Complexity Problem）有直接的结构对应。
1. 引言
1.1 从证明工具到度量工具
Paper 5给出了结构归纳。本文把Paper 3的守恒原理搬进ρ-算术，使ρ成为历史项上自底向上可计算的函数，然后用这个函数做具体计算。
1.2 核心发现的修正
在计算 ρ_E(n) 的过程中，一个初始的直觉是"素数没有乘法捷径，因此 ρ_E(p) = p"。这个直觉是错误的。虽然素数的 Hist*(p) 不包含以 mul 为根节点的项（Paper 4 §7.2），但乘法可以出现在更低层——例如 succ(mul(h_2^std, h_5^std)) 是值为11的紧凑项，代价 2+5+2+1=10<11。因此素数也可以从邻近合数的乘法结构中继承压缩。
正确的理解：ρ_E(n) 度量的是 n 在ρ-算术项文法中的最小表达式代价，不区分素数和合数。δ(n) = n - ρ_E(n) 是一个关于整数的表达式压缩复杂度的量，不是素性判据。
2. 紧凑历史项空间 Hist*(n) 【构造】
2.1 平凡操作的定义
加零： add(0, h) 或 add(h, 0)，对任意 h。
乘一： mul(succ(0), h) 或 mul(h, succ(0))，对任意 h。
含零乘法： mul(0, h) 或 mul(h, 0)，对任意 h（包括 h = 0）。
2.2 紧凑项
历史项 h 是紧凑的，当且仅当 h 的任何子项都不是平凡操作。
2.3 Hist*(n)
引理：Hist*(n) 非空且有限。
非空性：标准历史 h_n^std ∈ Hist*(n)（纯后继链不含平凡操作）。
有限性：紧凑项的每个 add 子项的两个输入val都 > 0，每个 mul 子项的两个输入val都 > 1。因此任何子项的val严格小于父项的val（对 succ 减1，对 add 和 mul 减至少1），树深度被 n 有界约束。有限深度、有限分支（每层至多两个子节点）保证了 Hist*(n) 有限。
3. ρ的递归定义 【递归定义】
3.1 C1：ρ(0) = 0
3.2 C2：ρ(succ(h)) = ρ(h) + 1
3.3 C3：ρ(add(h₁, h₂)) = ρ(h₁) + ρ(h₂) + α
3.4 C4：ρ(mul(h₁, h₂)) = ρ(h₁) + ρ(h₂) + β
默认参数：α = 1，β = 2，β ≥ α > 0。
3.5 守恒性
C1–C4把ρ定义为历史项上自底向上递归计算的函数。守恒是自动的：输出ρ = 输入ρ之和 + 运算代价。
4. ρ_E 的定义与递推公式 【定理】
4.1 定义
ρ_E(n) 是外延值 n 在所有紧凑生成路径中最便宜的代价。由 Hist*(n) 的非空性和有限性（§2.3引理），最小值存在。
4.2 核心定理：ρ_E 的递推公式
命题。 对 n > 0（α = 1, β = 2），
初始条件：ρ_E(0) = 0。
证明思路。 Hist*(n) 中任何紧凑项 h 的根节点是 succ、add 或 mul 之一（n > 0 排除了 0）。
若根节点是 succ(h')：ρ_w(h) = ρ_w(h') + 1，且 h' 是 Hist*({n-1}) 的紧凑项。最优时 ρ_w(h') = ρ_E(n-1)。
若根节点是 add(h₁, h₂)：紧凑性要求 val(h₁), val(h₂) > 0。设 a = val(h₁), b = val(h₂)，a+b=n。最优时 ρ_w(h) = ρ_E(a) + ρ_E(b) + 1。
若根节点是 mul(h₁, h₂)：紧凑性要求 val(h₁), val(h₂) > 1。设 a = val(h₁), b = val(h₂)，ab=n。最优时 ρ_w(h) = ρ_E(a) + ρ_E(b) + 2。这一分支仅当 n 有大于1的因子对时非空。
ρ_E(n) 取三条路径的全局最小值。∎
引理（加法支配）。 对所有 n ≥ 2，对所有 a, b > 0 且 a + b = n：ρ_E(a) + ρ_E(b) ≥ ρ_E(n-1)。
证明。 分两种情况。
情况一：b = 1（即 a = n-1）。ρ_E(n-1) + ρ_E(1) = ρ_E(n-1) + 1 > ρ_E(n-1)。直接成立。
情况二：b > 1。由子可加性（add 是合法紧凑路径，a > 0 且 b-1 > 0，a + (b-1) = n-1）：
由后继分支：ρ_E(b) ≤ ρ_E(b-1) + 1，即 ρ_E(b-1) ≥ ρ_E(b) - 1。代入：
∎
推论。 加法路径代价 ρ_E(a) + ρ_E(b) + 1 ≥ ρ_E(n-1) + 1 = 后继路径代价。加法路径永远不比后继路径便宜。因此递推简化为：
加法路径被后继路径支配——这是 α = 1 时的严格推论。
4.3 ρ_E 的小 n 值
| n | ρ_E(n) | 最优路径 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | succ(0) |
| 2 | 2 | 后继 |
| 3 | 3 | 后继 |
| 4 | 4 | 后继（mul(2,2): 2+2+2=6 > 4） |
| 5 | 5 | 后继 |
| 6 | 6 | 后继（mul(2,3): 2+3+2=7 > 6） |
| 7 | 7 | 后继 |
| 8 | 8 | 后继（mul(2,4): 2+4+2=8 = 8，平局） |
| 9 | 8 | mul(3,3): 3+3+2=8 < 9 |
| 10 | 9 | succ(mul(3,3)): 8+1=9 < 10；或 mul(2,5): 2+5+2=9 |
| 11 | 10 | succ(mul(2,5)): 9+1=10；或 succ²(mul(3,3)): 8+2=10 |
| 12 | 9 | mul(3,4): 3+4+2=9 |
| 13 | 10 | succ(mul(3,4)): 9+1=10 |
| 16 | 10 | mul(4,4): 4+4+2=10；或 mul(2,8): 2+8+2=12 > 10 |
关键观察：从 n=9 开始，ρ_E(n) < n。 而且 ρ_E(11) = 10 < 11，ρ_E(13) = 10 < 13——素数也有 δ > 0。
5. δ(n) 的性质 【定理】
5.1 定义
5.2 δ 的单调性
命题。 δ 是单调不减的：δ(n+1) ≥ δ(n) 对所有 n ≥ 0。
证明。 由递推公式的后继分支：ρ_E(n+1) ≤ ρ_E(n) + 1。因此 δ(n+1) = (n+1) - ρ_E(n+1) ≥ (n+1) - (ρ_E(n)+1) = n - ρ_E(n) = δ(n)。∎
5.3 δ 的小 n 值
| n | 素/合 | ρ_E(n) | δ(n) |
|---|---|---|---|
| 0–8 | — | n | 0 |
| 9 | 合 | 8 | 1 |
| 10 | 合 | 9 | 1 |
| 11 | 素 | 10 | 1 |
| 12 | 合 | 9 | 3 |
| 13 | 素 | 10 | 3 |
| 14 | 合 | 11 | 3 |
| 15 | 合 | 10 | 5 |
| 16 | 合 | 10 | 6 |
5.4 δ 不是素性判据
从表格可以看出：δ 不区分素数和合数。 δ(11) = 1（素数），δ(10) = 1（合数）。δ(13) = 3（素数），δ(12) = 3（合数）。
这是因为乘法子项可以出现在紧凑项的任何深度（不只是根节点）。素数 p 虽然不能被直接写成非平凡乘法，但可以被写成 succ^k(mul(...))——从邻近的合数通过后继步继承乘法压缩。
5.5 δ 度量的是什么
δ(n) 度量的是：在ρ-算术的项文法和给定权重下，n 相对于纯后继链能节省多少表达式代价。 这是一个表达式压缩复杂度——它反映了 n 能从乘法子结构中继承多少结构压缩。
δ 在合数处可能跳跃（当新的乘法分解变得可用时），然后通过后继路径向后续的数传播压缩收益。这就是为什么 δ 是单调不减的——一旦某个 n 获得了压缩收益，n+1 至少可以通过 succ 继承这个收益。
6. ρ_E 与整数复杂度问题 【讨论】
6.1 结构对应
ρ_E(n) 的递推公式与理论计算机科学中的整数复杂度问题（Integer Complexity Problem）有直接的结构对应。整数复杂度 ‖n‖ 定义为：用1和加法、乘法表达 n 所需的最少1的个数。已知 ‖n‖ 满足类似的递推：
ρ_E(n) 是同一类问题在ρ-算术项文法中的加权变体——区别在于基本构造器是 0 和 succ（而非1和加法），且运算代价有权重。
6.2 已知结果的可能导入
整数复杂度问题自Selfridge和Guy以来已有大量研究。已知 3log₃ n ≤ ‖n‖ ≤ 3log₂ n（Selfridge）。如果 ρ_E(n) 可以被精确地翻译为 ‖n‖ 的某个仿射变换，那么关于 ‖n‖ 的已知上下界可以直接导入ρ-算术。
6.3 计算复杂性
计算 ρ_E(n)（寻找最便宜生成路径）的算法复杂度本身值得研究。递推公式中的乘法分支需要遍历 n 的所有因子对，这依赖于 n 的因式分解。PA认为所有自然数是平坦的；ρ-算术揭示了：获取 n 的最小操作代价是 n 的因子结构的函数。
7. 投影性质 【定义】
7.1 P1–P3（从Paper 5继承）
P1：val满射。P2：val是Σ-同态。P3：≡_E = ker(val)。
7.2 投影恒等式
其中 ρ_val(h) = ρ(h) - ρ_E(val(h)) ≥ 0。
地位声明。 这是 ρ_E 定义的代数推论（投影恒等式），不是施加独立约束的公理。ρ_val(h) 度量的是：当前历史项 h 相对于同值最便宜紧凑项的超额表达代价。它不是"val映射本身造成的损失"，而是"h 的表达效率与最优表达的差距"。
8. 讨论
8.1 为什么初始直觉是错的
"素数没有乘法捷径"这个直觉的错误在于：它只考虑了根节点是否为mul，忽略了mul可以出现在任何深度。Paper 4 §7.2的素数刻画（Hist*_mul(p) = ∅）只约束了根节点，不约束内部子项。而 ρ_E 的递推允许通过后继和加法从内部的乘法子结构继承压缩。
8.2 δ 的跳跃点
δ 在合数处可能跳跃——当新的因子对使得乘法路径变得更便宜时。δ 的跳跃点与 n 的因子结构直接相关。高度合成数（highly composite numbers）可能是 δ 增长最快的位置。δ 的增长率与已知数论函数（d(n), Ω(n)等）的精确关系是M3的核心问题。
8.3 先导发现：素数是 δ 的不动点
定理。 若 n 是素数，则 δ(n) = δ(n-1)。
证明。 n 是素数意味着 n 没有非平凡因子对 a, b > 1 使得 ab = n。因此 ρ_E 递推中乘法分支为空。加法分支被后继分支支配（§4.2注）。唯一的竞争路径是后继分支：ρ_E(n) = ρ_E(n-1) + 1。因此 δ(n) = n - ρ_E(n) = n - ρ_E(n-1) - 1 = (n-1) - ρ_E(n-1) = δ(n-1)。∎
等价表述。 δ(n) > δ(n-1) 蕴含 n 是合数。即：δ 的跳跃只发生在合数处。素数是 δ 序列中的"平台"——它们不增加压缩效率，只从前一个数继承。
意义。 这是一个关于素数的正面动力学刻画：素数是 δ 函数的不动点。这个性质在PA的基础语言 {0, S, +, ×} 中不直接表达——它需要 ρ_E 和 δ 的概念，而这些是ρ-算术的固有对象。
8.4 反向不成立：某些合数也是不动点
δ(n) = δ(n-1) 不蕴含 n 是素数。合数 n 也可以是 δ 的不动点——当其所有乘法路径都不比后继路径便宜时。
例： n = 4 = 2 × 2。ρ_E(2) + ρ_E(2) + 2 = 6 > 4 = ρ_E(3) + 1。乘法路径更贵。δ(4) = δ(3) = 0。
更值得注意的例子：n = 21 = 3 × 7（两个素数的乘积）。ρ_E(3) + ρ_E(7) + 2 = 3 + 7 + 2 = 12。ρ_E(20) + 1 = 11 + 1 = 12。平局。δ(21) = δ(20)。
观察： 两个素数的乘积（半素数，semiprime）更容易成为 δ 的不动点。原因：素数的 ρ_E 没有从乘法根获得过压缩（素数没有非平凡乘法根项），它们的压缩全部来自对邻近合数的后继继承。因此，当两个素数 p, q 作为因子时，pq 的乘法路径代价 ρ_E(p) + ρ_E(q) + 2 相对较大，更可能不比后继路径便宜。
注意：这里不能说"ρ_E(p) = p"——大素数的 ρ_E 可以小于 p（通过从邻近合数继承压缩，如 ρ_E(11) = 10 < 11）。半素数成为不动点的原因不是因子"没有压缩"，而是因子的压缩来源于后继继承而非乘法根——继承来的压缩在拼合时的效率增益不够大。
相比之下，因子本身是合数的数（如 12 = 3 × 4，其中 ρ_E(4) = 4 但将来随着嵌套乘法的参与可能更低）更容易在乘法路径上获得压缩。
8.5 Paper 7 预告：δ 不动点的完整刻画
δ 的不动点集 {n : δ(n) = δ(n-1)} 包含所有素数，也包含某些合数（特别是因子结构"不够丰富"的合数，如半素数）。
Paper 7将研究以下问题：
- δ 不动点集的精确刻画：哪些合数是不动点？是否恰好是那些"乘法路径代价 ≥ 后继路径代价"的合数？
- 不动点集的密度：素数在不动点集中占多大比例？是否渐近地，"几乎所有"不动点都是素数？
- δ 跳跃点（δ(n) > δ(n-1)）的分布与因子结构的精确关系
- δ 不动点刻画与孪生素数问题的可能关联
这些问题将 δ 的动力学分析与经典数论直接对接，构成M3阶段的第一个测试案例。
8.6 ρ_w 没有给出新的素性判据——但给出了新的素数动力学性质
ρ_w 度量本身不区分素数和合数——δ 对两者都可以是正的。但"素数一定是 δ 不动点"是一条新的定理，它用 δ 的动力学语言刻画了素数的一个正面性质。这不是素性判据的替代品，但它是ρ-算术对"素数为什么特殊"的第一个非平凡回答：素数是那些不增加压缩效率的数。
9. 开放问题
- ρ_E(n) 的精确公式。 递推公式给出了动态规划，但是否存在闭合形式？与整数复杂度 ‖n‖ 的精确关系是什么？
- δ(n) 的增长率。 δ 单调不减，但增长多快？已知整数复杂度满足 ‖n‖ ≤ 3log₂ n，这意味着 δ(n) ≥ n - O(log n)——对大 n，几乎所有的后继代价都可以被压缩掉。
- δ 的跳跃点分布。 δ 在哪些 n 处严格增长？跳跃点与高度合素数或其他特殊数类的关系。
- 嵌套乘法的效率。 递推公式中，是否存在 n 使得嵌套乘法（mul(mul(...), ...)）比单层乘法更便宜？
- |Hist*(n)| 的精确计数。
- 更强正规形。 压缩预序的不可约条件能否将Hist*(n)压缩到更可控的大小？
- 规模依赖度量。 如果 β 依赖于输入规模（如 β(a,b) = log(ab)），δ 的行为如何变化？
参考文献
- Han Qin, "On the Remainder of Choice," 2026. DOI: 10.5281/zenodo.18914682
- Han Qin, "The Quantitative Identity of the Remainder," 2026. DOI: 10.5281/zenodo.18927658
- Han Qin, "ρ-Conservation," 2026. DOI: 10.5281/zenodo.18929819
- Han Qin, "A Draft Term Model for ρ-Arithmetic," 2026. DOI: 10.5281/zenodo.18930810
- Han Qin, "Generation Axioms and Structural Induction," ZFCρ Series Paper 5, 2026.
- Peano, G. Arithmetices principia, nova methodo exposita. Turin: Bocca, 1889.
- Guy, R. K. "Some suspiciously simple sequences." Amer. Math. Monthly 93 (1986): 186–190.
ZFCρ 系列 · 数学基础 · 返回论文列表