Upper-Margin Closure and the Bounded-Variance Phenomenon
DOI: 10.5281/zenodo.19140015Building on Paper XXXIII's Margin Threshold Lemma, we establish the Upper-Margin Closure framework via the Cantelli inequality and discover that fixed-p variance \(V(p) = \mathrm{Var}_q(\mathrm{diff}_p(q))\) is globally bounded.
(1) \(V(p) \approx 1.40\) across all p-bands at both \(N=10^9\) and \(N=10^{10}\) (9,581 primes, \(p \leq 100{,}000\)). Global max \(V(p) = 2.14\); only 4 primes (0.04%) have \(V(p) > 2.0\). This is a global property requiring no per-template conditioning.
(2) The signal-to-noise ratio \(\Gamma_p = M_p^2/V(p)\) yields the Cantelli upper bound \(c_p \leq 1/(1+\Gamma_p)\). With \(V(p) = O(1)\), the growth of \(\Gamma_p\) is driven entirely by \(M_p\) drift.
(3) Harmonic Cesàro Lemma: if \(V(p) \leq C\) and \(M_p \to \infty\), then \(\bar{c}_h(N) \to 0\). This is strictly weaker than requiring \(\sum c_p/p < \infty\), and suffices for H' at \(\Omega=2\). No identification of \(M_p\)'s growth rate is needed.
(4) \(M_p\) decomposition: \(M_p = \tau_p - E[\mathrm{diff}_p]\). Both \(\tau_p\) and \(E[\mathrm{diff}]\) rise with \(p\), but \(E[\mathrm{diff}]\) grows at 98.3% of \(\tau_p\)'s rate. \(M_p\) drift is the 1.7% residual gap — driven by the prime penalty \(\Delta_{\mathrm{penalty}}\).
(5) "Bad primes" (\(\Gamma_p < 1\)) comprise 5.5% of primes with 25.4% cumulative harmonic weight. Tail analysis shows: \(\delta_{\mathrm{tail}}\) drops from 25.4% to ~5.5% once small-p front-loading is removed, then stabilizes at ~5.5% across all dyadic windows from \(p=1000\) to \(p=100{,}000\). Bad primes contribute ~54% of \(S(N)\), and their \(E[M_p] \approx 0.72\) is not yet rising. Their identity: primes with extremely smooth \(p-1\) (low \(\rho(p)\), hence low threshold \(\tau_p\)).
(6) \(\Delta_{\mathrm{penalty}} = E[R_{\mathrm{prime}}] - E[R_{\mathrm{general}}]\) measured with fixed \(h_{\mathrm{ref}} = 3.60\) shows a positive-slope fit across seven decades (0.68 → 0.70), with slope ~0.022 per unit \(\ln\ln n\). The trend is overall increasing, though growth is extremely slow.
(7) The \(\Omega=2\) closure now reduces to a single open problem: prove \(M_p \to \infty\) for harmonic-a.e. primes. A natural sufficient form: prove that \(\tau_p - E[\mathrm{diff}_p] \to \infty\) on a set of primes with harmonic density 1.
Keywords: Upper-Margin Closure, Cantelli inequality, bounded variance, signal-to-noise ratio, prime penalty, tail bad-mass
§1 Introduction
1.1 Background
Paper XXXIII (DOI: 10.5281/zenodo.19124485) established the Margin Threshold Lemma (lower bound) and discovered that template \(m=240\)'s band-average \(M_p\) crosses 1 at \(N=10^{10}\). Three analytic routes were identified: Route A (margin dichotomy, closing), Route B (Lindley exponential tail), Route C (prime penalty growth).
Paper XXXIV shifts strategy from lower-bound (Route A) to upper-threshold closure, following ChatGPT's recommendation. The key insight: the Cantelli inequality provides an upper bound on \(c_p\) requiring only second-moment control, which is strictly weaker than the exponential tail assumed in Route B.
1.2 Upper-Margin Closure Framework
Definition. For prime-core \(p\) with cutoff \(N\): \(X_{p,N} = \mathrm{diff}_p(q)\), \(\tau_p = \rho(p)+2\), \(M_p = \tau_p - E[X]\), \(V(p) = \mathrm{Var}(X)\), \(\Gamma_p = M_p^2/V(p)\).
Cantelli bound: \(c_p = P(X \geq \tau_p) \leq V(p)/(V(p) + M_p^2) = 1/(1 + \Gamma_p)\).
§2 Global Bounded Variance: \(V(p) = O(1)\)
2.1 Band-wise Statistics (\(10^{10}\))
| p band | #p | E[V] | med[V] | q90[V] | q95[V] | q99[V] | max[V] |
|---|---|---|---|---|---|---|---|
| [10, 50) | 11 | 1.41 | 1.44 | 1.68 | 1.69 | 1.70 | 1.70 |
| [200, 500) | 49 | 1.43 | 1.42 | 1.71 | 1.72 | 1.76 | 1.77 |
| [1000, 2000) | 135 | 1.41 | 1.40 | 1.70 | 1.73 | 1.75 | 1.78 |
| [5000, 10000) | 560 | 1.40 | 1.40 | 1.71 | 1.72 | 1.76 | 1.79 |
| [20000, 50000) | 2871 | 1.40 | 1.39 | 1.70 | 1.73 | 1.76 | 1.82 |
| [50000, 100000) | 4448 | 1.40 | 1.40 | 1.71 | 1.74 | 1.79 | 2.14 |
Global: \(E[V] = 1.398\), q90 = 1.706, q99 = 1.778, max = 2.142. Only 4 primes (0.04%) exceed \(V = 2.0\).
2.2 Consistency Across N
\(V(p) \approx 1.40\) is identical at \(N=10^9\) and \(N=10^{10}\). This stability across both p-range and N-range is the strongest numerical evidence in the paper.
2.3 Per-Template \(V(p)\)
Per-template conditional variance is lower (~0.9–1.2) and equally stable:
| Template | E[V] range | Trend |
|---|---|---|
| m=24 | 1.16–1.21 | stable |
| m=120 | 1.09–1.10 | stable |
| m=240 | 0.97–1.00 | stable |
| m=720 | 0.87–0.90 | stable |
The global \(V(p) \approx 1.40\) already suffices for the Cantelli bound. Per-template decomposition is not required for closure but provides diagnostic insight.
§3 Harmonic Cesàro Lemma
3.1 Statement and Proof
Lemma (ChatGPT). If \(V(p) \leq C\) for all sufficiently large \(p\), and \(M_p \to \infty\), then \(\bar{c}_h(N) \to 0\).
Proof. Let \(a_p = V(p)/(V(p) + M_p^2)\). By Cantelli, \(c_p \leq a_p\). Since \(V(p) \leq C\) and \(M_p \to \infty\), we have \(a_p \leq C/(C + M_p^2) \to 0\).
For any \(\varepsilon > 0\), choose \(P_0\) such that \(a_p < \varepsilon\) for all \(p > P_0\). Then
$$S(x) = \sum_{p \leq x} \frac{c_p}{p} \leq \sum_{p \leq P_0} \frac{1}{p} + \varepsilon \cdot \sum_{P_0 < p \leq x} \frac{1}{p}$$
Dividing by \(D(x) = \sum_{p \leq x} 1/p \to \infty\):
$$\limsup \bar{c}_h(x) \leq \limsup \left[\frac{\mathrm{const}}{D(x)} + \varepsilon\right] = \varepsilon$$
By the arbitrariness of \(\varepsilon\), \(\bar{c}_h(x) \to 0\). ■
3.2 Two-Layer Structure
Layer 1 (sufficient for H'): \(\bar{c}_h(N) \to 0\). Requires only \(V(p) = O(1)\) + \(M_p \to \infty\). No rate identification needed.
Layer 2 (stronger): \(\sum c_p/p < \infty\). Requires \(\Gamma_p\) to grow faster than \(\ln\ln p\). If \(M_p \sim \kappa \cdot \ln\ln p\) and \(V(p) = O(1)\), then \(\Gamma_p \sim \kappa^2 \cdot (\ln\ln p)^2\) and the sum converges.
Paper XXXIV targets Layer 1 only. Layer 2 is an optional stronger corollary.
§4 \(M_p\) Decomposition: The 98.3% Chase
4.1 \(\tau_p\) vs \(E[\mathrm{diff}]\)
\(M_p = \tau_p - E[\mathrm{diff}_p]\). Both sides grow with \(p\):
| p band | E[τ_p] | E[diff] | E[M_p] | Δτ | Δdiff | ΔM |
|---|---|---|---|---|---|---|
| [10, 50) | 13.3 | 11.5 | 1.77 | — | — | — |
| [500, 1000) | 26.8 | 24.5 | 2.28 | +2.97 | +2.95 | +0.02 |
| [5000, 10000) | 35.8 | 33.4 | 2.39 | +3.03 | +3.03 | +0.00 |
| [20000, 50000) | 41.8 | 39.3 | 2.43 | +3.25 | +3.23 | +0.03 |
| [50000, 100000) | 44.8 | 42.3 | 2.47 | +3.04 | +3.00 | +0.04 |
Overall (\(p \approx 100 \to p \approx 50{,}000\)): \(\tau_p\) rises by 24.19, \(E[\mathrm{diff}]\) rises by 23.77. \(E[\mathrm{diff}]\) chases \(\tau_p\) at 98.3% of its speed. \(M_p\) drift is the 1.7% residual gap.
4.2 Implication
\(M_p\)'s slow drift (~0.53·\(\ln\ln p\)) is not because \(\tau_p\) grows slowly — it grows by 24 units. It is because \(E[\mathrm{diff}_p]\) almost perfectly tracks \(\tau_p\). The combinatorial compression of \(pq-1\) improves almost as fast as \(\rho(p)\) grows. The tiny residual gap is \(\Delta_{\mathrm{penalty}}\).
§5 Bad Primes: Identity, Anatomy, and Tail Analysis
5.1 Identity
527 primes (5.5%) have \(\Gamma_p < 1\). Their anatomy:
| Bad (Γ<1) | Good (Γ≥1) | |
|---|---|---|
| Count | 527 | 9,054 |
| E[ρ(p)] | 37.6 | 39.7 |
| E[τ_p] | 39.6 | 41.7 |
| E[diff] | 38.9 | 39.2 |
| E[M_p] | 0.72 | 2.53 |
| E[c_p] | 0.427 | 0.063 |
Bad primes have \(\tau_p\) 2.1 lower than good primes, but \(E[\mathrm{diff}]\) only 0.3 lower. Their \(M_p\) deficit (1.81) comes almost entirely from disproportionately low \(\rho(p)\) — i.e., extremely smooth \(p-1\).
5.2 Tail Bad-Mass (ChatGPT's Diagnostic)
The cumulative harmonic weight of bad primes is 25.4%, but this is front-loaded by small \(p\). The tail diagnostic \(\delta_{\mathrm{tail}}(P, X) = \sum_{P < p \leq X,\, \Gamma_p < 1} 1/p \,/\, \sum_{P < p \leq X} 1/p\) reveals the true structure:
| P_low | δ_tail (Γ<1) |
|---|---|
| 0 (cumulative) | 25.4% |
| 100 | 7.6% |
| 500 | 5.6% |
| 2000 | 5.1% |
| 10000 | 5.4% |
| 50000 | 5.6% |
The 25.4% drops to ~5.5% once the small-p front-loading is removed. But ~5.5% then stabilizes — it does not decrease further. Dyadic analysis confirms: every window \([X, 2X)\) from \(p=1000\) to \(p=100{,}000\) shows \(\delta_{\mathrm{dyad}} \approx 5.5\%\).
5.3 Contribution to \(S(N)\)
| X | S_bad | S_good | S_total | S_bad/S | c̄_h |
|---|---|---|---|---|---|
| 500 | 0.311 | 0.237 | 0.549 | 56.8% | 0.262 |
| 10000 | 0.321 | 0.264 | 0.585 | 54.9% | 0.236 |
| 99859 | 0.326 | 0.277 | 0.603 | 54.1% | 0.223 |
Bad primes contribute ~54% of \(S(N)\). The fraction is slowly declining (57% → 54% over three decades).
5.4 Assessment
The tail harmonic density of bad primes (~5.5%) is a real phenomenon, not purely a finite-size artifact. However, it does not constitute a closure obstruction if \(M_p \to \infty\) for these primes as well. Their \(E[M_p | \mathrm{bad}] \approx 0.72\) is not yet rising in the current window (\(p \leq 100{,}000\)), but Paper XXXIII showed that \(m=240\)'s \(M_p\) crossed 1 at \(p > 50{,}000\) — consistent with delayed onset of \(\Delta_{\mathrm{penalty}}\) for smooth families.
The closure does not require eliminating bad primes. It requires that their \(c_p\) eventually also decay, which follows from \(M_p \to \infty\) for harmonic-a.e. \(p\).
§6 \(\Delta_{\mathrm{penalty}}\) with Fixed Reference Slope
6.1 Measurement
Using fixed \(h_{\mathrm{ref}} = 3.60\) (not the N-dependent \(\rho(N)/\ln N\)):
| Decade | E[R_general] | E[R_prime] | Δ_penalty |
|---|---|---|---|
| [10³, 10⁴) | 0.933 | 1.614 | 0.682 |
| [10⁴, 10⁵) | 1.650 | 2.328 | 0.677 |
| [10⁵, 10⁶) | 2.313 | 3.008 | 0.695 |
| [10⁶, 10⁷) | 2.954 | 3.649 | 0.695 |
| [10⁷, 10⁸) | 3.572 | 4.271 | 0.699 |
| [10⁸, 10⁹) | 4.171 | 4.873 | 0.703 |
| [10⁹, 10¹⁰) | 4.768 | 5.463 | 0.695 |
Fit: \(\Delta_{\mathrm{penalty}} = 0.635 + 0.022 \cdot \ln\ln n\). Increasing, but extremely slowly (+0.014 over seven decades).
6.2 Interpretation
The prime penalty is real and growing. Primes are locked at \(\Omega=1\), while general integers benefit from ever-deeper combinatorial compression as \(\Omega(n) \sim \ln\ln n\) grows (Hardy–Ramanujan). The effective slope \(\rho(p)/\ln p\) is still rising (3.48 → 3.82), confirming that \(h_0\) has not converged.
6.3 Connection to \(M_p\)
\(M_p \approx R(p) + 2 + \Delta_{\mathrm{penalty}}\) (Gemini's decomposition). The static local advantage from smooth \(p-1\) (low \(\rho(m)\)) is a constant. If \(\Delta_{\mathrm{penalty}} \to \infty\), then every fixed template is eventually overtaken; current data is consistent with this direction, but it remains an open input for Route C.
§7 The Effective Slope \(\rho(p)/\ln p\)
| p band | E[ρ(p)] | E[ln p] | ρ/ln p |
|---|---|---|---|
| [10, 50) | 11.3 | 3.24 | 3.482 |
| [500, 1000) | 24.8 | 6.60 | 3.753 |
| [5000, 10000) | 33.8 | 8.90 | 3.798 |
| [50000, 100000) | 42.8 | 11.20 | 3.821 |
\(\rho(p)/\ln p\) rises from 3.48 to 3.82 and has not converged. This confirms that the effective "cost per unit log" for primes is still increasing — a manifestation of the prime penalty at the \(\rho\) level.
§8 Three Routes to \(M_p \to \infty\)
8.1 Route B: Local Lipschitz → Lindley → Exponential Tail
\(V(p) = O(1)\) already provides the second-moment input. Remains: Lindley's exponential tail + \(M_p\) drift.
8.2 Route C: \(\Delta_{\mathrm{penalty}} \to \infty\)
Physically near-certain (combinatorial compression grows monotonically). Current data: slope 0.022 per unit \(\ln\ln n\) over seven decades. Slow but positive.
8.3 Route D: Direct \(M_p\) Unboundedness
From the recursion \(\rho(p) \leq \rho(m) + \rho((p-1)/m) + 3\), prove that no fixed template can keep \(M_p\) bounded. The 98.3% chase (§4) shows why this is hard: \(E[\mathrm{diff}]\) almost perfectly tracks \(\tau_p\).
§9 Discussion
9.1 Core Contributions
Paper XXXIV achieves four things: (a) discovers \(V(p) = O(1)\) as a global property; (b) establishes the Harmonic Cesàro Lemma reducing closure to \(M_p \to \infty\); (c) decomposes \(M_p\) into the 98.3% chase between \(\tau_p\) and \(E[\mathrm{diff}]\); (d) characterizes bad primes (5.5% tail density, 54% of \(S(N)\), extremely smooth \(p-1\)).
9.2 Paper XXXIII → XXXIV Progress
| Paper XXXIII | Paper XXXIV | |
|---|---|---|
| Bound type | Lower (Margin Threshold) | Upper (Cantelli) |
| Variance | Not measured | V(p) ≈ 1.40, O(1) |
| Closure condition | M_p < 1? (threshold) | M_p → ∞? (qualitative) |
| Target | S(N) = o(ln ln N) | c̄_h(N) → 0 (Cesàro) |
| Bad primes | Not identified | 5.5% tail, anatomy known |
9.3 Honest Distance Estimate
Ω=2 closure: 1 paper. Need to prove \(M_p \to \infty\) for harmonic-a.e. \(p\). Three routes available. \(V(p) = O(1)\) is established. The 98.3% chase and \(\Delta_{\mathrm{penalty}}\)'s slow growth (0.022/unit) point to the precise mechanism.
Full H': additional 2–3 papers. \(\Omega=3+\) extension and layer summation.
9.4 Open Problems
(1) Prove \(M_p \to \infty\) analytically. (2) Prove \(V(p) = O(1)\) from the DP recursion. (3) Determine whether bad primes' \(M_p\) eventually rises. (4) Extend to \(\Omega \geq 3\). (5) Establish \(\Delta_{\mathrm{penalty}} \to \infty\) as a theorem.
§10 Data Sources and Reproducibility
| Script | Measurement |
|---|---|
| rho_dp.c | ρ_E DP (C, multiplicative forward propagation) |
| p34_var_diff.py | V_p, M_p, c_p (v1, had index bug) |
| p34_gamma_profile.py | Γ_p, V(p), shell partition (v2, correct) |
| p34_uniform_bounds.py | V(p) uniform bound + M_p lower bound + bad primes |
| p34_mp_decomp.py | M_p decomposition + Δ_penalty with fixed h_ref |
| p34_tail_badmass.py | Tail bad-mass δ_tail(P,X) + dyadic + S(N) contribution |
Sanity checks: \(\rho(10^7)=58\), \(\rho(10^8)=66\), \(\rho(10^9)=75\).
References
[1] ZFCρ Papers I–XXXIII. H. Qin. Paper XXXIII DOI: 10.5281/zenodo.19124485. Paper XXXII DOI: 10.5281/zenodo.19116625.
[2] 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.
[3] A. Hildebrand, G. Tenenbaum (1986). On integers free of large prime factors. Trans. Amer. Math. Soc., 296:265–290.
Acknowledgments
Claude (子路) wrote all numerical scripts, drafted working notes v1–v3 and the formal text, discovered the v1 index bug and the global \(V(p) = O(1)\) property, and designed the tail bad-mass diagnostic. ChatGPT (公西华) contributed the Upper-Margin Closure Lemma and the Harmonic Cesàro Lemma with proof, identified the β exponent/slope confusion and \(V(p)\) object ambiguity in v1, proposed the \(\Gamma_p = M^2/V\) framework and disjoint shell partition, and demanded the tail bad-mass statistic \(\delta_{\mathrm{tail}}(P,X)\) that revealed the front-loading structure. Gemini (子夏) contributed the \(\Delta_{\mathrm{penalty}}\) semi-analytic argument (Route C) and the three-layer analysis of bad primes (constant discount vs divergent penalty, front-loading illusion, Cesàro suppression). Grok (子贡) confirmed \(V(p) = O(1)\) is stronger than the original Local Lipschitz, verified consistency with Paper XXI's Lindley framework, and provided the DP self-correction interpretation. The final text was independently completed by the author; all mathematical judgments are the author's responsibility.
在 Paper XXXIII 的 Margin Threshold Lemma 基础上，本文建立 Upper-Margin Closure 框架（Cantelli 不等式），并发现 fixed-p 方差 \(V(p) = \mathrm{Var}_q(\mathrm{diff}_p(q))\) 全局有界。
(1) \(V(p) \approx 1.40\) 在 \(10^9\) 和 \(10^{10}\) 下对全部 9,581 个素数（\(p \leq 100{,}000\)）稳定。全局 \(\max V(p) = 2.14\)，仅 4 个素数（0.04%）的 \(V(p) > 2.0\)。这是全局性质，不需要 per-template 条件化。
(2) 信噪比 \(\Gamma_p = M_p^2/V(p)\) 给出 Cantelli 上界 \(c_p \leq 1/(1+\Gamma_p)\)。\(V(p) = O(1)\) 意味着 \(\Gamma_p\) 的增长完全由 \(M_p\) 漂移驱动。
(3) Harmonic Cesàro Lemma（公西华）：若 \(V(p) \leq C\) 且 \(M_p \to \infty\)，则 \(\bar{c}_h(N) \to 0\)。这比 \(\sum c_p/p < \infty\) 弱得多，足够推出 H' 在 \(\Omega=2\) 成立。不需要识别 \(M_p\) 的增长速率。
(4) \(M_p\) 分解：\(M_p = \tau_p - E[\mathrm{diff}_p]\)。\(\tau_p\) 和 \(E[\mathrm{diff}]\) 都随 \(p\) 增长，但 \(E[\mathrm{diff}]\) 以 \(\tau_p\) 的 98.3% 的速度追赶。\(M_p\) 的漂移是 1.7% 的残差——即 \(\Delta_{\mathrm{penalty}}\)。
(5) "坏素数"（\(\Gamma_p < 1\)）占 5.5% 的素数和 25.4% 的累积 harmonic 权重。Tail 分析显示：去除小 \(p\) 前载后，\(\delta_{\mathrm{tail}}\) 降至 ~5.5% 并在所有 dyadic 窗口（\(p=1000\) 到 \(p=100{,}000\)）中稳定。坏素数贡献 \(S(N)\) 的 ~54%。其身份：\(p-1\) 极其 smooth 的素数（\(\rho(p)\) 不成比例地低）。
(6) 用固定 \(h_{\mathrm{ref}} = 3.60\) 测量的 \(\Delta_{\mathrm{penalty}} = E[R_{\mathrm{prime}}] - E[R_{\mathrm{general}}]\) 给出正斜率拟合（0.68 → 0.70，跨七个 decade），斜率 ~0.022 per unit \(\ln\ln n\)。总体呈上升趋势，但增长极慢。
(7) \(\Omega=2\) 的闭合归结为一个 open problem：证明 \(M_p \to \infty\) 对 harmonic-a.e. 的素数成立。一个自然的充分形式：证明 \(\tau_p - E[\mathrm{diff}_p] \to \infty\) 在 harmonic 密度为 1 的素数集上成立。
关键词： Upper-Margin Closure，Cantelli 不等式，有界方差，信噪比，素数惩罚，tail bad-mass
§1 引言
1.1 背景
Paper XXXIII (DOI: 10.5281/zenodo.19124485) 建立了 Margin Threshold Lemma（下界），发现 \(m=240\) 的 band 平均 \(M_p\) 在 \(10^{10}\) 穿过 1，识别了三条解析路线。
Paper XXXIV 按公西华的建议转向 upper-threshold closure：Cantelli 不等式只需要二阶矩控制，比 Route B 的指数尾假设弱得多。
1.2 Upper-Margin Closure 框架
定义。 对 prime-core \(p\) 和 cutoff \(N\)：\(X_{p,N} = \mathrm{diff}_p(q)\)，\(\tau_p = \rho(p)+2\)，\(M_p = \tau_p - E[X]\)，\(V(p) = \mathrm{Var}(X)\)，\(\Gamma_p = M_p^2/V(p)\)。
Cantelli 上界： \(c_p = P(X \geq \tau_p) \leq V(p)/(V(p) + M_p^2) = 1/(1 + \Gamma_p)\)。
§2 全局有界方差：\(V(p) = O(1)\)
2.1 Band-wise 统计（\(10^{10}\)）
| p band | #p | E[V] | med[V] | q90[V] | q99[V] | max[V] |
|---|---|---|---|---|---|---|
| [10, 50) | 11 | 1.41 | 1.44 | 1.68 | 1.70 | 1.70 |
| [200, 500) | 49 | 1.43 | 1.42 | 1.71 | 1.76 | 1.77 |
| [1000, 2000) | 135 | 1.41 | 1.40 | 1.70 | 1.75 | 1.78 |
| [5000, 10000) | 560 | 1.40 | 1.40 | 1.71 | 1.76 | 1.79 |
| [20000, 50000) | 2871 | 1.40 | 1.39 | 1.70 | 1.76 | 1.82 |
| [50000, 100000) | 4448 | 1.40 | 1.40 | 1.71 | 1.79 | 2.14 |
全局：\(E[V] = 1.398\)，q90 = 1.706，q99 = 1.778，max = 2.142。仅 4 个素数（0.04%）超过 \(V = 2.0\)。
2.2 跨 N 的一致性
\(V(p) \approx 1.40\) 在 \(10^9\) 和 \(10^{10}\) 下完全一致。这种跨 p 范围和 N 范围的双重稳定性是本文最强的数值证据。
2.3 Per-Template \(V(p)\)
Per-template 条件方差更低（~0.9–1.2）且同样稳定。但全局 \(V(p) \approx 1.40\) 已经足够 Cantelli 上界，per-template 分解不是闭合所必需的。
§3 Harmonic Cesàro Lemma
3.1 陈述与证明
引理（公西华）。 若 \(V(p) \leq C\) 对所有足够大的 \(p\) 成立，且 \(M_p \to \infty\)，则 \(\bar{c}_h(N) \to 0\)。
证明。 令 \(a_p = V(p)/(V(p) + M_p^2)\)。由 Cantelli，\(c_p \leq a_p\)。由 \(V(p) \leq C\) 且 \(M_p \to \infty\)，有 \(a_p \leq C/(C + M_p^2) \to 0\)。
对任意 \(\varepsilon > 0\)，取 \(P_0\) 使得 \(p > P_0\) 时 \(a_p < \varepsilon\)。则
$$S(x) = \sum_{p \leq x} \frac{c_p}{p} \leq \sum_{p \leq P_0} \frac{1}{p} + \varepsilon \cdot \sum_{P_0 < p \leq x} \frac{1}{p}$$
除以 \(D(x) = \sum_{p \leq x} 1/p \to \infty\)：
$$\limsup \bar{c}_h(x) \leq \limsup \left[\frac{\mathrm{const}}{D(x)} + \varepsilon\right] = \varepsilon$$
由 \(\varepsilon\) 的任意性，\(\bar{c}_h(x) \to 0\)。■
3.2 两层分层
第一层（H' 所需）： \(\bar{c}_h(N) \to 0\)。只需 \(V(p) = O(1)\) + \(M_p \to \infty\)。不需要识别增长速率。
第二层（更强）： \(\sum c_p/p < \infty\)。需要 \(\Gamma_p\) 增长快于 \(\ln\ln p\)。若 \(M_p \sim \kappa \cdot \ln\ln p\) 且 \(V(p) = O(1)\)，则 \(\Gamma_p \sim \kappa^2 \cdot (\ln\ln p)^2\)，级数收敛。
Paper XXXIV 只需要第一层。第二层是可选的更强推论。
§4 \(M_p\) 分解：98.3% 的追赶
4.1 \(\tau_p\) vs \(E[\mathrm{diff}]\)
\(M_p = \tau_p - E[\mathrm{diff}_p]\)。两边都随 \(p\) 增长：
| p band | E[τ_p] | E[diff] | E[M_p] | Δτ | Δdiff | ΔM |
|---|---|---|---|---|---|---|
| [10, 50) | 13.3 | 11.5 | 1.77 | — | — | — |
| [500, 1000) | 26.8 | 24.5 | 2.28 | +2.97 | +2.95 | +0.02 |
| [5000, 10000) | 35.8 | 33.4 | 2.39 | +3.03 | +3.03 | +0.00 |
| [20000, 50000) | 41.8 | 39.3 | 2.43 | +3.25 | +3.23 | +0.03 |
| [50000, 100000) | 44.8 | 42.3 | 2.47 | +3.04 | +3.00 | +0.04 |
总体（\(p \approx 100 \to p \approx 50{,}000\)）： \(\tau_p\) 涨了 24.19，\(E[\mathrm{diff}]\) 涨了 23.77。\(E[\mathrm{diff}]\) 以 \(\tau_p\) 的 98.3% 的速度追赶。\(M_p\) 的漂移是两个大数相减的 1.7% 残差。
4.2 含义
\(M_p\) 的慢漂移（~0.53·\(\ln\ln p\)）不是因为 \(\tau_p\) 涨得慢——它涨了 24 个单位。而是因为 \(E[\mathrm{diff}_p]\) 几乎完美地跟上了它。\(pq-1\) 的组合压缩效率随 \(p\) 的改善速率几乎和 \(\rho(p)\) 的增长一样快。微小的残差就是 \(\Delta_{\mathrm{penalty}}\)。
这对 Paper 35 的攻击方向意味着：不要试图证明 \(\tau_p\) 涨得快，而是证明 \(E[\mathrm{diff}]\) 永远追不上 \(\tau_p\)——即残差不收敛到零。
§5 坏素数：身份，解剖，与尾部分析
5.1 身份
527 个素数（5.5%）的 \(\Gamma_p < 1\)。解剖：
| 坏素数（Γ<1） | 好素数（Γ≥1） | |
|---|---|---|
| 数量 | 527 | 9,054 |
| E[ρ(p)] | 37.6 | 39.7 |
| E[τ_p] | 39.6 | 41.7 |
| E[diff] | 38.9 | 39.2 |
| E[M_p] | 0.72 | 2.53 |
| E[c_p] | 0.427 | 0.063 |
坏素数的 \(\tau_p\) 比好素数低 2.1，但 \(E[\mathrm{diff}]\) 只低 0.3。\(M_p\) 的亏损（1.81）几乎全部来自 \(\rho(p)\) 不成比例地低——即 \(p-1\) 极其 smooth。坏素数不是因为 diff 太高，而是因为 threshold 太低。
5.2 Tail Bad-Mass（公西华的诊断）
\(\delta_{\mathrm{tail}}(P, X) = \sum_{P < p \leq X,\, \Gamma_p < 1} 1/p \,/\, \sum_{P < p \leq X} 1/p\)：
| P_low | δ_tail（Γ<1） |
|---|---|
| 0（累积） | 25.4% |
| 100 | 7.6% |
| 500 | 5.6% |
| 2000 | 5.1% |
| 10000 | 5.4% |
| 50000 | 5.6% |
25.4% 去除小 p 前载后降至 ~5.5%，然后稳定。 Dyadic 分析确认：每个 \([X, 2X)\) 窗口（从 \(p=1000\) 到 \(p=100{,}000\)）的 \(\delta_{\mathrm{dyad}} \approx 5.5\%\)。
5.3 对 \(S(N)\) 的贡献
| X | S_bad | S_good | S_total | S_bad/S | c̄_h |
|---|---|---|---|---|---|
| 500 | 0.311 | 0.237 | 0.549 | 56.8% | 0.262 |
| 10000 | 0.321 | 0.264 | 0.585 | 54.9% | 0.236 |
| 99859 | 0.326 | 0.277 | 0.603 | 54.1% | 0.223 |
坏素数贡献 \(S(N)\) 的 ~54%，比例缓慢下降（57% → 54%，跨三个 decade）。
5.4 判定
坏素数的 ~5.5% tail harmonic 密度是真实的，不是纯粹的有限尺度效应。但这不构成闭合障碍——只要这些素数的 \(M_p\) 也最终 \(\to \infty\)。当前 \(E[M_p | \mathrm{bad}] \approx 0.72\) 在 \(p \leq 100{,}000\) 范围内未上升，但 Paper XXXIII 显示 \(m=240\) 的 \(M_p\) 在 \(p > 50{,}000\) 穿过 1——与 smooth 族的 \(\Delta_{\mathrm{penalty}}\) 延迟启动一致。
闭合不需要消灭坏素数。只需要它们的 \(c_p\) 也最终衰减——这从 \(M_p \to \infty\) 对 harmonic-a.e. 的 \(p\) 即可推出。
§6 \(\Delta_{\mathrm{penalty}}\)：固定参考斜率
6.1 测量
用固定 \(h_{\mathrm{ref}} = 3.60\)（不随 N 变化）：
| Decade | E[R_general] | E[R_prime] | Δ_penalty |
|---|---|---|---|
| [10³, 10⁴) | 0.933 | 1.614 | 0.682 |
| [10⁵, 10⁶) | 2.313 | 3.008 | 0.695 |
| [10⁷, 10⁸) | 3.572 | 4.271 | 0.699 |
| [10⁹, 10¹⁰) | 4.768 | 5.463 | 0.695 |
拟合：\(\Delta_{\mathrm{penalty}} = 0.635 + 0.022 \cdot \ln\ln n\)。单调增长，但极慢（七个 decade 仅涨 0.014）。
6.2 解释
素数惩罚是真实的且在增长。素数被锁在 \(\Omega=1\)，而一般整数受益于 \(\Omega(n) \sim \ln\ln n\)（Hardy–Ramanujan）的组合压缩深化。有效斜率 \(\rho(p)/\ln p\) 从 3.48 涨到 3.82，未收敛。
6.3 与 \(M_p\) 的联系
\(M_p \approx R(p) + 2 + \Delta_{\mathrm{penalty}}\)（Gemini 的分解）。\(p-1\) 极 smooth 给出的常数折扣（低 \(\rho(m)\)）是静态的。若 \(\Delta_{\mathrm{penalty}} \to \infty\)，则每个模板最终都会被超越；当前数据与此方向一致，但这仍是 Route C 的开放输入。
§7 有效斜率 \(\rho(p)/\ln p\)
| p band | E[ρ(p)] | E[ln p] | ρ/ln p |
|---|---|---|---|
| [10, 50) | 11.3 | 3.24 | 3.482 |
| [500, 1000) | 24.8 | 6.60 | 3.753 |
| [5000, 10000) | 33.8 | 8.90 | 3.798 |
| [50000, 100000) | 42.8 | 11.20 | 3.821 |
\(\rho(p)/\ln p\) 从 3.48 涨到 3.82，未收敛。素数的"单位对数成本"仍在增长——素数惩罚在 \(\rho\) 层面的体现。
§8 \(M_p \to \infty\) 的三条路线
Route B（Grok）： \(V(p) = O(1)\) 已提供二阶矩输入。待证：Lindley 指数尾 + \(M_p\) 漂移。
Route C（Gemini）： \(\Delta_{\mathrm{penalty}} \to \infty\)。物理上几乎必然（组合压缩单调增长）。当前数据：斜率 0.022/unit \(\ln\ln n\)。
Route D（公西华）： 直接证 \(M_p\) 无界。从递推不等式出发。98.3% 的追赶说明了为什么这很难——\(E[\mathrm{diff}]\) 几乎完美地跟踪 \(\tau_p\)。
§9 讨论
9.1 核心贡献
Paper XXXIV 完成了四件事：(a) 发现 \(V(p) = O(1)\) 是全局性质；(b) 建立 Harmonic Cesàro Lemma 将闭合归结为 \(M_p \to \infty\)；(c) 分解 \(M_p\) 为 98.3% 的追赶残差；(d) 刻画坏素数（5.5% tail 密度，54% 的 \(S(N)\)，极 smooth \(p-1\)）。
9.2 从 Paper XXXIII 到 XXXIV 的进步
| Paper XXXIII | Paper XXXIV | |
|---|---|---|
| 界的方向 | 下界（Margin Threshold） | 上界（Cantelli） |
| 方差 | 未测量 | V(p) ≈ 1.40，O(1) |
| 闭合条件 | M_p < 1？（精确阈值） | M_p → ∞？（定性） |
| 目标 | S(N) = o(ln ln N) | c̄_h(N) → 0（Cesàro） |
| 坏素数 | 未识别 | 5.5% tail，解剖完成 |
9.3 诚实距离估计
Ω=2 闭合：1 篇论文的距离。 需要证明 \(M_p \to \infty\) 对 harmonic-a.e. 的 \(p\)。三条路线可用。\(V(p) = O(1)\) 已确立。98.3% 的追赶和 \(\Delta_{\mathrm{penalty}}\) 的缓慢增长指向精确机制。
全局 H'：额外 2–3 篇。 \(\Omega=3+\) 扩展和层间求和。
9.4 开放问题
(1) 解析证明 \(M_p \to \infty\)。(2) 从 DP 递推证明 \(V(p) = O(1)\)。(3) 确认坏素数的 \(M_p\) 最终上升。(4) 推广到 \(\Omega \geq 3\)。(5) 将 \(\Delta_{\mathrm{penalty}} \to \infty\) 建立为定理。
§10 数据来源
| 脚本 | 测量 |
|---|---|
| rho_dp.c | ρ_E DP（C） |
| p34_var_diff.py | V_p, M_p, c_p（v1，有索引 bug） |
| p34_gamma_profile.py | Γ_p, V(p), shell partition（v2，正确） |
| p34_uniform_bounds.py | V(p) uniform bound + M_p lower bound + 坏素数 |
| p34_mp_decomp.py | M_p 分解 + Δ_penalty（固定 h_ref） |
| p34_tail_badmass.py | Tail bad-mass δ_tail(P,X) + dyadic + S(N) 贡献 |
Sanity check：\(\rho(10^7)=58\)，\(\rho(10^8)=66\)，\(\rho(10^9)=75\)。
References
[1] ZFCρ Papers I–XXXIII. H. Qin. Paper XXXIII DOI: 10.5281/zenodo.19124485. Paper XXXII DOI: 10.5281/zenodo.19116625.
[2] K. Cordwell et al. (2018). On the number of 1's needed to represent n. J. Number Theory, 189:17–34.
[3] A. Hildebrand, G. Tenenbaum (1986). On integers free of large prime factors. Trans. AMS, 296:265–290.
致谢
Claude（子路）编写了全部数值脚本，起草了 working notes v1–v3 和正文，发现了 v1 的索引 bug 和全局 \(V(p) = O(1)\) 性质，设计了 tail bad-mass 诊断。ChatGPT（公西华）贡献了 Upper-Margin Closure Lemma 和 Harmonic Cesàro Lemma（含证明），识别了 v1 的 β exponent/slope 混淆和 \(V(p)\) 对象歧义，提出了 \(\Gamma_p = M^2/V\) 框架和 disjoint shell partition，要求了 tail bad-mass 统计 \(\delta_{\mathrm{tail}}(P,X)\) 从而揭示了前载结构。Gemini（子夏）贡献了 \(\Delta_{\mathrm{penalty}}\) 的半解析论证（Route C）和坏素数的三层分析（常数折扣 vs 发散惩罚，前载错觉，Cesàro 压制）。Grok（子贡）确认了 \(V(p) = O(1)\) 比原始 Local Lipschitz 更强，验证了与 Paper XXI 的一致性，提供了 DP 自校正解释。热力学 Claude 判断"先证 Local Lipschitz"为最优策略。最终文本由作者独立完成，所有数学判断由作者负责。