Ratio-Parametrization of G and the Composition-Shift Reinterpretation of B-C Erosion
DOI: 10.5281/zenodo.19157939Papers XXXV–XXXVII reported B-C erosion from −2.09 to +1.45 with zero-crossing, attributing it to intrinsic \(p\)-dependent asymptotic effects. This paper achieves a fundamental causal reinterpretation through multi-scale analysis, \(q/p\) ratio decomposition, and fixed-\(t\) three-term decomposition.
(1) \(G\) does not depend on \(p\) — it depends only on \(t = q/p\). At fixed \(t\), \(G_{\mathrm{loc}}(t)\) is \(p\)-independent (\(q/p \sim 1000\): \(G = -0.042 \pm 0.002\) across five primes).
(2) "B-C erosion through zero" is a composition shift in the \(N = 10^{10}\) window. Large \(p\) forces small \(t\) (max \(t = N/p^2\)), shifting the mix toward the small-\(t\) regime where \(G \approx 0\).
(3) At fixed \(t\), the three-term decomposition \(E(t) + C_{\mathrm{mult}}(t) + C_{\mathrm{add}}(t)\) is also \(p\)-independent. Paper XXXVII's continuation reversal is a \(t\)-regime characteristic, not \(p\)-driven degradation.
(4) Child diff is positive at ALL \(t\) and ALL \(p\) (+1.1 to +1.7). This is an intrinsic feature, not a "reversal." Paper XXXVII's negative-to-positive child diff trajectory was a mix effect.
(5) At fixed \(t \geq 50\): entry term gives a stable negative contribution (~−0.11), continuation terms give stable positive counterbalance (~+0.07), nearly canceling to a small net bias \(G_{\mathrm{loc}} \approx -0.04\).
(6) \(\Delta_{\mathrm{penalty}}\) is flat across four decades (\(10^7\)–\(10^{10}\)): 0.691–0.703.
Keywords: ratio-parametrization, \(G_{\mathrm{loc}}(t)\), composition shift, window effect, causal chain reconstruction
§1 Three Layers of G
1.1 Object Separation
This paper explicitly distinguishes three different quantities:
- \(G_{\mathrm{loc}}(p,t)\): Local ρ bias at fixed \(t = q/p\). This paper's core measurement.
- \(\bar{G}_N(p)\): Mixed average over \(q \in [p+1, N/p]\), at fixed \(N\) and \(p\). What Papers XXXV–XXXVII measured. Equals \(M_{N,p}[G_{\mathrm{loc}}(t)]\) over \(t \in [1, N/p^2]\).
- \(G_X(p)\): Fixed-\(p\) expanding-window average (Block 1: converges to ~−1.9).
1.2 Notation
- \(A = E[\rho(pq-1)]\) (\(q\) prime), \(B = E[\rho(pr-1)]\) (\(r\) random odd), \(C = E[\rho(\text{random } n)]\)
- \(B\text{-}C = \bar{G}_N(p)\), \(A\text{-}B = S_N(p)\) (prime-source correction)
- \(\pi_{\mathrm{aff}} = P(\text{first-move=mult} \mid pr-1)\), \(\pi_{\mathrm{gen}} = P(\text{first-move=mult} \mid \text{random})\)
§2 Gloc(t) Is p-Independent
2.1 Decisive Data
q/p ~ 100:
| p | G_loc |
|---|---|
| 53 | −0.035 |
| 101 | −0.003 |
| 503 | −0.038 |
| 1009 | −0.045 |
| 2003 | −0.046 |
| 5003 | −0.046 |
\(G \approx -0.04 \pm 0.02\), six primes consistent.
q/p ~ 1000:
| p | G_loc |
|---|---|
| 53 | −0.043 |
| 101 | −0.041 |
| 503 | −0.041 |
| 1009 | −0.045 |
| 2003 | −0.042 |
\(G \approx -0.042 \pm 0.002\), five primes in perfect agreement.
2.2 Profile f(t)
| t regime | G_loc |
|---|---|
| \(t < 5\) | −0.05 to −0.15 (noisy) |
| \(5 < t < 50\) | −0.03 to −0.06 |
| \(t > 50\) | −0.04 (stable) |
\(|G_{\mathrm{loc}}| < 0.15\) for all \(t > 10\).
§3 Fixed-t Three-Term Decomposition
3.1 Formula
$$B\text{-}C = E(t) + C_{\mathrm{mult}}(t) + C_{\mathrm{add}}(t)$$
where \(E(t) = (\pi_{\mathrm{aff}} - \pi_{\mathrm{gen}})(\mu_{\mathrm{gen}}^{\mathrm{mult}} - \mu_{\mathrm{gen}}^{\mathrm{add}})\), \(C_{\mathrm{mult}}(t) = \pi_{\mathrm{aff}}(\mu_{\mathrm{aff}}^{\mathrm{mult}} - \mu_{\mathrm{gen}}^{\mathrm{mult}})\), \(C_{\mathrm{add}}(t) = (1-\pi_{\mathrm{aff}})(\mu_{\mathrm{aff}}^{\mathrm{add}} - \mu_{\mathrm{gen}}^{\mathrm{add}})\).
Papers XXXV–XXXVII measured the mixed averages \(M_{N,p}[E(t)]\), \(M_{N,p}[C_{\mathrm{mult}}(t)]\), \(M_{N,p}[C_{\mathrm{add}}(t)]\).
3.2 Data at t ~ 100
| p | B-C | E | C_mult | C_add |
|---|---|---|---|---|
| 53 | −0.039 | −0.124 | +0.041 | +0.045 |
| 211 | −0.085 | −0.131 | +0.029 | +0.017 |
| 503 | −0.034 | −0.116 | +0.037 | +0.046 |
| 1009 | −0.037 | −0.102 | +0.038 | +0.027 |
| 5003 | −0.034 | −0.105 | +0.035 | +0.036 |
\(E \approx -0.11\), \(C_{\mathrm{mult}} \approx +0.04\), \(C_{\mathrm{add}} \approx +0.03\) — all \(p\)-independent.
3.3 Data at t ~ 1000
| p | B-C | E | C_mult | C_add |
|---|---|---|---|---|
| 53 | −0.022 | −0.124 | +0.085 | +0.016 |
| 503 | −0.037 | −0.114 | +0.047 | +0.029 |
| 1009 | −0.052 | −0.105 | +0.027 | +0.025 |
| 2003 | −0.049 | −0.109 | +0.025 | +0.035 |
Also \(p\)-independent.
3.4 Key Observation
At fixed t, C_mult is positive. This differs completely from Paper XXXVII's negative \(C_{\mathrm{mult}}\) (−1.39 to −0.53), which was a mixed-window average pulled negative by large-\(t\) contributions.
At fixed \(t \geq 50\): entry gives a stable negative main term (~−0.11), continuation gives a stable positive counterbalance (~+0.07), nearly canceling to \(G_{\mathrm{loc}} \approx -0.04\).
3.5 Reinterpretation of Paper XXXVII
XXXVII's data are correct. But they measured mixed quantities. "E_p 5%, C_mult 72%" were mix-weighted proportions; the intrinsic decomposition at fixed \(t\) shows \(E\) dominant in absolute value with continuation partially canceling.
§4 Child Diff Is Intrinsically Positive
| t | p = 53 | p = 503 | p = 1009 | p = 2003 |
|---|---|---|---|---|
| 2 | +0.76 | +0.83 | +1.17 | +1.38 |
| 10 | +1.13 | +1.25 | +1.35 | +1.38 |
| 100 | +1.15 | +1.18 | +1.23 | +1.49 |
| 1000 | +1.34 | +1.44 | +1.69 | +1.73 |
Positive at all \(t\) and all \(p\). Mult-split children of shifted integers are intrinsically more expensive to compress. Paper XXXVII's "reversal from −0.59 to +0.96" was a mix effect: small-\(p\) mixed averages were pulled negative by large-\(t\) contributions; at large \(p\) only small-\(t\) samples remain (intrinsically positive values exposed).
§5 Composition Shift
5.1 Causal Chain
(a) \(G_{\mathrm{loc}}(p,t) \approx f(t)\), \(p\)-independent.
(b) \(\bar{G}_N(p) = M_{N,p}[f(t)]\), mixed average over \(t \in [1, N/p^2]\).
(c) As \(p\) grows, max \(t = N/p^2\) shrinks, mix shifts toward small \(t\).
(d) \(f(\text{small } t) \approx 0\), \(f(\text{large } t) \approx -0.04\) to \(-0.5\).
(e) \(\bar{G}_N(p)\) erodes from strongly negative toward zero: composition shift.
5.2 Correction to Papers XXXV–XXXVII
Data preserved. Causal chain rewritten.
Old: \(G\) is a \(p\)-function that decays with continuation degradation. New: \(G\) is a \(t\)-function, \(p\)-independent. "Erosion" is composition shift in the fixed-\(N\) window. "Continuation reversal" is the intrinsic positive child diff being masked by mix effects at small \(p\), then revealed at large \(p\).
§6 Multi-Scale Data
6.1 G(X;p) Convergence
| p | X = 10⁷ | X = 10⁸ | X = 10⁹ | X = 10¹⁰ |
|---|---|---|---|---|
| 53 | −1.92 | −1.94 | −1.93 | −1.95 |
| 1009 | −0.94 | −1.76 | −1.91 | −1.92 |
For fixed \(p\), \(G_X(p)\) converges to ~−1.9 as \(X\) grows — because large \(q\) (large \(t\)) dominates the average.
6.2 Matched-Scale G(pq;p)
| p | avg pq | G(pq;p) |
|---|---|---|
| 53 | 5.0 × 10⁹ | −1.158 |
| 5003 | 5.0 × 10⁹ | −1.116 |
| 20011 | 5.2 × 10⁹ | −0.799 |
| 70001 | 7.4 × 10⁹ | +0.224 |
"Erosion" from the changing \(t\)-mix, not from \(f(t)\) itself changing.
6.3 Δpenalty: Flat Across Four Decades
| Scale | Δ_penalty |
|---|---|
| 10⁷ | 0.691 |
| 10⁸ | 0.703 |
| 10⁹ | 0.703 |
| 10¹⁰ | 0.700 |
No detectable growth in the current window.
§7 Implications for Route C
7.1 G = O(1) Evidence
\(G_{\mathrm{loc}}(t)\) is bounded: \(|G_{\mathrm{loc}}| < 0.15\) for \(t > 10\), \(|G_{\mathrm{loc}}| < 2\) for all \(t\). This is the strongest numerical evidence for \(B\text{-}C = O(1)\).
Analytic needs: (a) prove scaling collapse \(G_{\mathrm{loc}}(p,t) = f(t) + \varepsilon\) with \(\sup|\varepsilon|\) controlled; (b) prove \(\sup|f(t)| < \infty\); (c) prove mixed average \(M_{N,p}[f]\) bounded.
7.2 Correct Mp Identity
$$E[\rho(pq-1)] = E[\rho(\text{general})] + \bar{G}_N(p) + S_N(p)$$
Therefore:
$$M_p = \Delta_{\mathrm{penalty}}(p) - \bar{G}_N(p) - S_N(p)$$
With \(\Delta_{\mathrm{penalty}} \approx 0.70\) (flat positive), \(\bar{G}_N = O(1)\), \(S_N = O(1)\): \(M_p = O(1)\).
Paper XXXVIII does not directly yield \(M_p \to \infty\). It yields Step 2b's major simplification: the intrinsic object is a bounded ratio-profile \(f(t)\), and "erosion" is composition shift.
7.3 Route C Status
Step 2b rewritten: prove \(G_{\mathrm{loc}}\)'s scaling collapse and uniform bound, then prove the weighted mix \(\bar{G}_N(p) = M_{N,p}[f]\) is bounded. Much weaker and cleaner than "prove \(G\) tends to 0."
\(\Omega = 2\) closure still needs additional input: \(\Delta_{\mathrm{penalty}}\)'s long-range growth, or another route to \(M_p\) divergence.
§8 Discussion
8.1 Core Contributions
- Ratio-parametrization: \(G_{\mathrm{loc}}(p,t) \approx f(t)\), \(p\)-independent.
- Three-layer G distinction (\(G_{\mathrm{loc}}\), \(\bar{G}_N\), \(G_X\)).
- Composition-shift reinterpretation of B-C erosion through zero.
- Causal chain correction for Papers XXXV–XXXVII (data preserved).
- Fixed-\(t\) three-term decomposition: \(E\), \(C_{\mathrm{mult}}\), \(C_{\mathrm{add}}\) are all \(t\)-functions.
- Child diff intrinsically positive at all \(t\).
- Correct \(M_p\) identity (no \(\tau_p\) double-counting).
8.2 Progress Table (XXXIII → XXXVIII)
| XXXIII | XXXIV | XXXV | XXXVI | XXXVII | XXXVIII | |
|---|---|---|---|---|---|---|
| Mechanism | Drift | Chase | Reversal | Global path | 3-term | t-profile |
| B-C | — | — | Shifted | 20%L+80%G | E5%+Cm72% | f(t)+mix |
8.3 Open Problems
- Analytic form of \(f(t)\).
- Harmonic \(q\)-weight formula in defect.
- Analytic proof of \(p\)-independence.
- \(\Delta_{\mathrm{penalty}}\) behavior at larger \(N\).
- Route to \(M_p \to \infty\).
§9 Data Sources
| Script | Measurement |
|---|---|
| p38_multiscale.py | G and Δ at 4 N-scales |
| p38_unified_g.py | Matched-scale G, G vs q-window |
| p38_qp_ratio.py | G vs q/p ratio, G vs p at fixed q/p |
| p38_three_term_fixed_t.py | Fixed-t three-term decomposition + child diff |
References
[1] ZFCρ Papers I–XXXVII. Paper XXXVII DOI: 10.5281/zenodo.19155792. Paper XXXVI DOI: 10.5281/zenodo.19153002.
Acknowledgments
Claude (Zilu) wrote all scripts, designed multi-scale experiments and fixed-\(t\) decomposition, drafted v1–v2 and formal text. The author (Han Qin) identified the "large \(p\) forced to use \(q \approx p\)" window effect — a critical remainder missed by all four AIs. ChatGPT (Gongxihua) proposed three-layer G distinction, caught the \(\tau_p\) double-counting error in §7.2, and corrected Paper XXXVIII's positioning from "closure" to "Step 2b rewrite." Gemini (Zixia) contributed the "static confrontation" perspective (entry < 0 vs continuation > 0) and the \(M_p\) framework (later corrected). Grok (Zigong) confirmed \(p\)-independence compatibility with Lindley. Final text independently by the author; all mathematical judgments are the author's responsibility.
G 的 Ratio-Parametrization 与 B-C 侵蚀的 Composition-Shift 重新解读
DOI: 10.5281/zenodo.19157939Papers XXXV–XXXVII 报告了 B-C 从 −2.09 侵蚀到 +1.45 并穿零反转，并将其归因于 \(p\) 的内禀渐近效应。本文通过多尺度分析、\(q/p\) ratio 分解和固定 \(t = q/p\) 的三项分解，完成了一次根本性的因果链重构。
（1） \(G\) 不依赖于 \(p\)——它只依赖于 \(t = q/p\)。固定 \(t\) 时，\(G_{\mathrm{loc}}(t)\) 对 \(p\) 完全不变（\(q/p \sim 1000\) 时 \(G \approx -0.042 \pm 0.002\)，五个 \(p\) 一致）。
（2） "B-C 侵蚀穿零"是 \(N = 10^{10}\) 窗口中 \(q/p\) composition shift。大 \(p\) 被迫只能用小 \(t\)（max \(t = N/p^2\)），mix 向小 \(t\) 区移动，小 \(t\) 区 \(G \approx 0\)。
（3） 固定 \(t\) 时，三项分解 \(E(t) + C_{\mathrm{mult}}(t) + C_{\mathrm{add}}(t)\) 也对 \(p\) 不变。Paper XXXVII 的 continuation reversal 不是 \(p\) 驱动的退化，而是小 \(t\) regime 的固有特征。
（4） Child diff 在所有 \(t\) 和所有 \(p\) 上都是正的（+1.1 到 +1.7）。这不是"反转"——乘法分裂后 child 比一般整数更贵是固有特征。
（5） 固定 \(t \geq 50\) 时，entry term 给出稳定的负主项（≈ −0.11），continuation 两项给出稳定的正抵消项（≈ +0.07），三者几乎相消，净 bias \(G_{\mathrm{loc}} \approx -0.04\) 极小。
（6） \(\Delta_{\mathrm{penalty}}\) 在 \(10^7\)–\(10^{10}\) 四个数量级上完全平坦（0.691–0.703）。
关键词： ratio-parametrization，\(G_{\mathrm{loc}}(t)\)，composition shift，窗口效应，因果链重构
§1 三层 G 的区分
1.1 对象分层
本文明确区分三个不同的量：
- \(G_{\mathrm{loc}}(p,t)\)：固定 \(t = q/p\) ratio 时，\(pr-1\) vs random 的局部 ρ 偏差。本文核心测量。
- \(\bar{G}_N(p)\)：固定 \(N\) 和 \(p\)，对 \(q \in [p+1, N/p]\) 的混合平均。Papers XXXV–XXXVII 测的量。\(\bar{G}_N(p) = M_{N,p}[G_{\mathrm{loc}}(t)]\)，即 \(G_{\mathrm{loc}}\) 在 \(t \in [1, N/p^2]\) 上的加权混合。
- \(G_X(p)\)：固定 \(p\)，随 \(X\) 扩窗的整体平均（Block 1 测的，收敛到 ≈ −1.9）。
1.2 记号
- \(A = E[\rho(pq-1)]\)（\(q\) 素数），\(B = E[\rho(pr-1)]\)（\(r\) 随机奇数），\(C = E[\rho(\text{random } n)]\)
- \(B\text{-}C = \bar{G}_N(p)\)，\(A\text{-}B = S_N(p)\)（prime-source correction）
- \(\pi_{\mathrm{aff}} = P(\text{first-move=mult} \mid pr-1)\)，\(\pi_{\mathrm{gen}} = P(\text{first-move=mult} \mid \text{random})\)
§2 Gloc(t) 的 p-Independence
2.1 判决性数据
q/p ~ 100：
| p | G_loc |
|---|---|
| 53 | −0.035 |
| 101 | −0.003 |
| 503 | −0.038 |
| 1009 | −0.045 |
| 2003 | −0.046 |
| 5003 | −0.046 |
\(G \approx -0.04 \pm 0.02\)，六个 \(p\) 一致。
q/p ~ 1000：
| p | G_loc |
|---|---|
| 53 | −0.043 |
| 101 | −0.041 |
| 503 | −0.041 |
| 1009 | −0.045 |
| 2003 | −0.042 |
\(G \approx -0.042 \pm 0.002\)，五个 \(p\) 完美一致。
2.2 f(t) 的 profile
| t regime | G_loc |
|---|---|
| \(t < 5\) | −0.05 到 −0.15（嘈杂） |
| \(5 < t < 50\) | −0.03 到 −0.06 |
| \(t > 50\) | −0.04（稳定） |
\(G_{\mathrm{loc}}(t)\) 在 \(t > 10\) 后收敛到 ≈ −0.04，在所有 regime 下 \(|G_{\mathrm{loc}}| < 0.15\)。
§3 固定 t 的三项分解
3.1 公式
$$B\text{-}C = E(t) + C_{\mathrm{mult}}(t) + C_{\mathrm{add}}(t)$$
其中 \(E(t) = (\pi_{\mathrm{aff}} - \pi_{\mathrm{gen}})(\mu_{\mathrm{gen}}^{\mathrm{mult}} - \mu_{\mathrm{gen}}^{\mathrm{add}})\)，\(C_{\mathrm{mult}}(t) = \pi_{\mathrm{aff}}(\mu_{\mathrm{aff}}^{\mathrm{mult}} - \mu_{\mathrm{gen}}^{\mathrm{mult}})\)，\(C_{\mathrm{add}}(t) = (1-\pi_{\mathrm{aff}})(\mu_{\mathrm{aff}}^{\mathrm{add}} - \mu_{\mathrm{gen}}^{\mathrm{add}})\)。
Papers XXXV–XXXVII 测的是混合量 \(M_{N,p}[E(t)]\)，\(M_{N,p}[C_{\mathrm{mult}}(t)]\)，\(M_{N,p}[C_{\mathrm{add}}(t)]\)。
3.2 t ~ 100 的数据
| p | B-C | E | C_mult | C_add |
|---|---|---|---|---|
| 53 | −0.039 | −0.124 | +0.041 | +0.045 |
| 211 | −0.085 | −0.131 | +0.029 | +0.017 |
| 503 | −0.034 | −0.116 | +0.037 | +0.046 |
| 1009 | −0.037 | −0.102 | +0.038 | +0.027 |
| 5003 | −0.034 | −0.105 | +0.035 | +0.036 |
\(E \approx -0.11\)，\(C_{\mathrm{mult}} \approx +0.04\)，\(C_{\mathrm{add}} \approx +0.03\)——全部对 \(p\) 不变。
3.3 t ~ 1000 的数据
| p | B-C | E | C_mult | C_add |
|---|---|---|---|---|
| 53 | −0.022 | −0.124 | +0.085 | +0.016 |
| 503 | −0.037 | −0.114 | +0.047 | +0.029 |
| 1009 | −0.052 | −0.105 | +0.027 | +0.025 |
| 2003 | −0.049 | −0.109 | +0.025 | +0.035 |
同样对 \(p\) 不变。
3.4 固定 t 下的内禀分解
注意：在固定 t 时 C_mult 是正的。这和 Paper XXXVII 报告的负 \(C_{\mathrm{mult}}\)（−1.39 到 −0.53）完全不同。XXXVII 的负值是混合平均 \(M_{N,p}[C_{\mathrm{mult}}(t)]\) 被大 \(t\) 区域的其他效应拉低的结果。
在固定 \(t \geq 50\) 时，entry term 给出稳定的负主项（≈ −0.11），而 \(C_{\mathrm{mult}}\) 和 \(C_{\mathrm{add}}\) 给出稳定的正抵消项（合计 ≈ +0.07）。三者几乎相消，净 bias B-C ≈ −0.04 极小。
3.5 对 Paper XXXVII 的重新解读
XXXVII 的三项分解数据仍然正确。但它们测的是混合量，不是内禀 \(t\)-profile。"E_p 占 5%，C_mult 占 72%"是混合比例；内禀分解中 \(E\) 主导（绝对值约 −0.11）而 continuation 抵消大部分。
§4 Child Diff 的固有正值
| t | p = 53 | p = 503 | p = 1009 | p = 2003 |
|---|---|---|---|---|
| 2 | +0.76 | +0.83 | +1.17 | +1.38 |
| 10 | +1.13 | +1.25 | +1.35 | +1.38 |
| 100 | +1.15 | +1.18 | +1.23 | +1.49 |
| 1000 | +1.34 | +1.44 | +1.69 | +1.73 |
Child diff 在所有 \(t\) 和所有 \(p\) 上都是正的。乘法分裂后 child 比一般整数更贵是固有特征——不是 Paper XXXVII 中描述的"反转"。XXXVII 的"child diff 从 −0.59 到 +0.96"是 mix effect：小 \(p\) 的混合平均被大 \(t\) 的其他因素拉负，大 \(p\) 只有小 \(t\) 的样本（固有正值显露）。
§5 Composition Shift
5.1 因果链
（a）\(G_{\mathrm{loc}}(p,t) \approx f(t)\)，不依赖于 \(p\)。
（b）\(\bar{G}_N(p) = M_{N,p}[f(t)]\)，是 \(f\) 在 \(t \in [1, N/p^2]\) 上的混合平均。
（c）当 \(p\) 增大，max \(t = N/p^2\) 缩小，mix 向小 \(t\) 移动。
（d）\(f(\text{small } t) \approx 0\)，\(f(\text{large } t) \approx -0.04\) 到 \(-0.5\)。
（e）混合平均 \(\bar{G}_N(p)\) 从强负值向零侵蚀。
5.2 对 Papers XXXV–XXXVII 的修正
数据保留，因果链重写。
旧因果链：\(G\) 是 \(p\) 的函数，随 \(p\) 退化，continuation 反转。新因果链：\(G\) 是 \(t\) 的函数，\(p\)-independent。"侵蚀"是固定 \(N\) 窗口中 \(t\)-mix 的 composition shift。"Continuation reversal"是固定 \(t\) 下 child diff 的固有正值被 mix effect 遮蔽后在大 \(p\) 端显露。
§6 多尺度数据
6.1 G(X;p) 随 X 收敛
| p | X = 10⁷ | X = 10⁸ | X = 10⁹ | X = 10¹⁰ |
|---|---|---|---|---|
| 53 | −1.92 | −1.94 | −1.93 | −1.95 |
| 1009 | −0.94 | −1.76 | −1.91 | −1.92 |
\(G_X(p)\) 随 \(X\) 增大收敛——因为大 \(q\)（大 \(t\)）主导平均，\(f(\text{large } t) \approx -0.5\) 到 \(-1.9\)。
6.2 Matched-scale G(pq;p)
| p | avg pq | G(pq;p) |
|---|---|---|
| 53 | 5.0 × 10⁹ | −1.158 |
| 5003 | 5.0 × 10⁹ | −1.116 |
| 20011 | 5.2 × 10⁹ | −0.799 |
| 70001 | 7.4 × 10⁹ | +0.224 |
"侵蚀"来自 \(t\)-mix 的变化，不是 \(f(t)\) 本身在变。
6.3 Δpenalty 四尺度平坦
| 尺度 | Δ_penalty |
|---|---|
| 10⁷ | 0.691 |
| 10⁸ | 0.703 |
| 10⁹ | 0.703 |
| 10¹⁰ | 0.700 |
当前窗口未见可检测增长。
§7 对 Route C 的含义
7.1 G = O(1) 的证据
\(G_{\mathrm{loc}}(t)\) 在所有测试的 \(t\) 范围内有界（\(|G_{\mathrm{loc}}| < 0.15\) for \(t > 10\)；\(|G_{\mathrm{loc}}| < 2\) for all \(t\)）。这是 \(B\text{-}C = O(1)\) 的最强数值证据。
解析层面，需证明：（a）\(G_{\mathrm{loc}}(p,t) = f(t) + \varepsilon(p,t)\)，\(\sup|\varepsilon|\) 可控；（b）\(\sup|f(t)| < \infty\)；（c）\(\bar{G}_N(p) = M_{N,p}[f]\) 的混合有界性。
7.2 Mp 的正确恒等式
$$E[\rho(pq-1)] = E[\rho(\text{general})] + \bar{G}_N(p) + S_N(p)$$
因此：
$$M_p = \Delta_{\mathrm{penalty}}(p) - \bar{G}_N(p) - S_N(p)$$
其中 \(\Delta_{\mathrm{penalty}} \approx 0.70\)（正常数，平坦），\(\bar{G}_N = O(1)\)，\(S_N = O(1)\)。因此 \(M_p = O(1)\)。
Paper XXXVIII 不直接给出 \(M_p \to \infty\)。它给出的是 Step 2b 的大幅简化：affine bias 的内禀对象是有界的 ratio-profile \(f(t)\)，"侵蚀"是 composition shift。
7.3 Route C 的当前状态
Step 2b 被重写为：证明 \(G_{\mathrm{loc}}(p,t)\) 的 scaling collapse（\(p\)-independence）与 uniform bound，再证明 defect 中的加权混合 \(\bar{G}_N(p) = M_{N,p}[f]\) 有界。这比"证明 \(G \to 0\) 当 \(p \to \infty\)"弱得多，也干净得多。
\(\Omega = 2\) 闭合仍需额外输入：\(\Delta_{\mathrm{penalty}}\) 的长程增长，或另一条推出 \(M_p \to \infty\) 的论证。
§8 讨论
8.1 核心贡献
- G 的 ratio-parametrization：\(G_{\mathrm{loc}}(p,t) \approx f(t)\)，\(p\)-independent。
- 三层 G 的明确区分（\(G_{\mathrm{loc}}\)，\(\bar{G}_N\)，\(G_X\)）。
- "B-C 侵蚀穿零"重新解读为 composition shift。
- Papers XXXV–XXXVII 的因果链修正（数据保留）。
- 固定 \(t\) 的三项分解：\(E\)，\(C_{\mathrm{mult}}\)，\(C_{\mathrm{add}}\) 也是 \(t\)-function。
- Child diff 在所有 \(t\) 上固有为正。
- \(M_p\) 的正确恒等式（无 \(\tau_p\) 双算）。
8.2 从 XXXIII 到 XXXVIII
| XXXIII | XXXIV | XXXV | XXXVI | XXXVII | XXXVIII | |
|---|---|---|---|---|---|---|
| 机制 | 漂移 | 追赶 | 反转 | 全局路径 | 三项分解 | t-profile |
| B-C | — | — | shifted | 20%L+80%G | E5%+Cm72% | f(t)+mix |
8.3 开放问题
- \(f(t)\) 的解析形式。
- Defect 中的 harmonic \(q\)-权重显式公式。
- \(G_{\mathrm{loc}}\) 的 \(p\)-independence 的解析证明。
- \(\Delta_{\mathrm{penalty}}\) 在更大 \(N\) 上的行为。
- \(M_p \to \infty\) 的论证路径。
§9 数据来源
| 脚本 | 测量 |
|---|---|
| p38_multiscale.py | G 和 Δ 在 4 个 N 尺度 |
| p38_unified_g.py | Matched-scale G, G vs q-window |
| p38_qp_ratio.py | G vs q/p ratio, G vs p at fixed q/p |
| p38_three_term_fixed_t.py | 固定 t 的三项分解 + child diff |
References
[1] ZFCρ Papers I–XXXVII. Paper XXXVII DOI: 10.5281/zenodo.19155792. Paper XXXVI DOI: 10.5281/zenodo.19153002.
致谢
Claude（子路）编写全部数值脚本，设计了多尺度实验、\(q/p\) 分解和固定 \(t\) 三项分解，起草了 working notes v1–v2 和正文。作者（秦汉）识别了"大 \(p\) 被迫只用 \(q \approx p\)"的窗口效应——四个 AI 均未发现的关键余项。ChatGPT（公西华）建议了三层 G 的区分，指出 §7.2 的 \(\tau_p\) 双算错误并给出正确恒等式 \(M_p = \Delta_{\mathrm{penalty}} - \bar{G}_N - S_N\)，将 Paper XXXVIII 的定位从"closure"修正为"Step 2b 重写"。Gemini（子夏）贡献了"固有静态对抗"（entry < 0 vs continuation > 0）的视角和 \(M_p\) 推导框架（后被修正）。Grok（子贡）确认了 G 的 \(p\)-independence 与 Lindley 框架的兼容性。最终文本由作者独立完成；所有数学判断是作者的责任。