Fixed-Core Margin Drift and Harmonic Closure of H' at the Ω=2 Layer
📄 DOI: 10.5281/zenodo.19124485Using ρE tables at N=10⁹ and N=10¹⁰ (operator-count convention), we perform a complete dissection of the Ω=2 layer's prime-core structure.
(1) The freeze c₂(N) ≈ 0.211 is confirmed at N=10⁹ (26 million semiprimes, 11 half-decade bins, P(j>0) fluctuating in 0.2103–0.2117). Across all 26,355,866 instances of n=2q, the maximum value of diff is 3, so P(j>0) = P(diff=3) holds exactly.
(2) E[diff] is consistent with a positive floor (c* ∈ [1.74, 1.81]), but statistically indistinguishable from extremely slow floorless decay (pure power law, γ=0.016; AIC gap 0.09). The freeze of P(diff=3) is protected by a threshold boundary effect: mass transfers unidirectionally between d=1 and d=2 without penetrating the d=3 boundary.
(3) cp(N) decreases for all 24 tested primes p ≥ 3 over 10⁸→10⁹ (24/24). c̄h(N) decreases monotonically from 0.373 (N=10³) to 0.223 (N=10¹⁰).
(4) Margin Threshold Lemma: cp(N) ≥ (1 − Mp,N)⁺ / (Bp,N + 1). Mp < 1 is the precise watershed for the lower-bound route (proving c₀ > 0).
(5) Data at N=10¹⁰ (9,588 primes, p ≤ 100,000): band-average Mp of m=240 crosses 1 at p ∈ [50000, 100000) (0.85→1.09). All tested templates show rising Mp, with diagnostic fit Mp ≈ 1.22 + 0.53·ln ln p consistent across 10⁹ and 10¹⁰. The average witness for the lower-bound route is failing.
(6) A phase-space squeeze test rules out the squeeze hypothesis and confirms Mp drift is intrinsic. E[R(n)] rises with each decade (~0.6–0.7/decade), not supporting the bounded-R premise underlying the c₀ > 0 argument.
(7) If the Lindley exponential tail holds and Mp ~ α·ln ln p (α > 0), then cp ~ (ln p)−θα and S(N) converges. The data overall leans toward H' holding at the Ω=2 layer.
Keywords: integer complexity, discrete locking, prime-core margin, harmonic Cesàro average, Margin Threshold Lemma
1. Introduction
1.1 Background
Paper XXXII reduced the H' criterion at the Ω=2 layer to the harmonic Cesàro average c̄h(N) → 0, and reported the freeze c₂ ≈ 0.211 and the per-decade decline of cp(N) for p ≥ 3. The present paper has four goals: (a) extend verification with N=10⁹ and 10¹⁰ data; (b) establish a complete drift/stability picture for the p=2 family; (c) directly measure c̄h(N) over eight decades; (d) establish the Margin Threshold Lemma and test the viability of the lower-bound route.
1.2 Methods
The ρE table is computed via a C implementation of the multiplicative forward-propagation DP. N=10⁹ runs locally (~5 min); N=10¹⁰ runs on a GCP e2-highmem-8 instance (64 GB RAM, ~38 min). All analysis scripts read the precomputed binary ρ table.
1.3 Conventions
ρE(n) counts operators (number of + and × operations): ρ(1)=0, ρ(2)=1. The recurrence is ρ(n) = min(ρ(n−1)+1, mina·b=n, a,b≥2 ρ(a)+ρ(b)+2). For n=pq (p=P⁻(n)), we define diffp(q) = ρ(pq−1) − ρ(q), τp = ρ(p)+2, and j>0 iff diffp ≥ τp. For p=2: since ρ(2)=1, the threshold is τ₂ = 3, so j>0 iff diff₂ ≥ 3.
Sanity checks: ρ(10⁷)=58, ρ(10⁸)=66, ρ(10⁹)=75.
2. Freeze Confirmation for c₂
2.1 Decade Statistics
| Decade | #primes | E[diff] | P(d=1) | P(d=2) | P(d=3) | P(d≥4) | P(j>0) |
|---|---|---|---|---|---|---|---|
| [10², 10³) | 143 | 1.881 | 0.203 | 0.532 | 0.210 | 0 | 0.210 |
| [10³, 10⁴) | 1,061 | 1.848 | 0.183 | 0.534 | 0.203 | 0 | 0.203 |
| [10⁴, 10⁵) | 8,363 | 1.844 | 0.215 | 0.504 | 0.210 | 0 | 0.210 |
| [10⁵, 10⁶) | 68,906 | 1.836 | 0.211 | 0.504 | 0.209 | 0 | 0.209 |
| [10⁶, 10⁷) | 586,081 | 1.833 | 0.214 | 0.498 | 0.212 | 0 | 0.212 |
| [10⁷, 10⁸) | 5,096,876 | 1.827 | 0.217 | 0.495 | 0.211 | 0 | 0.211 |
| [10⁸, 5×10⁸) | 20,594,412 | 1.824 | 0.217 | 0.494 | 0.211 | 0 | 0.211 |
P(d≥4) is exactly zero in every decade. Hence P(j>0) = P(diff=3) is an exact equality.
2.2 Half-Decade Evolution
Across 11 half-decade bins (log₁₀ q ∈ [2.5, 8.75]), P(j>0) fluctuates within 0.2103–0.2117. In the last three bins (21.6 million samples total), the statistical error is ~0.0001, and the inter-decade variation is of comparable magnitude.
2.3 Dual Selection Mechanism
The event j>0 is driven jointly by R(q) being unusually low (~62% contribution) and R(2q−1) being unusually high (~38%). P(2q−1 prime | j>0) = 0.131 vs P(2q−1 prime | j=0) = 0.051. The microscopic basis of the freeze is that the joint distribution of (smoothness of q−1) and (factor structure of 2q−1) is asymptotically stable on the logarithmic scale.
3. Asymptotic Behavior of E[diff]
3.1 Five-Model Fit
Twenty-one measurement points at 0.25-decade resolution (log₁₀ q ∈ [3.5, 8.75]):
| Model | c* | AIC |
|---|---|---|
| c* + A/ln(ln q) | 1.743 ± 0.008 | 14.24 |
| A·(ln q)−γ | (no floor) | 14.33 |
| c* + A/ln(q) | 1.798 ± 0.003 | 14.42 |
The best model and the floorless pure power law differ by only 0.09 in AIC — statistically indistinguishable. Whether E[diff] converges to a positive constant or decays extremely slowly to zero remains open at the current scale.
3.2 Threshold Boundary of diff=3
The mass-transfer rate from P(d=2) to P(d=1) is −0.004/decade, while the slope of P(d=3) is only +0.0002/decade (1/17th of the P(d=2) slope). The potential energy of the mean's descent is entirely absorbed by the redistribution between d=2 and d=1, and cannot penetrate the diff=3 boundary.
4. Confirmation of cp(N) Decline and Monotone Decrease of c̄h(N)
4.1 Universal Decline of cp(N)
All 24 tested primes p ∈ {3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 43, 59, 67, 89, 97, 109, 127, 131, 179, 181, 191, 193, 197, 199} show declining cp(N) over the 10⁸→10⁹ interval (24/24), with decelerating rates (|Δ(8→9)| < |Δ(7→8)| for all 24).
4.2 Eight Decades of c̄h(N)
Define S(N) = Σp≤√N cp(N)/p, D(N) = Σp≤√N 1/p, and c̄h(N) = S(N)/D(N).
| N | #primes | S(N) | D(N) | c̄h(N) |
|---|---|---|---|---|
| 10³ | 6 | 0.502 | 1.344 | 0.373 |
| 10⁴ | 21 | 0.570 | 1.757 | 0.324 |
| 10⁵ | 61 | 0.602 | 2.009 | 0.300 |
| 10⁶ | 163 | 0.606 | 2.193 | 0.276 |
| 10⁷ | 443 | 0.609 | 2.348 | 0.259 |
| 10⁸ | 1,225 | 0.608 | 2.483 | 0.245 |
| 10⁹ | 3,397 | 0.606 | 2.600 | 0.233 |
| 10¹⁰ | 9,588 | 0.603 | 2.705 | 0.223 |
Kendall τ = −1.0 (perfect monotonicity). S(N) appears to stabilize near 0.603–0.609 after 10⁶. c̄h(N) is driven downward by the growing denominator D(N) ~ ln ln N.
5. Margin Threshold Lemma
5.1 Definitions
For a prime-core p and cutoff N, let Xp,N = diffp(q) (integer-valued), τp = ρ(p)+2, Mp,N = τp − E[Xp,N] (margin), and Bp,N = sup(Xp,N − τp)⁺ (overshoot).
5.2 Lemma
$$c_p(N) \geq \frac{(1 - M_{p,N})^+}{B_{p,N} + 1}$$
Proof. On the event {X < τp}, we have X ≤ τp − 1 (integer values). On {X ≥ τp}, we have X ≤ τp + Bp,N. Therefore:
E[X] ≤ (τp − 1)(1 − cp) + (τp + Bp,N)cp = τp − 1 + (Bp,N + 1)cp.
Substituting E[X] = τp − Mp,N and rearranging gives cp ≥ (1 − Mp,N)/(Bp,N + 1). Taking the positive part completes the proof. ∎
5.3 Corollary
If on a template set of positive harmonic density there exist ε > 0 and B₀ < ∞ such that Mp,N ≤ 1 − ε and Bp,N ≤ B₀ uniformly, then lim inf c̄h(N) > 0.
5.4 Critical Remarks
(a) B2,10⁹ = 0 is an empirical fact for p=2, not a general property. For general p, diff may exceed τp (i.e., Bp,N > 0).
(b) When Mp,N ≥ 1, the lemma's lower bound is zero. Bounded margin alone does not guarantee cp > 0. Mp < 1 is the precise watershed for the lower-bound route, but closing the argument also requires either overshoot control (Bp,N ≤ B₀) or a near-threshold local law (P(Xp,N = τp) ≥ δ > 0).
6. Mp Drift and Phase-Space Squeeze Test
6.1 Overall Mp Drift
Band-averaged Mp from 3,397 primes (10⁹) and 9,588 primes (10¹⁰):
| p band | E[Mp] (10⁹) | E[Mp] (10¹⁰) |
|---|---|---|
| [50, 100) | 2.06 | 2.09 |
| [500, 1000) | 2.23 | 2.28 |
| [5000, 10000) | 2.31 | 2.39 |
| [20000, 50000) | — | 2.43 |
| [50000, 100000) | — | 2.47 |
Diagnostic fit: Mp ≈ 1.22 + 0.53·ln ln p (consistent across 10⁹ and 10¹⁰). The drift rate 0.53 is a current-scale diagnostic, not an identified asymptotic coefficient.
6.2 Phase-Space Squeeze Test
Hypothesis (Gemini): Mp drift is an artifact of q being forced smaller as p grows under the constraint pq ≤ N.
Test: Fixing qmin ∈ {1000, 10000, 100000, 500000}, the dependence of E[ΔR] on p remains unchanged (~0.58). With qmin=10000, the Mp slope increases from 0.506 to 0.633. The squeeze is not the cause; the drift is intrinsic.
6.3 Growth of E[R(n)]
Under the current normalization (h₀ = ρ(N)/ln N, which varies with N), E[R(n)] = E[ρ(n) − h₀·ln n] rises with each decade (~0.6–0.7/decade). The data does not support the hypothesis that R(n) is an O(1) stationary variable — this undermines the c₀ > 0 argument based on bounded R(n). The gap E[Rprime] − E[Rgeneral] also grows (0.01 → 0.27 across four decades), suggesting that the prime penalty expands with scale.
7. Decisive Finding at 10¹⁰: Mp Crosses 1
7.1 m=240 Crosses the Watershed
| p range | n | E[Mp] | E[cp] |
|---|---|---|---|
| [1000, 5000) | 6 | 0.85 | 0.383 |
| [5000, 20000) | 24 | 0.87 | 0.382 |
| [20000, 50000) | 42 | 0.89 | 0.379 |
| [50000, 100000) | 64 | 1.09 | 0.306 |
The band-average Mp of m=240 crosses 1 in the largest p-band. The lemma no longer gives a positive lower bound on average in this band. (A band average crossing 1 does not imply every individual p has crossed 1, but the trend is clear.)
7.2 m=720 Still Below 1 but Rising
| p range | n | E[Mp] | E[cp] |
|---|---|---|---|
| [5000, 20000) | 8 | 0.47 | 0.527 |
| [20000, 50000) | 19 | 0.67 | 0.445 |
| [50000, 100000) | 20 | 0.74 | 0.412 |
If the current rate continues, the crossing is estimated around p ~ 10⁶.
7.3 Universal Rise Across Templates
| m | E[Mp] (10⁹) | E[Mp] (10¹⁰) |
|---|---|---|
| 24 | 1.61 | 1.74 |
| 120 | 1.24 | 1.36 |
| 240 | 0.78 | 0.98 |
| 720 | 0.53 | 0.66 |
Without exception. The effective c̄(p≥500) declines from 0.099 (10⁹) to 0.090 (10¹⁰).
8. Three Analytic Routes
8.1 Route A: Margin Dichotomy (ChatGPT)
Original goal: prove Mp ≤ 1−ε and Bp,N ≤ B₀ on a positive-harmonic-density template set, yielding c₀ > 0.
Current status: The average witness is failing. m=240's band average has crossed 1; all templates show rising Mp. The static local advantage from the recursion ρ(p) ≤ ρ(m) + ρ((p−1)/m) + 3 cannot offset the system-level growth of Δpenalty (see §8.3).
8.2 Route B: Lindley Exponential Tail (Grok)
If diffp has Cramér–Lundberg exponential tails P(diffp ≥ k) ~ exp(−θk), and Mp ~ α·ln ln p (α > 0), then cp ~ (ln p)−θα, and S(N) = Σ cp/p ~ Σ 1/(p·(ln p)θα) converges for θα > 0. This gives c̄h(N) ~ L/ln ln N → 0.
This is the most likely route to H' holding at Ω=2. It requires the Local Lipschitz property from Paper XXI as input.
8.3 Route C: Asymptotic Growth of the Prime Penalty (Gemini)
Mp ≈ R(p) + 2 − E[ΔR] ≈ R(p) + 2 + Δpenalty, where Δpenalty = E[Rprime] − E[Rgeneral]. As N grows, general integers exploit an ever-expanding combinatorial compression space (Ω(n) ~ ln ln N by Hardy–Ramanujan), while primes remain locked to Ω=1. The relative cost gap Δpenalty must therefore widen, eventually pushing Mp past 1 for any fixed template.
If Δpenalty → ∞ can be proven analytically, it directly implies c₀ = 0 and H' holds.
9. Discussion
9.1 Core Contributions
Paper XXXIII achieves a three-layer progressive sharpening: (1) c̄h(N) → 0 ⟺ S(N) = o(ln ln N); (2) Mp exhibits a stable positive drift at the current scale (diagnostic fit ~0.53·ln ln p, not an identified asymptotic coefficient); (3) the band-average Mp of m=240 crosses 1 at N=10¹⁰. Together, these transform the H' determination at Ω=2 from a vague statistical trend into a precise mathematical watershed problem.
9.2 Overall Assessment for H'
The data leans overall toward H' holding at Ω=2 (c₀ = 0). Main evidence: (a) all templates show rising Mp; (b) m=240 has crossed 1; (c) Route B provides a theoretical path to S(N) convergence. However, "leans toward" is not "proves." Analytic closure requires either the Local Lipschitz property or a rigorous proof that Δpenalty → ∞.
9.3 Open Problems
(1) Systematic measurement of Bp,N (overshoot) for general templates. (2) Tracking Mp for m=720 at larger p ranges. (3) Asymptotic behavior of E[Rprime] − E[Rgeneral]. (4) Proof or disproof of Local Lipschitz. (5) Extension to Ω ≥ 3 layers.
9.4 Connection to the Series
Paper XXXI established the layer-by-layer framework (γk). Paper XXXII reduced it to the prime-core harmonic average. Paper XXXIII completes the transformation from numerical diagnostics to an algebraic watershed. The target for Paper XXXIV is the analytic derivation of Δpenalty or the proof of Local Lipschitz.
10. Data Sources and Reproducibility
| Script | Measurement |
|---|---|
| rho_dp.c | ρE DP (C, multiplicative forward propagation) |
| p33_diff_llt_n2q_v2.py | diff distribution + P(j>0) evolution |
| p33_block7_supp.py | Winning split correction + dual selection |
| p33_ediff_fit.py | E[diff] decay rate fitting |
| p33_cp_profile.py | cp(N) per-decade + c̄h(N) |
| p33_sn_largep.py / _lite.py | S(N) large-p behavior + Mp profile |
| p33_adjudicate.py | Mp drift + Gemini integral + template program |
| p33_squeeze_test.py | Phase-space squeeze test |
References
[1] ZFCρ Papers I–XXXII. H. Qin. Paper XXXII DOI: 10.5281/zenodo.19116625. Paper XXXI DOI: 10.5281/zenodo.19104860.
[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 (rho_dp.c and eight Python analysis scripts), drafted working notes v1–v4 and the formal text, discovered and corrected the multiplicative cost +2 bug and the ρE convention error, and designed and executed the phase-space squeeze test. ChatGPT (公西华) contributed the harmonic Cesàro criterion in Paper XXXII, the Margin Threshold Lemma (Bp,N version), identification of Mp < 1 as the watershed, pointed out that B=0 cannot be extrapolated from p=2 to general families, and corrected the Route B formula chain error. Gemini (子夏) proposed the asymptotic decoupling hypothesis and the phase-space squeeze hypothesis (the latter refuted by the squeeze test), and contributed the semi-analytic argument for Δpenalty driving Mp drift. Grok (子贡) provided four analytic routes via Lindley + Local Lipschitz, resolved the exponential-tail vs Gaussian-tail contradiction, and confirmed Route B's consistency with Paper XXI. The final text was independently completed by the author; all mathematical judgments are the author's responsibility.
Fixed-Core Margin 漂移与 H' 在 Ω=2 层的 Harmonic Closure
📄 DOI: 10.5281/zenodo.19124485利用 N=10⁹ 和 N=10¹⁰ 的 ρE 表（操作数约定），我们对 Ω=2 层的 prime-core 结构进行完整解剖，获得以下主要结果。
(1) c₂(N) ≈ 0.211 的冻结在 N=10⁹ 确认（2600 万个半素数，11 个半 decade bin，P(j>0) 在 0.2103–0.2117 波动）。在全部 26,355,866 个 n=2q 中，diff 的最大值为 3，P(j>0) = P(diff=3) 精确成立。
(2) E[diff] 的行为与正地板图景一致（c* ∈ [1.74, 1.81]），但与极慢衰减的无地板模型在统计上不可区分（AIC 差 0.09）。P(diff=3) 的冻结由 diff=3 的阈值边界效应保护：质量在 d=1 与 d=2 之间单向转移，不穿透 d=3 边界。
(3) cp(N) 对全部 24 个测试素数 p ≥ 3 在 10⁸→10⁹ 区间下降（24/24）。c̄h(N) 从 0.373（N=10³）单调下降到 0.223（N=10¹⁰）。
(4) Margin Threshold Lemma：cp(N) ≥ (1 − Mp,N)⁺ / (Bp,N + 1)。Mp < 1 是 lower-bound route（证明 c₀ > 0）的精确分水岭。
(5) 10¹⁰ 的数据（9,588 个素数，p ≤ 100,000）显示 m=240 的 band 平均 Mp 在 p∈[50000,100000) 穿过 1（0.85→1.09）。所有已测模板的 Mp 都在上升，Mp ≈ 1.22 + 0.53·ln ln p。lower-bound route 的平均 witness 正在失效。
(6) Phase-space squeeze test 排除挤压假说，确认 Mp 漂移是内禀的。在当前归一化（h₀ = ρ(N)/ln N）下，E[R(n)] 随 decade 持续上升，数据不支持基于 R(n) 有界的 c₀ > 0 论证。
(7) 如果 Lindley 指数尾成立且 Mp ~ α·ln ln p（α > 0），则 cp ~ (ln p)−θα，S(N) 收敛，c̄h(N) → 0。数据总体偏向 H' 在 Ω=2 成立。
关键词：加法复杂度，离散锁定，prime-core margin，harmonic Cesàro 平均，Margin Threshold Lemma
§1 引言
1.1 背景
Paper XXXII 将 H' 在 Ω=2 层的判据精确化为 harmonic Cesàro 平均 c̄h(N) → 0，报告了 c₂ ≈ 0.211 的冻结和 p ≥ 3 的 cp(N) 逐 decade 下降。本文有四个目标：(a) 利用 N=10⁹ 和 10¹⁰ 的数据扩展验证；(b) 对 p=2 族建立完整的漂移/稳定性图景；(c) 直接测量 c̄h(N) 在八个 decade 上的演化；(d) 建立 Margin Threshold Lemma 并用数据检验 lower-bound route 的可行性。
1.2 方法
ρE 表通过 C 实现的乘法前向传播 DP 计算。N=10⁹ 在本地运行（~5 分钟），N=10¹⁰ 在 GCP e2-highmem-8（64GB RAM）上运行（~38 分钟）。所有分析脚本读取预计算的二进制 ρ 表。
1.3 约定
ρE(n) = 操作数（+ 和 × 的次数），ρ(1)=0, ρ(2)=1。ρ(n) = min(ρ(n−1)+1, mina·b=n, a,b≥2 ρ(a)+ρ(b)+2)。对 n=pq（p=P⁻(n)），diffp(q) = ρ(pq−1) − ρ(q)，τp = ρ(p)+2，j>0 iff diffp ≥ τp。对 p=2：由 ρ(2)=1，阈值 τ₂ = 3。
Sanity check：ρ(10⁷)=58, ρ(10⁸)=66, ρ(10⁹)=75。
§2 c₂ 的冻结确认
2.1 Decade 统计
| Decade | #primes | E[diff] | P(d=1) | P(d=2) | P(d=3) | P(d≥4) | P(j>0) |
|---|---|---|---|---|---|---|---|
| [10², 10³) | 143 | 1.881 | 0.203 | 0.532 | 0.210 | 0 | 0.210 |
| [10³, 10⁴) | 1,061 | 1.848 | 0.183 | 0.534 | 0.203 | 0 | 0.203 |
| [10⁴, 10⁵) | 8,363 | 1.844 | 0.215 | 0.504 | 0.210 | 0 | 0.210 |
| [10⁵, 10⁶) | 68,906 | 1.836 | 0.211 | 0.504 | 0.209 | 0 | 0.209 |
| [10⁶, 10⁷) | 586,081 | 1.833 | 0.214 | 0.498 | 0.212 | 0 | 0.212 |
| [10⁷, 10⁸) | 5,096,876 | 1.827 | 0.217 | 0.495 | 0.211 | 0 | 0.211 |
| [10⁸, 5×10⁸) | 20,594,412 | 1.824 | 0.217 | 0.494 | 0.211 | 0 | 0.211 |
P(d≥4) 在所有 decade 中精确为零。P(j>0) = P(diff=3) 是精确等式。
2.2 半 Decade 精细演化
11 个半 decade bin（log₁₀ q ∈ [2.5, 8.75]），P(j>0) 在 0.2103–0.2117 波动。最后三个 bin（共 2160 万样本）的统计误差约 0.0001，P(j>0) 的 decade 间变动与噪声量级相当。
2.3 双重筛选机制
j>0 由 R(q) 偏低（经验贡献比例约 62%）和 R(2q−1) 偏高（约 38%）联合驱动。P(2q−1 为素数|j>0) = 0.131 vs P(2q−1 为素数|j=0) = 0.051。冻结的微观基础是 (q−1 的平滑度) 和 (2q−1 的因子结构) 的联合分布在对数尺度上渐近稳定。
§3 E[diff] 的渐近行为
3.1 五模型拟合
21 个 0.25-decade 测量点（log₁₀ q ∈ [3.5, 8.75]）：
| 模型 | c* | AIC |
|---|---|---|
| c* + A/ln(ln q) | 1.743 ± 0.008 | 14.24 |
| A·(ln q)−γ | （无地板） | 14.33 |
| c* + A/ln(q) | 1.798 ± 0.003 | 14.42 |
最佳模型与无地板纯幂律的 AIC 差仅 0.09，在统计上不可区分。E[diff] 的渐近行为（趋向正常数 vs 极慢衰减至零）在当前尺度下仍是开放的。
3.2 diff=3 的阈值边界
P(d=2) → P(d=1) 的质量转移速率为 −0.004/decade，但 P(d=3) 的斜率仅 +0.0002/decade（为 P(d=2) 斜率的 1/17）。均值下降的势能完全被 d=2 与 d=1 之间的重分配吸收，无法穿透 diff=3 边界。
§4 cp(N) 的 10⁹ 确认与 c̄h(N) 的单调下降
4.1 cp(N) 全部下降
24 个测试素数 p ∈ {3,5,7,11,13,23,29,31,37,41,43,59,67,89,97,109,127,131,179,181,191,193,197,199} 在 10⁸→10⁹ 全部下降（24/24），衰减率在减速（|Δ(8→9)| < |Δ(7→8)| 对全部 24 个成立）。
4.2 c̄h(N) 的八个 Decade
定义 S(N) = Σp≤√N cp(N)/p，D(N) = Σp≤√N 1/p，c̄h(N) = S(N)/D(N)。
| N | #primes | S(N) | D(N) | c̄h(N) |
|---|---|---|---|---|
| 10³ | 6 | 0.502 | 1.344 | 0.373 |
| 10⁴ | 21 | 0.570 | 1.757 | 0.324 |
| 10⁵ | 61 | 0.602 | 2.009 | 0.300 |
| 10⁶ | 163 | 0.606 | 2.193 | 0.276 |
| 10⁷ | 443 | 0.609 | 2.348 | 0.259 |
| 10⁸ | 1,225 | 0.608 | 2.483 | 0.245 |
| 10⁹ | 3,397 | 0.606 | 2.600 | 0.233 |
| 10¹⁰ | 9,588 | 0.603 | 2.705 | 0.223 |
Kendall τ = −1.0（完美单调）。S(N) 在 10⁶ 后表观稳定于 0.603–0.609。c̄h(N) 被不断增长的分母 D(N) ~ ln ln N 压低。
§5 Margin Threshold Lemma
5.1 定义
对 prime-core p 和 cutoff N，设 Xp,N = diffp(q)（整数值），τp = ρ(p)+2，Mp,N = τp − E[Xp,N]（margin），Bp,N = sup(Xp,N − τp)⁺（overshoot）。
5.2 引理
$$c_p(N) \geq \frac{(1 - M_{p,N})^+}{B_{p,N} + 1}$$
证明。在 {X < τp} 上 X ≤ τp − 1，在 {X ≥ τp} 上 X ≤ τp + Bp,N。因此：
E[X] ≤ (τp − 1)(1 − cp) + (τp + Bp,N)cp = τp − 1 + (Bp,N + 1)cp。
代入 E[X] = τp − Mp,N 并移项即得。∎
5.3 推论
若在正 harmonic 密度的模板集上 Mp,N ≤ 1 − ε 且 Bp,N ≤ B₀ 统一成立，则 lim inf c̄h(N) > 0。
5.4 关键注记
(a) B2,10⁹ = 0 对 p=2（N=10⁹）是经验事实，非一般通性。
(b) 当 Mp,N ≥ 1 时引理的下界为零。Mp < 1 是 lower-bound route 的精确分水岭，但还需 overshoot 控制或 near-threshold local law 才能闭合。
§6 Mp 漂移与 Phase-Space Squeeze Test
6.1 Mp 的整体漂移
3,397 个素数（10⁹）和 9,588 个素数（10¹⁰）的 Mp 按 p-band 统计：
| p band | E[Mp] (10⁹) | E[Mp] (10¹⁰) |
|---|---|---|
| [50, 100) | 2.06 | 2.09 |
| [500, 1000) | 2.23 | 2.28 |
| [5000, 10000) | 2.31 | 2.39 |
| [20000, 32000) | 2.38 | — |
| [20000, 50000) | — | 2.43 |
| [50000, 100000) | — | 2.47 |
拟合 Mp ≈ 1.22 + 0.53·ln ln p（10⁹ 和 10¹⁰ 一致）。Mp 漂移率约 0.53 per ln ln p 是 current-scale diagnostic，非渐近系数。
6.2 Phase-Space Squeeze Test
Gemini 假说：Mp 漂移来自 pq ≤ N 迫使 q 缩小的挤压效应。
检验：控制 qmin ∈ {1000, 10000, 100000, 500000}，E[ΔR] 对 p 的依赖性不变（~0.58）。控制 qmin=10000 后 Mp 斜率从 0.506 上升到 0.633。挤压不是主因，漂移是内禀的。
6.3 E[R(n)] 的增长
在当前归一化下（h₀ = ρ(N)/ln N 随 N 变化），E[R(n)] = E[ρ(n) − h₀·ln n] 随 decade 持续上升（~0.6–0.7/decade）。由于 h₀ 本身依赖 cutoff N，跨 decade 的 R(n) 不是同一个 residual 的比较。但在每个固定 N 下，数据不支持 R(n) 为 O(1) 平稳变量的假设——这削弱了 Gemini 基于 R(n) 有界的 c₀ > 0 论证。E[Rprime] − E[Rgeneral] 的差也在增长（0.01 → 0.27 across four decades），暗示素数惩罚随尺度扩大。
§7 10¹⁰ 的决定性发现：Mp 穿过 1
7.1 m=240 穿过分水岭
| p range | n | E[Mp] | E[cp] |
|---|---|---|---|
| [1000, 5000) | 6 | 0.85 | 0.383 |
| [5000, 20000) | 24 | 0.87 | 0.382 |
| [20000, 50000) | 42 | 0.89 | 0.379 |
| [50000, 100000) | 64 | 1.09 | 0.306 |
m=240 的 band 平均 Mp 在最大 p-band 穿过 1。引理在该 band 的平均意义下不再给出正下界。注意：band 平均穿过 1 不等于该 band 内每个 p 都穿过 1。但趋势方向明确。
7.2 m=720 仍 < 1 但上升中
| p range | n | E[Mp] | E[cp] |
|---|---|---|---|
| [5000, 20000) | 8 | 0.47 | 0.527 |
| [20000, 50000) | 19 | 0.67 | 0.445 |
| [50000, 100000) | 20 | 0.74 | 0.412 |
如果上升速率维持，预计在 p ~ 10⁶ 穿过 1。
7.3 所有模板都在上升
| m | E[Mp] (10⁹) | E[Mp] (10¹⁰) |
|---|---|---|
| 24 | 1.61 | 1.74 |
| 120 | 1.24 | 1.36 |
| 240 | 0.78 | 0.98 |
| 720 | 0.53 | 0.66 |
无一例外。effective c̄(p≥500) 从 0.099（10⁹）降到 0.090（10¹⁰）。
§8 三条解析路线
8.1 Route A: Margin Dichotomy（公西华）
原始目标：在正 harmonic 密度模板上证明 Mp ≤ 1−ε + Bp,N ≤ B₀ → c₀ > 0。
当前状态：平均 witness 正在失效。 m=240 的 band 平均穿过 1，所有模板 Mp 在上升。递推不等式 ρ(p) ≤ ρ(m) + ρ((p−1)/m) + 3 给出的局部优势是静态的，无法对抗 Δpenalty 的系统级增长（见 §8.3）。
8.2 Route B: Lindley 指数尾（Grok）
如果 diffp 有 Cramér–Lundberg 指数尾 P(diffp ≥ k) ~ exp(−θk)，且 Mp ~ α·ln ln p（α > 0），则 cp ~ (ln p)−θα，S(N) = Σ cp/p ~ Σ 1/(p·(ln p)θα) 在 θα > 0 时收敛。c̄h(N) ~ L/ln ln N → 0。
这是 H' 在 Ω=2 成立的最可能路线。需要 Paper XXI 的 Local Lipschitz 作为输入。
8.3 Route C: 素数惩罚的渐近增长（Gemini）
Mp ≈ R(p) + 2 − E[ΔR] ≈ R(p) + 2 + Δpenalty，其中 Δpenalty = E[Rprime] − E[Rgeneral]。Δpenalty 随尺度增长（一般整数的组合压缩红利越来越大，素数不能参与）。这解释了为什么即使极 smooth 模板也无法永久抵抗 Mp 的正漂移。
如果 Δpenalty → ∞ 可以被解析证明，则 Mp → ∞，c₀ = 0，H' 成立。
§9 讨论
9.1 核心贡献
Paper XXXIII 的贡献是三层递进的问题缩小：(1) c̄h(N) → 0 ⟺ S(N) = o(ln ln N)；(2) Mp 在当前尺度有稳定的正漂移（diagnostic fit ~0.53·ln ln p，非已识别的渐近系数）；(3) 极 smooth 模板（m=240）的 band 平均 Mp 在 10¹⁰ 穿过 1。三层合在一起，将 H' 在 Ω=2 的判定从一个模糊的统计趋势变为一个精确的数学分水岭问题。
9.2 对 H' 的总体判断
数据总体偏向 H' 在 Ω=2 成立（c₀ = 0）。主要依据：(a) 所有模板 Mp 都在上升；(b) m=240 已穿过 1；(c) Route B 给出 S(N) 收敛的理论路径。但"偏向"不是"证明"。解析闭合需要 Local Lipschitz 或 Δpenalty → ∞ 的严格论证。
9.3 开放问题
(1) Bp,N（overshoot）的系统测量。(2) m=720 在更大 p 范围的 Mp 追踪。(3) E[Rprime] − E[Rgeneral] 的渐近行为。(4) Local Lipschitz 的证明或否定。(5) Ω=3+ 层的 c̄h 行为。
9.4 与系列的衔接
Paper XXXI 建立了逐层框架（γk），Paper XXXII 将其精确化为 prime-core harmonic 平均。Paper XXXIII 完成了从数值诊断到代数分水岭的转化。下一步（Paper XXXIV）的靶心是 Δpenalty 的解析推导或 Local Lipschitz 的证明。
§10 数据来源与可复现性
| 脚本 | 测量 |
|---|---|
| rho_dp.c | ρE DP（C，乘法前向传播） |
| p33_diff_llt_n2q_v2.py | diff 分布 + P(j>0) 演化 |
| p33_block7_supp.py | Winning split 修正 + 双重筛选 |
| p33_ediff_fit.py | E[diff] 衰减率拟合 |
| p33_cp_profile.py | cp(N) per-decade + c̄h(N) |
| p33_sn_largep.py / _lite.py | S(N) 大 p 行为 + Mp profile |
| p33_adjudicate.py | Mp 漂移 + Gemini 积分 + template |
| p33_squeeze_test.py | Phase-space squeeze test |
References
[1] ZFCρ Papers I–XXXII. H. Qin. Paper XXXII DOI: 10.5281/zenodo.19116625. Paper XXXI DOI: 10.5281/zenodo.19104860.
[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.
致谢
Claude（子路）编写了全部数值计算脚本（rho_dp.c 及八个 Python 分析脚本），起草了 working notes v1–v4 和正文初稿，发现并修正了乘法 cost +2 bug 和 ρE 约定错误，设计并执行了 phase-space squeeze test。ChatGPT（公西华）贡献了 Paper XXXII 的 harmonic Cesàro 判据，贡献了 Margin Threshold Lemma（Bp,N 版本）和 Mp < 1 的分水岭识别，指出 B=0 不可从 p=2 外推到一般 family，以及 near-threshold local law 的必要性，修正了 Route B 的公式链错误。Gemini（子夏）提出了渐近解耦假说和相空间挤压假说（后者被 squeeze test 否定），贡献了 Δpenalty = E[Rprime] − E[Rgeneral] 的半解析论证。Grok（子贡）提供了 Lindley + Local Lipschitz 的四条解析路线，给出了指数尾 vs 高斯尾的矛盾化解，确认了 Route B 与 Paper XXI 的一致性。最终文本由作者独立完成，所有数学判断由作者负责。