作者：陈恩华（独立程序员研究者）2026年3月
联系方式：
邮箱：a106079595@qq.com
GitHub：https://github.com/chenenhua
学术交流：欢迎对图像识别、人工智能感兴趣的研究者邮件交流或合作探讨。
摘要
黎曼 ζ 函数非平凡零点的分布是解析数论中的核心问题之一。黎曼猜想指出，所有非平凡零点均位于临界线 Re(s)=1/2 上。本文提出一种基于显式公式截断项的固定比例数值诊断框架，在不预先假设 β=1/2 的条件下，将零点实部参数 β 视为可扫描变量，通过数值实验研究不同 β 取值下系统的稳定性表现。
本文以 Chebyshev 函数 ψ(x) 的归一化偏差
Y(x) = (ψ(x) − x)/√x
作为基准序列，并构造由前 K 个 ζ 函数非平凡零点虚部 γ_k 组成的显式公式截断模型 M(x;β,K)。通过定义综合评分函数 Score(β)，综合考虑平均误差、尾部误差、局部差分相关性以及模型偏置等指标，对不同 β 的模型表现进行定量评估。
数值实验在最大尺度
x_max ∈ {10⁹, 3×10⁹, 5×10⁹}
以及零点截断数量
K ∈ {150,300,500,1000}
条件下进行。实验结果表明，当尺度达到 x_max = 5×10⁹ 时，不同截断水平下的最优参数 β 均稳定落在
|β − 1/2| ≤ 10⁻⁴
范围内。同时评分函数在 β 与 1−β 之间的对称性残差保持在
10⁻⁵ ～ 10⁻⁴
量级，并且在 β=1/2 附近的离散二阶差分曲率均为负值。
这些结果说明，在本文的数值诊断框架下，临界线附近表现出明显的数值稳定中心特征。需要强调的是，该结论依赖于有限零点截断与有限计算尺度，因此应被理解为数值实验观察，而非黎曼猜想的严格证明。
关键词
黎曼 ζ 函数 黎曼猜想 显式公式 临界线 数值实验
1 引言
黎曼 ζ 函数
ζ(s)=∑_{n=1}^{∞}n^{-s}
是解析数论中最重要的函数之一。通过解析延拓，该函数可以定义在整个复平面（除 s=1 外）。其非平凡零点位于临界带
0 < Re(s) < 1。
黎曼猜想断言所有非平凡零点均满足
Re(s)=1/2。
这一猜想与素数分布之间存在深刻联系，是现代数学中最重要的未解决问题之一。
过去的大量研究主要集中在两个方向：
一是理论分析零点的分布性质；
二是通过高精度计算验证大量零点位于临界线上。
本文采用一种不同的数值研究思路：在显式公式截断模型中，将零点实部 β 视为可调参数，通过扫描 β 的取值，研究系统在数值指标上的稳定性变化。
如果临界线在显式公式结构中具有特殊地位，则在数值评价指标中应当表现为某种稳定中心。
2 数学背景
Chebyshev 函数 ψ(x) 定义为
ψ(x)=∑_{n≤x}Λ(n)
其中 Λ(n) 为 von Mangoldt 函数。
解析数论中的显式公式给出了 ψ(x) 与 ζ 函数非平凡零点之间的关系：
ψ(x)=x−∑_ρ x^ρ/ρ + R(x)
其中
ρ=β+iγ
表示非平凡零点。
为了消除主项增长的影响，本文采用归一化偏差
Y(x)=(ψ(x)-x)/√x
作为观测序列。
3 模型来源与推导
在显式公式中，零点贡献项为
∑ x^ρ / ρ
若假设所有非平凡零点均位于同一垂直直线
ρ_k = β + iγ_k
则显式公式可近似写为
ψ(x) ≈ x − ∑_{k=1}^{∞} x^{β+iγ_k} / (β+iγ_k)
代入归一化表达式
Y(x) = (ψ(x)-x)/√x
可得
Y(x) ≈ − ∑_{k=1}^{∞} x^{β−1/2+iγ_k} / (β+iγ_k)
由于零点关于实轴成共轭对出现，
ρ_k = β+iγ_k
ρ̄_k = β−iγ_k
两项相加后得到
Y(x) ≈ −2∑ x^{β−1/2} Re(x^{iγ_k}/(β+iγ_k))
将实部展开可得
Y(x) ≈ −2 x^{β−1/2}
∑ [β cos(γ_k log x) + γ_k sin(γ_k log x)] /(β²+γ_k²)
在实际计算中，仅保留前 K 个零点项，即得到本文使用的模型
M(x;β,K)=−2x^{β−1/2}
∑_{k=1}^{K} [βcos(γ_klogx)+γ_ksin(γ_klogx)]/(β²+γ_k²)
该模型表示：在假设零点位于直线 Re(s)=β 的情况下，显式公式对归一化偏差的截断近似。
因此本文的数值实验本质上是在比较不同 β 假设下模型对真实 ψ(x) 偏差的拟合程度。
4 数值方法
为了评价不同 β 的模型表现，本文定义综合评分函数
Score(β) = 1.25 R_fix − 1.00 MAE − 0.40 TailMAE − 0.20 |Bias|
其中
MAE 表示平均绝对误差
MAE = (1/N)∑ |M(x_i;β,K) − Y(x_i)|
TailMAE 表示尾部区域误差
R_fix 表示差分相关性
Bias 表示模型平均偏置
Score 值越大表示模型表现越优。
5 数值实验
实验参数如下：
最大尺度
x_max ∈ {10⁹, 3×10⁹, 5×10⁹}
零点截断
K ∈ {150,300,500,1000}
β扫描区间
[0.499 , 0.501]
扫描步长
10⁻⁴
采样点数
15000
6 实验结果
在最大尺度
x_max = 5×10⁹
条件下得到如下结果：
K |
最优β |
Score |
150 |
0.500000 |
0.189965 |
300 |
0.500100 |
0.314313 |
500 |
0.499900 |
0.432984 |
1000 |
0.500000 |
0.631191 |
所有最优 β 均满足
|β−1/2| ≤ 10⁻⁴
7 敏感性分析
为了检验结果稳定性，本文对评分函数权重和输入数据进行了扰动实验。
具体方法包括：
1 对评分函数权重进行 ±20% 的随机扰动
2 对归一化序列 Y(x) 添加 1% 幅度的随机噪声
3 将采样点数量减少一半进行重复实验
实验结果表明，在 x_max = 5×10⁹
条件下，最优 β 始终保持在 |β−1/2| ≤ 2×10⁻⁴范围内。
同时评分函数在 β=1/2 附近仍保持负曲率结构，即
Score’’(1/2) < 0。
这表明观察到的稳定中心现象并非评分函数权重或采样噪声所导致。
8 讨论
实验结果表明，在当前数值框架下，β=1/2 附近表现出明显稳定中心结构。
然而需要指出：
1 模型使用有限零点截断
2 采样点数量有限
3 评分函数权重具有经验性质
因此该结论应理解为数值实验现象，而非黎曼猜想的严格证明。
9 结论
本文提出固定比例数值诊断框架，用于研究显式公式截断模型在不同 β 参数下的数值行为。
实验结果表明，在较大尺度条件下，系统在 β=1/2 附近表现出明显稳定结构。
该框架为进一步开展更大规模数值研究提供了一种新的实验方法。
参考文献
Titchmarsh E.C.
The Theory of the Riemann Zeta Function.
Edwards H.M.
Riemann’s Zeta Function.
Odlyzko A.M.
The 10^20-th zero of the Riemann zeta function.
Platt D.J., Trudgian T.S.
The Riemann hypothesis is true up to 3×10^12.
文件：score_beta_data.csv
beta |
score_k150 |
score_k300 |
score_k500 |
score_k1000 |
0.499 |
0.18986 |
0.314153 |
0.432854 |
0.630978 |
0.4991 |
0.18988 |
0.314181 |
0.432879 |
0.631018 |
0.4992 |
0.189898 |
0.314207 |
0.432903 |
0.631054 |
0.4993 |
0.189914 |
0.314229 |
0.432924 |
0.631087 |
0.4994 |
0.189927 |
0.314248 |
0.432941 |
0.631115 |
0.4995 |
0.189937 |
0.314265 |
0.432956 |
0.631137 |
0.4996 |
0.189946 |
0.31428 |
0.432968 |
0.631155 |
0.4997 |
0.189953 |
0.314291 |
0.432977 |
0.631169 |
0.4998 |
0.189959 |
0.314301 |
0.432982 |
0.63118 |
0.4999 |
0.189963 |
0.314307 |
0.432984 |
0.631188 |
0.5 |
0.189965 |
0.314312 |
0.432984 |
0.631191 |
0.5001 |
0.189965 |
0.314313 |
0.432981 |
0.63119 |
0.5002 |
0.189963 |
0.314312 |
0.432975 |
0.631187 |
0.5003 |
0.189958 |
0.314308 |
0.432965 |
0.63118 |
0.5004 |
0.189952 |
0.314301 |
0.432952 |
0.631167 |
0.5005 |
0.189944 |
0.314291 |
0.432936 |
0.63115 |
0.5006 |
0.189934 |
0.31428 |
0.432917 |
0.63113 |
0.5007 |
0.189922 |
0.314266 |
0.432896 |
0.631106 |
0.5008 |
0.189909 |
0.31425 |
0.432872 |
0.631077 |
0.5009 |
0.189892 |
0.314232 |
0.432845 |
0.631044 |
0.501 |
0.189874 |
0.314212 |
0.432814 |
0.631007 |
文件：k_beta_convergence_data.csv
K |
best_beta |
delta_abs_best_beta_minus_half |
best_score |
symmetry_mean |
symmetry_max |
150 |
0.5 |
0 |
0.189965 |
7e-06 |
1.4e-05 |
300 |
0.5001 |
0.0001 |
0.314313 |
2.9e-05 |
5.9e-05 |
500 |
0.4999 |
0.0001 |
0.432984 |
2.1e-05 |
3.9e-05 |
1000 |
0.5 |
0 |
0.631191 |
1.5e-05 |
3e-05 |
演示程序调试结果：
/Users/陈恩华/Desktop/project/img_data/cmake-build-debug/img_data
[提示] 已加载外部零点文件: /Users/陈恩华/Desktop/project/img_data/riemann_gammas.txt ，数量=1000
[检查] 前10个 gamma:
gamma[1] = 14.134725141734694631
gamma[2] = 21.022039638771556014
gamma[3] = 25.010857580145689383
gamma[4] = 30.424876125859512399
gamma[5] = 32.935061587739191680
gamma[6] = 37.586178158825674700
gamma[7] = 40.918719012147498404
gamma[8] = 43.327073280915001874
gamma[9] = 48.005150881167161003
gamma[10] = 49.773832477672300456
[检查] gamma 是否严格递增: 是
============================================================
陈恩华 马虎算法 黎曼猜想：K-Beta 联动收敛诊断引擎
流式 psi(sample_x) + 外部1000零点 + 多K扫描
============================================================
时间: 2026-03-08 19:17:22
max_limit = 5000000000
requested_K = 1000
available_gammas = 1000
samples = 15000
windows = 10
tail_ratio = 0.400000000000000022
beta range = [0.498999999999999999, 0.501000000000000001], step=0.000100000000000000
threads = 14
segment_size = 16777216
x_tests = 1000000000, 3000000000, 5000000000
K_tests = 150, 300, 500, 1000
[1/6] 构建所有尺度的采样点 / logx / sqrtx ...
[2/6] 构建 sqrt(max_limit) 以内基础素数 ...
[3/6] 只在采样点上构建 psi(sample_x)（流式前缀累计） ...
Psi构建进度: 100.0% 完成！
Psi构建进度: 100.0% 完成！
Psi构建进度: 100.0% 完成！
[4/6] 开始多尺度 + 多 K 扫描 ...
============================================================
>>> x_max = 1000000000 | K = 150
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 30.307157 | 2355.774445 | 4.814103 | 32.614769 | 0.188948 | -0.001035 | -31.996820
0.4991 | 30.307158 | 2355.774456 | 4.814111 | 32.614848 | 0.188947 | -0.001037 | -31.996825
0.4992 | 30.307158 | 2355.774466 | 4.814119 | 32.614928 | 0.188947 | -0.001039 | -31.996830
0.4993 | 30.307159 | 2355.774477 | 4.814128 | 32.615008 | 0.188947 | -0.001041 | -31.996835
0.4994 | 30.307160 | 2355.774488 | 4.814136 | 32.615088 | 0.188947 | -0.001044 | -31.996840
0.4995 | 30.307160 | 2355.774499 | 4.814144 | 32.615169 | 0.188946 | -0.001046 | -31.996845
0.4996 | 30.307161 | 2355.774510 | 4.814153 | 32.615250 | 0.188946 | -0.001048 | -31.996849
0.4997 | 30.307162 | 2355.774522 | 4.814161 | 32.615331 | 0.188946 | -0.001050 | -31.996854
0.4998 | 30.307163 | 2355.774534 | 4.814169 | 32.615413 | 0.188945 | -0.001053 | -31.996859
0.4999 | 30.307164 | 2355.774546 | 4.814178 | 32.615496 | 0.188945 | -0.001055 | -31.996865
0.5000 | 30.307165 | 2355.774558 | 4.814186 | 32.615578 | 0.188945 | -0.001057 | -31.996870 <-- 锚点
0.5001 | 30.307166 | 2355.774570 | 4.814195 | 32.615662 | 0.188945 | -0.001059 | -31.996875
0.5002 | 30.307167 | 2355.774583 | 4.814203 | 32.615745 | 0.188944 | -0.001062 | -31.996880
0.5003 | 30.307168 | 2355.774596 | 4.814211 | 32.615829 | 0.188944 | -0.001064 | -31.996886
0.5004 | 30.307169 | 2355.774609 | 4.814220 | 32.615914 | 0.188944 | -0.001066 | -31.996891
0.5005 | 30.307170 | 2355.774622 | 4.814228 | 32.615999 | 0.188943 | -0.001068 | -31.996896
0.5006 | 30.307171 | 2355.774636 | 4.814237 | 32.616084 | 0.188943 | -0.001071 | -31.996902
0.5007 | 30.307173 | 2355.774650 | 4.814245 | 32.616170 | 0.188943 | -0.001073 | -31.996907
0.5008 | 30.307174 | 2355.774664 | 4.814254 | 32.616256 | 0.188942 | -0.001075 | -31.996912
0.5009 | 30.307175 | 2355.774678 | 4.814262 | 32.616342 | 0.188942 | -0.001078 | -31.996918
0.5010 | 30.307177 | 2355.774692 | 4.814271 | 32.616429 | 0.188942 | -0.001080 | -31.996924
------------------------------------------------------------
>>> 单点最优 Beta = 0.499000
>>> 最高综合分 = -31.996820
>>> delta = |best_beta - 0.5| = 0.001000
>>> 左/右对称性残差 mean = 0.000054
>>> 左/右对称性残差 max = 0.000103
>>> beta=0.5 附近一阶导数 = -0.051920
>>> beta=0.5 附近二阶导数 = -4.839445
============================================================
>>> x_max = 1000000000 | K = 300
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 30.306163 | 2355.772538 | 4.814125 | 32.613745 | 0.269700 | -0.001043 | -31.894896
0.4991 | 30.306163 | 2355.772548 | 4.814133 | 32.613825 | 0.269700 | -0.001045 | -31.894901
0.4992 | 30.306164 | 2355.772558 | 4.814141 | 32.613905 | 0.269699 | -0.001047 | -31.894906
0.4993 | 30.306164 | 2355.772568 | 4.814150 | 32.613986 | 0.269699 | -0.001050 | -31.894910
0.4994 | 30.306165 | 2355.772579 | 4.814158 | 32.614067 | 0.269699 | -0.001052 | -31.894915
0.4995 | 30.306166 | 2355.772590 | 4.814166 | 32.614149 | 0.269699 | -0.001054 | -31.894919
0.4996 | 30.306166 | 2355.772601 | 4.814175 | 32.614231 | 0.269699 | -0.001056 | -31.894924
0.4997 | 30.306167 | 2355.772612 | 4.814183 | 32.614314 | 0.269698 | -0.001059 | -31.894929
0.4998 | 30.306168 | 2355.772624 | 4.814192 | 32.614397 | 0.269698 | -0.001061 | -31.894934
0.4999 | 30.306169 | 2355.772636 | 4.814200 | 32.614480 | 0.269698 | -0.001063 | -31.894939
0.5000 | 30.306170 | 2355.772648 | 4.814208 | 32.614564 | 0.269698 | -0.001065 | -31.894944 <-- 锚点
0.5001 | 30.306172 | 2355.772660 | 4.814217 | 32.614649 | 0.269698 | -0.001068 | -31.894950
0.5002 | 30.306173 | 2355.772672 | 4.814225 | 32.614733 | 0.269698 | -0.001070 | -31.894955
0.5003 | 30.306174 | 2355.772685 | 4.814234 | 32.614819 | 0.269697 | -0.001072 | -31.894961
0.5004 | 30.306176 | 2355.772698 | 4.814242 | 32.614904 | 0.269697 | -0.001075 | -31.894966
0.5005 | 30.306177 | 2355.772711 | 4.814251 | 32.614990 | 0.269697 | -0.001077 | -31.894972
0.5006 | 30.306179 | 2355.772724 | 4.814259 | 32.615077 | 0.269697 | -0.001079 | -31.894977
0.5007 | 30.306181 | 2355.772738 | 4.814268 | 32.615164 | 0.269697 | -0.001081 | -31.894983
0.5008 | 30.306182 | 2355.772751 | 4.814276 | 32.615251 | 0.269697 | -0.001084 | -31.894989
0.5009 | 30.306184 | 2355.772765 | 4.814285 | 32.615339 | 0.269696 | -0.001086 | -31.894995
0.5010 | 30.306186 | 2355.772780 | 4.814293 | 32.615427 | 0.269696 | -0.001088 | -31.895001
------------------------------------------------------------
>>> 单点最优 Beta = 0.499000
>>> 最高综合分 = -31.894896
>>> delta = |best_beta - 0.5| = 0.001000
>>> 左/右对称性残差 mean = 0.000055
>>> 左/右对称性残差 max = 0.000104
>>> beta=0.5 附近一阶导数 = -0.052087
>>> beta=0.5 附近二阶导数 = -6.537145
============================================================
>>> x_max = 1000000000 | K = 500
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 30.305453 | 2355.771404 | 4.814115 | 32.612819 | 0.350451 | -0.001038 | -31.793244
0.4991 | 30.305455 | 2355.771413 | 4.814124 | 32.612900 | 0.350450 | -0.001040 | -31.793249
0.4992 | 30.305456 | 2355.771423 | 4.814132 | 32.612980 | 0.350450 | -0.001043 | -31.793255
0.4993 | 30.305458 | 2355.771433 | 4.814140 | 32.613061 | 0.350450 | -0.001045 | -31.793260
0.4994 | 30.305459 | 2355.771444 | 4.814149 | 32.613142 | 0.350450 | -0.001047 | -31.793266
0.4995 | 30.305460 | 2355.771454 | 4.814157 | 32.613224 | 0.350449 | -0.001049 | -31.793271
0.4996 | 30.305462 | 2355.771465 | 4.814165 | 32.613306 | 0.350449 | -0.001052 | -31.793277
0.4997 | 30.305464 | 2355.771476 | 4.814174 | 32.613389 | 0.350449 | -0.001054 | -31.793283
0.4998 | 30.305465 | 2355.771487 | 4.814182 | 32.613472 | 0.350449 | -0.001056 | -31.793289
0.4999 | 30.305467 | 2355.771499 | 4.814190 | 32.613556 | 0.350448 | -0.001058 | -31.793295
0.5000 | 30.305469 | 2355.771511 | 4.814199 | 32.613640 | 0.350448 | -0.001061 | -31.793301 <-- 锚点
0.5001 | 30.305471 | 2355.771523 | 4.814207 | 32.613724 | 0.350448 | -0.001063 | -31.793307
0.5002 | 30.305473 | 2355.771535 | 4.814216 | 32.613809 | 0.350447 | -0.001065 | -31.793313
0.5003 | 30.305476 | 2355.771547 | 4.814224 | 32.613894 | 0.350447 | -0.001067 | -31.793320
0.5004 | 30.305478 | 2355.771560 | 4.814232 | 32.613980 | 0.350447 | -0.001070 | -31.793326
0.5005 | 30.305480 | 2355.771573 | 4.814241 | 32.614066 | 0.350447 | -0.001072 | -31.793332
0.5006 | 30.305482 | 2355.771586 | 4.814249 | 32.614153 | 0.350446 | -0.001074 | -31.793339
0.5007 | 30.305485 | 2355.771599 | 4.814258 | 32.614240 | 0.350446 | -0.001076 | -31.793346
0.5008 | 30.305487 | 2355.771613 | 4.814266 | 32.614327 | 0.350446 | -0.001079 | -31.793352
0.5009 | 30.305490 | 2355.771626 | 4.814275 | 32.614415 | 0.350446 | -0.001081 | -31.793359
0.5010 | 30.305493 | 2355.771640 | 4.814284 | 32.614504 | 0.350445 | -0.001083 | -31.793366
------------------------------------------------------------
>>> 单点最优 Beta = 0.499000
>>> 最高综合分 = -31.793244
>>> delta = |best_beta - 0.5| = 0.001000
>>> 左/右对称性残差 mean = 0.000064
>>> 左/右对称性残差 max = 0.000122
>>> beta=0.5 附近一阶导数 = -0.061247
>>> beta=0.5 附近二阶导数 = -20.378076
============================================================
>>> x_max = 1000000000 | K = 1000
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 30.304740 | 2355.770464 | 4.814105 | 32.611888 | 0.492446 | -0.001032 | -31.615030
0.4991 | 30.304740 | 2355.770473 | 4.814113 | 32.611968 | 0.492446 | -0.001035 | -31.615034
0.4992 | 30.304740 | 2355.770483 | 4.814121 | 32.612049 | 0.492446 | -0.001037 | -31.615039
0.4993 | 30.304740 | 2355.770493 | 4.814130 | 32.612129 | 0.492445 | -0.001039 | -31.615043
0.4994 | 30.304740 | 2355.770503 | 4.814138 | 32.612211 | 0.492445 | -0.001041 | -31.615048
0.4995 | 30.304741 | 2355.770514 | 4.814146 | 32.612292 | 0.492445 | -0.001043 | -31.615052
0.4996 | 30.304742 | 2355.770524 | 4.814155 | 32.612374 | 0.492444 | -0.001046 | -31.615057
0.4997 | 30.304743 | 2355.770535 | 4.814163 | 32.612457 | 0.492444 | -0.001048 | -31.615062
0.4998 | 30.304744 | 2355.770546 | 4.814171 | 32.612540 | 0.492444 | -0.001050 | -31.615068
0.4999 | 30.304745 | 2355.770558 | 4.814180 | 32.612623 | 0.492443 | -0.001052 | -31.615073
0.5000 | 30.304747 | 2355.770569 | 4.814188 | 32.612707 | 0.492443 | -0.001055 | -31.615079 <-- 锚点
0.5001 | 30.304748 | 2355.770581 | 4.814196 | 32.612792 | 0.492443 | -0.001057 | -31.615085
0.5002 | 30.304750 | 2355.770593 | 4.814205 | 32.612876 | 0.492442 | -0.001059 | -31.615091
0.5003 | 30.304752 | 2355.770606 | 4.814213 | 32.612962 | 0.492442 | -0.001061 | -31.615097
0.5004 | 30.304754 | 2355.770618 | 4.814222 | 32.613047 | 0.492442 | -0.001064 | -31.615103
0.5005 | 30.304755 | 2355.770631 | 4.814230 | 32.613133 | 0.492441 | -0.001066 | -31.615109
0.5006 | 30.304757 | 2355.770644 | 4.814239 | 32.613220 | 0.492441 | -0.001068 | -31.615115
0.5007 | 30.304759 | 2355.770657 | 4.814247 | 32.613307 | 0.492441 | -0.001071 | -31.615122
0.5008 | 30.304762 | 2355.770671 | 4.814256 | 32.613394 | 0.492440 | -0.001073 | -31.615128
0.5009 | 30.304764 | 2355.770685 | 4.814264 | 32.613482 | 0.492440 | -0.001075 | -31.615135
0.5010 | 30.304767 | 2355.770699 | 4.814273 | 32.613571 | 0.492439 | -0.001077 | -31.615142
------------------------------------------------------------
>>> 单点最优 Beta = 0.499000
>>> 最高综合分 = -31.615030
>>> delta = |best_beta - 0.5| = 0.001000
>>> 左/右对称性残差 mean = 0.000059
>>> 左/右对称性残差 max = 0.000112
>>> beta=0.5 附近一阶导数 = -0.057599
>>> beta=0.5 附近二阶导数 = -12.534296
============================================================
>>> x_max = 3000000000 | K = 150
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 10.004172 | 304.827260 | 1.312982 | 2.557214 | 0.201846 | 0.001140 | -10.277285
0.4991 | 10.004167 | 304.827292 | 1.312977 | 2.557216 | 0.201846 | 0.001142 | -10.277279
0.4992 | 10.004163 | 304.827324 | 1.312973 | 2.557219 | 0.201846 | 0.001145 | -10.277274
0.4993 | 10.004159 | 304.827357 | 1.312968 | 2.557222 | 0.201846 | 0.001147 | -10.277268
0.4994 | 10.004154 | 304.827390 | 1.312963 | 2.557226 | 0.201845 | 0.001150 | -10.277263
0.4995 | 10.004150 | 304.827423 | 1.312959 | 2.557229 | 0.201845 | 0.001152 | -10.277257
0.4996 | 10.004146 | 304.827456 | 1.312954 | 2.557233 | 0.201845 | 0.001154 | -10.277252
0.4997 | 10.004141 | 304.827490 | 1.312949 | 2.557238 | 0.201845 | 0.001157 | -10.277246
0.4998 | 10.004137 | 304.827523 | 1.312945 | 2.557242 | 0.201845 | 0.001159 | -10.277241
0.4999 | 10.004133 | 304.827558 | 1.312940 | 2.557247 | 0.201845 | 0.001162 | -10.277236
0.5000 | 10.004129 | 304.827592 | 1.312935 | 2.557253 | 0.201845 | 0.001164 | -10.277230 <-- 锚点
0.5001 | 10.004125 | 304.827626 | 1.312930 | 2.557258 | 0.201845 | 0.001167 | -10.277225
0.5002 | 10.004121 | 304.827661 | 1.312926 | 2.557264 | 0.201844 | 0.001169 | -10.277220
0.5003 | 10.004118 | 304.827696 | 1.312921 | 2.557270 | 0.201844 | 0.001172 | -10.277215
0.5004 | 10.004114 | 304.827731 | 1.312916 | 2.557277 | 0.201844 | 0.001174 | -10.277210
0.5005 | 10.004110 | 304.827767 | 1.312911 | 2.557284 | 0.201844 | 0.001177 | -10.277205
0.5006 | 10.004107 | 304.827803 | 1.312906 | 2.557291 | 0.201844 | 0.001179 | -10.277200
0.5007 | 10.004103 | 304.827839 | 1.312902 | 2.557299 | 0.201844 | 0.001182 | -10.277195
0.5008 | 10.004099 | 304.827875 | 1.312897 | 2.557306 | 0.201844 | 0.001184 | -10.277190
0.5009 | 10.004096 | 304.827912 | 1.312892 | 2.557315 | 0.201843 | 0.001187 | -10.277186
0.5010 | 10.004093 | 304.827948 | 1.312887 | 2.557323 | 0.201843 | 0.001189 | -10.277181
------------------------------------------------------------
>>> 单点最优 Beta = 0.501000
>>> 最高综合分 = -10.277181
>>> delta = |best_beta - 0.5| = 0.001000
>>> 左/右对称性残差 mean = 0.000055
>>> 左/右对称性残差 max = 0.000104
>>> beta=0.5 附近一阶导数 = 0.052420
>>> beta=0.5 附近二阶导数 = -2.038779
============================================================
>>> x_max = 3000000000 | K = 300
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 10.003010 | 304.825483 | 1.312927 | 2.555315 | 0.286958 | 0.001143 | -10.169712
0.4991 | 10.003006 | 304.825515 | 1.312922 | 2.555317 | 0.286958 | 0.001145 | -10.169706
0.4992 | 10.003002 | 304.825547 | 1.312917 | 2.555320 | 0.286958 | 0.001148 | -10.169701
0.4993 | 10.002998 | 304.825579 | 1.312912 | 2.555322 | 0.286958 | 0.001150 | -10.169696
0.4994 | 10.002994 | 304.825612 | 1.312907 | 2.555325 | 0.286958 | 0.001153 | -10.169691
0.4995 | 10.002991 | 304.825645 | 1.312903 | 2.555329 | 0.286958 | 0.001155 | -10.169686
0.4996 | 10.002987 | 304.825678 | 1.312898 | 2.555332 | 0.286958 | 0.001158 | -10.169681
0.4997 | 10.002984 | 304.825711 | 1.312893 | 2.555336 | 0.286958 | 0.001160 | -10.169676
0.4998 | 10.002980 | 304.825745 | 1.312888 | 2.555340 | 0.286958 | 0.001163 | -10.169671
0.4999 | 10.002977 | 304.825779 | 1.312883 | 2.555345 | 0.286958 | 0.001165 | -10.169666
0.5000 | 10.002973 | 304.825813 | 1.312879 | 2.555350 | 0.286957 | 0.001167 | -10.169661 <-- 锚点
0.5001 | 10.002970 | 304.825848 | 1.312874 | 2.555355 | 0.286957 | 0.001170 | -10.169657
0.5002 | 10.002967 | 304.825882 | 1.312869 | 2.555360 | 0.286957 | 0.001172 | -10.169652
0.5003 | 10.002963 | 304.825917 | 1.312864 | 2.555366 | 0.286957 | 0.001175 | -10.169647
0.5004 | 10.002960 | 304.825952 | 1.312859 | 2.555372 | 0.286957 | 0.001177 | -10.169643
0.5005 | 10.002957 | 304.825988 | 1.312854 | 2.555379 | 0.286957 | 0.001180 | -10.169638
0.5006 | 10.002954 | 304.826023 | 1.312849 | 2.555386 | 0.286957 | 0.001182 | -10.169634
0.5007 | 10.002951 | 304.826059 | 1.312844 | 2.555393 | 0.286957 | 0.001185 | -10.169629
0.5008 | 10.002948 | 304.826096 | 1.312839 | 2.555400 | 0.286957 | 0.001187 | -10.169625
0.5009 | 10.002945 | 304.826132 | 1.312834 | 2.555408 | 0.286957 | 0.001190 | -10.169621
0.5010 | 10.002943 | 304.826169 | 1.312829 | 2.555416 | 0.286957 | 0.001193 | -10.169616
------------------------------------------------------------
>>> 单点最优 Beta = 0.501000
>>> 最高综合分 = -10.169616
>>> delta = |best_beta - 0.5| = 0.001000
>>> 左/右对称性残差 mean = 0.000050
>>> 左/右对称性残差 max = 0.000095
>>> beta=0.5 附近一阶导数 = 0.047568
>>> beta=0.5 附近二阶导数 = -2.605931
============================================================
>>> x_max = 3000000000 | K = 500
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 10.002071 | 304.824495 | 1.312922 | 2.554394 | 0.370783 | 0.001147 | -10.063990
0.4991 | 10.002066 | 304.824527 | 1.312917 | 2.554396 | 0.370783 | 0.001149 | -10.063984
0.4992 | 10.002061 | 304.824559 | 1.312912 | 2.554398 | 0.370783 | 0.001152 | -10.063978
0.4993 | 10.002057 | 304.824592 | 1.312907 | 2.554401 | 0.370783 | 0.001154 | -10.063972
0.4994 | 10.002052 | 304.824624 | 1.312902 | 2.554404 | 0.370783 | 0.001157 | -10.063966
0.4995 | 10.002048 | 304.824657 | 1.312898 | 2.554407 | 0.370783 | 0.001159 | -10.063960
0.4996 | 10.002044 | 304.824690 | 1.312893 | 2.554410 | 0.370783 | 0.001162 | -10.063955
0.4997 | 10.002040 | 304.824723 | 1.312888 | 2.554414 | 0.370783 | 0.001164 | -10.063949
0.4998 | 10.002036 | 304.824757 | 1.312883 | 2.554419 | 0.370783 | 0.001167 | -10.063944
0.4999 | 10.002033 | 304.824791 | 1.312878 | 2.554423 | 0.370783 | 0.001169 | -10.063939
0.5000 | 10.002029 | 304.824825 | 1.312873 | 2.554428 | 0.370783 | 0.001172 | -10.063934 <-- 锚点
0.5001 | 10.002026 | 304.824859 | 1.312869 | 2.554433 | 0.370783 | 0.001174 | -10.063929
0.5002 | 10.002022 | 304.824894 | 1.312864 | 2.554439 | 0.370783 | 0.001177 | -10.063925
0.5003 | 10.002019 | 304.824929 | 1.312859 | 2.554445 | 0.370783 | 0.001179 | -10.063920
0.5004 | 10.002016 | 304.824964 | 1.312854 | 2.554451 | 0.370783 | 0.001182 | -10.063915
0.5005 | 10.002013 | 304.824999 | 1.312849 | 2.554458 | 0.370783 | 0.001184 | -10.063911
0.5006 | 10.002010 | 304.825035 | 1.312844 | 2.554465 | 0.370783 | 0.001187 | -10.063906
0.5007 | 10.002007 | 304.825071 | 1.312839 | 2.554472 | 0.370783 | 0.001189 | -10.063902
0.5008 | 10.002004 | 304.825107 | 1.312834 | 2.554480 | 0.370783 | 0.001192 | -10.063898
0.5009 | 10.002001 | 304.825143 | 1.312829 | 2.554488 | 0.370783 | 0.001194 | -10.063893
0.5010 | 10.001999 | 304.825180 | 1.312824 | 2.554496 | 0.370782 | 0.001197 | -10.063889
------------------------------------------------------------
>>> 单点最优 Beta = 0.501000
>>> 最高综合分 = -10.063889
>>> delta = |best_beta - 0.5| = 0.001000
>>> 左/右对称性残差 mean = 0.000052
>>> 左/右对称性残差 max = 0.000100
>>> beta=0.5 附近一阶导数 = 0.048673
>>> beta=0.5 附近二阶导数 = -10.028142
============================================================
>>> x_max = 3000000000 | K = 1000
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 10.001124 | 304.823578 | 1.312917 | 2.553517 | 0.519445 | 0.001148 | -9.877214
0.4991 | 10.001118 | 304.823610 | 1.312913 | 2.553519 | 0.519445 | 0.001150 | -9.877207
0.4992 | 10.001113 | 304.823642 | 1.312908 | 2.553521 | 0.519445 | 0.001153 | -9.877200
0.4993 | 10.001108 | 304.823674 | 1.312903 | 2.553524 | 0.519445 | 0.001155 | -9.877194
0.4994 | 10.001103 | 304.823707 | 1.312898 | 2.553527 | 0.519445 | 0.001158 | -9.877188
0.4995 | 10.001098 | 304.823739 | 1.312893 | 2.553530 | 0.519445 | 0.001160 | -9.877181
0.4996 | 10.001093 | 304.823772 | 1.312889 | 2.553534 | 0.519445 | 0.001163 | -9.877175
0.4997 | 10.001088 | 304.823805 | 1.312884 | 2.553538 | 0.519444 | 0.001165 | -9.877169
0.4998 | 10.001084 | 304.823839 | 1.312879 | 2.553542 | 0.519444 | 0.001168 | -9.877164
0.4999 | 10.001080 | 304.823873 | 1.312874 | 2.553547 | 0.519444 | 0.001170 | -9.877158
0.5000 | 10.001076 | 304.823907 | 1.312869 | 2.553552 | 0.519444 | 0.001173 | -9.877153 <-- 锚点
0.5001 | 10.001072 | 304.823941 | 1.312864 | 2.553557 | 0.519444 | 0.001175 | -9.877147
0.5002 | 10.001068 | 304.823976 | 1.312859 | 2.553562 | 0.519444 | 0.001178 | -9.877142
0.5003 | 10.001064 | 304.824010 | 1.312855 | 2.553568 | 0.519444 | 0.001180 | -9.877137
0.5004 | 10.001061 | 304.824045 | 1.312850 | 2.553575 | 0.519444 | 0.001183 | -9.877132
0.5005 | 10.001057 | 304.824081 | 1.312845 | 2.553581 | 0.519444 | 0.001185 | -9.877128
0.5006 | 10.001054 | 304.824116 | 1.312840 | 2.553588 | 0.519444 | 0.001188 | -9.877123
0.5007 | 10.001051 | 304.824152 | 1.312835 | 2.553596 | 0.519443 | 0.001190 | -9.877118
0.5008 | 10.001048 | 304.824188 | 1.312830 | 2.553603 | 0.519443 | 0.001193 | -9.877114
0.5009 | 10.001045 | 304.824225 | 1.312825 | 2.553611 | 0.519443 | 0.001195 | -9.877110
0.5010 | 10.001042 | 304.824262 | 1.312820 | 2.553620 | 0.519443 | 0.001198 | -9.877106
------------------------------------------------------------
>>> 单点最优 Beta = 0.501000
>>> 最高综合分 = -9.877106
>>> delta = |best_beta - 0.5| = 0.001000
>>> 左/右对称性残差 mean = 0.000056
>>> 左/右对称性残差 max = 0.000108
>>> beta=0.5 附近一阶导数 = 0.054090
>>> beta=0.5 附近二阶导数 = -22.389472
============================================================
>>> x_max = 5000000000 | K = 150
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 0.055753 | 0.004879 | 0.056023 | 0.004894 | 0.214473 | 0.000348 | 0.189860
0.4991 | 0.055741 | 0.004877 | 0.056002 | 0.004891 | 0.214473 | 0.000349 | 0.189880
0.4992 | 0.055730 | 0.004875 | 0.055984 | 0.004888 | 0.214473 | 0.000350 | 0.189898
0.4993 | 0.055720 | 0.004873 | 0.055968 | 0.004885 | 0.214473 | 0.000350 | 0.189914
0.4994 | 0.055712 | 0.004872 | 0.055955 | 0.004883 | 0.214473 | 0.000351 | 0.189927
0.4995 | 0.055706 | 0.004871 | 0.055945 | 0.004880 | 0.214473 | 0.000352 | 0.189937
0.4996 | 0.055700 | 0.004870 | 0.055937 | 0.004879 | 0.214473 | 0.000353 | 0.189946
0.4997 | 0.055696 | 0.004869 | 0.055930 | 0.004877 | 0.214473 | 0.000353 | 0.189953
0.4998 | 0.055692 | 0.004868 | 0.055925 | 0.004876 | 0.214473 | 0.000354 | 0.189959
0.4999 | 0.055689 | 0.004868 | 0.055921 | 0.004875 | 0.214473 | 0.000355 | 0.189963
0.5000 | 0.055688 | 0.004868 | 0.055919 | 0.004875 | 0.214473 | 0.000355 | 0.189965 <-- 锚点
0.5001 | 0.055688 | 0.004868 | 0.055919 | 0.004875 | 0.214473 | 0.000356 | 0.189965
0.5002 | 0.055688 | 0.004868 | 0.055922 | 0.004875 | 0.214473 | 0.000357 | 0.189963
0.5003 | 0.055691 | 0.004869 | 0.055927 | 0.004875 | 0.214473 | 0.000358 | 0.189958
0.5004 | 0.055694 | 0.004869 | 0.055934 | 0.004876 | 0.214473 | 0.000358 | 0.189952
0.5005 | 0.055699 | 0.004870 | 0.055942 | 0.004877 | 0.214473 | 0.000359 | 0.189944
0.5006 | 0.055705 | 0.004872 | 0.055952 | 0.004879 | 0.214473 | 0.000360 | 0.189934
0.5007 | 0.055712 | 0.004873 | 0.055963 | 0.004881 | 0.214473 | 0.000360 | 0.189922
0.5008 | 0.055720 | 0.004875 | 0.055976 | 0.004883 | 0.214473 | 0.000361 | 0.189909
0.5009 | 0.055730 | 0.004876 | 0.055991 | 0.004886 | 0.214473 | 0.000362 | 0.189892
0.5010 | 0.055741 | 0.004878 | 0.056009 | 0.004889 | 0.214473 | 0.000363 | 0.189874
------------------------------------------------------------
>>> 单点最优 Beta = 0.500000
>>> 最高综合分 = 0.189965
>>> delta = |best_beta - 0.5| = 0.000000
>>> 左/右对称性残差 mean = 0.000007
>>> 左/右对称性残差 max = 0.000014
>>> beta=0.5 附近一阶导数 = 0.010429
>>> beta=0.5 附近二阶导数 = -233.186399
============================================================
>>> x_max = 5000000000 | K = 300
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 0.045289 | 0.003200 | 0.045312 | 0.003198 | 0.302107 | 0.000336 | 0.314153
0.4991 | 0.045271 | 0.003198 | 0.045287 | 0.003194 | 0.302107 | 0.000336 | 0.314181
0.4992 | 0.045254 | 0.003196 | 0.045265 | 0.003191 | 0.302107 | 0.000337 | 0.314207
0.4993 | 0.045240 | 0.003195 | 0.045245 | 0.003188 | 0.302107 | 0.000338 | 0.314229
0.4994 | 0.045227 | 0.003193 | 0.045228 | 0.003186 | 0.302108 | 0.000338 | 0.314248
0.4995 | 0.045216 | 0.003192 | 0.045214 | 0.003184 | 0.302108 | 0.000339 | 0.314265
0.4996 | 0.045206 | 0.003191 | 0.045202 | 0.003182 | 0.302108 | 0.000340 | 0.314280
0.4997 | 0.045198 | 0.003190 | 0.045193 | 0.003180 | 0.302108 | 0.000341 | 0.314291
0.4998 | 0.045191 | 0.003190 | 0.045186 | 0.003179 | 0.302108 | 0.000341 | 0.314301
0.4999 | 0.045187 | 0.003189 | 0.045182 | 0.003178 | 0.302108 | 0.000342 | 0.314307
0.5000 | 0.045183 | 0.003189 | 0.045180 | 0.003178 | 0.302108 | 0.000343 | 0.314312 <-- 锚点
0.5001 | 0.045181 | 0.003189 | 0.045181 | 0.003178 | 0.302108 | 0.000343 | 0.314313
0.5002 | 0.045181 | 0.003190 | 0.045184 | 0.003178 | 0.302108 | 0.000344 | 0.314312
0.5003 | 0.045183 | 0.003190 | 0.045190 | 0.003179 | 0.302108 | 0.000345 | 0.314308
0.5004 | 0.045186 | 0.003191 | 0.045198 | 0.003180 | 0.302108 | 0.000345 | 0.314301
0.5005 | 0.045191 | 0.003192 | 0.045209 | 0.003181 | 0.302108 | 0.000346 | 0.314291
0.5006 | 0.045198 | 0.003193 | 0.045222 | 0.003183 | 0.302108 | 0.000347 | 0.314280
0.5007 | 0.045205 | 0.003195 | 0.045237 | 0.003185 | 0.302109 | 0.000347 | 0.314266
0.5008 | 0.045215 | 0.003197 | 0.045253 | 0.003187 | 0.302109 | 0.000348 | 0.314250
0.5009 | 0.045225 | 0.003199 | 0.045271 | 0.003190 | 0.302109 | 0.000349 | 0.314232
0.5010 | 0.045237 | 0.003201 | 0.045291 | 0.003193 | 0.302109 | 0.000350 | 0.314212
------------------------------------------------------------
>>> 单点最优 Beta = 0.500100
>>> 最高综合分 = 0.314313
>>> delta = |best_beta - 0.5| = 0.000100
>>> 左/右对称性残差 mean = 0.000029
>>> 左/右对称性残差 max = 0.000059
>>> beta=0.5 附近一阶导数 = 0.029454
>>> beta=0.5 附近二阶导数 = -307.483177
============================================================
>>> x_max = 5000000000 | K = 500
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 0.038423 | 0.002303 | 0.037962 | 0.002258 | 0.389223 | 0.000334 | 0.432854
0.4991 | 0.038406 | 0.002301 | 0.037939 | 0.002254 | 0.389223 | 0.000335 | 0.432879
0.4992 | 0.038391 | 0.002299 | 0.037918 | 0.002251 | 0.389223 | 0.000335 | 0.432903
0.4993 | 0.038378 | 0.002297 | 0.037899 | 0.002248 | 0.389223 | 0.000336 | 0.432924
0.4994 | 0.038366 | 0.002296 | 0.037884 | 0.002246 | 0.389223 | 0.000337 | 0.432941
0.4995 | 0.038356 | 0.002294 | 0.037871 | 0.002244 | 0.389223 | 0.000337 | 0.432956
0.4996 | 0.038348 | 0.002293 | 0.037860 | 0.002242 | 0.389223 | 0.000338 | 0.432968
0.4997 | 0.038343 | 0.002293 | 0.037853 | 0.002241 | 0.389223 | 0.000339 | 0.432977
0.4998 | 0.038339 | 0.002292 | 0.037849 | 0.002240 | 0.389223 | 0.000340 | 0.432982
0.4999 | 0.038338 | 0.002292 | 0.037847 | 0.002239 | 0.389223 | 0.000340 | 0.432984
0.5000 | 0.038338 | 0.002292 | 0.037848 | 0.002239 | 0.389223 | 0.000341 | 0.432984 <-- 锚点
0.5001 | 0.038339 | 0.002292 | 0.037850 | 0.002239 | 0.389223 | 0.000342 | 0.432981
0.5002 | 0.038343 | 0.002292 | 0.037857 | 0.002239 | 0.389223 | 0.000342 | 0.432975
0.5003 | 0.038348 | 0.002293 | 0.037866 | 0.002240 | 0.389223 | 0.000343 | 0.432965
0.5004 | 0.038356 | 0.002293 | 0.037880 | 0.002241 | 0.389223 | 0.000344 | 0.432952
0.5005 | 0.038366 | 0.002295 | 0.037895 | 0.002242 | 0.389223 | 0.000344 | 0.432936
0.5006 | 0.038378 | 0.002296 | 0.037913 | 0.002244 | 0.389223 | 0.000345 | 0.432917
0.5007 | 0.038391 | 0.002297 | 0.037933 | 0.002246 | 0.389223 | 0.000346 | 0.432896
0.5008 | 0.038405 | 0.002299 | 0.037956 | 0.002248 | 0.389223 | 0.000346 | 0.432872
0.5009 | 0.038422 | 0.002301 | 0.037981 | 0.002251 | 0.389223 | 0.000347 | 0.432845
0.5010 | 0.038440 | 0.002304 | 0.038011 | 0.002255 | 0.389223 | 0.000348 | 0.432814
------------------------------------------------------------
>>> 单点最优 Beta = 0.499900
>>> 最高综合分 = 0.432984
>>> delta = |best_beta - 0.5| = 0.000100
>>> 左/右对称性残差 mean = 0.000021
>>> 左/右对称性残差 max = 0.000039
>>> beta=0.5 附近一阶导数 = -0.013820
>>> beta=0.5 附近二阶导数 = -250.885214
============================================================
>>> x_max = 5000000000 | K = 1000
[Beta] | [MAE_fix] | [MSE_fix] | [Tail_MAE_fix] | [Tail_MSE_fix] | [R_fix] | [EnergyBias] | [Score]
-------------------------------------------------------------------------------------------------------------
0.4990 | 0.030563 | 0.001471 | 0.030214 | 0.001439 | 0.538955 | 0.000334 | 0.630978
0.4991 | 0.030538 | 0.001469 | 0.030175 | 0.001436 | 0.538955 | 0.000335 | 0.631018
0.4992 | 0.030516 | 0.001467 | 0.030139 | 0.001433 | 0.538955 | 0.000335 | 0.631054
0.4993 | 0.030497 | 0.001465 | 0.030107 | 0.001430 | 0.538955 | 0.000336 | 0.631087
0.4994 | 0.030480 | 0.001463 | 0.030079 | 0.001427 | 0.538955 | 0.000337 | 0.631115
0.4995 | 0.030467 | 0.001462 | 0.030056 | 0.001425 | 0.538955 | 0.000337 | 0.631137
0.4996 | 0.030456 | 0.001461 | 0.030038 | 0.001424 | 0.538955 | 0.000338 | 0.631155
0.4997 | 0.030448 | 0.001461 | 0.030023 | 0.001422 | 0.538955 | 0.000339 | 0.631169
0.4998 | 0.030441 | 0.001460 | 0.030010 | 0.001421 | 0.538955 | 0.000339 | 0.631180
0.4999 | 0.030438 | 0.001460 | 0.030001 | 0.001421 | 0.538955 | 0.000340 | 0.631188
0.5000 | 0.030436 | 0.001460 | 0.029996 | 0.001420 | 0.538955 | 0.000341 | 0.631191 <-- 锚点
0.5001 | 0.030437 | 0.001460 | 0.029995 | 0.001420 | 0.538955 | 0.000341 | 0.631190
0.5002 | 0.030440 | 0.001460 | 0.029996 | 0.001421 | 0.538955 | 0.000342 | 0.631187
0.5003 | 0.030445 | 0.001461 | 0.030001 | 0.001422 | 0.538955 | 0.000343 | 0.631180
0.5004 | 0.030454 | 0.001462 | 0.030011 | 0.001423 | 0.538955 | 0.000343 | 0.631167
0.5005 | 0.030465 | 0.001463 | 0.030025 | 0.001424 | 0.538955 | 0.000344 | 0.631150
0.5006 | 0.030478 | 0.001464 | 0.030042 | 0.001426 | 0.538955 | 0.000345 | 0.631130
0.5007 | 0.030493 | 0.001466 | 0.030063 | 0.001428 | 0.538955 | 0.000346 | 0.631106
0.5008 | 0.030512 | 0.001468 | 0.030089 | 0.001431 | 0.538955 | 0.000346 | 0.631077
0.5009 | 0.030533 | 0.001470 | 0.030118 | 0.001434 | 0.538955 | 0.000347 | 0.631044
0.5010 | 0.030557 | 0.001472 | 0.030151 | 0.001437 | 0.538955 | 0.000348 | 0.631007
------------------------------------------------------------
>>> 单点最优 Beta = 0.500000
>>> 最高综合分 = 0.631191
>>> delta = |best_beta - 0.5| = 0.000000
>>> 左/右对称性残差 mean = 0.000015
>>> 左/右对称性残差 max = 0.000030
>>> beta=0.5 附近一阶导数 = 0.013670
>>> beta=0.5 附近二阶导数 = -358.190759
[5/6] 生成 K-beta 收敛诊断 ...
============================================================
>>> K-Beta 联动收敛诊断（取最大尺度 x_max=5000000000）
============================================================
[K] | [best_beta] | [delta=|best_beta-0.5|] | [best_score] | [sym_mean] | [sym_max]
--------------------------------------------------------------------------------------
150 | 0.500000 | 0.000000 | 0.189965 | 0.000007 | 0.000014
300 | 0.500100 | 0.000100 | 0.314313 | 0.000029 | 0.000059
500 | 0.499900 | 0.000100 | 0.432984 | 0.000021 | 0.000039
1000 | 0.500000 | 0.000000 | 0.631191 | 0.000015 | 0.000030
[6/6] 写出 CSV ...
结果已输出到：/Users/陈恩华/Desktop/project/img_data/ceh_k_beta_convergence_v12.csv
总耗时：171.61 秒
进程已结束，退出代码为 0
