The Layer-by-Layer Bottleneck Map for H' — Stratified Framework, Moving Bottleneck, and Ω-Universal Scale Attenuation
DOI: 10.5281/zenodo.19104860We conduct quantitative analysis on the "ultimate battleground" established in Paper XXX (intra-layer scale attenuation vs inter-layer $\Omega$ drift), constructing a layer-by-layer bottleneck map for H'.
A stratified framework is established: $q_N = \sum_k P(\Omega=k|\text{comp}) \cdot q_k(N)$, where $q_k(N) = 1 - P(j=0|\Omega=k,N)$. This decomposes H' ($q_N \to 0$) into the weighted competition between intra-layer failure rates and inter-layer mass.
The additive advantage growth rate $\Delta \approx +0.10/\text{decade}$ is highly consistent across all tested $\Omega$ layers ($k=2,\ldots,5$) over the first five decades — exhibiting significant $\Omega$ universality. However, in the latest decade $[10^8, 10^9)$, $\Delta$ slows to $\approx +0.065$. The per-layer growth rate $b_k$ peaks at $\Omega=3$ ($b_3 = 0.023$) — $\Omega=3$ is the phenomenological engine layer at current scales.
The current bottleneck layer is $\Omega=4$. The Sathe–Selberg density peak lies at $k \approx \ln\ln N \approx 3$ ($N=10^8$), pushed rightward to $k=4$ by the monotone increase of $q_k$. The bottleneck transitions from $\Omega=3$ to $\Omega=4$ at $N \approx 2\times10^5$ (moving bottleneck).
Per-layer decay exponents $\gamma_k$ are extracted: $q_k(N) \sim C_k \cdot (\ln N)^{-\gamma_k}$. Model A wins every layer over Model B (constant limit), supporting $q_k \to 0$ for each fixed $\Omega$. However, $\gamma_k$ decays rapidly ($\gamma_2=0.45,\ \gamma_7=0.03$), with fit $\gamma_k \sim 14/k^3$ ($\beta \approx 3$). The effective index $\Gamma = \lambda \cdot \gamma_k$ at the bottleneck is decreasing — an unfavorable signal, but based on five-point fitting and possibly reflecting transient curvature.
Composite diagnostic: two favorable ($q_k \to 0$, $\Delta$ universal), one undetermined ($\Delta$ slowdown), four unfavorable ($\beta \gg 1$, bottleneck rightward push, $r_N \downarrow$, $E[j|j \ge 1] \downarrow$). Data more compatible with H' failing at $N \le 10^9$, but $\ln\ln N$-scale phenomena exceed any finite-scale numerics.
Keywords: additive complexity, $\Omega$ stratification, scale attenuation, bottleneck map, moving bottleneck, Sathe–Selberg, decay exponent
§1 Introduction
1.1 Background
Paper XXX established H' $\Leftrightarrow$ $q_N \to 0$ and positioned the ultimate battleground as "intra-layer attenuation vs inter-layer $\Omega$ drift." This paper conducts quantitative analysis on that battleground, building the layer-by-layer bottleneck map.
1.2 Contributions
- (A) $\Omega$-universal additive advantage growth with slowdown detection (§2).
- (B) Per-layer growth rates $b_k$: $\Omega=3$ as phenomenological engine (§3).
- (C) Stratified framework $q_N = \sum P(\Omega=k) \cdot q_k(N)$ (§4).
- (D) Bottleneck identification and moving bottleneck picture (§4–§5).
- (E) $\Omega$ drift quantification (§5).
- (F) Decay exponents $\gamma_k$ and effective index $\Gamma$ criterion (§6).
- (G) Bottleneck dynamics and intermittency tests (§7).
1.3 Notation
Series conventions apply. New: $q_k(N) = 1 - P(j=0|\Omega=k, n \le N)$; $b_k = dP(j=0|\Omega=k)/d(\ln N)$ (local slope); $\gamma_k$: decay exponent in $q_k \sim C_k(\ln N)^{-\gamma_k}$; $\Gamma_N(k) = (\ln\ln N) \cdot \gamma_k$.
§2 Ω-Universal Additive Advantage Growth
2.1 Data
Numerical Observation 1. Mean of $M_n - (\rho_E(n-1)+1)$ by $\Omega$ and decade (positive = additive wins, $N = 10^9$):
| Ω | [10³,10⁴) | [10⁵,10⁶) | [10⁷,10⁸) | [10⁸,10⁹) | Δ/dec (first 5) | Δ (latest) |
|---|---|---|---|---|---|---|
| 2 | +0.092 | +0.347 | +0.531 | +0.605 | ~+0.11 | +0.07 |
| 3 | −0.800 | −0.563 | −0.405 | −0.342 | ~+0.10 | +0.06 |
| 4 | −1.351 | −1.109 | −0.952 | −0.890 | ~+0.10 | +0.06 |
| 5 | −1.869 | −1.569 | −1.398 | −1.333 | ~+0.12 | +0.07 |
First five decades: per-layer increment $\approx +0.10$, exhibiting $\Omega$ universality. Latest decade: increment $\approx +0.065$ — growth rate slowing. This may be a $1/\ln N$ effect or genuine deceleration. $\Omega$ universality persists throughout.
2.2 Current Layer Status ($N = 10^9$)
- $\Omega=2$: gap $+0.61$, $P(j=0) = 0.78$. Additive dominant.
- $\Omega=3$: gap $-0.34$, $P(j=0) = 0.54$. Transitioning.
- $\Omega=4$: gap $-0.89$, $P(j=0) = 0.32$. Multiplicative still dominant.
- $\Omega=5$: gap $-1.33$, $P(j=0) = 0.18$. Multiplicative strongly dominant.
§3 Per-Layer Growth Rates and Engine Layer
3.1 Linear Fit
Numerical Observation 2. $P(j=0|\Omega=k) = a_k + b_k \cdot \ln N$ (local, $N \le 10^8$):
| Ω | b_k (per ln N) | per decade | current P(j=0) | R² |
|---|---|---|---|---|
| 2 | 0.0120 | 0.028 | 0.752 | 0.979 |
| 3 | 0.0225 | 0.052 | 0.483 | 0.985 |
| 4 | 0.0179 | 0.041 | 0.267 | 0.992 |
| 5 | 0.0137 | 0.032 | 0.133 | 0.979 |
| 6 | 0.0060 | 0.014 | 0.064 | 0.975 |
| 7 | 0.0020 | 0.005 | 0.032 | 0.802 |
3.2 Ω=3 as Phenomenological Engine
$b_k$ peaks at $\Omega=3$: sweet spot between saturation ($\Omega=2$: already near 0.75, marginal returns diminishing) and combinatorial resistance ($\Omega \ge 4$: factorization options suppress $b_k$). $\Omega=3$ is the phenomenological engine layer at current scales. True asymptotic engine may shift once $\gamma_k$ are analytically determined.
§4 The Layer-by-Layer Bottleneck Map
4.1 Stratified Framework
$$q_N = \sum_{k \ge 2} P(\Omega=k \mid \text{comp}) \cdot q_k(N)$$ H' requires $q_N \to 0$. This decomposes the problem precisely into the weighted competition between intra-layer failure rates $q_k(N)$ and inter-layer mass $P(\Omega=k)$.
4.2 Current Bottleneck
Numerical Observation 3. Stratified contributions at $N = 10^8$ (qualitative bottleneck identification):
| Ω | P(Ω=k|comp) | q_k | contribution P·q_k |
|---|---|---|---|
| 2 | 0.185 | 0.248 | 0.046 |
| 3 | 0.252 | 0.517 | 0.130 |
| 4 | 0.222 | 0.733 | 0.163 |
| 5 | 0.153 | 0.867 | 0.132 |
| 6 | 0.090 | 0.936 | 0.084 |
| 7+ | 0.098 | ~0.97 | ~0.095 |
Largest contribution: $\Omega=4$.
4.3 Rightward Shift Mechanism
The Sathe–Selberg density $P(\Omega=k)$ peaks at $k \approx \ln\ln N \approx 3$ ($N=10^8$). Since $q_k$ increases monotonically with $k$, the weighted bottleneck distribution shifts rightward of the density peak — pushing the bottleneck from $\Omega=3$ to $\Omega=4$.
4.4 Ω=4 Double Disadvantage
At $\ln\ln N \approx 2.9$: $\Omega=4$ has $k-1=3 > 2.9$, so $P(\Omega=4)$ is still growing. Simultaneously, $q_4 = 0.73$ remains high. Both forces point unfavorably for H'.
4.5 Moving Bottleneck
H' requires control only within the moving bottleneck window near $k \approx \ln\ln N$. For fixed $k$, $P(\Omega=k)$ eventually decreases on its own — so q_k convergence is only critical in the moving window.
§5 Moving Bottleneck Evolution and Ω Drift
5.1 Bottleneck Trajectory
Numerical Observation 4. Bottleneck $\Omega$ (peak of $P(\Omega=k|\text{comp}) \cdot q_k$) as $N$ grows:
| N | ln ln N | Bottleneck Ω | Density peak Ω | Bottleneck contribution |
|---|---|---|---|---|
| 10³ | 1.93 | 3 | 2 | 0.215 |
| 10⁴ | 2.22 | 3 | 2 | 0.192 |
| 10⁵ | 2.44 | 3 | 3 | 0.169 |
| 2×10⁵ | 2.50 | 3→4 | 3 | 0.167 |
| 10⁶ | 2.63 | 4 | 3 | 0.165 |
| 10⁷ | 2.78 | 4 | 3 | 0.161 |
| 10⁸ | 2.91 | 4 | 3 | 0.157 |
Bottleneck transitions $\Omega=3 \to 4$ at $N \approx 2\times10^5$ ($\ln\ln N \approx 2.5$), stable at $\Omega=4$ thereafter. Bottleneck contribution slowly declining (0.215 → 0.157).
5.2 E[Ω|comp] ≈ ln ln N + 1.2
| N | E[Ω|comp] | ln ln N | E[Ω] − ln ln N |
|---|---|---|---|
| 10³ | 3.260 | 1.933 | 1.33 |
| 10⁵ | 3.695 | 2.444 | 1.25 |
| 10⁷ | 3.917 | 2.780 | 1.14 |
| 10⁸ | 4.102 | 2.914 | 1.19 |
$E[\Omega|\text{comp}] \approx \ln\ln N + 1.2$, compatible with Sathe–Selberg. Drift extremely slow.
§6 Decay Exponents γ_k and the Effective Index Criterion
6.1 γ_k Extraction ($N = 10^9$)
Numerical Observation 5. Model A: $q_k(N) \sim C_k \cdot (\ln N)^{-\gamma_k}$. Model B: $q_k(N) = a_k + b_k/\ln N$ (constant limit if $a_k > 0$).
| Ω | γ_k | R²(A) | R²(B) | q_∞(B) | winner |
|---|---|---|---|---|---|
| 2 | 0.449 | 0.993 | 0.974 | 0.023 | A |
| 3 | 0.413 | 0.999 | 0.988 | 0.362 | A |
| 4 | 0.253 | 0.988 | 0.963 | 0.612 | A |
| 5 | 0.165 | 0.997 | 0.981 | 0.762 | A |
| 6 | 0.076 | 0.973 | 0.936 | 0.887 | A |
| 7 | 0.030 | 0.825 | 0.748 | 0.949 | A |
Model A wins every layer. Strongly supports $q_k \to 0$ for each fixed $\Omega$, but does not constitute asymptotic confirmation.
6.2 Effective Index Γ = λ · γ_k
At the bottleneck window $k \approx \lambda = \ln\ln N$: $q_k(N) \sim (\ln N)^{-\gamma_k} = e^{-\lambda \cdot \gamma_k} = e^{-\Gamma}$. The true criterion for H' is whether $\Gamma_N(k) = \lambda \cdot \gamma_k \to \infty$ at the bottleneck.
- $\Gamma \to \infty$: $q_k \to 0$, H' compatible.
- $\Gamma \to \text{const}$: $q_k \to \text{const} > 0$, H' fails.
- $\Gamma \to 0$: $q_k \to 1$, H' completely fails.
6.3 β ≈ 3 and the Unfavorable Signal
Fit $\gamma_k \approx 13.9/k^{3.0}$ ($R^2 = 0.92$, $k=3,\ldots,7$). At bottleneck $k \approx \lambda$:
If $\gamma_k \sim C/k^\beta$: $\beta < 1 \Rightarrow \Gamma \to \infty$ (H' favorable); $\beta > 1 \Rightarrow \Gamma \to 0$ (H' unfavorable). With $\beta \approx 3$, $\Gamma$ is decreasing:
| ln N | λ | k_bott | γ_bott | Γ = λ·γ |
|---|---|---|---|---|
| 20 | 3.0 | 3 | 0.53 | 1.58 |
| 50 | 3.9 | 4 | 0.22 | 0.88 |
| 100 | 4.6 | 5 | 0.12 | 0.53 |
| 200 | 5.3 | 6 | 0.07 | 0.36 |
| 500 | 6.2 | 7 | 0.04 | 0.26 |
Γ is decreasing — an unfavorable signal for H'. However: (1) five-point fit only; (2) $k \in [3,7]$ is the onset of combinatorial explosion — $\beta \approx 3$ may be transient curvature; (3) high-$\Omega$ layers ($k \gg 7$) may have thick-tailed $\gamma_k$ (effective $\beta < 1$ asymptotically). Far from refutation.
§7 Bottleneck Dynamics and Intermittency Tests
7.1 Bottleneck Centroid
Numerical Observation 6. Weighted centroid $\bar{k}_B = \sum k \cdot B_k / \sum B_k$, rightward push $\Delta_N = \bar{k}_B - \ln\ln N$:
| N | ln ln N | k̄_B | σ_B | k_peak | Δ_N | q_N |
|---|---|---|---|---|---|---|
| 10³ | 1.93 | 3.62 | 1.27 | 3 | +1.69 | 0.657 |
| 10⁴ | 2.22 | 3.99 | 1.53 | 3 | +1.77 | 0.660 |
| 10⁵ | 2.44 | 4.23 | 1.65 | 3 | +1.79 | 0.650 |
| 10⁶ | 2.63 | 4.43 | 1.71 | 4 | +1.80 | 0.641 |
| 10⁷ | 2.78 | 4.59 | 1.75 | 4 | +1.81 | 0.634 |
| 10⁸ | 2.91 | 4.73 | 1.79 | 4 | +1.81 | 0.629 |
$\Delta_N$ slowly increasing (1.69 → 1.81): bottleneck centroid slightly outruns $\Omega$ drift. $\sigma_B$ widening (1.27 → 1.79): bottleneck spreading across more layers.
7.2 r_N — Intermittency Test
Numerical Observation 7. H' requires $r_N = E[j|\text{comp}]/q_N \to \infty$.
| Decade | E[j|comp] | q | r = E[j]/q | P(j≥2) |
|---|---|---|---|---|
| [10³, 10⁴) | 1.133 | 0.660 | 1.715 | 0.330 |
| [10⁴, 10⁵) | 1.102 | 0.649 | 1.698 | 0.322 |
| [10⁵, 10⁶) | 1.083 | 0.641 | 1.691 | 0.316 |
| [10⁶, 10⁷) | 1.070 | 0.634 | 1.687 | 0.312 |
| [10⁷, 10⁸) | 1.060 | 0.630 | 1.683 | 0.309 |
| [10⁸, 10⁹) | 1.053 | 0.626 | 1.682 | 0.307 |
$r_N$ declining from 1.715 to 1.682. Decline slowing but no reversal. H' requires $r \to \infty$.
However: as failure mass migrates to higher-$\Omega$ layers (where $E[j|j \ge 1]$ is intrinsically larger), global $r_N$ may eventually rise through compositional change — even if each per-layer $r_k$ falls. This "compositional intermittency" scenario is not ruled out.
7.3 Per-Layer Jump Depth
Numerical Observation 8. $E[j|j \ge 1, \Omega=k]$ by decade:
| Decade | Ω=2 | Ω=3 | Ω=4 | Ω=5 | Ω=6 |
|---|---|---|---|---|---|
| [10³,10⁴) | 1.228 | 1.369 | 1.602 | 1.972 | 2.378 |
| [10⁵,10⁶) | 1.209 | 1.313 | 1.511 | 1.781 | 2.093 |
| [10⁷,10⁸) | 1.203 | 1.294 | 1.454 | 1.685 | 1.959 |
| [10⁸,10⁹) | 1.201 | 1.287 | 1.435 | 1.651 | 1.912 |
Every layer, every decade: $E[j|j \ge 1, \Omega=k]$ decreasing. Zero exceptions. Jumps becoming shallower, not deeper — opposite of what H' (intermittency) requires at fixed layers. But decline rate slowing (especially $\Omega=2$: 1.203 → 1.201).
7.4 Composite Diagnostic
| Signal | Direction | H' implication |
|---|---|---|
| Per-layer $q_k \to 0$ (Model A wins all) | ✓ favorable | Necessary condition supported |
| $\Delta > 0$ and $\Omega$-universal | ✓ favorable | Attenuation at work |
| $\Delta$ slowing (0.10 → 0.065) | — undetermined | May be $1/\ln N$ or genuine decay |
| $\gamma_k$ decay $\beta \approx 3$ | ✗ unfavorable | Effective index $\Gamma$ decreasing |
| $\Delta_N$ slowly increasing (1.69→1.81) | ✗ unfavorable | Bottleneck outruns $\Omega$ drift |
| $r_N$ declining (1.715 → 1.682) | ✗ unfavorable | Intermittency direction reversed |
| $E[j|j \ge 1, \Omega=k]$ falling all layers | ✗ unfavorable | Jumps shallower, not deeper |
Two favorable, one undetermined, four unfavorable. At $N \le 10^9$, data more compatible with H' failing. But $\ln\ln N$-scale phenomena exceed any finite-scale numerics. $\beta \approx 3$ may be transient curvature of the combinatorial explosion onset, not the asymptotic regime.
§8 Discussion
8.1 Core Contributions
- $\Delta \approx +0.10/\text{decade}$: $\Omega$ universality — strongly supports macroscopic log-scale driving.
- Engine layer $\Omega=3$ ($b_k$ peak) vs bottleneck layer $\Omega=4$ ($P \cdot q$ peak) — separated.
- Stratified framework $q_N = \sum P(\Omega=k) \cdot q_k(N)$.
- Moving bottleneck: $\Omega=3 \to 4$ at $N \approx 2\times10^5$, stable through $N=10^9$.
- Decay exponents $\gamma_k$: Model A (power-law to 0) universally beats Model B (constant limit). $\gamma_k$ decays from 0.45 to 0.03.
- Quantitative H' boundary: $\gamma_k \sim 14/k^3$ ($\beta \approx 3 \gg 1$), effective index $\Gamma$ decreasing — unfavorable signal.
- Intermittency tests: $r_N$ declining, per-layer jump depth declining — both opposite to H'.
- Bottleneck centroid rightward push $\Delta_N$: slowly increasing (1.69 → 1.81), $\sigma_B$ widening.
- Composite: 2 favorable, 4 unfavorable — data currently more compatible with $\neg$H'.
8.2 Open Problems
- Analytic $\gamma_k$ form — from DP recursion structure.
- Uniform window control: $\sup_{k \in W_N} q_k \to 0$.
- Analytic proof $P(j=0|\Omega=2) \to 1$ — easiest starting point.
- Explanation for $\Omega$ universality of $\Delta \approx +0.10$.
- Bottleneck $\Omega=4 \to 5$ transition timing.
- $r_N$ reversal timing if H' holds.
- $\Delta_N$ convergence — $O(1)$ or persistent growth?
8.3 Acknowledgments
ChatGPT contributed: stratified framework, $P(\Omega=k)$ direction correction, density peak location ($k \approx \ln\ln N$, not $1+\ln\ln N$), moving bottleneck definition, model misspecification detection, and the effective index $\Gamma = \lambda \cdot \gamma_k$ criterion with $\beta$ threshold. Gemini contributed: $\Omega$ universality upgrade, $\Omega=3$ as engine layer, transient curvature caveat for $\beta \approx 3$, Erdős–Kac kernel suggestion. Grok ensured series consistency. Claude wrote all computation scripts and drafted working notes v1–v2. Final text completed independently by the author; all mathematical judgments are the author's responsibility.
§9 Data Sources
| Script | Measurement | Section |
|---|---|---|
| p31_block123.py | M_n gap + P(j=0|Ω) fit + Ω drift | §2–§5 |
| p31_block45.py | q_k decay (γ_k) + bottleneck evolution | §5–§6 |
| p31_block67.py | Bottleneck profile + r_N + E[j|j≥1,Ω] | §7 |
Sanity checks: $\rho_E(10^7) = 58$, $\rho_E(10^8) = 66$, $\rho_E(10^9) = 74$.
References
- ZFCρ Papers I–XXX. H. Qin. Paper XXVIII DOI: 10.5281/zenodo.19078654. Paper XXIX DOI: 10.5281/zenodo.19083884. Paper XXX DOI: 10.5281/zenodo.19101810.
- G. H. Hardy, S. Ramanujan (1917). The normal number of prime factors of a number n. Quart. J. Math., 48:76–92.
- A. Selberg (1954). Note on a paper by L. G. Sathe. J. Indian Math. Soc., 18:83–87.
- P. Erdős, M. Kac (1940). The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math., 62:738–742.
- K. Cordwell, S. Epstein, A. Hemmady, S. J. Miller, E. Steiner (2018). On the number of 1's needed to represent n. J. Number Theory, 189:17–34.
H' 的逐层瓶颈地图——分层加权框架、Moving Bottleneck 与 Scale Attenuation 的 Ω 普适性
DOI: 10.5281/zenodo.19104860我们在 Paper XXX 确立的"终极战场"（层内 scale attenuation vs 层间 $\Omega$ 漂移）上展开定量分析，构建 H' 的逐层瓶颈地图。
建立分层加权框架：$q_N = \sum_k P(\Omega=k|\text{comp}) \cdot q_k(N)$，其中 $q_k(N) = 1 - P(j=0|\Omega=k,N)$。这把 H'（$q_N \to 0$）精确分解为层内失败率与层间质量的加权竞争。
Additive 优势的增长速率 $\Delta \approx +0.10/\text{decade}$ 在所有测试 $\Omega$ 层（$k=2,\ldots,5$）上高度一致——显示出显著的 $\Omega$ 普适性。但在最新 decade $[10^8, 10^9)$，$\Delta$ 放缓至 $\approx +0.065$。逐层增速 $b_k$ 在 $\Omega=3$ 达到峰值（$b_3 = 0.023$）——$\Omega=3$ 是当前尺度下的唯象引擎层。
识别当前瓶颈层。$q_N$ 的最大分层贡献来自 $\Omega=4$——Sathe-Selberg 密度峰在 $k \approx \ln\ln N \approx 3$（$N=10^8$），被 $q_k$ 的单调递增右推到 $k=4$。瓶颈在 $N \approx 2\times10^5$ 从 $\Omega=3$ 跳到 $\Omega=4$（moving bottleneck）。
提取逐层衰减指数 $\gamma_k$：$q_k(N) \sim C_k \cdot (\ln N)^{-\gamma_k}$。Model A 在所有层上胜出，支持每个固定 $\Omega$ 层的 $q_k \to 0$。但 $\gamma_k$ 急剧衰减（$\gamma_2=0.45, \gamma_7=0.03$），拟合 $\gamma_k \sim 14/k^3$（$\beta \approx 3$），bottleneck 上的有效指数 $\Gamma = \lambda \cdot \gamma_k$ 递减——不利方向信号。
综合诊断：两个有利信号（$q_k \to 0$、$\Delta$ 普适）、一个待定（$\Delta$ 放缓）、四个不利信号（$\beta \gg 1$、瓶颈右推、$r_N \downarrow$、$E[j|j\ge1] \downarrow$）。在 $N \le 10^9$ 上数据与 H' 不成立更相容，但 $\ln\ln N$ 级现象超越任何有限尺度数值。
关键词：加法复杂度，$\Omega$ 分层，scale attenuation，瓶颈地图，moving bottleneck，Sathe-Selberg，衰减指数
§1 引言
1.1 背景
Paper XXX 将 H' 精确等价为 $q_N \to 0$，并将终极战场定位为"层内 attenuation vs 层间 $\Omega$ 漂移"。本文在这个战场上展开定量分析，构建逐层瓶颈地图。
1.2 本文贡献
- (A) Additive 优势增长的 $\Omega$ 普适性：$\Delta \approx +0.10/\text{decade}$，附带放缓检测（§2）。
- (B) 逐层增速 $b_k$：$\Omega=3$ 是唯象引擎层（§3）。
- (C) 分层加权框架 $q_N = \sum P(\Omega=k) \cdot q_k(N)$（§4）。
- (D) 瓶颈识别与 moving bottleneck 演化（§4–§5）。
- (E) $\Omega$ 漂移量化（§5）。
- (F) 逐层衰减指数 $\gamma_k$ 提取与有效指数 $\Gamma$ 判据（§6）。
- (G) Bottleneck 动态与间歇性检验（§7）。
1.3 记号
沿用系列约定。新增：$q_k(N) = 1 - P(j=0|\Omega=k, n \le N)$；$b_k = dP(j=0|\Omega=k)/d(\ln N)$（当前尺度局部斜率）；$\gamma_k$：对数幂律衰减指数；$\Gamma_N(k) = (\ln\ln N) \cdot \gamma_k$。
§2 Additive 优势增长的 Ω 普适性
2.1 数据
数值观察 1. $M_n - (\rho_E(n-1)+1)$ 的均值（正 = additive 赢），按 $\Omega$ 和 decade：
| Ω | [10³,10⁴) | [10⁵,10⁶) | [10⁷,10⁸) | [10⁸,10⁹) | Δ/decade(前5) | Δ(最新) |
|---|---|---|---|---|---|---|
| 2 | +0.092 | +0.347 | +0.531 | +0.605 | ~+0.11 | +0.07 |
| 3 | −0.800 | −0.563 | −0.405 | −0.342 | ~+0.10 | +0.06 |
| 4 | −1.351 | −1.109 | −0.952 | −0.890 | ~+0.10 | +0.06 |
| 5 | −1.869 | −1.569 | −1.398 | −1.333 | ~+0.12 | +0.07 |
前五个 decade：每层每 decade 增量约 $+0.10$，显示出显著的 $\Omega$ 普适性。最新 decade：增量降至约 $+0.065$——additive 优势的增速在放缓。可能是 $1/\ln N$ 效应，也可能是真实衰减，当前尺度无法区分。$\Omega$ 普适性仍然成立。
2.2 各层当前状态（$N = 10^9$）
- $\Omega=2$：gap 均值 $+0.61$，$P(j=0) = 0.78$。Additive 明确占优。
- $\Omega=3$：gap 均值 $-0.34$，$P(j=0) = 0.54$。正在翻转。
- $\Omega=4$：gap 均值 $-0.89$，$P(j=0) = 0.32$。乘法仍占优但在缩小。
- $\Omega=5$：gap 均值 $-1.33$，$P(j=0) = 0.18$。乘法强势占优。
§3 逐层增速与引擎层
3.1 线性拟合
数值观察 2. $P(j=0|\Omega=k) = a_k + b_k \cdot \ln N$（当前尺度局部近似）：
| Ω | b_k (per ln N) | per decade | 当前 P(j=0) | R² |
|---|---|---|---|---|
| 2 | 0.0120 | 0.028 | 0.752 | 0.979 |
| 3 | 0.0225 | 0.052 | 0.483 | 0.985 |
| 4 | 0.0179 | 0.041 | 0.267 | 0.992 |
| 5 | 0.0137 | 0.032 | 0.133 | 0.979 |
| 6 | 0.0060 | 0.014 | 0.064 | 0.975 |
| 7 | 0.0020 | 0.005 | 0.032 | 0.802 |
3.2 Ω=3 是当前的唯象引擎层
$b_k$ 在 $\Omega=3$ 达到峰值（0.0225），不是单调递减。$\Omega=2$：接近饱和（0.75），边际效用递减。$\Omega=3$：饱和与组合抵抗之间的甜蜜点——因子分解选项少，但基数足够大，处于最陡爬升期。$\Omega \ge 4$：组合爆炸使乘法抵抗力增强，$b_k$ 被压制。$\Omega=3$ 是当前尺度下驱动 $p_N$ 上升的唯象引擎层。
§4 H' 的逐层瓶颈地图
4.1 分层加权框架
$$q_N = \sum_{k \ge 2} P(\Omega=k \mid \text{comp}) \cdot q_k(N)$$
H' 要求 $q_N \to 0$。这把问题精确分解为层内失败率 $q_k(N)$ 与层间质量 $P(\Omega=k)$ 的加权竞争。
4.2 当前瓶颈层
数值观察 3. $N = 10^8$ 的分层示意图（定性瓶颈识别）：
| Ω | P(Ω=k|comp) | q_k | 贡献 P·q_k |
|---|---|---|---|
| 2 | 0.185 | 0.248 | 0.046 |
| 3 | 0.252 | 0.517 | 0.130 |
| 4 | 0.222 | 0.733 | 0.163 |
| 5 | 0.153 | 0.867 | 0.132 |
| 6 | 0.090 | 0.936 | 0.084 |
| 7+ | 0.098 | ~0.97 | ~0.095 |
最大贡献来自 $\Omega=4$。Sathe-Selberg 密度峰在 $k \approx \ln\ln N \approx 3$，被 $q_k$ 的单调递增右推到 $k=4$。$\Omega=4$ 的双重不利：$q_4 = 0.73$ 仍然很高，且 $P(\Omega=4)$ 还在增长（因为 $k-1=3 > \ln\ln N \approx 2.9$）。
4.3 Moving Bottleneck
H' 不要求每个固定 $k$ 的 $q_k$ 都趋于 0（固定 $k$ 的 $P(\Omega=k)$ 最终会自行下降）。真正需要控制的是 moving bottleneck window——$k$ 在 $\ln\ln N$ 附近的那一带层的 $q_k$ 是否下降得足够快。
§5 Moving Bottleneck 演化与 Ω 漂移
5.1 瓶颈迁移精确追踪
数值观察 4. $P(\Omega=k|\text{comp}) \cdot q_k$ 的峰值位置随 $N$ 的迁移：
| N | ln ln N | 瓶颈 Ω | 密度峰 Ω | 瓶颈贡献 |
|---|---|---|---|---|
| 10³ | 1.93 | 3 | 2 | 0.215 |
| 10⁴ | 2.22 | 3 | 2 | 0.192 |
| 10⁵ | 2.44 | 3 | 3 | 0.169 |
| 2×10⁵ | 2.50 | 3→4 | 3 | 0.167 |
| 10⁶ | 2.63 | 4 | 3 | 0.165 |
| 10⁷ | 2.78 | 4 | 3 | 0.161 |
| 10⁸ | 2.91 | 4 | 3 | 0.157 |
瓶颈在 $N \approx 2\times10^5$ 从 $\Omega=3$ 跳到 $\Omega=4$，此后稳定。瓶颈贡献在缓慢下降（0.215 → 0.157）。
5.2 E[Ω|comp] ≈ ln ln N + 1.2
| N | E[Ω|comp] | ln ln N | E[Ω] − ln ln N |
|---|---|---|---|
| 10³ | 3.260 | 1.933 | 1.33 |
| 10⁵ | 3.695 | 2.444 | 1.25 |
| 10⁷ | 3.917 | 2.780 | 1.14 |
| 10⁸ | 4.102 | 2.914 | 1.19 |
$E[\Omega|\text{comp}] \approx \ln\ln N + 1.2$，与 Sathe-Selberg 的 $\ln\ln N + O(1)$ 图景相容。漂移极其缓慢。
§6 逐层衰减指数 γ_k
6.1 模型与拟合
数值观察 5. Model A（对数幂律）vs Model B（常数极限）：
| Ω | γ_k | R²(A) | a_k(B) | R²(B) | 胜者 |
|---|---|---|---|---|---|
| 2 | 0.449 | 0.997 | 0.167 | 0.985 | A 胜 |
| 3 | 0.413 | 0.999 | 0.362 | 0.988 | A 胜 |
| 4 | 0.253 | 0.988 | 0.612 | 0.963 | A 胜 |
| 5 | 0.165 | 0.997 | 0.762 | 0.981 | A 胜 |
| 6 | 0.076 | 0.973 | 0.887 | 0.936 | A 胜 |
| 7 | 0.030 | 0.825 | 0.949 | 0.748 | A 胜 |
Model A 在每层都赢。当前数据强烈支持：对每个固定 $\Omega=k$，$q_k \to 0$（不趋向正常数）。但这是有限尺度的模型比较，不构成渐近确认。
6.2 有效指数 Γ_N = λ · γ_k
在 bottleneck window $k \approx \lambda = \ln\ln N$ 处：$q_k(N) \sim e^{-\lambda \cdot \gamma_k} = e^{-\Gamma}$。H' 的真正判据是 bottleneck window 上的 $\Gamma_N(k) = \lambda \cdot \gamma_k$ 是否趋于无穷。
拟合 $\gamma_k \approx 13.9/k^{3.0}$（$R^2 = 0.92$，$\beta \approx 3 \gg 1$）。外推显示 $\Gamma$ 在递减（见主要数据表），对 H' 不利。
| ln N | λ | k_bott | γ_bott | Γ = λ·γ |
|---|---|---|---|---|
| 20 | 3.0 | 3 | 0.53 | 1.58 |
| 50 | 3.9 | 4 | 0.22 | 0.88 |
| 100 | 4.6 | 5 | 0.12 | 0.53 |
| 200 | 5.3 | 6 | 0.07 | 0.36 |
| 500 | 6.2 | 7 | 0.04 | 0.26 |
$\Gamma$ 在递减——不利方向信号。但：（1）五个点的拟合，不代表渐近律；（2）$k \in [3,7]$ 是组合爆炸初期，$\beta \approx 3$ 可能是瞬态曲率；（3）高 $\Omega$ 层（$k \gg 7$）可能有渐近厚尾（有效 $\beta < 1$）。远未构成否定。
§7 Bottleneck 动态与间歇性检验
7.1 瓶颈质心演化
数值观察 6. 加权质心 $\bar{k}_B$，右推量 $\Delta_N = \bar{k}_B - \ln\ln N$：
| N | ln ln N | k̄_B | σ_B | k_peak | Δ_N | q_N |
|---|---|---|---|---|---|---|
| 10³ | 1.93 | 3.62 | 1.27 | 3 | +1.69 | 0.657 |
| 10⁴ | 2.22 | 3.99 | 1.53 | 3 | +1.77 | 0.660 |
| 10⁵ | 2.44 | 4.23 | 1.65 | 3 | +1.79 | 0.650 |
| 10⁶ | 2.63 | 4.43 | 1.71 | 4 | +1.80 | 0.641 |
| 10⁷ | 2.78 | 4.59 | 1.75 | 4 | +1.81 | 0.634 |
| 10⁸ | 2.91 | 4.73 | 1.79 | 4 | +1.81 | 0.629 |
$\Delta_N$ 缓慢增大（1.69 → 1.81），$\sigma_B$ 增大（1.27 → 1.79）——瓶颈窗口在变宽，贡献越来越分散。
7.2 r_N 间歇性检验
数值观察 7. H' 要求 $r_N = E[j|\text{comp}]/q_N \to \infty$：
| Decade | E[j|comp] | q | r = E[j]/q | P(j≥2) |
|---|---|---|---|---|
| [10³, 10⁴) | 1.133 | 0.660 | 1.715 | 0.330 |
| [10⁴, 10⁵) | 1.102 | 0.649 | 1.698 | 0.322 |
| [10⁵, 10⁶) | 1.083 | 0.641 | 1.691 | 0.316 |
| [10⁶, 10⁷) | 1.070 | 0.634 | 1.687 | 0.312 |
| [10⁷, 10⁸) | 1.060 | 0.630 | 1.683 | 0.309 |
| [10⁸, 10⁹) | 1.053 | 0.626 | 1.682 | 0.307 |
$r_N$ 从 1.715 缓慢下降到 1.682，下降速率放缓，但没有反转信号。但"由瓶颈右移驱动的全局间歇性"尚未被排除——失败质量向高 $\Omega$ 层迁移，而高 $\Omega$ 层的 $E[j|j \ge 1]$ 本身更大，可能最终通过层间组成变化使全局 $r_N$ 上升。
7.3 逐层跳跃深度
数值观察 8. $E[j|j \ge 1, \Omega=k]$ 按 decade：
| Decade | Ω=2 | Ω=3 | Ω=4 | Ω=5 | Ω=6 |
|---|---|---|---|---|---|
| [10³,10⁴) | 1.228 | 1.369 | 1.602 | 1.972 | 2.378 |
| [10⁵,10⁶) | 1.209 | 1.313 | 1.511 | 1.781 | 2.093 |
| [10⁷,10⁸) | 1.203 | 1.294 | 1.454 | 1.685 | 1.959 |
| [10⁸,10⁹) | 1.201 | 1.287 | 1.435 | 1.651 | 1.912 |
每一层、每一个 decade，$E[j|j \ge 1, \Omega=k]$ 都在下降。零例外。跳跃在变浅，与 H'（间歇性要求深度增加）方向相反。但下降速率在放缓。
7.4 综合诊断
| 信号 | 方向 | H' 意义 |
|---|---|---|
| 每层 $q_k \to 0$（Model A 全胜） | ✓ 有利 | 必要条件受数据支持 |
| $\Delta > 0$ 且 $\Omega$ 普适 | ✓ 有利 | attenuation 在工作 |
| $\Delta$ 从 0.10 放缓到 0.065 | — 待定 | 可能是 $1/\ln N$ 效应或真实衰减 |
| $\gamma_k$ 衰减 $\beta \approx 3$ | ✗ 不利 | 有效指数 $\Gamma$ 递减 |
| $\Delta_N$ 缓慢增大（1.69→1.81） | ✗ 不利 | 瓶颈比 $\Omega$ 漂移跑得快 |
| $r_N$ 在降（1.715 → 1.682） | ✗ 不利 | 间歇性方向相反 |
| $E[j|j \ge 1, \Omega=k]$ 每层都降 | ✗ 不利 | 跳跃变浅不变深 |
两个有利、一个待定、四个不利。在有限尺度上数据与 H' 不成立更相容。但 $N = 10^9$ 对于 $\ln\ln N$ 级现象来说几乎是零——所有趋势随时可能在远超当前的尺度上反转。$\beta \approx 3$ 可能只是组合爆炸初期的瞬态曲率。
§8 讨论
8.1 核心贡献
- $\Delta \approx +0.10/\text{decade}$ 的 $\Omega$ 普适性——强烈支持 attenuation 由宏观标度驱动。
- 引擎层 $\Omega=3$（$b_k$ 峰值）vs 瓶颈层 $\Omega=4$（$P \cdot q$ 峰值）的分离。
- 分层加权框架 $q_N = \sum P(\Omega=k) \cdot q_k(N)$。
- Moving bottleneck：$\Omega=3 \to 4$ at $N \approx 2\times10^5$，此后稳定。
- $\gamma_k$：Model A 全面胜出；$\gamma_k$ 从 0.45 急剧衰减到 0.03。
- H' 定量边界：$\gamma_k \sim 14/k^3$（$\beta \approx 3 \gg 1$），有效指数 $\Gamma$ 递减——不利信号。
- 间歇性检验：$r_N$ 降、$E[j|j \ge 1, \Omega=k]$ 每层都降——方向与 H' 相反。
- 瓶颈质心右推量 $\Delta_N$ 缓慢增大（1.69 → 1.81）。
- 综合诊断：两个有利、四个不利；有限尺度数据更相容于 $\neg$H'。
8.2 开放问题
- $\gamma_k$ 的解析形式——从 DP 递推结构推导。
- Uniform window control：$\sup_{k \in W_N} q_k \to 0$ 的精确条件。
- 对 $\Omega=2$ 的解析证明 $q_2 \sim C \cdot (\ln N)^{-0.45} \to 0$——最容易的起点。
- $\Delta \approx +0.10$ 的 $\Omega$ 无关性的解析解释。
- 瓶颈从 $\Omega=4$ 何时让位于 $\Omega=5$？
- $r_N$ 何时反转上升？
- $\Delta_N$ 是否收敛到 $O(1)$？
8.3 致谢
ChatGPT 贡献了分层加权框架、$P(\Omega=k)$ 单调方向修正、密度峰位置精确化（$k \approx \ln\ln N$ 而非 $1+\ln\ln N$）、moving bottleneck 精确定义、$N^{-b_k}$ 模型误设识别、以及有效指数 $\Gamma = \lambda \cdot \gamma_k$ 判据和 $\beta$ 判据。Gemini 贡献了 $\Delta \approx +0.10$ 的 $\Omega$ 普适性升格、$\Omega=3$ 作为引擎层的定位、$\beta \approx 3$ 的瞬态曲率警告、以及 Erdős-Kac 高斯核建议。Grok 保证了与前 30 篇的一致性。Claude 编写了全部数值计算脚本并起草了 working notes v1-v2。最终文本由作者独立完成，所有数学判断由作者负责。
§9 数据来源与可复现性
| 脚本 | 测量 | §引用 |
|---|---|---|
| p31_block123.py | M_n gap + P(j=0|Ω) 线性拟合 + Ω 漂移 | §2–§5 |
| p31_block45.py | q_k 衰减拟合（γ_k 提取）+ bottleneck 演化 | §5–§6 |
| p31_block67.py | Bottleneck profile + r_N 演化 + E[j|j≥1,Ω] | §7 |
Sanity check：$\rho_E(10^7) = 58$，$\rho_E(10^8) = 66$，$\rho_E(10^9) = 74$。
参考文献
- ZFCρ Papers I–XXX. H. Qin. Paper XXVIII DOI: 10.5281/zenodo.19078654. Paper XXIX DOI: 10.5281/zenodo.19083884. Paper XXX DOI: 10.5281/zenodo.19101810.
- G. H. Hardy, S. Ramanujan (1917). The normal number of prime factors of a number n. Quart. J. Math., 48:76–92.
- A. Selberg (1954). Note on a paper by L. G. Sathe. J. Indian Math. Soc., 18:83–87.
- P. Erdős, M. Kac (1940). The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math., 62:738–742.
- K. Cordwell, S. Epstein, A. Hemmady, S. J. Miller, E. Steiner (2018). On the number of 1's needed to represent n. J. Number Theory, 189:17–34.