Why Primes Have No Shortcuts: An O(1) Prime Pre-Sieve from Precomputed ρE Tables
DOI: 10.5281/zenodo.19246377This is an applied paper in the ZFCρ series. We prove a structural theorem on integer complexity \(\rho_E\) (the Successor Dominance Lemma): in the \(\rho_E\) dynamic program, every prime's optimal representation is determined by the successor path (+1); multiplicative forward propagation never reaches a prime. Equivalently, primes are zero-indegree source nodes in the multiplicative directed graph defined by \(\rho_E\) — they radiate outward (their multiples receive propagation) but receive no multiplicative information themselves. This is the precise formalization of what the ZFCρ framework calls "absolute zero."
From this lemma we obtain a table-side screening predicate: given a precomputed \(\rho_E\) table, check whether \(\rho_E(n) = \rho_E(n-1) + 1\) for each candidate \(n\) (O(1) lookup). Failure implies \(n\) is composite. Experiments at scale \(10^{10}\) confirm the pre-sieve stably eliminates approximately 59.5% of the search space with 100% recall.
The real value of this trivial theorem lies in its byproducts — three previously invisible structural constants: the elimination rate converges to ~59.5%; the fraction of "reverse-temperature primes" satisfying \(\rho_E(p+1) > \rho_E(p)\) stabilizes at ~10.9%; and the local convexity \(h(n)\), averaged by Ω layer, has a zero crossing that stabilizes at Ω ≈ 4 for \(N = 10^{10}\), consistent with the Ω = 3→4 self-organization phase boundary identified in the main ZFCρ series.
§0 Relationship to the Main ZFCρ Series
The ZFCρ paper series (Papers 1–42) builds a number-theoretic framework centered on integer complexity ρ. This paper departs from two main-line conclusions:
(1) Paper 41 defines primes as "absolute zero" in the multiplicative network. This paper formalizes that intuition as a provable theorem.
(2) Papers 33–42 establish Ω = 3→4 as the critical boundary of self-organization. This paper provides an independent numerical verification of that phase transition via the zero crossing of \(h(\Omega)\).
§1 The ρE Dynamic Program
1.1 Definition
The \(\rho_E\) DP executes in order \(n = 2, 3, 4, \ldots\):
Base cases: \(\rho_E(0) = 0\), \(\rho_E(1) = 0\)
$$\rho_E(n) = \min\bigl(\underbrace{\rho_E(n-1) + 1}_{\text{successor path}}, \;\; \underbrace{M(n)}_{\text{multiplicative path}}\bigr)$$
$$M(n) = \min_{a \cdot b = n,\; 2 \leq a \leq b} \bigl(\rho_E(a) + \rho_E(b) + 2\bigr)$$
When \(n\) has no nontrivial factor pair with \(2 \leq a \leq b\), we set \(M(n) = +\infty\).
1.2 Semantics of the Two Paths
The successor path is unconditionally available for all \(n \geq 2\), at cost +1. The multiplicative path corresponds to factorization \(n = a \times b\), at cost \(\rho_E(a) + \rho_E(b) + 2\), available only when \(n\) has a nontrivial factorization.
1.3 Successor Difference
Define \(\delta(n) = \rho_E(n) - \rho_E(n-1)\). From the DP structure: \(\delta(n) \leq 1\) for all \(n \geq 2\). When \(\delta(n) = 1\), the successor path is optimal. When \(\delta(n) \leq 0\), the multiplicative path is strictly better.
§2 The Successor Dominance Lemma
2.1 Statement
Lemma (Successor Dominance at Primes). For every prime \(p \geq 2\):
$$\rho_E(p) = \rho_E(p-1) + 1$$
Equivalently, \(\delta(p) = 1\).
2.2 Proof
Upper bound. By the DP definition, \(\rho_E(n) \leq \rho_E(n-1) + 1\) for all \(n \geq 2\). Hence \(\delta(p) \leq 1\).
Lower bound. \(M(p)\) requires the existence of \(a, b \geq 2\) with \(a \cdot b = p\). Since \(p\) is prime, no such pair exists. Therefore \(M(p) = +\infty\).
It follows that \(\rho_E(p) = \min(\rho_E(p-1) + 1, +\infty) = \rho_E(p-1) + 1\). \(\blacksquare\)
In graph-theoretic terms: primes are zero-indegree source nodes in the multiplicative directed graph defined by \(\rho_E\). No multiplicative edge points to them.
On the triviality of the proof. This proof takes three lines, and that is itself part of the conclusion. The weight of this paper lies not in the proof's difficulty, but in the computational application and structural constants it reveals.
2.3 Corollaries
Corollary 1 (Necessary condition for composites). For \(n \geq 2\): if \(\delta(n) \leq 0\), then \(n\) is composite. (Sufficient but not necessary criterion for compositeness.)
Corollary 2 (Pre-sieve). Given a \(\rho_E\) table, primes can only occur at positions where \(\delta(n) = 1\). All positions with \(\delta(n) \leq 0\) can be excluded without further primality testing.
§3 The Pre-Sieve Algorithm
3.1 Description
Input: A precomputed \(\rho_E\) table (for \(n \leq N\)) and a set of candidate integers \(S \subseteq \{2, \ldots, N\}\).
Pre-sieve step: For each \(n \in S\), check whether \(\rho_E(n) = \rho_E(n-1) + 1\) holds (O(1) array access). If not (\(\delta(n) \leq 0\)): remove from candidates. If yes: retain as prime candidate.
Follow-up: Run deterministic primality test (e.g. Miller–Rabin or AKS) on the retained set.
3.2 Guarantees
Recall = 100%: No prime is ever discarded. No false negatives: Every removed \(n\) is genuinely composite.
The only cost is false positives: some composites satisfy \(\delta(n) = 1\) and are retained. Their proportion is analyzed in §4.
3.3 Positioning and Significance
This pre-sieve differs fundamentally from existing sieves (Eratosthenes, Miller–Rabin, AKS): (1) it is a byproduct of structural analysis, not a purpose-built sieving tool — zero marginal cost given an existing \(\rho_E\) table; (2) it answers why, not just which — primes remain in the candidate set because they have no incoming edges in the multiplicative network; (3) its output includes previously unknown structural constants invisible to classical sieves.
More precisely, this paper's algorithmic contribution is a table-side screening predicate — a criterion attached to an existing data table — not a standalone primality algorithm.
§4 Experimental Results
4.1 Experimental Design
The \(\rho_E\) table was precomputed to \(N = 10^{10}\) (~20 GB, int16 format) using the DP implementation from Paper 32 (rho_dp.c). A streaming analysis program processed the entire dataset on a MacBook Pro (Apple Silicon), ~3 hours total runtime. A segmented prime sieve provides ground truth.
4.2 Universality of \(\delta(p) = 1\)
| \(N\) | Primes in analysis range | Standard \(\pi(N)\) | \(\delta(p) = 1\) universal |
|---|---|---|---|
| \(10^3\) | 167 | 168 | ✓ |
| \(10^5\) | 9,591 | 9,592 | ✓ |
| \(10^6\) | 78,497 | 78,498 | ✓ |
| \(10^8\) | 5,761,454 | 5,761,455 | ✓ |
| \(10^9\) | 50,847,533 | 50,847,534 | ✓ |
| \(10^{10}\) | 455,052,503 | 455,052,511 | ✓ |
Across 7 orders of magnitude and 455 million primes, not a single exception. The theoretical proof guarantees this holds for all primes; the experimental data is verification, not discovery.
4.3 Pre-Sieve Performance
| \(N\) | \(\delta=1\) set size | Kept (%) | Elimination (%) | Precision |
|---|---|---|---|---|
| \(10^3\) | 443 | 44.4% | 55.6% | 37.7% |
| \(10^5\) | 41,206 | 41.2% | 58.8% | 23.3% |
| \(10^6\) | 408,897 | 40.9% | 59.1% | 19.2% |
| \(10^8\) | 40,606,840 | 40.6% | 59.4% | 14.2% |
| \(10^9\) | 405,501,998 | 40.6% | 59.4% | 12.5% |
| \(10^{10}\) | 4,050,243,194 | 40.5% | 59.5% | 11.2% |
The elimination rate converges to approximately 59.5% for \(N \geq 10^8\). The pre-sieve stably removes roughly 60% of candidates at any scale. Precision declines as ~1/ln N, consistent with the Prime Number Theorem.
4.4 Three-Temperature Classification
Integers partition naturally into three classes:
(1) Hot composites (\(\delta(n) \leq 0\), ~59.5%). The multiplicative path is strictly better. These have efficient factorization routes in the multiplicative network.
(2) Warm composites (\(\delta(n) = 1\), composite, ~36.0%). A multiplicative path exists but does not beat the successor path.
(3) Absolute zero (primes, ~1/ln N). No multiplicative path exists. These are the precise mathematical realization of absolute zero: multiplicative self-organization is completely frozen.
4.5 Structure of \(\delta = 1\) Composites
At \(N = 10^{10}\): 84.2% are squarefree; 15.8% have repeated prime factors; only 4.6% are adjacent to a prime. Ω distribution: Ω=2 (32.9%), Ω=3 (34.5%), Ω=4 (20.4%), Ω=5 (8.3%), Ω≥6 (4.0%). The δ=1 signal detects "insufficient multiplicative path efficiency," not "proximity to primes."
4.6 Core Type Analysis: sf vs ss
| Type | Count (\(N = 10^{10}\)) | mean \(h(n)\) | mean \(\rho_E / \log_3 n\) |
|---|---|---|---|
| prime | 455,052,503 | +0.973 | 4.225 |
| squarefree (sf) | 5,624,218,367 | +0.291 | 4.201 |
| repeated factor (ss) | 3,920,729,027 | −0.530 | 4.168 |
The ss class has lower \(\rho_E/\log_3 n\) than sf by 0.033. Repeated prime factors (e.g. \(p^2\)) provide path reuse in the DP: the representation of \(p\) is computed once and invoked twice in \(p^2 = p \times p\). This is the operational counterpart of what the ZFCρ framework calls 8D (the memory layer) — repeated factors serve as "memory anchors."
4.7 Hierarchical Structure of \(h(\Omega)\) and Phase Transition
Local convexity: \(h(n) = \rho_E(n) - [\rho_E(n-1) + \rho_E(n+1)]/2\). Across \(N = 10^{10}\):
| Ω | Count | mean \(h\) | std |
|---|---|---|---|
| 1 | 455,052,503 | +0.973 | 0.597 |
| 2 | 1,493,776,420 | +0.697 | 0.646 |
| 3 | 2,227,121,973 | +0.361 | 0.680 |
| 4 | 2,139,236,858 | +0.004 | 0.685 |
| 5 | 1,570,678,124 | −0.329 | 0.667 |
| 6 | 977,694,271 | −0.615 | 0.648 |
| 7 | 550,454,750 | −0.855 | 0.634 |
| 8 | 291,646,795 | −1.056 | 0.625 |
| 9 | 148,930,535 | −1.227 | 0.619 |
| 10 | 74,342,563 | −1.376 | 0.614 |
The zero crossing stabilizes at Ω ≈ 4 at the tested scale. At Ω=4 the mean \(h\) is nearly zero (+0.004); by Ω=5 it is clearly negative. This is consistent with the Ω = 3→4 self-organization phase boundary identified in the main ZFCρ series.
The zero crossing separates two regimes: at Ω ≤ 3 integers are locally "convex" (successor path competitive); at Ω ≥ 4 they are locally "concave" (multiplicative path dominates). The std ≈ 0.60–0.68 is nearly constant across Ω values — only the center shifts.
§5 Identities and Structural Constraints
5.1 The h–δ Identity
$$h(n) = \frac{\delta_\mathrm{back}(n) - \delta_\mathrm{fwd}(n)}{2}$$
Verified pointwise on the \(10^{10}\) dataset (maximum error < \(10^{-6}\), from floating-point rounding).
5.2 Consequences for Primes
By the Lemma, \(\delta_\mathrm{back}(p) = 1\) for all primes, giving \(h(p) = [1 - \delta_\mathrm{fwd}(p)]/2\).
| \(\delta_\mathrm{fwd}(p)\) range | Prime fraction | Meaning |
|---|---|---|
| < 0 | 62.8% | \(p+1\) is cheaper than \(p\) |
| = 0 | 26.3% | \(p+1\) costs the same as \(p\) |
| > 0 | 10.9% | \(p+1\) is more expensive than \(p\) |
The 10.9% fraction is completely stable between \(10^9\) and \(10^{10}\), marking it as a structural constant. (Connection to safe primes: when \((p+1)/2\) is prime, the path to \(p+1\) via ×2 is expensive.)
5.3 h(p) > 0 is Not Universal
While \(\delta(p) = 1\) is universal (a theorem), \(h(p) > 0\) is not — 10.9% of primes violate it. The causation runs one way: \(\delta(p) = 1\) is the theorem, and the value of \(h(p)\) follows.
§6 Open Problems
6.1 Successor dominance under standard \(\rho\) (full additive DP). Verified for \(N = 10^5\), 9,591 primes, zero exceptions. Proof remains open — requires ruling out anomalously small \(\rho(a) + \rho(p-a)\) for all \(a \in [1, p/2]\).
6.2 Analytic form of the 59.5% elimination rate. The Dickman ρ function may provide an entry point. Note: \(1 - 1/e \approx 0.632\) is close but does not match precisely.
6.3 Arithmetic characterization of the 10.9% constant. Conjectured connection to safe primes or the size of the largest prime factor of \(p+1\).
6.4 Precise functional form of \(h(\Omega)\). Residuals of the linear fit show systematic concave curvature. Candidates: \(h(\Omega) = a - b\ln(\Omega + c)\) or \(h(\Omega) = a - b\Omega^c\).
6.5 Graph-theoretic interface. Do the ZFCρ Ω-layer classifications align with community boundaries in the multiplicative network? Does the percolation threshold correspond to Ω ≈ 3?
§7 Interpretation within the SAE Framework
7.1 Primes as Genesis Positions
The \(\rho_E\) DP maps onto SAE's two routes of dimensional unfolding: the multiplicative path is recombination (decomposing \(n\) into existing components); the successor path is incrementation (minimal forward step on existing structure). The Successor Dominance Lemma says: primes are positions where recombination is entirely unreachable. They can only be produced by incrementation. In SAE language, primes correspond to genesis positions — the irreducible dimensions of the integer world.
7.2 Three Temperatures and Three Construct-Generation Modes
Hot composites (59.5%): Recombination is more efficient than incrementation — the "normal unfolding" mode of SAE's construct space. Warm composites (36.0%): Recombination is possible but not economical. Absolute zero (primes, ~1/ln N): Recombination is impossible — the precise locus of genesis. The sparsity of primes (density ~1/ln N → 0) corresponds to a basic SAE insight: truly irreducible dimensional emergence grows ever rarer but never ceases entirely.
7.3 Memory Anchors and 8D
Repeated prime factors (ss-cores) have lower \(\rho_E/\log_3 n\) — path reuse in the DP. This is an operational definition of memory: direct invocation of an existing representation rather than recomputation. In the SAE framework, 8D (the memory layer) functions in exactly this way. The "memory anchor" role of ss-cores is not metaphor but precise structural correspondence.
7.4 The Ω Phase Transition and Critical Points of Dimensional Unfolding
The zero crossing of \(h(\Omega)\) at Ω ≈ 4 marks the point at which a construct's internal complexity exceeds a critical value, making recombination far more efficient than incrementation. Three to four components suffice to cover most targets — the onset of combinatorial explosion.
§8 Conclusion
The Successor Dominance Lemma is formally trivial. But its implications are deep: it converts the ZFCρ framework's intuition that "primes are absolute zero" into a precise theorem, and immediately yields a pre-sieve of practical value.
Experiments at scale \(10^{10}\) reveal three stable structural constants: a 59.5% elimination rate, a 10.9% reverse-temperature prime fraction, and a phase-transition zero crossing at Ω ≈ 4. These connect ZFCρ's pure number-theoretic structure with computable primality discrimination.
From the SAE perspective: primes are the irreducible emergences of the integer world — they mark the positions where recombination fails, and only pure incrementation can reach them.
References
[1] Han Qin, ZFCρ Paper 1. Zenodo, DOI: 10.5281/zenodo.18914682.
[2] Han Qin, ZFCρ Paper 32. Zenodo.
[3] Han Qin, ZFCρ Paper 41. Zenodo.
[4] Han Qin, ZFCρ Paper 42. Zenodo, DOI: 10.5281/zenodo.19226607.
[5] Han Qin, ZFCρ Paper 43. Zenodo, DOI: 10.5281/zenodo.19240183.
[6] J. Arias de Reyna & J. van de Lune, arXiv:1404.1850, 2014.
[7] R. K. Guy, Amer. Math. Monthly, 93:186–190, 1986.
[8] K. Dickman, Ark. Mat. Astron. Fys., 22:1–14, 1930.
Appendix A. Experimental Code
The \(\rho_E\) DP computation uses rho_dp.c (Paper 32 convention); the streaming analysis uses rho_presieve_from_bin.c. All code and raw data are available from the author upon request.
Appendix B. Symbol Table
| Symbol | Meaning |
|---|---|
| \(\rho_E(n)\) | Integer complexity (Paper 32 convention) |
| \(\delta(n) = \delta_\mathrm{back}(n)\) | \(\rho_E(n) - \rho_E(n-1)\) |
| \(\delta_\mathrm{fwd}(n)\) | \(\rho_E(n+1) - \rho_E(n)\) |
| \(h(n)\) | Local convexity, \(= [\delta_\mathrm{back}(n) - \delta_\mathrm{fwd}(n)] / 2\) |
| \(\Omega(n)\) | Number of prime factors of \(n\) with multiplicity |
| \(M(n)\) | Optimal multiplicative path cost |
| sf | Squarefree composite |
| ss | Composite with at least one repeated prime factor |
本文是 ZFCρ 系列的应用论文。我们证明了一个关于整数复杂度 \(\rho_E\) 的结构性定理（后继支配引理）: 在 \(\rho_E\) 的动态规划中，素数的最优表示永远由后继路径（+1）给出，乘法前向传播永远无法到达素数位置。素数在 \(\rho_E\) 的乘法有向图中是入度为零的源节点——它们向外辐射，但自身不接收任何乘法信息。这是 ZFCρ 框架所称的"绝对零度"的精确形式化。
基于这个引理，我们得到一个 table-side screening predicate: 给定预计算的 \(\rho_E\) 表，对任意 \(n\) 检查 \(\rho_E(n) = \rho_E(n-1)+1\) 是否成立（O(1) 查表）。不成立则 \(n\) 必为合数。\(10^{10}\) 规模的实验验证了该预筛在 100% recall 下稳定消除约 59.5% 的搜索空间。
这个 trivial theorem 的真正价值在于它的副产品——三个此前不可见的结构常数: 消除率收敛至 ~59.5%，满足 \(\rho_E(p+1) > \rho_E(p)\) 的"逆温素数"占 ~10.9%，以及局部凸度 \(h(n)\) 按 Ω 层的均值零交叉在 \(N = 10^{10}\) 下稳定落在 Ω ≈ 4，与 ZFCρ 主线的 Ω = 3→4 自组织相变边界吻合。
§0 与 ZFCρ 主线的关系
ZFCρ 系列论文（Papers 1-42）建立了以整数复杂度 ρ 为核心的数论框架。本文从主线的两个结论出发：（一）Paper 41 将素数定义为"绝对零度"，本文将其形式化为可证明的定理；（二）Papers 33-42 的 Ω 层分析给出 Ω = 3→4 的临界边界，本文通过 \(h(\Omega)\) 零交叉提供独立数值验证。
§1 ρE 的动态规划结构
1.1 定义
$$\rho_E(n) = \min\bigl(\underbrace{\rho_E(n-1) + 1}_{\text{后继路径}}, \;\; \underbrace{M(n)}_{\text{乘法路径}}\bigr)$$
$$M(n) = \min_{a \cdot b = n,\; 2 \leq a \leq b} \bigl(\rho_E(a) + \rho_E(b) + 2\bigr)$$
当 \(n\) 没有满足 \(2 \leq a \leq b\) 的非平凡因子对时，\(M(n) = +\infty\)。
1.2 两条路径的语义
后继路径对所有 \(n \geq 2\) 无条件可用，代价固定为 +1。乘法路径对应因子分解 \(n = a \times b\)，代价为 \(\rho_E(a) + \rho_E(b) + 2\)，仅当 \(n\) 有非平凡因子时可用。
1.3 后继差分
定义 \(\delta(n) = \rho_E(n) - \rho_E(n-1)\)。由 DP 结构：\(\delta(n) \leq 1\) 对所有 \(n \geq 2\) 成立。
§2 后继支配引理
2.1 引理陈述
Lemma（Successor Dominance at Primes）。对所有素数 \(p \geq 2\)：
$$\rho_E(p) = \rho_E(p-1) + 1$$
等价地，\(\delta(p) = 1\)。
2.2 证明
上界。\(\rho_E(n) \leq \rho_E(n-1) + 1\) 对所有 \(n \geq 2\) 成立，故 \(\delta(p) \leq 1\)。
下界。\(M(p)\) 要求存在 \(a, b \geq 2\) 使 \(a \cdot b = p\)。但 \(p\) 是素数，不存在这样的因子对，故 \(M(p) = +\infty\)。从而 \(\rho_E(p) = \min(\rho_E(p-1)+1, +\infty) = \rho_E(p-1)+1\)。\(\blacksquare\)
素数在 \(\rho_E\) 定义的乘法有向图中是入度为零的源节点。
2.3 推论
Corollary 1（合数必要条件）。对 \(n \geq 2\)：若 \(\delta(n) \leq 0\)，则 \(n\) 是合数。
Corollary 2（Pre-sieve）。给定 \(\rho_E\) 表，素数只可能出现在 \(\delta(n) = 1\) 的位置。所有 \(\delta(n) \leq 0\) 的位置可以直接排除。
§3 预筛算法
3.1 算法描述
对每个候选 \(n\)，检查 \(\rho_E(n) = \rho_E(n-1)+1\) 是否成立（O(1) 数组访问）。不成立则移除；成立则保留为候选素数。后续对候选集执行 Miller-Rabin 等确定性测试。
3.2 保证
Recall = 100%：不会漏掉任何素数。无假阴性：被移除的 \(n\) 都确实是合数。
3.3 定位与意义
本预筛是（一）结构分析的副产品，边际成本为零；（二）回答"为什么"，揭示因果机制；（三）输出不只是素数列表，还有此前未知的结构常数。这是一个 table-side screening predicate，不是 standalone primality algorithm。
§4 实验结果
4.1 实验设计
使用 Paper 32 的 DP 实现（rho_dp.c）预计算 \(\rho_E\) 表到 \(N = 10^{10}\)（约 20GB，int16 格式）。在 MacBook Pro（M 系列芯片）上流式处理全部数据，总耗时约 3 小时。
4.2 δ(p) = 1 的 Universality 验证
| \(N\) | 分析范围内素数 | 标准 \(\pi(N)\) | \(\delta(p) = 1\) universal |
|---|---|---|---|
| \(10^3\) | 167 | 168 | ✓ |
| \(10^5\) | 9,591 | 9,592 | ✓ |
| \(10^6\) | 78,497 | 78,498 | ✓ |
| \(10^8\) | 5,761,454 | 5,761,455 | ✓ |
| \(10^9\) | 50,847,533 | 50,847,534 | ✓ |
| \(10^{10}\) | 455,052,503 | 455,052,511 | ✓ |
跨 7 个数量级，4.55 亿个素数，无一例外。
4.3 预筛性能
| \(N\) | \(\delta=1\) 集合大小 | 占比 | 消除率 | Precision |
|---|---|---|---|---|
| \(10^3\) | 443 | 44.4% | 55.6% | 37.7% |
| \(10^5\) | 41,206 | 41.2% | 58.8% | 23.3% |
| \(10^6\) | 408,897 | 40.9% | 59.1% | 19.2% |
| \(10^8\) | 40,606,840 | 40.6% | 59.4% | 14.2% |
| \(10^9\) | 405,501,998 | 40.6% | 59.4% | 12.5% |
| \(10^{10}\) | 4,050,243,194 | 40.5% | 59.5% | 11.2% |
消除率在 \(N \geq 10^8\) 后收敛至约 59.5%。在任意大的 \(N\) 下稳定消除约 60% 的搜索空间。Precision 按 ~1/ln N 下降，与素数定理一致。
4.4 三温度分类
（一）热合数（\(\delta(n) \leq 0\)，~59.5%）：乘法路径严格更优。
（二）温合数（\(\delta(n) = 1\) 且为合数，~36.0%）：乘法路径存在但不优于后继路径。
（三）绝对零度（素数，~1/ln N）：无乘法路径——乘法自组织完全冻结。
4.5 δ=1 合数的结构
在 \(N = 10^{10}\)：84.2% 是 squarefree，15.8% 含重复素因子，只有 4.6% 与某个素数相邻。Ω 分布：Ω=2（32.9%），Ω=3（34.5%），Ω=4（20.4%），Ω=5（8.3%），Ω≥6（4.0%）。δ=1 信号检测的是"乘法路径效率不足"，不是"离素数近"。
4.6 核心类型分析：sf vs ss
| 类型 | 数量（\(N=10^{10}\)） | mean \(h(n)\) | mean \(\rho_E/\log_3 n\) |
|---|---|---|---|
| prime | 455,052,503 | +0.973 | 4.225 |
| squarefree (sf) | 5,624,218,367 | +0.291 | 4.201 |
| 含重复因子 (ss) | 3,920,729,027 | −0.530 | 4.168 |
ss 类的 \(\rho_E/\log_3 n\) 比 sf 低 0.033。重复素因子（如 \(p^2\)）在 DP 中提供路径复用——这是 ZFCρ 框架中 8D（记忆层）的操作性对应，重复因子 = DP 的"记忆锚点"。
4.7 h(Ω) 的层级结构与相变
局部凸度 \(h(n) = \rho_E(n) - [\rho_E(n-1)+\rho_E(n+1)]/2\)：
| Ω | 数量 | mean h | std |
|---|---|---|---|
| 1 | 455,052,503 | +0.973 | 0.597 |
| 2 | 1,493,776,420 | +0.697 | 0.646 |
| 3 | 2,227,121,973 | +0.361 | 0.680 |
| 4 | 2,139,236,858 | +0.004 | 0.685 |
| 5 | 1,570,678,124 | −0.329 | 0.667 |
| 6 | 977,694,271 | −0.615 | 0.648 |
| 7 | 550,454,750 | −0.855 | 0.634 |
| 8 | 291,646,795 | −1.056 | 0.625 |
| 9 | 148,930,535 | −1.227 | 0.619 |
| 10 | 74,342,563 | −1.376 | 0.614 |
零交叉在测试尺度上稳定落在 Ω ≈ 4。Ω = 4 的 mean h 已接近零（+0.004）；Ω = 5 则明确为负。与 ZFCρ 主线的 Ω = 3→4 自组织相变边界吻合。
§5 恒等式与结构约束
5.1 h 与 δ 的恒等式
$$h(n) = \frac{\delta_\mathrm{back}(n) - \delta_\mathrm{fwd}(n)}{2}$$
在 \(10^{10}\) 数据上逐点验证（最大误差 < \(10^{-6}\)，来自浮点舍入）。
5.2 对素数的推论
由 Lemma，\(\delta_\mathrm{back}(p) = 1\)，故 \(h(p) = [1 - \delta_\mathrm{fwd}(p)]/2\)。
| \(\delta_\mathrm{fwd}(p)\) 范围 | 素数比例 | 含义 |
|---|---|---|
| < 0 | 62.8% | \(p+1\) 比 \(p\) 便宜 |
| = 0 | 26.3% | \(p+1\) 和 \(p\) 等价 |
| > 0 | 10.9% | \(p+1\) 比 \(p\) 贵 |
10.9% 的比例在 \(10^9\) 和 \(10^{10}\) 之间完全稳定，是一个结构常数。
5.3 h(p) > 0 不是 universal 的
虽然 \(\delta(p) = 1\) 是 universal 的（定理），但 \(h(p) > 0\) 不是——10.9% 的素数违反它。因果关系是：\(\delta(p) = 1\) 是定理，\(h(p)\) 的值从中推出。
§6 开放问题
6.1 标准 \(\rho\) 下的后继支配（完整加法 DP）。在 \(N = 10^5\) 验证，9,591 个素数，零例外。证明仍然开放。
6.2 59.5% 消除率的解析形式。Dickman ρ function 可能是入口。\(1 - 1/e \approx 0.632\) 接近但不精确匹配。
6.3 10.9% 常数的算术特征。猜测与 safe primes 或 \(p+1\) 的最大素因子有关。
6.4 \(h(\Omega)\) 的精确函数形式。残差有系统性凹弯曲。候选：\(h(\Omega) = a - b\ln(\Omega+c)\) 或 \(h(\Omega) = a - b\Omega^c\)。
6.5 图论接口。ZFCρ 的 Ω 层分类是否与乘法网络的 community 边界对齐？渗透阈值是否对应 Ω ≈ 3？
§7 SAE 框架下的解读
7.1 素数作为 Genesis 位置
\(\rho_E\) 的 DP 对应 SAE 的两种维度展开途径：乘法路径 = 重组，后继路径 = 递增。后继支配引理说明：素数是重组完全不可达的位置，只能通过递增到达。在 SAE 语言中，素数对应的是 genesis 位置——整数世界中的不可还原维度。
7.2 三温度与三种 Construct 生成模式
热合数（59.5%）：重组比递增更高效——SAE 的"常规展开"。温合数（36.0%）：重组可能但不经济。绝对零度（素数，~1/ln N）：重组不可能——genesis 的精确位置。素数的稀疏性（密度 ~1/ln N → 0）对应 SAE 的基本洞察：真正不可还原的维度涌现越来越稀少，但永远不会停止。
7.3 记忆锚点与 8D
ss-core 的低 \(\rho_E/\log_3 n\) 来自路径复用——这是记忆的操作性定义：对已有表示的直接调用，而非重新计算。SAE 的 8D（记忆层）功能正是这个。ss-core 的"记忆锚点"角色是精确的结构对应，不是比喻。
7.4 Ω 相变与维度展开的临界点
\(h(\Omega)\) 的零交叉（Ω ≈ 4）标记了 construct 内部复杂度超过临界值后重组变得更高效的临界点——组合爆炸的起点：三到四个组件已足以覆盖大部分目标。
§8 结论
后继支配引理在形式上是 trivial 的。但它将 ZFCρ 中"素数是绝对零度"的直觉转化为精确定理，并立即产生了具有实际价值的预筛算法。\(10^{10}\) 规模的实验揭示了三个稳定的结构常数：59.5% 的消除率，10.9% 的逆温素数比例，Ω ≈ 4 的相变零交叉。
在 SAE 的视角下：素数是整数世界中的不可还原涌现——它们标记了重组失效的位置，只有纯粹的递增能到达它们。
参考文献
[1] Han Qin, ZFCρ Paper 1. Zenodo, DOI: 10.5281/zenodo.18914682.
[2] Han Qin, ZFCρ Paper 32. Zenodo.
[3] Han Qin, ZFCρ Paper 41. Zenodo.
[4] Han Qin, ZFCρ Paper 42. Zenodo, DOI: 10.5281/zenodo.19226607.
[5] Han Qin, ZFCρ Paper 43. Zenodo, DOI: 10.5281/zenodo.19240183.
[6] J. Arias de Reyna & J. van de Lune, arXiv:1404.1850, 2014.
[7] R. K. Guy, Amer. Math. Monthly, 93:186–190, 1986.
[8] K. Dickman, Ark. Mat. Astron. Fys., 22:1–14, 1930.
附录 A. 实验代码
\(\rho_E\) 的 DP 计算使用 rho_dp.c（Paper 32），流式分析使用 rho_presieve_from_bin.c。所有代码和原始数据可从作者处获取。
附录 B. 符号表
| 符号 | 含义 |
|---|---|
| \(\rho_E(n)\) | 整数复杂度（Paper 32 convention） |
| \(\delta(n) = \delta_\mathrm{back}(n)\) | \(\rho_E(n) - \rho_E(n-1)\) |
| \(\delta_\mathrm{fwd}(n)\) | \(\rho_E(n+1) - \rho_E(n)\) |
| \(h(n)\) | 局部凸度，\(= [\delta_\mathrm{back}(n) - \delta_\mathrm{fwd}(n)]/2\) |
| \(\Omega(n)\) | \(n\) 的素因子个数（含重复） |
| \(M(n)\) | 乘法路径最优值 |
| sf | squarefree 合数 |
| ss | 含重复素因子的合数 |