ZFCρ Paper LIX: Spectral Ratio Boundedness and the Two-Conjecture Closure of H'

Qin, Han

doi:10.5281/zenodo.19480519

ZFCρ Paper LIX: Spectral Ratio Boundedness and the Two-Conjecture Closure of H'

Han Qin · ORCID 0009-0009-9583-0018

DOI: 10.5281/zenodo.19480519 · CC BY 4.0

§1. Introduction

1.1. Status after Paper 58

Paper 58 reduced H' on the deterministic spectral/Dirichlet route to two explicit conjectures:

Conjecture 58.1 (Cross-product screening): For all sufficiently large j, sub-blocks at screening scale have nonpositive cross-product sum.
Conjecture 58.2 (Spectral derivative bound): 2^j · (B^{(1)})² ≤ C for all large j.

Under 58.1 + 58.2 + SD (unconditional, Paper 50): H' follows.

1.2. What this paper does

Three rounds of experiments and multi-AI review lead to:

(i) Falsification of Conjecture 58.1. The cross-product sign is oscillatory, not eventually negative.

(ii) Replacement. Conjecture 59.1 (R_wt = O(1)) replaces 58.1 as the canonical DSF conjecture, supported by 19 consecutive j-values.

(iii) BL mechanism correction. The spectral derivative is small because of exponential baseline decay (analytic), not antisymmetric cancellation (false).

(iv) Refinement of 58.2. Conjecture 59.2 sharpens the BL condition with a uniform-in-t formulation.

(v) Complete closure chain. 59.1 + 59.2 + SD → H' is proved with no hidden assumptions.

1.3. Notation

All notation follows Papers 50–51. In particular:

h(p) = η(p) − μ₁₂(p mod 12) (character-stripped residual)
B_j = Σ_{p∈I_j} h(p)/p (dyadic block sum), I_j = [2^j, 2^{j+1})
D_j = Σ_{p∈I_j} (h(p)/p)² (diagonal)
R_wt = B_j² / (j · D_j) (weighted spectral ratio, Paper 56)
E_j(t) = Σ_{p∈I_j} h(p)/p · (p^{-it} − e^{-it·c_j}), c_j = (j+½)ln 2
B_j^{(1)} = Σ_{p∈I_j} h(p)/p · (log p − c_j) (first spectral derivative)

§2. Fine-grained j-scan

2.1. Experimental design

Previous papers (53–58) tested only j = 14, 17, 20, 23, 26, 29, 32 (spacing 3). This paper scans every integer j from 14 to 33, using N = 10¹⁰ with the rho binary (P32 convention, λ = 3.856763).

Each block is processed in a single streaming pass computing B_j, D_j, B_j^{(1)}, R_wt, and cross-product sums at five sub-block scales (n* = 500, 2000, 10000, 50000, 200000).

2.2. Data

Table 1. Complete j-scan (j = 14–32, excluding truncated j = 33).

j	m_j	R_wt	2^j·(B¹)²	Cross@10K	Cross@50K	B⁰ sign
14	1,612	8.73	0.0042	—	—	−
15	3,030	13.78	0.0003	—	—	−
16	5,709	10.94	0.0001	—	—	−
17	10,749	15.86	0.0010	—	—	−
18	20,390	18.12	0.0035	—	—	−
19	38,635	19.58	0.0011	+0.21	—	−
20	73,586	21.99	0.0020	+0.40	—	−
21	140,336	25.80	0.0121	+0.46	—	−
22	268,216	24.32	0.0154	+0.43	+0.39	−
23	513,708	25.30	0.0329	+0.49	+0.42	−
24	985,818	18.07	0.0307	+0.47	+0.47	−
25	1,894,120	12.75	0.0240	+0.42	+0.39	−
26	3,645,744	4.21	0.0428	+0.15	+0.26	−
27	7,027,290	0.79	0.0686	−1.92	−2.94	−
28	13,561,907	1.94	0.0542	−0.61	−1.69	+
29	26,207,278	8.87	0.0523	+0.22	−0.11	+
30	50,697,537	20.30	0.0553	+0.37	+0.18	+
31	98,182,656	9.84	0.0478	+0.22	−0.20	+
32	190,335,585	2.70	0.0109	−0.55	−2.21	−

Note: j = 33 excluded (truncated block: N = 10¹⁰ < 2³⁴, m_j = 61M vs expected ~370M).

2.3. R_wt damped oscillation

R_wt exhibits clear damped oscillation:

Peak 1: j ≈ 21–23, R_wt ≈ 25.
Trough 1: j = 27, R_wt = 0.79.
Peak 2: j = 30, R_wt = 20.3.
Trough 2: j = 32, R_wt = 2.70.

Period ≈ 7–9 in j, consistent with Paper 55's T = 8 = 2³. Amplitude decays: peaks 25 → 20, troughs 0.8 → 2.7. All 19 values fall within [0.79, 25.3].

§3. Falsification of Conjecture 58.1

3.1. Cross-product sign is oscillatory

Conjecture 58.1 stated that for sufficiently large j, the cross-product sum at screening scale is nonpositive. The j-scan reveals:

Negative: j = 27, 28, 31, 32 (R_wt troughs).
Positive: j = 29, 30 (R_wt peaks), and all j ≤ 26.

The cross-product sign tracks the R_wt oscillation cycle. It is not eventually negative.

3.2. B⁰ sign also oscillates

B_j = Σ h(p)/p changes sign at j = 28: negative for j ≤ 27, positive for j = 28–31, negative again at j = 32. This is the damped oscillation in the cumulative sum of h/p.

3.3. Consequence

Conjecture 58.1 is false in its stated form. The "eventual screening" and "adaptive screening scale" reformulations are also invalidated: cross-product negativity is a phase-dependent phenomenon, not an asymptotic one. The Paper 52 Theorem 3.1 route (cross-product → deterministic DSF) cannot be applied at every large j.

§4. Conjecture 59.1: Spectral Ratio Boundedness

4.1. Statement

Conjecture 59.1. There exists a constant C > 0 such that for all j ≥ 1:

R_wt := B_j² / (j · D_j) ≤ C.

Equivalently: I_j(0) ≤ C · j · D_j (Paper 51 spectral target).

4.2. Relation to prior work

Paper 56 first identified R_wt = O(1) as the front-door conjecture. Paper 57 provided a conditional bridge from screened covariance to expectation-level R_wt decay. The present paper confirms R_wt = O(1) with 19 data points and elevates it to the canonical DSF statement.

4.3. Mechanism vs statement

The damped oscillation of Paper 55 (rebound ratio r ≈ 0.80, absorption rate η ≈ 0.20, period T = 8 = 2³) provides the structural explanation for why R_wt is bounded. But Conjecture 59.1 is the statement; the damped oscillation is the mechanism. Following the lesson of Papers 53–56, mechanism parameters are not confused with the theorem target.

The oscillation unifies four observables: R_wt oscillation (Table 1), cross-product sign oscillation (§3.1), B⁰ sign changes (§3.2), and z/√j peak and decline (Paper 57).

§5. BL mechanism: exponential baseline, not antisymmetric cancellation

5.1. Falsification of antisymmetric cancellation

Paper 58 hypothesized that B^{(1)} = Σ h(p)/p · (log p − c_j) is small due to cancellation between left (log p < c_j) and right (log p > c_j) halves. The cancellation ratio |B^{(1)}| / max(|B^{(1)}_L|, |B^{(1)}_R|) at j ≥ 26 is 0.72–0.97, meaning essentially no cancellation. Yet 2^j · (B^{(1)})² < 0.07 everywhere.

5.2. The true mechanism

The iid baseline 2^j · Var_iid := 2^j · Σ(x_r · dlp_r)² decreases monotonically from 0.0011 (j=14) to 0.0005 (j=32), approximately O(1/j). This is the dominant effect: each individual contribution x_r · dlp_r decays as 1/p ≈ 2^{-j}, and 2^j times the sum of their squares yields O(1/j) from prime counting.

The excess ratio (B^{(1)})²/Var_iid peaks at 112 (j=27) but stays sub-exponential (likely oscillatory, tracking the R_wt cycle). Polynomial amplification cannot overcome exponential baseline decay.

5.3. Two-layer baseline

Unconditional (coarse): 2^j · Σ(x_r · dlp_r)² ≤ C · j (from Guy + Chebyshev).

Conditional (sharp): 2^j · Σ(x_r · dlp_r)² ≤ C / j (if D_j ≈ σ²/(j·2^j), empirically confirmed).

The sharp bound motivates Conjecture 59.2 but is not in the closure chain.

§6. Conjecture 59.2: Spectral Derivative Bound

6.1. Statement

Conjecture 59.2. For all sufficiently large j:

sup_{|t| ≤ 1} |E_j(t)|² / t² ≤ C_j · 2^{-j}, where log C_j = o(j).

6.2. Relation to Paper 58

Paper 58's Conjecture 58.2 (2^j · (B^{(1)})² ≤ C) controls the leading Taylor coefficient. Conjecture 59.2 strengthens this to uniform-in-t control on |t| ≤ 1, as required by Paper 50's BL_α definition.

6.3. Empirical support

Leading-order: 2^j · (B^{(1)})² has maximum 0.069 (j=27) across 19 j-values. Exact E_j(t) at t = 0.1, 0.3, 1.0 confirms 2^j · |E_j(t)|²/t² < 0.06 for |t| ≤ 1 at all tested blocks.

§7. Complete closure chain

7.1. Conjecture 59.1 → DSF_α for all α > 0

Proof. By 59.1: B_j² ≤ C · j · D_j. By Guy's bound and Chebyshev (unconditional): D_j ≤ C' · j / 2^j. Combining: B_j² ≤ C'' · j² / 2^j. Therefore DSF_α = Σ (1+j)^{1+2α} |B_j|² ≤ Σ j^{3+2α} / 2^j < ∞ for all α > 0. □

7.2. Conjecture 59.2 → BL_α for all α < 1

Proof. From 59.2: |E_j(t)| ≤ √C_j · 2^{-j/2} · |t| for |t| ≤ 1. For α < 1: sup_{|t|≤1} |E_j(t)|/|t|^α ≤ √C_j · 2^{-j/2}. Since log C_j = o(j), the sum Σ √C_j · 2^{-j/2} converges. □

7.3. Assembly

SD is unconditional (Paper 50 Prop 4.1). DSF_α + BL_α → Hölder (Paper 50 Thm 6.2). Hölder + SD → A_h converges (Paper 50 Prop 6.3, Karamata). A_h → D(s) boundary (Paper 50 Prop 6.4). Prime power layer analytic (Paper 50 Prop 7.1, unconditional). → H'.

Conj 59.1 + Guy + Chebyshev → DSF_α ∀α > 0
Conj 59.2 → BL_α ∀α < 1
SD (unconditional)
  → Hölder → A_h converges → D(s) boundary + P(s) analytic → H'

Input: 59.1 + 59.2 + SD. Output: H'. Hidden assumptions: None.

§8. Role reassignment

Paper	Final role
50	Foundation: SD, DSF_α/BL_α definitions, chain Steps 4–8
51	Foundation: spectral target, WN rate, gap quantification
52	Alternative route (backup); Thm 3.1 superseded
53	Mechanism: screened long memory
54	Mechanism: power-law autocorrelation
55	Core mechanism: damped oscillation
56	Statement origin: R_wt = O(1) first identified
57	Conditional bridge (subsumed by 59.1)
58	Refinement: 58.2 → 59.2; 58.1 killed
59	Statement: canonical conjectures, complete chain

§9. Conclusion

The distance from H' is exactly two conjectures:

Conjecture 59.1 (R_wt = O(1)): supported by 19 j-values in [0.79, 25.3].

Conjecture 59.2 (spectral derivative bound): supported by 19 j-values with 2^j·(B¹)² < 0.07.

The closure chain 59.1 + 59.2 + SD → H' has no hidden assumptions. Conjecture 59.2 is closer to provable. Conjecture 59.1 encapsulates the deep arithmetic of IC recursion.

References

[50] Paper L. Character stripping, DSF_α, BL_α, SD, closure chain. DOI: 10.5281/zenodo.19381111.

[51] Paper LI. WN-rate correction, spectral reformulation. DOI: 10.5281/zenodo.19393954.

[52] Paper LII. Probabilistic framework. DOI: 10.5281/zenodo.19407328.

[53] Paper LIII. Screened long memory. DOI: 10.5281/zenodo.19415440.

[54] Paper LIV. Power-law autocorrelation. DOI: 10.5281/zenodo.19426094.

[55] Paper LV. Damped oscillation. DOI: 10.5281/zenodo.19447349.

[56] Paper LVI. Affine Trough, direct spectral ratio. DOI: 10.5281/zenodo.19464605.

[57] Paper LVII. Screened Spectral Bound. DOI: 10.5281/zenodo.19469013.

[58] Paper LVIII. Deterministic DSF + BL. DOI: 10.5281/zenodo.19476508.

AI Contributions

Claude (子路): All numerical experiments (paper59_exp.c, paper59_jscan.c, N = 10¹⁰), j-scan discovery, antisymmetric cancellation falsification, cross-product oscillation identification, closure chain construction and verification, Step 1 fix (Guy + Chebyshev unconditional bound), working notes v1–v3. ChatGPT (公西华): Chain review identifying hidden third assumption (Hypothesis (D) in Step 1), Step 1 fix validation, statement-vs-mechanism separation principle, Conjecture 59.2 formulation (uniform-in-t, log C_j = o(j)), two-layer baseline recommendation, final accept of working notes. Gemini (子夏): Critical screening scale concept (n*_crit), "exponential baseline vs polynomial amplification" framework, j=29 window as structural (damped oscillation trough), excess ratio oscillation interpretation. Grok (子贡): Paper 57 consistency review, 59.2 priority recommendation, B^{(1)} antisymmetric suppression hypothesis (later falsified by data).

ZFCρ 第LIX篇：谱比有界性与H'的双猜想闭合

秦汉 · ORCID 0009-0009-9583-0018

DOI: 10.5281/zenodo.19480519 · CC BY 4.0

§1. 引言

1.1. 第58篇之后的状态

第58篇在确定性谱/Dirichlet路线上将H'归结为两个显式猜想：

猜想58.1（交叉积屏蔽）： 对所有足够大的j，屏蔽尺度下的子块交叉积和为非正。
猜想58.2（谱导数界）： 2^j · (B^{(1)})² ≤ C 对所有大j成立。

在58.1 + 58.2 + SD（无条件，第50篇）下：H'成立。

1.2. 本文的贡献

三轮实验和多AI审查导致：

(i) 猜想58.1被证伪。 交叉积符号是振荡的，不是最终为负。

(ii) 替代。 猜想59.1（R_wt = O(1)）取代58.1成为典范DSF猜想，获19个连续j值支持。

(iii) BL机制修正。 谱导数之所以小，是因为指数基底衰减（解析的），不是反对称抵消（错误的）。

(iv) 58.2的精化。 猜想59.2以uniform-in-t的形式加强了BL条件。

(v) 完整闭合链。 59.1 + 59.2 + SD → H'被证明，无隐藏假设。

1.3. 记号

沿用第50–51篇。特别地：

h(p) = η(p) − μ₁₂(p mod 12)（特征剥离残差）
B_j = Σ_{p∈I_j} h(p)/p（二进块和），I_j = [2^j, 2^{j+1})
D_j = Σ_{p∈I_j} (h(p)/p)²（对角线）
R_wt = B_j² / (j · D_j)（加权谱比，第56篇）
E_j(t) = Σ_{p∈I_j} h(p)/p · (p^{-it} − e^{-it·c_j})，c_j = (j+½)ln 2
B_j^{(1)} = Σ_{p∈I_j} h(p)/p · (log p − c_j)（一阶谱导数）

§2. 精细j扫描

2.1. 实验设计

此前论文（53–58）仅测试 j = 14, 17, 20, 23, 26, 29, 32（间距3）。本文扫描 j = 14 至 33 的每一个整数，使用 N = 10¹⁰，P32约定（λ = 3.856763）。

每个块在单次流式遍历中计算 B_j, D_j, B_j^{(1)}, R_wt，以及五个子块尺度（n* = 500, 2000, 10000, 50000, 200000）的交叉积。

2.2. 数据

表1. 完整j扫描（j = 14–32，排除截断的j = 33）。

j	m_j	R_wt	2^j·(B¹)²	Cross@10K	Cross@50K	B⁰符号
14	1,612	8.73	0.0042	—	—	−
15	3,030	13.78	0.0003	—	—	−
16	5,709	10.94	0.0001	—	—	−
17	10,749	15.86	0.0010	—	—	−
18	20,390	18.12	0.0035	—	—	−
19	38,635	19.58	0.0011	+0.21	—	−
20	73,586	21.99	0.0020	+0.40	—	−
21	140,336	25.80	0.0121	+0.46	—	−
22	268,216	24.32	0.0154	+0.43	+0.39	−
23	513,708	25.30	0.0329	+0.49	+0.42	−
24	985,818	18.07	0.0307	+0.47	+0.47	−
25	1,894,120	12.75	0.0240	+0.42	+0.39	−
26	3,645,744	4.21	0.0428	+0.15	+0.26	−
27	7,027,290	0.79	0.0686	−1.92	−2.94	−
28	13,561,907	1.94	0.0542	−0.61	−1.69	+
29	26,207,278	8.87	0.0523	+0.22	−0.11	+
30	50,697,537	20.30	0.0553	+0.37	+0.18	+
31	98,182,656	9.84	0.0478	+0.22	−0.20	+
32	190,335,585	2.70	0.0109	−0.55	−2.21	−

注：j = 33排除（截断块：N = 10¹⁰ < 2³⁴，m_j = 61M，远低于预期~370M）。

2.3. R_wt的阻尼振荡

R_wt呈现清晰的阻尼振荡：

波峰1： j ≈ 21–23，R_wt ≈ 25。
波谷1： j = 27，R_wt = 0.79。
波峰2： j = 30，R_wt = 20.3。
波谷2： j = 32，R_wt = 2.70。

周期约 7–9（j为单位），与第55篇的 T = 8 = 2³ 一致。振幅衰减：波峰 25 → 20，波谷 0.8 → 2.7。所有19个值落在 [0.79, 25.3] 内。

§3. 猜想58.1的证伪

3.1. 交叉积符号是振荡的

猜想58.1断言：对足够大的j，屏蔽尺度下的交叉积和为非正。j扫描揭示：

负值： j = 27, 28, 31, 32（R_wt波谷）。
正值： j = 29, 30（R_wt波峰），以及所有j ≤ 26。

交叉积符号追踪R_wt的振荡周期。它不是最终为负的。

3.2. B⁰符号也振荡

B_j = Σ h(p)/p在j = 28处变号：j ≤ 27为负，j = 28–31为正，j = 32再次为负。这是h/p累积和的阻尼振荡。

3.3. 后果

猜想58.1在其当前形式下为假。此前提出的"最终屏蔽"和"自适应屏蔽尺度"重构方案也被否决：交叉积的负性是相位依赖的现象，不是渐近现象。第52篇定理3.1的路线（交叉积 → 确定性DSF）不能在每个大j上应用。

§4. 猜想59.1：谱比有界性

4.1. 陈述

猜想59.1。 存在常数 C > 0 使得对所有 j ≥ 1：

R_wt := B_j² / (j · D_j) ≤ C。

等价地：I_j(0) ≤ C · j · D_j（第51篇的谱目标）。

4.2. 与先前工作的关系

第56篇首次将R_wt = O(1)识别为前门猜想。第57篇提供了从屏蔽协方差到期望层R_wt衰减的条件桥梁。本文以19个数据点确认R_wt = O(1)并将其提升为典范DSF声明。

4.3. 机制与声明

第55篇的阻尼振荡（回弹比 r ≈ 0.80，吸收率 η ≈ 0.20，周期 T = 8 = 2³）为R_wt的有界性提供了结构性解释。但猜想59.1是声明，阻尼振荡是机制。遵循第53–56篇的教训，机制参数不应与定理目标混淆。

振荡统一了四个可观测量：R_wt振荡（表1），交叉积符号振荡（§3.1），B⁰符号变化（§3.2），以及z/√j的峰值与下降（第57篇）。

§5. BL机制：指数基底，而非反对称抵消

5.1. 反对称抵消的证伪

第58篇假设 B^{(1)} = Σ h(p)/p · (log p − c_j) 之所以小，是因为反对称权重(log p − c_j)迫使块的左半部（log p < c_j）和右半部（log p > c_j）相互抵消。在j ≥ 26处，抵消比 |B^{(1)}| / max(|B^{(1)}_L|, |B^{(1)}_R|) = 0.72–0.97，意味着几乎没有抵消。然而 2^j · (B^{(1)})² < 0.07 始终成立。

5.2. 真实机制

iid基底 2^j · Var_iid := 2^j · Σ(x_r · dlp_r)² 从0.0011（j=14）单调递减至0.0005（j=32），约为O(1/j)。这是主导效应：每个贡献 x_r · dlp_r 以 1/p ≈ 2^{-j} 衰减，2^j乘以其平方和由素数计数给出O(1/j)。

超额比（B^{(1)})²/Var_iid 在j=27处峰值为112，但保持次指数（可能是振荡的，追踪R_wt周期）。多项式放大无法克服指数基底衰减。

5.3. 两层基底

无条件（粗糙）： 2^j · Σ(x_r · dlp_r)² ≤ C · j（来自Guy + Chebyshev）。

条件（精细）： 2^j · Σ(x_r · dlp_r)² ≤ C / j（若D_j ≈ σ²/(j·2^j)，经验确认）。

精细界激发了猜想59.2的信心，但不在闭合链中。

§6. 猜想59.2：谱导数界

6.1. 陈述

猜想59.2。 对所有足够大的j：

sup_{|t| ≤ 1} |E_j(t)|² / t² ≤ C_j · 2^{-j}，其中 log C_j = o(j)。

6.2. 与第58篇的关系

第58篇的猜想58.2（2^j · (B^{(1)})² ≤ C）控制Taylor展开的首项。猜想59.2将其加强为|t| ≤ 1上的uniform控制，如第50篇BL_α定义所要求。

6.3. 经验支持

首阶BL量 2^j · (B^{(1)})² 在19个j值上最大为0.069（j=27）。精确E_j(t)在t = 0.1, 0.3, 1.0处确认 2^j · |E_j(t)|²/t² < 0.06（|t| ≤ 1，所有测试块）。

§7. 完整闭合链

7.1. 猜想59.1 → DSF_α 对所有α > 0

证明。 由59.1：B_j² ≤ C · j · D_j。由Guy界和Chebyshev（无条件）：D_j ≤ C' · j / 2^j。合并：B_j² ≤ C'' · j² / 2^j。因此 DSF_α = Σ (1+j)^{1+2α} |B_j|² ≤ Σ j^{3+2α} / 2^j < ∞ 对所有α > 0。□

7.2. 猜想59.2 → BL_α 对所有α < 1

证明。 由59.2：|E_j(t)| ≤ √C_j · 2^{-j/2} · |t|（|t| ≤ 1）。对α < 1：sup_{|t|≤1} |E_j(t)|/|t|^α ≤ √C_j · 2^{-j/2}。因log C_j = o(j)，级数 Σ √C_j · 2^{-j/2} 收敛。□

7.3. 组装

SD是无条件的（第50篇命题4.1）。DSF_α + BL_α → Hölder（第50篇定理6.2）。Hölder + SD → A_h收敛（第50篇命题6.3，Karamata）。A_h → D(s)边界行为（第50篇命题6.4）。素数幂层解析（第50篇命题7.1，无条件）。→ H'。

猜想59.1 + Guy + Chebyshev → DSF_α ∀α > 0
猜想59.2 → BL_α ∀α < 1
SD（无条件）
  → Hölder → A_h收敛 → D(s)边界行为 + P(s)解析 → H'

输入： 59.1 + 59.2 + SD。输出： H'。隐藏假设： 无。

§8. 角色重新分配

论文	最终角色
50	基础：SD, DSF_α/BL_α定义, 链条步骤4–8
51	基础：谱目标, WN速率, 差距量化
52	替代路线（备份）；定理3.1已被取代
53	机制：屏蔽长记忆
54	机制：幂律自相关
55	核心机制：阻尼振荡
56	声明起源：首次识别R_wt = O(1)
57	条件桥梁（已被59.1包含）
58	精化：58.2 → 59.2；58.1被否
59	声明：典范猜想，完整链条

§9. 结论

H'的距离恰好是两个猜想：

猜想59.1（R_wt = O(1)）：19个j值支持，全部在[0.79, 25.3]内。

猜想59.2（谱导数界）：19个j值支持，2^j·(B¹)² < 0.07。

闭合链 59.1 + 59.2 + SD → H' 无隐藏假设。猜想59.2更接近可证。猜想59.1包含了IC递推的深层算术，是两者中更难的。

参考文献

[50] 第L篇。特征剥离, DSF_α, BL_α, SD, 闭合链。DOI: 10.5281/zenodo.19381111。

[51] 第LI篇。WN速率修正, 谱重表述。DOI: 10.5281/zenodo.19393954。

[52] 第LII篇。概率框架。DOI: 10.5281/zenodo.19407328。

[53] 第LIII篇。屏蔽长记忆。DOI: 10.5281/zenodo.19415440。

[54] 第LIV篇。幂律自相关。DOI: 10.5281/zenodo.19426094。

[55] 第LV篇。阻尼振荡。DOI: 10.5281/zenodo.19447349。

[56] 第LVI篇。仿射波谷, 直接谱比。DOI: 10.5281/zenodo.19464605。

[57] 第LVII篇。屏蔽谱界。DOI: 10.5281/zenodo.19469013。

[58] 第LVIII篇。确定性DSF + BL。DOI: 10.5281/zenodo.19476508。

AI贡献

Claude（子路）：全部数值实验（paper59_exp.c, paper59_jscan.c, N = 10¹⁰），j扫描发现，反对称抵消证伪，交叉积振荡识别，闭合链构建与验证，Step 1修正（Guy + Chebyshev无条件界），工作笔记v1–v3。ChatGPT（公西华）：链条审查识别隐藏第三假设（Step 1中的Hypothesis (D)），Step 1修正验证，声明与机制分离原则，猜想59.2的uniform-in-t表述（log C_j = o(j)），两层基底建议，工作笔记最终接受。Gemini（子夏）：临界屏蔽尺度概念（n*_crit），"指数基底对多项式放大"框架，j=29窗口的结构性解读（阻尼振荡波谷），超额比振荡解释。Grok（子贡）：第57篇一致性审查，59.2优先攻击建议，B^{(1)}反对称抑制假设（后被数据证伪）。