The 1/φ(m) Local Model, Explanation Rate Plateau, and Global DP Path Effects
DOI: 10.5281/zenodo.19153002Paper XXXV confirmed that predecessor advantage erosion originates from the shifted multiplicative structure (B-C dominant, D-C ≈ 0). This paper precisely quantifies the shifted-prime local model through four progressive experiments and locates the true source of the B-C bias.
(1) \(P(m \mid pq-1) = 1/\varphi(m)\) holds exactly for all smooth moduli (joint distribution), invariant across all \(p\). Not just marginals — the joint valuation vector \((v_2, v_3, v_5, v_7)\) follows the shifted-prime local model exactly.
(2) \(v_2(pq-1) = 2.000\) exactly. Correct derivation: \(\Pr(v_2 \geq k) = 1/\varphi(2^k) = 1/2^{k-1}\), hence \(E[v_2] = 2\). Extra expected valuation formula: \(\sum_\ell 1/(\ell-1)^2 \approx 1.375\) (full prime sum), with \(\ell = 2, 3\) contributing ~91%.
(3) Bias/Δ ratio crosses zero: \((B\text{-}C)/\Delta\) from \(-1.48\) to \(+0.35\).
(4) Controlling \(v_2, v_3\) reduces B-C by only 18–21%. Weighted contribution explains only 44% of \(|B\text{-}C|\). \(\rho\)'s response to the small-prime profile is highly nonlinear.
(5) Explanation rate vs \(y\) curve plateaus completely: from controlling \(v_2\) (19.2%) to controlling \(v_2, \ldots, v_{13}\) (19.5%) — virtually no increase. Smooth kernel \(K_y\) matching confirms the same. Within the tested range (\(y \leq 13\)), small-prime valuation conditioning explains only ~20% of B-C, with no further gains as \(y\) increases.
(6) Cofactor test: after matching \(K_7\), cofactor \(\rho\) difference remains 0.49–0.65 (significant). ~80% of B-C comes from cofactor's global DP path differences.
(7) Multi-scale cofactor gap follows the same erosion-and-reversal trajectory as B-C. Cofactor accounts for 80–90% of B-C at \(p < 20000\). The global path effect is not "residual" — it IS the body of B-C.
Keywords: 1/φ(m) local model, explanation rate plateau, cofactor test, global DP path effect, fixed-y localization failure
§1 Introduction
1.1 Background
Paper XXXV (DOI: 10.5281/zenodo.19143732): 16-band erosion and reversal (gap from −2.089 to +1.451), three-way bias decomposition (D-C ≈ 0, B-C dominant 70–74%, A-B secondary 26–30%), shifted-prime local model framework.
1.2 Notation
Following Paper XXXV's three-way decomposition, for a given prime \(p\) we define four ensemble means:
- A = \(E[\rho(pq-1)]\) (\(q\) prime — actual diff predecessor)
- B = \(E[\rho(pr-1)]\) (\(r\) random odd — shifted-composite model, i.e., \(\mu_{\mathrm{aff}}\))
- C = \(E[\rho(\text{random } n)]\) (same-scale general integer, i.e., \(\mu_{\mathrm{general}}\))
- D = \(E[\rho(n \equiv -1 \bmod p)]\) (exact residue class)
Total gap \(= A\text{-}C = (A\text{-}B) + (B\text{-}C)\), where A-B is the prime-source bias and B-C is the residue-class bias (shifted structure bias).
1.3 Four Progressive Layers
Layer 1 (Block 1): 1/φ(m) joint profile verification + bias/Δ ratio decline.
Layer 2 (Block 2): \(v_2, v_3\) control test — discovers nonlinear \(\rho\) response.
Layer 3 (Block 3): Explanation rate vs \(y\) curve + cofactor test — confirms "truly global."
Layer 4 (Block 4): Multi-scale cofactor gap — confirms synchronous erosion trajectory.
§2 Exact Verification of the 1/φ(m) Joint Profile
2.1 Algebraic Foundation
For \(\ell \neq p\) and smooth modulus \(m\) (all prime factors \(\neq p\)):
$$m \mid (pq-1) \iff q \equiv p^{-1} \pmod{m}$$
Primes \(q\) are asymptotically equidistributed among the \(\varphi(m)\) reduced residue classes mod \(m\) (Dirichlet / Siegel-Walfisz), hence \(P(m \mid pq-1) = 1/\varphi(m) + o(1)\).
2.2 Marginal Distribution
\(P(\ell \mid pq-1) = 1/(\ell-1)\) to four significant figures, for \(\ell = 2, 3, 5, 7, 11, 13\) and all \(p\) (101–70001):
| ℓ | P(ℓ | pq-1) | P(ℓ | random) | 1/ℓ | 1/φ(ℓ) |
|---|---|---|---|---|
| 2 | 1.0000 | 0.5000 | 0.5000 | 1.0000 |
| 3 | 0.5000 | 0.3336 | 0.3333 | 0.5000 |
| 5 | 0.2500 | 0.2000 | 0.2000 | 0.2500 |
| 7 | 0.1667 | 0.1432 | 0.1429 | 0.1667 |
| 11 | 0.1000 | 0.0909 | 0.0909 | 0.1000 |
| 13 | 0.0833 | 0.0769 | 0.0769 | 0.0833 |
2.3 Joint Distribution
\(P(m \mid pq-1) / (1/\varphi(m))\) ranges from 0.997 to 1.005 for all smooth moduli. The joint valuation vector distribution follows the 1/φ(m) local model exactly.
| m | Description | Ratio (p=1009) | Ratio (p=50021) |
|---|---|---|---|
| 4 | 2² | 1.000 | 1.000 |
| 12 | 2²·3 | 0.999 | 1.002 |
| 30 | 2·3·5 | 0.999 | 1.002 |
| 60 | 2²·3·5 | 0.998 | 1.007 |
| 120 | 2³·3·5 | 0.994 | 0.997 |
| 210 | 2·3·5·7 | 0.998 | 0.975 |
§3 \(v_2(pq-1) = 2.000\) and the Extra Valuation Formula
3.1 Correct Derivation
\(2^k \mid (pq-1) \iff q \equiv p^{-1} \pmod{2^k}\). Odd primes \(q\) are equidistributed among the \(\varphi(2^k) = 2^{k-1}\) odd residue classes mod \(2^k\). Therefore:
$$\Pr(v_2(pq-1) \geq k) = \frac{1}{2^{k-1}}$$
$$E[v_2(pq-1)] = \sum_{k \geq 1} \frac{1}{2^{k-1}} = 2.000$$
Data: \(E[v_2(pq-1)] = 2.000\) (all \(p\)), \(E[v_2(\text{random})] = 1.000\). Difference = +1.000.
3.2 Extra Expected Valuation
For any prime \(\ell\):
$$\sum_{k \geq 1} \left(\frac{1}{\varphi(\ell^k)} - \frac{1}{\ell^k}\right) = \frac{1}{(\ell-1)^2}$$
Total extra valuation (full prime sum) \(\approx 1.375\). \(\ell = 2\) contributes 1.000, \(\ell = 3\) contributes 0.250, together ~91%.
§4 Bias/Δ Ratio Decline and Zero Crossing
Define \(\Delta = \Delta_{\mathrm{penalty}}\) (measured at \(pq\) scale, ≈ 0.706). Data from Block 1 (independent sampling).
| p | A-C | B-C | A-B | (A-C)/Δ | (B-C)/Δ | (A-B)/Δ |
|---|---|---|---|---|---|---|
| 29 | −1.42 | −1.05 | −0.37 | −2.01 | −1.48 | −0.52 |
| 503 | −1.46 | −1.03 | −0.43 | −2.06 | −1.45 | −0.61 |
| 5003 | −1.40 | −0.96 | −0.44 | −1.99 | −1.36 | −0.62 |
| 20011 | −1.00 | −0.68 | −0.32 | −1.42 | −0.96 | −0.46 |
| 50021 | −0.39 | −0.22 | −0.17 | −0.55 | −0.31 | −0.24 |
| 70001 | +0.08 | +0.25 | −0.17 | +0.11 | +0.35 | −0.24 |
\((B\text{-}C)/\Delta\) crosses zero from −1.48 to +0.35. Numerical evidence for Step 2b (\(\mu_{\mathrm{shift}} - \mu_{\mathrm{general}} = o(\Delta_{\mathrm{penalty}})\)) continues to strengthen.
§5 Nonlinear ρ Response: \(v_2, v_3\) Control Test
5.1 Results
| p | B-C uncond | B-C | v₂=2, v₃=0 | Reduction |
|---|---|---|---|
| 1009 | −0.819 | −0.669 | 18% |
| 10007 | −0.646 | −0.510 | 21% |
| 50021 | +0.177 | +0.311 | −76% |
5.2 Weighted Contribution
Define \(\lambda_{\ell,k} = E[\rho \mid v_\ell \geq k-1] - E[\rho \mid v_\ell \geq k]\) (ρ-saving from one extra level of \(\ell\)-adic valuation, positive). \(\Delta\mathrm{prob} = 1/\varphi(\ell^k) - 1/\ell^k\) (extra divisibility probability, positive). \(\lambda \times \Delta\mathrm{prob}\) measures each \((\ell, k)\)'s contribution to \(|B\text{-}C|\) under a linear additive model:
| (ℓ, k) | λ | Δprob | λ × Δprob |
|---|---|---|---|
| (2, 1) | 0.154 | +0.500 | +0.077 |
| (2, 2) | 0.290 | +0.250 | +0.073 |
| (3, 1) | 0.337 | +0.167 | +0.056 |
| (5, 1) | 0.363 | +0.050 | +0.018 |
| Total | +0.281 |
Total weighted contribution = 0.281, while \(|B\text{-}C| = 0.646\) (\(p = 10007\)). Linear weighting explains only 44% of \(|B\text{-}C|\). This shows \(\rho\)'s response to the small-prime profile is not additive — interaction effects and global path selection contribute the majority.
5.3 Interpretation
Raw probability gap is 91% from \(\ell = 2, 3\), but \(\rho\)-weighted gap is only ~20% (amount eliminated by conditioning). \(\rho\)'s response to the small-prime profile is highly nonlinear.
§6 Explanation Rate vs \(y\): The Decisive Experiment
6.1 Core Results
Progressively expand control variables and measure B-C explanation rate.
p = 1009:
| y | Controlled | B-C | Z_y | expl% |
|---|---|---|---|
| 2 | v₂ | −0.661 | 19.2% |
| 3 | v₂, v₃ | −0.661 | 19.3% |
| 5 | v₂, v₃, v₅ | −0.661 | 19.3% |
| 7 | v₂, …, v₇ | −0.660 | 19.4% |
| 13 | v₂, …, v₁₃ | −0.659 | 19.5% |
p = 10007: from 23.7% (y=2) to 24.0% (y=13).
Explanation rate from y=2 to y=13 is virtually flat (increase < 0.5%).
6.2 Smooth Kernel \(K_y\) Matching Confirms
| y | K_y levels (p=1009) | B-C | K_y | expl% |
|---|---|---|---|
| 2 | 20 | −0.661 | 19.2% |
| 7 | 1110 | −0.660 | 19.4% |
| 13 | 3369 | −0.659 | 19.5% |
Exact smooth kernel matching also plateaus at 19–24%.
6.3 Verdict
Within the tested range (\(y \leq 13\)), small-prime valuation conditioning explains only ~20% of B-C, with no gains as \(y\) increases. The remaining ~80% comes from structure beyond the smooth kernel. The possibility of slowly growing \(y(X) \to \infty\) cannot yet be ruled out, but the fixed-\(y\) localization approach has failed in current data.
§7 Cofactor Test: Direct Evidence for Global Path Effects
7.1 Design
After matching \(K_7\) (controlling all valuations for 2, 3, 5, 7), compare cofactor \(\rho = \rho(n/K_7)\).
7.2 Results
| p | E[ρ(cofactor_pr) − ρ(cofactor_rand)] | Significant |
|---|---|---|
| 1009 | −0.649 | Yes |
| 10007 | −0.487 | Yes |
After matching the smooth kernel, cofactor \(\rho\) still differs by 0.49–0.65.
7.3 Implications
The advantage of \(pr-1\) is not in its small factors — it is in its large factors. Even when the small-factor part is identical, \(pr-1\)'s cofactor is still easier to compress than a general integer's. "Multiply by \(p\) then subtract 1" affects not just the small-prime profile, but the global topology of the factorization network.
§8 Multi-Scale Cofactor Gap: Synchronous Trajectory with B-C
8.1 Cofactor Gap (Block 4, independent sampling)
| p | B-C | cof gap | cof/B-C |
|---|---|---|---|
| 53 | −0.97 | −0.80 | 82% |
| 211 | −0.96 | −0.85 | 88% |
| 1009 | −0.96 | −0.76 | 80% |
| 5003 | −0.92 | −0.78 | 85% |
| 10007 | −0.82 | −0.67 | 82% |
| 20011 | −0.59 | −0.40 | 67% |
| 50021 | +0.14 | +0.28 | — |
| 70001 | +0.81 | +0.92 | — |
8.2 Key Observations
(a) Cofactor gap and B-C follow the same erosion-and-reversal trajectory. Both cross zero synchronously. The cofactor gap is not "residual" — it IS the body of B-C.
(b) cof/B-C is stable at 80–90% for \(p < 20000\), consistent with §6's explanation rate (~20% local).
(c) Both cross zero synchronously at \(p > 50000\). The global path effect is also decaying.
8.3 Full Three-Way Decomposition (Block 4, representative p)
| p | A-C | A-B | B-C | %prime | %resid |
|---|---|---|---|---|---|
| 53 | −1.23 | −0.38 | −0.84 | 31% | 69% |
| 503 | −1.25 | −0.43 | −0.82 | 34% | 66% |
| 5003 | −1.21 | −0.43 | −0.78 | 36% | 64% |
| 10007 | −1.08 | −0.41 | −0.67 | 38% | 62% |
| 20011 | −0.80 | −0.35 | −0.45 | 44% | 56% |
| 50021 | −0.05 | −0.20 | +0.15 | — | — |
| 70001 | +0.67 | −0.15 | +0.82 | — | — |
Prime-source share rises from 31% to 44%, then inverts at the zero-crossing.
§9 Complete Picture of B-C and Route C Repositioning
9.1 Three-Layer Decomposition
| Layer | Share | Mechanism | Behavior with p |
|---|---|---|---|
| Small-prime profile (1/φ vs 1/ℓ) | ~20% | Extra v₂, v₃ divisibility | Profile fixed, leverage decays |
| Cofactor global path | ~80% | DP optimal path differences | Synchronous erosion, crosses zero |
| Total B-C | 100% | Shifted multiplicative structure | Erosion and reversal |
9.2 Why the Global Effect Also Decays (Heuristic Interpretations)
The following are heuristic, awaiting rigorous formalization:
(a) As \(p\) grows, \(pq\) becomes enormous; the cofactor's factor space is vast, and the shifted structure's path advantage is statistically diluted.
(b) DP optimal paths may approach a form of "universal uniformity" at large \(N\) (Gemini's "Markov barrier" hypothesis: shifted memory persists for only finite depth; deep subtrees revert to general-integer statistics).
(c) Thermodynamic analogy: in a sufficiently large system, the memory of the local generation mechanism is washed out by the ever-growing combinatorial complexity.
9.3 Route C Step 2b Repositioned
Old strategy (failed in current data): Use finite-dimensional small-prime profile differences (fixed-\(y\) localization) to prove \(B\text{-}C = O(1)\). The explanation rate plateau shows this route covers only ~20%.
New strategy: Directly prove that the matched-scale affine-ensemble bias is bounded. Define \(\mu_{\mathrm{aff}}(X; p) := E[\rho(pr-1)]\) (\(r\) random odd), \(\mu_{\mathrm{gen}}(X) := E[\rho(n)]\) (same-scale general integers, \(X \sim pr\)).
Step 2b new target: Prove \(\mu_{\mathrm{aff}}(X; p) - \mu_{\mathrm{gen}}(X) = O(1)\), or at least \(o(\Delta_{\mathrm{penalty}}(X))\).
This approach does not require decomposing B-C's microscopic sources — it bypasses the fixed-\(y\) localization failure. What must be shown is: DP's average memory of the shifted vs general ensemble does not grow without bound.
Numerical support: \((B\text{-}C)/\Delta\) crosses zero; cofactor gap erodes synchronously with B-C; global path effect accounts for ~80% of B-C but also decays. These support the boundedness conjecture but do not yet constitute proof.
§10 Discussion
10.1 Core Contributions
(a) 1/φ(m) joint profile — exact verification. (b) \(v_2 = 2.000\) — correct algebraic derivation. (c) Extra valuation formula \(\sum 1/(\ell-1)^2 \approx 1.375\). (d) Nonlinear \(\rho\) response discovery (\(v_2, v_3\) control eliminates only 18–21%). (e) Explanation rate vs \(y\) plateau — small-prime explains only ~20%. (f) Cofactor test confirms global DP path effect (~80%). (g) Multi-scale cofactor gap synchronous with B-C erosion. (h) Route C strategic repositioning.
10.2 Progress Table
| XXXIII | XXXIV | XXXV | XXXVI | |
|---|---|---|---|---|
| Bound | Lower | Upper | — | — |
| V(p) | — | O(1) | — | — |
| Closure condition | M_p < 1? | M_p → ∞? | Non-plateauing | — |
| Mechanism | Drift | 98.3% chase | Advantage reversal | 1/φ(m) + global path |
| Depth | Threshold | Qualitative div. | Bias decomposition | Fixed-y localization failure |
| B-C source | — | — | Shifted structure | 20% local + 80% global |
10.3 Distance Estimate
Paper XXXVI reveals that B-C's body is a global DP path effect, requiring a new approach for Step 2b — not fixed-\(y\) localization, but directly proving that the matched-scale affine-ensemble bias (\(\mu_{\mathrm{aff}} - \mu_{\mathrm{gen}}\)) is bounded or \(o(\Delta_{\mathrm{penalty}})\). Ω=2 closure may require 2 papers: one establishing matched-scale affine bias boundedness, one closing with \(\Delta_{\mathrm{penalty}} \to \infty\).
10.4 Open Problems
(1) Why does the cofactor \(\rho\) difference also decay with \(p\)? Does this point to "large-number universal uniformity" of DP paths?
(2) Does \(\rho(n)\)'s sensitivity to generation mechanism decay as \(o(\ln n)\)?
(3) Can the matched-scale affine-ensemble bias \(\mu_{\mathrm{aff}}(X;p) - \mu_{\mathrm{gen}}(X) = O(1)\) be proved directly, bypassing B-C's microscopic decomposition?
(4) What are the systematic topological differences between DP optimal paths for shifted vs general integers?
§11 Data Sources
| Script | Measurement |
|---|---|
| p36_bias_ratios.py | Bias/Δ, small-prime profile, v₂ distribution |
| p36_joint_profile.py | Joint profile, weighted contribution, v₂v₃ control |
| p36_explanation_curve.py | Explanation rate vs y, smooth kernel matching, cofactor test |
| p36_cofactor_decay.py | Multi-scale cofactor gap, three-way decomposition |
Sanity check: ρ(10⁷)=58, ρ(10⁸)=66, ρ(10⁹)=75.
References
[1] ZFCρ Papers I–XXXV. Paper XXXV DOI: 10.5281/zenodo.19143732. Paper XXXIV DOI: 10.5281/zenodo.19140015. Paper XXXIII DOI: 10.5281/zenodo.19124485.
[2] K. Cordwell et al. (2018). J. Number Theory, 189:17–34.
[3] A. Hildebrand, G. Tenenbaum (1986). Trans. AMS, 296:265–290.
Acknowledgments
Claude (子路/Zilu) wrote all numerical scripts, designed the explanation rate vs y curve and cofactor test, and drafted working notes v1–v3 and the formal text. ChatGPT (公西华/Gongxihua) suggested the bias/Δ ratio measurement, corrected the v₂ derivation error, proposed the v₂v₃ killer test and the "explanation rate vs y" decisive experiment, identified the remaining 80% as likely global DP path effects, and proposed the "directly prove matched-scale affine bias boundedness" strategy. Gemini (子夏/Zixia) contributed the semi-analytic framework (later corrected by the killer test), proposed the "Markov barrier" and "shortcut chain reaction" interpretations, and the shift from local profile to friable core. Grok (子贡/Zigong) confirmed the nonlinear response is compatible with Paper 18–19's linear self-correction (different levels), and that cofactor decay is consistent with Lindley's stationary distribution. The final text was independently completed by the author; all mathematical judgments are the author's responsibility.
Paper XXXV 确认前驱优势的侵蚀与反转来自 shifted 乘法结构（B-C 主导，D-C ≈ 0）。本文通过四轮递进实验精确量化 shifted-prime local model，并定位 B-C 偏差的真实来源。
(1) \(P(m \mid pq-1) = 1/\varphi(m)\) 对所有 smooth moduli（联合分布）精确成立，对所有 \(p\) 不变。不只是边缘分布——联合 valuation vector \((v_2, v_3, v_5, v_7)\) 的分布精确遵循 shifted-prime local model。
(2) \(v_2(pq-1) = 2.000\) 精确成立。正确推导：\(\Pr(v_2 \geq k) = 1/\varphi(2^k) = 1/2^{k-1}\)，故 \(E[v_2] = 2\)。额外 expected valuation 公式：\(\sum_\ell 1/(\ell-1)^2 \approx 1.375\)，其中 \(\ell=2,3\) 贡献约 91%。
(3) Bias/Δ 比值穿零：\((B\text{-}C)/\Delta\) 从 \(-1.48\) 降到 \(+0.35\)。
(4) \(v_2, v_3\) 控制后 B-C 仅减 18–21%。Weighted contribution 仅解释 44%。\(\rho\) 对 small-prime profile 的响应高度非线性。
(5) 解释率 vs \(y\) 曲线完全平台化：从控制 \(v_2\)（19.2%）到控制 \(v_2, \ldots, v_{13}\)（19.5%）几乎不动。Smooth kernel \(K_y\) 精确匹配确认同样结果。在当前测试范围（\(y \leq 13\)）内，small-prime valuation conditioning 只能解释 ~20% 的 B-C，且随 \(y\) 不再增长。
(6) Cofactor 测试：匹配 \(K_7\) 后，cofactor \(\rho\) 差仍 0.49–0.65（显著）。B-C 的主体（~80%）来自 cofactor 的全局 DP 路径差异。
(7) 11 个 \(p\) 尺度的 cofactor gap 走与 B-C 完全相同的侵蚀-穿零轨迹。Cofactor 占 B-C 的 80–90%（\(p < 20000\)）。全局路径效应不是 B-C 的"额外"部分——它就是 B-C 的主体。
关键词：1/φ(m) local model，解释率平台化，cofactor 测试，全局 DP 路径效应，fixed-y 局部化失效
§1 引言
1.1 背景
Paper XXXV (DOI: 10.5281/zenodo.19143732)：16-band 侵蚀与反转（gap 从 −2.089 到 +1.451），三路偏差分解（D-C ≈ 0，B-C 主导 70–74%，A-B 次要 26–30%），shifted-prime local model 框架。
1.2 记号约定
本文沿用 Paper XXXV 的三路偏差分解。对给定的素数 \(p\)，定义四个 ensemble mean：
- A = \(E[\rho(pq-1)]\)（\(q\) 为素数——实际的 diff predecessor）
- B = \(E[\rho(pr-1)]\)（\(r\) 为随机奇数——shifted-composite model，即 \(\mu_{\mathrm{aff}}\)）
- C = \(E[\rho(\text{random } n)]\)（同尺度一般整数，即 \(\mu_{\mathrm{general}}\)）
- D = \(E[\rho(n \equiv -1 \bmod p)]\)（精确同余类）
Total gap \(= A\text{-}C = (A\text{-}B) + (B\text{-}C)\)，其中 A-B 是 prime-source bias，B-C 是 residue-class bias（shifted 结构偏差）。
1.3 四层递进
Layer 1（Block 1）：1/φ(m) 联合 profile 验证 + bias/Δ 比值衰减。
Layer 2（Block 2）：\(v_2, v_3\) 控制测试——发现 \(\rho\) 的非线性响应。
Layer 3（Block 3）：解释率 vs \(y\) 曲线 + cofactor 测试——确认"truly global"。
Layer 4（Block 4）：多 \(p\) 尺度 cofactor gap——确认同步侵蚀轨迹。
§2 1/φ(m) 联合 Profile 的精确验证
2.1 代数基础
对 \(\ell \neq p\) 和 smooth modulus \(m\)（所有素因子 \(\neq p\)）：
$$m \mid (pq-1) \iff q \equiv p^{-1} \pmod{m}$$
素数 \(q\) 在 mod \(m\) 的 \(\varphi(m)\) 个 reduced residue classes 中均匀分布（Dirichlet / Siegel-Walfisz），因此 \(P(m \mid pq-1) = 1/\varphi(m) + o(1)\)。
2.2 边缘分布
\(P(\ell \mid pq-1) = 1/(\ell-1)\) 精确到四位，对 \(\ell = 2, 3, 5, 7, 11, 13\) 和所有 \(p\)（101–70001）不变：
| ℓ | P(ℓ | pq-1) | P(ℓ | random) | 1/ℓ | 1/φ(ℓ) |
|---|---|---|---|---|
| 2 | 1.0000 | 0.5000 | 0.5000 | 1.0000 |
| 3 | 0.5000 | 0.3336 | 0.3333 | 0.5000 |
| 5 | 0.2500 | 0.2000 | 0.2000 | 0.2500 |
| 7 | 0.1667 | 0.1432 | 0.1429 | 0.1667 |
| 11 | 0.1000 | 0.0909 | 0.0909 | 0.1000 |
| 13 | 0.0833 | 0.0769 | 0.0769 | 0.0833 |
2.3 联合分布
\(P(m \mid pq-1) / (1/\varphi(m))\) 在 0.997–1.005 之间，对所有 smooth moduli。联合 valuation vector 的分布精确遵循 1/φ(m) local model。
| m | 描述 | ratio（p=1009） | ratio（p=50021） |
|---|---|---|---|
| 4 | 2² | 1.000 | 1.000 |
| 12 | 2²·3 | 0.999 | 1.002 |
| 30 | 2·3·5 | 0.999 | 1.002 |
| 60 | 2²·3·5 | 0.998 | 1.007 |
| 120 | 2³·3·5 | 0.994 | 0.997 |
| 210 | 2·3·5·7 | 0.998 | 0.975 |
§3 \(v_2(pq-1) = 2.000\) 与额外 Valuation 公式
3.1 正确推导
\(2^k \mid (pq-1) \iff q \equiv p^{-1} \pmod{2^k}\)。奇素数 \(q\) 在 mod \(2^k\) 的 \(\varphi(2^k) = 2^{k-1}\) 个奇剩余类中等分。因此：
$$\Pr(v_2(pq-1) \geq k) = \frac{1}{2^{k-1}}$$
$$E[v_2(pq-1)] = \sum_{k \geq 1} \frac{1}{2^{k-1}} = 2.000$$
数据：\(E[v_2(pq-1)] = 2.000\)（所有 \(p\)），\(E[v_2(\text{random})] = 1.000\)。差 = +1.000。
3.2 额外 Expected Valuation
对任意素数 \(\ell\)：
$$\sum_{k \geq 1} \left(\frac{1}{\varphi(\ell^k)} - \frac{1}{\ell^k}\right) = \frac{1}{(\ell-1)^2}$$
总额外 valuation（截断到 \(\ell \leq 31\)）\(\approx 1.369\)。完整素数和 \(\approx 1.375\)。\(\ell=2\) 贡献 1.000，\(\ell=3\) 贡献 0.250，合计占完整和的约 91%。
§4 Bias/Δ 比值的衰减与穿零
定义 \(\Delta = \Delta_{\mathrm{penalty}}\)（在 \(pq\) 尺度测量，≈ 0.706）。以下数据来自 Block 1 独立采样。
| p | A-C | B-C | A-B | (A-C)/Δ | (B-C)/Δ | (A-B)/Δ |
|---|---|---|---|---|---|---|
| 29 | −1.42 | −1.05 | −0.37 | −2.01 | −1.48 | −0.52 |
| 503 | −1.46 | −1.03 | −0.43 | −2.06 | −1.45 | −0.61 |
| 5003 | −1.40 | −0.96 | −0.44 | −1.99 | −1.36 | −0.62 |
| 20011 | −1.00 | −0.68 | −0.32 | −1.42 | −0.96 | −0.46 |
| 50021 | −0.39 | −0.22 | −0.17 | −0.55 | −0.31 | −0.24 |
| 70001 | +0.08 | +0.25 | −0.17 | +0.11 | +0.35 | −0.24 |
\((B\text{-}C)/\Delta\) 从 −1.48 穿零到 +0.35。Step 2b（\(\mu_{\mathrm{shift}} - \mu_{\mathrm{general}} = o(\Delta_{\mathrm{penalty}})\)）的数值证据持续增强。
§5 ρ 的非线性响应：\(v_2, v_3\) 控制测试
5.1 结果
| p | B-C uncond | B-C | v₂=2, v₃=0 | Reduction |
|---|---|---|---|
| 1009 | −0.819 | −0.669 | 18% |
| 10007 | −0.646 | −0.510 | 21% |
| 50021 | +0.177 | +0.311 | −76% |
5.2 Weighted Contribution
定义 \(\lambda_{\ell,k} = E[\rho \mid v_\ell \geq k-1] - E[\rho \mid v_\ell \geq k]\)（额外一层 \(\ell\)-adic valuation 带来的 \(\rho\) 节省量，正值）。\(\Delta\mathrm{prob} = 1/\varphi(\ell^k) - 1/\ell^k\)（shifted profile 相对一般整数的额外整除概率，正值）。\(\lambda \times \Delta\mathrm{prob}\) 衡量每个 \((\ell, k)\) 通过线性加权对 \(|B\text{-}C|\) 的贡献量级：
| (ℓ, k) | λ | Δprob | λ × Δprob |
|---|---|---|---|
| (2, 1) | 0.154 | +0.500 | +0.077 |
| (2, 2) | 0.290 | +0.250 | +0.073 |
| (3, 1) | 0.337 | +0.167 | +0.056 |
| (5, 1) | 0.363 | +0.050 | +0.018 |
| 总计 | +0.281 |
总 weighted contribution = 0.281，而 \(|B\text{-}C| = 0.646\)（\(p=10007\)）。线性加权只解释 \(|B\text{-}C|\) 的 44%。这表明 \(\rho\) 对 small-prime profile 的响应不是加性的——交互效应和全局路径选择贡献了大部分。
5.3 解读
Raw probability gap 91% 来自 \(\ell=2, 3\)，但 \(\rho\)-weighted gap 只有 ~20%（控制后消除量）。\(\rho\) 对 small-prime profile 的响应高度非线性——固定局部量无法消除全局路径依赖。
§6 解释率 vs \(y\)：决定性实验
6.1 核心结果
逐步增加控制变量，看 B-C 的解释率。
p = 1009：
| y | 控制变量 | B-C | Z_y | expl% |
|---|---|---|---|
| 2 | v₂ | −0.661 | 19.2% |
| 3 | v₂, v₃ | −0.661 | 19.3% |
| 5 | v₂, v₃, v₅ | −0.661 | 19.3% |
| 7 | v₂, …, v₇ | −0.660 | 19.4% |
| 13 | v₂, …, v₁₃ | −0.659 | 19.5% |
p = 10007：从 23.7%（y=2）到 24.0%（y=13）。
解释率从 y=2 到 y=13 几乎完全不动（增幅 < 0.5%）。
6.2 Smooth Kernel \(K_y\) 匹配确认
| y | K_y levels（p=1009） | B-C | K_y | expl% |
|---|---|---|---|
| 2 | 20 | −0.661 | 19.2% |
| 7 | 1110 | −0.660 | 19.4% |
| 13 | 3369 | −0.659 | 19.5% |
精确匹配 smooth kernel 后，解释率同样卡在 19–24%。
6.3 判决
在当前测试范围内（\(y \leq 13\)），small-prime valuation conditioning 只能解释 ~20% 的 B-C，且解释率随 \(y\) 增大不再上升。剩余 ~80% 来自 smooth kernel 之外的全局结构。尚不能排除 slowly growing \(y(X) \to \infty\) 的可能性，但 fixed-\(y\) 局部化路线在当前数据中已失效。
§7 Cofactor 测试：全局路径效应的铁证
7.1 设计
匹配 \(K_7\)（控制 2,3,5,7 的全部 valuation）后，比较 cofactor \(\rho = \rho(n/K_7)\)。
7.2 结果
| p | E[ρ(cofactor_pr) − ρ(cofactor_rand)] | 显著 |
|---|---|---|
| 1009 | −0.649 | 是 |
| 10007 | −0.487 | 是 |
匹配 smooth kernel 后，cofactor 的 \(\rho\) 仍差 0.49–0.65。
7.3 含义
\(pr-1\) 的优势不在小因子——在大因子。即使小因子部分完全相同，\(pr-1\) 的 cofactor 仍比一般整数更容易压缩。"multiply by \(p\) then subtract 1"影响的不只是 small-prime profile，而是整个因子分解网络的全局拓扑。
§8 多尺度 Cofactor Gap：与 B-C 的同步轨迹
8.1 Cofactor gap（Block 4 独立采样）
| p | B-C | cof gap | cof/B-C |
|---|---|---|---|
| 53 | −0.97 | −0.80 | 82% |
| 211 | −0.96 | −0.85 | 88% |
| 1009 | −0.96 | −0.76 | 80% |
| 5003 | −0.92 | −0.78 | 85% |
| 10007 | −0.82 | −0.67 | 82% |
| 20011 | −0.59 | −0.40 | 67% |
| 50021 | +0.14 | +0.28 | — |
| 70001 | +0.81 | +0.92 | — |
8.2 关键观察
(a) Cofactor gap 和 B-C 走完全相同的侵蚀-穿零轨迹。两者同步从负变正。Cofactor gap 不是 B-C 的"残余"——它就是 B-C 的主体。
(b) cof/B-C 在 \(p < 20000\) 时稳定在 80–90%。这和 §6 的解释率（~20% local）完全一致：small-prime 贡献 ~20%，cofactor 贡献 ~80%。
(c) 两者在 \(p > 50000\) 同步穿零反转。全局路径效应也在退化。在足够大的尺度上，shifted 结构的"记忆"被冲刷。
8.3 完整三路分解（Block 4，代表性 p）
| p | A-C | A-B | B-C | %prime | %resid |
|---|---|---|---|---|---|
| 53 | −1.23 | −0.38 | −0.84 | 31% | 69% |
| 503 | −1.25 | −0.43 | −0.82 | 34% | 66% |
| 5003 | −1.21 | −0.43 | −0.78 | 36% | 64% |
| 10007 | −1.08 | −0.41 | −0.67 | 38% | 62% |
| 20011 | −0.80 | −0.35 | −0.45 | 44% | 56% |
| 50021 | −0.05 | −0.20 | +0.15 | — | — |
| 70001 | +0.67 | −0.15 | +0.82 | — | — |
Prime-source 占比从 31% 升到 44%，然后在穿零区翻转。
§9 B-C 的完整图景与 Route C 的重新定位
9.1 三层分解
| 层 | 贡献占比 | 机制 | 随 p 变化 |
|---|---|---|---|
| Small-prime profile（1/φ vs 1/ℓ） | ~20% | 额外 v₂, v₃ 整除 | Profile 不变，杠杆率衰减 |
| Cofactor 全局路径 | ~80% | DP 最优路径差异 | 同步侵蚀穿零 |
| 总 B-C | 100% | Shifted 乘法结构 | 侵蚀反转 |
9.2 为什么全局效应也在退化（启发式解释）
以下三种解释均为 heuristic，有待严格化：
(a) 随 \(p\) 增大，\(pq\) 越来越大，cofactor 的因子空间庞大，shifted 结构的路径优势被统计稀释。
(b) DP 最优路径可能在极大 \(N\) 下趋向某种"普遍均匀性"（Gemini 的"马尔可夫屏障"假说：shifted 记忆只维持有限深度，深层子树退化为一般整数的统计行为）。
(c) 热力学类比：系统足够大时，局部生成方式的记忆被不断增长的组合复杂度冲刷。
9.3 Route C Step 2b 的重新定位
旧策略（在当前数据中已失效）：用有限维 small-prime profile 差异（fixed-\(y\) localization）来证明 \(B\text{-}C = O(1)\)。解释率平台化已证明 fixed-\(y\) 路线至多走 ~20%。
新策略：直接证明 matched-scale ensemble bias 有界。定义 \(\mu_{\mathrm{aff}}(X; p) := E[\rho(pr-1)]\)（\(r\) 随机奇数），\(\mu_{\mathrm{gen}}(X) := E[\rho(n)]\)（同尺度一般整数，\(X \sim pr\)）。
Step 2b 新目标：证明 \(\mu_{\mathrm{aff}}(X; p) - \mu_{\mathrm{gen}}(X) = O(1)\)，或至少 \(o(\Delta_{\mathrm{penalty}}(X))\)。
这条路不需要分解 B-C 的微观来源——它绕过了 fixed-\(y\) 局部化的失效。需要证明的是：DP 对 shifted vs general ensemble 的平均记忆不会随尺度无限放大。
数值支持：\((B\text{-}C)/\Delta\) 从 −1.48 穿零到 +0.35；cofactor gap 与 B-C 同步侵蚀；全局路径效应虽占 B-C 的 ~80%，但也在退化。这些支持 boundedness conjecture，但尚不构成证明。
§10 讨论
10.1 核心贡献
(a) 1/φ(m) 联合 profile 精确验证。(b) \(v_2 = 2.000\) 的正确代数推导。(c) 额外 valuation 公式 \(\sum 1/(\ell-1)^2 \approx 1.375\)。(d) \(\rho\) 的非线性响应发现（\(v_2, v_3\) 控制仅消 18–21%）。(e) 解释率 vs \(y\) 曲线平台化——small-prime 只解释 ~20%。(f) Cofactor 测试确认全局 DP 路径效应（~80%）。(g) 多尺度 cofactor gap 与 B-C 的同步侵蚀轨迹。(h) Route C 的战略重新定位。
10.2 从 XXXIII 到 XXXVI
| XXXIII | XXXIV | XXXV | XXXVI | |
|---|---|---|---|---|
| 界 | 下界 | 上界 | — | — |
| V(p) | — | O(1) | — | — |
| 闭合条件 | M_p < 1? | M_p → ∞? | 非平台化 | — |
| 机制 | 漂移 | 98.3% 追赶 | 优势反转 | 1/φ(m) + 全局路径 |
| 深度 | 阈值 | 定性发散 | 偏差分解 | Fixed-y 局部化失效 |
| B-C 来源 | — | — | Shifted 结构 | 20% local + 80% global |
10.3 距离估计
Paper XXXVI 揭示 B-C 的主体是全局 DP 路径效应，使得 Step 2b 的严格化需要新路线——不是 fixed-\(y\) 局部化，而是直接证明 matched-scale affine-ensemble bias（\(\mu_{\mathrm{aff}} - \mu_{\mathrm{gen}}\)）有界或 \(o(\Delta_{\mathrm{penalty}})\)。Ω=2 闭合可能需要 2 篇：一篇建立 matched-scale affine bias 的有界性，一篇用 \(\Delta_{\mathrm{penalty}} \to \infty\) 收尾。
10.4 开放问题
(1) 为什么 cofactor \(\rho\) 差也在随 \(p\) 衰减？是否指向"DP 路径的大数普遍均匀性"？
(2) \(\rho(n)\) 对生成方式的敏感度是否以 \(o(\ln n)\) 衰减？
(3) 能否绕过 B-C 的微观分解，直接证明 matched-scale affine-ensemble bias \(\mu_{\mathrm{aff}}(X;p) - \mu_{\mathrm{gen}}(X) = O(1)\)？
(4) Shifted 和一般整数的 DP 最优路径有哪些系统性拓扑差异？
§11 数据来源
| 脚本 | 测量 |
|---|---|
| p36_bias_ratios.py | Bias/Δ, small-prime profile, v₂ 分布 |
| p36_joint_profile.py | 联合 profile, weighted contribution, v₂v₃ 控制 |
| p36_explanation_curve.py | 解释率 vs y, smooth kernel 匹配, cofactor 测试 |
| p36_cofactor_decay.py | 多尺度 cofactor gap, 三路分解（11 个 p） |
Sanity check：ρ(10⁷)=58, ρ(10⁸)=66, ρ(10⁹)=75。
References
[1] ZFCρ Papers I–XXXV. Paper XXXV DOI: 10.5281/zenodo.19143732. Paper XXXIV DOI: 10.5281/zenodo.19140015. Paper XXXIII DOI: 10.5281/zenodo.19124485.
[2] K. Cordwell et al. (2018). J. Number Theory, 189:17–34.
[3] A. Hildebrand, G. Tenenbaum (1986). Trans. AMS, 296:265–290.
致谢
Claude（子路）编写全部数值脚本，设计了解释率 vs y 曲线和 cofactor 测试，起草了 working notes v1–v3 和正文。ChatGPT（公西华）建议了 bias/Δ 比值测量，修正了 v₂ 的错误推导，建议了 v₂v₃ 杀手测试和"解释率随 y 增长"这个决定性实验方向，指出剩余 80% 更可能是全局 DP 路径效应，提出"直接证明 shifted 效应 = O(1)"的新策略。Gemini（子夏）贡献了半解析论证框架（后被杀手测试修正），提出"马尔可夫屏障"和"捷径连锁反应"解释，以及从 local profile 转向 friable core 的建议。Grok（子贡）确认了非线性响应与 Paper 18–19 线性自校正不矛盾（不同层面），cofactor 衰减与 Lindley 平稳分布的一致性。最终文本由作者独立完成。