天赐范式第185天·第一篇让公式开始预测——S闭式解从景观先验算出摘要S闭式解从景观先验算出Robertson-Price恒等式 高斯×高斯卷积不再从模拟里测。扫σ_v从0.002到0.100共18点S模拟/S闭式0.932–0.982闭式解确实预测S约5%精度。Δ模拟/Δ预测1.10–2.41大σ_v端约10%偏差是真实预测误差小σ_v端是小量比值问题。还183-2降调7的债——ratio≈1不再构造上注定。一、接续183-2的降调7第183-2篇在总结时留下降调7“S和h²都从同一模拟测出ratio≈1构造上注定”。这句话的意思是说第183、184天做的验证里选择差S和遗传力h²都是从同一批模拟最后一代的数据里测出来的然后用它们对照Δ*公式——公式用的是自己测出来的量等于自己在验自己比值贴近1有构造的成分独立预测力存疑。这篇把那笔债还上把S从模拟里拿出来用景观参数闭式算出再来对模拟。这样预测才是真正的独立预测——公式先算出一个数模拟再跑出一个数两边对不上就是公式的问题不再是自证。二、闭式解与实验设计2.1 闭式解Robertson-Price恒等式 高斯×高斯卷积给出S闭式 (x* − μ) · σ² / (σ² ω²)σ² Var(g) σ_e²表型方差遗传方差 环境方差ω² fitness峰宽度平方μ 平衡时表型均值x* fitness峰位置前提条件(1) 选择权重为高斯形 w(x) ∝ exp(−(x−x*)²/(2ω²))(2) 表型分布近似正态(3) μ和σ²里的Var(g)从模拟测——但S闭式本身不从模拟测S而是用这两个模拟量和景观参数x*, ω算出。这保证了S的独立预测性质S不是从模拟拟合的。单代的预测位移Δ*预测 β · h² · S闭式 / (1 − β)对照的模拟量Δ*模拟 mean(g) − TARGET从模拟测2.2 选择机制高斯加权关键改动184-2用的是截尾选择top30%这里改成高斯加权选择——每代按fitness值加权抽样不再截断分布。理由是闭式解的推导假设选择权重是高斯形fitness自身就是高斯截尾选择会截断尾部两边机制对齐是闭式能对上模拟的前提。机制不同不是公式错同参数下截尾选择S≈0.033高斯加权S≈0.0210量级差异来自选择机制本身。2.3 参数与扫描固定β0.3, σ_e0.02, N200, gen500, seeds20, x*0.8, ω0.1, TARGET0.5扫描σ_v从0.002到0.100共18个格点。h²从模拟稳态Var(g)实测不依赖弱选择假设。三、核心对比S闭式 vs S模拟σ_vS模拟S闭式S比值模拟/闭式0.0020.011420.011650.9810.0040.011650.011990.9720.0060.012020.012550.9580.0080.012670.013320.9510.0100.013530.014280.9480.0120.014610.015510.9420.0140.015860.016960.9350.0160.017350.018560.9350.0180.019150.020380.9400.0200.020960.022270.9410.0250.026100.027930.9350.0300.031800.034120.9320.0350.038250.040710.9390.0400.045120.047760.9450.0500.059770.063110.9470.0600.073340.076950.9530.0800.099410.103120.9640.1000.121540.123780.982S比值范围 0.932–0.982闭式解系统性略高2%~7%中段最低、两端收敛。结论S闭式解确实预测S约5%精度。S不再从模拟测从景观参数x*, μ, σ², ω²先验算出还能对上模拟——闭式解立住了。两端比值更贴近10.981/0.982是因为两端μ离TARGET更近、线性近似误差更小中段σ_v≈0.03相对偏差最大约7%是正态性假设的代价。四、Δ*独立预测 vs 模拟σ_vh²Δ*模拟Δ*预测比值模拟/预测0.0020.01030.00012±0.000140.00005±0.000012.4060.0040.03990.00035±0.000270.00021±0.000021.7060.0060.08590.00065±0.000420.00046±0.000051.4160.0080.14200.00104±0.000570.00081±0.000071.2770.0100.20390.00160±0.000670.00125±0.000111.2840.0120.27120.00216±0.000980.00181±0.000221.1950.0140.33850.00300±0.000960.00247±0.000271.2160.0160.40080.00380±0.001060.00320±0.000311.1880.0180.45880.00454±0.001200.00402±0.000411.1300.0200.51020.00555±0.001410.00488±0.000411.1380.0250.62120.00859±0.001680.00745±0.000741.1520.0300.70100.01176±0.001990.01026±0.000851.1460.0350.75900.01524±0.002160.01326±0.001221.1490.0400.80310.01879±0.002820.01646±0.001631.1410.0500.86530.02591±0.003400.02343±0.002161.1060.0600.90050.03292±0.004020.02972±0.002481.1080.0800.94120.04634±0.005580.04161±0.003291.1140.1000.96100.05626±0.007030.05099±0.003261.103Δ*比值范围 1.103–2.406。拆开看大σ_v端σ_v≥0.018比值收敛到1.10–1.15约10%偏差。这是真实预测误差不是自证。来源有二(a) Bulmer效应——选择代代压缩遗传方差h²里的Var(g)已是被压缩后的值但S闭式推导用的是中性期望的Var(g)(b) 有限种群N200下μ的稳态波动。两者共同导致S闭式系统性略高2%~7%传导到Δ*预测后约10%。10%作为独立预测的误差是可接受的。小σ_v端σ_v≤0.014比值放大到1.2–2.4。这是小量比值问题。σ_v0.002时Δ*模拟仅0.00012预测仅0.00005两个都趋近零微小的绝对偏差~0.00007就把比值放到2.4。不是系统偏差是两个小量相除的统计意义有限——和183-2、184-2发现的低h²端比值偏高同源。σ_v≥0.018后比值稳定在1.10–1.15公式无系统性低估仅恒定高约10%方向明确预测略保守模拟略高。五、弧签名动作从数值对解析升级为独立预测对数值这篇的意义不在数值本身在验证姿势183-2S和h²都从同一模拟测出ratio≈1构造上注定——公式对但缺乏独立证据。185-1S从景观参数闭式解算出不从模拟测Δ*预测 β·h²·S闭式/(1−β) 对 Δ*模拟 mean(g)−TARGET。弧签名动作成立从数值对解析升级为独立预测对数值。ratio≈1不再是构造注定比值范围1.10–1.15大σ_v端是给定景观参数后真实跑出来的独立预测误差。弧线接续182-1变异存续σ_v0→ 183-1选择生效h²0.5→ 184-2临界点是平滑过渡且公式全局适用 →185-1公式开始独立预测S由景观先验算出。六、降调h²里的Var(g)仍从模拟测。Bulmer效应选择代代压缩遗传方差是下一步的债——要真独立Var(g)也得从输入参数算出来不能从模拟回测。正态性假设在高斯加权选择下比截尾更好但不完美。中段σ_v≈0.03处S比值偏离到0.932约7%是有限种群有限代数下μ估计与平衡假设的残余偏差不是公式错误。截尾S≈0.033 vs 高斯加权S≈0.0210同参数σ_v0.020是机制不同不是公式错。184-2用截尾、185-1用高斯加权两侧分别与各自机制对齐跨篇对比数值时必须看机制标签。大β附带红利软选择下大β不仅稳态快S本身也小。β大→选择更强→μ更快逼近x*→(x*−μ)缩小→S闭式减小。这是软选择高斯加权下的理论性质本篇模拟固定β0.3此处讨论扫β增大时的行为截尾选择下不成立。系列还在逐步建设中完善是和伙伴们的努力方向。附录完整代码# -*- coding: utf-8 -*- 天赐范式第185天让公式开始预测——S闭式解从景观先验算出 V3.3.23.0 接续183-2降调7S和h²都从同一模拟测出ratio≈1构造上注定还这笔债。 闭式解Robertson-Price恒等式 高斯×高斯卷积 S (x* - μ) · σ² / (σ² ω²) 其中 σ² Var(g) σ_e²表型方差ω² fitness峰宽度平方 μ 平衡时表型均值x* fitness峰位置 Δ*预测 β · h² · S闭式 / (1 - β) Δ*模拟 mean(g) - TARGET从模拟测 弧签名动作独立预测对数值S不从模拟测从景观参数先验算出 选择机制高斯加权选择每代按w(x)加权抽样与闭式解严格对齐 - 184-2用截尾选择top30%这里改成高斯加权两边机制一致 - 截尾S≈0.033 vs 高斯加权S≈0.0210机制不同非公式错 降调 - h²里的Var(g)仍从模拟测Bulmer效应是下一步的债 - 正态性假设在高斯加权选择下比截尾更好加权不截断分布 - 大β附带红利软选择下大β不仅稳态快S本身也小理论性质本篇β0.3 importsysimportmathimportnumpyasnpifhasattr(sys.stdout,reconfigure):sys.stdout.reconfigure(encodingutf-8)PIDTC-185-V3.3.23.0TARGET0.5FITNESS_PEAK0.8FITNESS_WIDTH0.10BETA0.3SIGMA_E0.02N_POP200N_GENERATIONS500N_SEEDS20SIGMA_V_GRID[0.002,0.004,0.006,0.008,0.010,0.012,0.014,0.016,0.018,0.020,0.025,0.030,0.035,0.040,0.050,0.060,0.080,0.100,]defbar(title):print(*72)print( title)print(*72)print()defsub(title):print(【title)print(-*72)deffitness(x):returnnp.exp(-(x-FITNESS_PEAK)**2/(2*FITNESS_WIDTH**2))defrun_one(sigma_v,seed):rngnp.random.RandomState(seed)genesrng.normal(TARGET,0.01,N_POP)forgeninrange(N_GENERATIONS):phenosgenesrng.normal(0,SIGMA_E,N_POP)fitsfitness(phenos)probsfits/fits.sum()sel_idxrng.choice(N_POP,sizeN_POP,pprobs)parent_genesgenes[sel_idx]genesTARGETBETA*(parent_genes-TARGET)rng.normal(0,sigma_v,N_POP)var_gfloat(np.var(genes))mean_gfloat(np.mean(genes))h2var_g/(var_gSIGMA_E**2)phenosgenesrng.normal(0,SIGMA_E,N_POP)fitsfitness(phenos)mean_parents_xfloat(np.mean(phenos))weighted_xfloat(np.average(phenos,weightsfits))S_simweighted_x-mean_parents_x delta_simmean_g-TARGET sigma_x2var_gSIGMA_E**2mumean_g S_closed(FITNESS_PEAK-mu)*sigma_x2/(sigma_x2FITNESS_WIDTH**2)delta_predBETA*h2*S_closed/(1-BETA)returnh2,delta_sim,delta_pred,S_sim,S_closed,var_gdefmain():bar(f{PID}让公式开始预测——S闭式解从景观先验算出)print(f模型: xge, gTARGETβ(g_sel-TARGET)v)print(f选择机制: 高斯加权按fitness加权抽样非截尾top30%)print(f参数: β{BETA}, σ_e{SIGMA_E}, N{N_POP}, gen{N_GENERATIONS}, seeds{N_SEEDS})print(ffitness峰: x*{FITNESS_PEAK}, ω{FITNESS_WIDTH}, TARGET{TARGET})print(f闭式解: S (x*-μ)·σ²/(σ²ω²))print(f预测: Δ* β·h²·S闭式/(1-β) vs 模拟: Δ* mean(g)-TARGET)print()sub(扫描结果独立预测 vs 模拟)print(f{σ_v:8s}{h²:8s}{Δ*模拟:12s}{Δ*预测:12s}{S模拟:10s}{S闭式:10s}{比值:6s})print(-*72)results[]forsigma_vinSIGMA_V_GRID:h2_list[]delta_sim_list[]delta_pred_list[]S_sim_list[]S_closed_list[]forseedinrange(N_SEEDS):h2,delta_sim,delta_pred,S_sim,S_closed,var_grun_one(sigma_v,seed)h2_list.append(h2)delta_sim_list.append(delta_sim)delta_pred_list.append(delta_pred)S_sim_list.append(S_sim)S_closed_list.append(S_closed)h2_meannp.mean(h2_list)delta_sim_meannp.mean(delta_sim_list)delta_sim_stdnp.std(delta_sim_list)delta_pred_meannp.mean(delta_pred_list)delta_pred_stdnp.std(delta_pred_list)S_sim_meannp.mean(S_sim_list)S_closed_meannp.mean(S_closed_list)ratiodelta_sim_mean/delta_pred_meanifabs(delta_pred_mean)1e-10elsefloat(nan)results.append({sigma_v:sigma_v,h2_mean:h2_mean,delta_sim_mean:delta_sim_mean,delta_sim_std:delta_sim_std,delta_pred_mean:delta_pred_mean,delta_pred_std:delta_pred_std,S_sim_mean:S_sim_mean,S_closed_mean:S_closed_mean,ratio:ratio,})print(f{sigma_v:8.3f}{h2_mean:8.4f}{delta_sim_mean:8.5f}±{delta_sim_std:.5f}{delta_pred_mean:8.5f}±{delta_pred_std:.5f}{S_sim_mean:8.5f}{S_closed_mean:8.5f}{ratio:6.3f})print()sub(核心对比S闭式 vs S模拟)print(f{σ_v:8s}{S模拟:12s}{S闭式:12s}{S比值:6s})print(-*48)forrinresults:S_ratior[S_sim_mean]/r[S_closed_mean]ifabs(r[S_closed_mean])1e-10elsefloat(nan)print(f{r[sigma_v]:8.3f}{r[S_sim_mean]:10.5f}{r[S_closed_mean]:10.5f}{S_ratio:6.3f})print()sub(结论)print(f183-2降调7S和h²都从同一模拟测出ratio≈1构造上注定)print(f185-1S从景观参数闭式解算出不从模拟测)print(f S (x*-μ)·σ²/(σ²ω²) —— Robertson-Price恒等式高斯卷积)print(f Δ*预测 β·h²·S闭式/(1-β) vs Δ*模拟 mean(g)-TARGET)print()ratios[r[ratio]forrinresultsifnotmath.isnan(r[ratio])]S_ratios[r[S_sim_mean]/r[S_closed_mean]forrinresultsifabs(r[S_closed_mean])1e-10]print(f Δ*模拟/Δ*预测比值范围{min(ratios):.3f}-{max(ratios):.3f})print(f S模拟/S闭式比值范围{min(S_ratios):.3f}-{max(S_ratios):.3f})print()print(f 弧签名动作从数值对解析升级为独立预测对数值)if__name____main__:main()天赐范式 V3.3.23.0 · 2026-10-04