天赐范式第188天·第一篇让环境开始轮回——周期峰的低通滤波响应摘要周期峰驱动下AR(1)均值响应是低通滤波幅值衰减|H(Ω)|相位滞后φ(Ω)离散z域闭式。临界周期T*2π/c≈523代数值比值1.00。一、接续187-1定向移动做完了周期呢187-1让峰开始移动——但那是单调趋势x*(t)x*_0v·t峰一直往一个方向跑。种群学会了就是常数滞后追得很稳。真实栖息地的环境变化有个更基本的形态周期。昼夜、季节、潮汐——环境不是一直变好或一直变坏是来回摆。这是环境弧的第二分量。数学上区别很大定向移动下种群对峰是常数滞后187-1的lag*周期驱动下种群均值是对峰的低通滤波——响应是幅值衰减相位滞后。这是AR(1)对正弦输入的标准稳态解理论结构全新不是187的延伸。二、理论推导——AR(1)低通滤波均值更新与187-1同μ(t1) (1−c)·μ(t) c·x*(t)c β·h²·K周期峰x*(t) x*_0 A·sin(Ω·t)Ω是角频率弧度/代。z域传递函数H(z) c / (z − (1−c))令z e^(jΩ)离散频率响应|H(Ω)| c / √(1 − 2(1−c)cos(Ω) (1−c)²)φ(Ω) −atan2(sin(Ω), cos(Ω) − 1 c)连续极限Ω→0, c≪1|H(Ω)| → c / √(c² Ω²)φ(Ω) → −arctan(Ω / c)临界周期半功率点 |H| 1/√2 对应 Ω* c连续近似临界周期T* 2π / c ≈ 2π / 0.012 ≈ 523代这是数字生命的时间尺度——环境周期比这短种群就感知不到。三、频率响应曲线理论vs数值固定σ_v0.02c0.0120振幅A0.3扫Ω ∈ {0.005, 0.01, 0.02, 0.03, 0.05, 0.1, 0.2}。正弦拟合提取种群均值的幅值和相位。ΩT(代)|H|离散|H|连续|H|数值比值φ离散φ数值平均fitness状态0.00512570.9240.9230.9231.00−0.3950.3770.729跟踪0.0106280.7700.7680.7731.00−0.6970.6730.498衰减0.0125240.7090.7070.7141.01−0.7880.7690.442衰减0.0203140.5170.5150.5181.00−1.0381.0140.353衰减0.0302090.3740.3720.3751.00−1.2031.1840.318衰减0.0501260.2350.2340.2361.00−1.3591.3490.298衰减0.100630.1200.1190.1201.00−1.5011.4920.291衰减0.200310.0600.0600.0601.00−1.6111.6110.288躺平幅值衰减|H|离散理论vs数值比值全部1.00-1.01——闭式精确。相位φ理论为负滞后数值返回atan2(−a,b)取绝对值两者符号约定不同下文比较绝对值。四、跟踪区、衰减区与躺平区跟踪区T ≥ 1257代Ω ≤ 0.005|H|≈0.92种群均值振幅接近峰振幅。峰慢种群跟得上只有相位滞后φ≈0.4弧度约79代延迟数值约75代。衰减区T 63~628代Ω 0.01~0.10 |H| 1种群均值振幅衰减。半功率点|H|1/√2≈0.707在临界周期T*2π/c≈523代附近见表中Ω0.012行T524代|H|离散0.709。T314代时|H|0.52T63代时|H|0.12——种群均值几乎不动峰在摆种群不摆。躺平区T ≤ 31代Ω ≥ 0.2|H|→0种群均值完全躺平。峰在动种群不动平均fitness≈0.29——反而可能不如假装峰不动。这是滤波特性的必然快于T*的周期种群感知不到。临界周期T*≈523代是分界线。cβ·h²·K0.012是跟踪速率T*2π/c是种群响应时间尺度。环境周期比T*长种群跟得上比T*短种群感知不到。五、结论与弧签名动作问题答案周期峰下种群怎么响应低通滤波幅值衰减|H(Ω)|相位滞后φ(Ω)AR(1)标准稳态解临界周期多长T*2π/c≈523代——数字生命感知环境变化的时间尺度下限理论准吗幅值比值全部1.00离散z域闭式精确和187-1什么关系187-1定向移动常数滞后188-1周期驱动低通滤波环境弧第二分量叙事弧187-1定向移动——常数滞后lag*v(1−c)/c188-1周期驱动——低通滤波|H(Ω)|φ(Ω)从追踪升级到滤波弧签名动作187定向追踪 → 188周期滤波。主线意义T*≈523代是数字生命感知环境变化的时间尺度下限。环境弧全貌定向187趋势分量→ 周期188-1季节分量→ 随机游走不可预测分量留给189。降调线性化近似。低通滤波公式是AR(1)线性系统的稳态解假设h²和K当常数用σ_v0.02自洽解。大振幅下种群偏离峰远h²和K有微小漂移187-1降调2同源。振幅A0.3下影响小更大振幅需验证。相位拟合有0~5%偏差随Ω减小而增大。数值相位绝对值比理论略小如0.377 vs 0.395偏差4.6%1.611 vs 1.611偏差0%。偏差源于低频端测量窗覆盖周期少、暂态残留多高频端周期多、暂态影响小。幅值精确匹配是核心结果相位偏差在降调里如实报。测量窗含整数个周期。gen4000、warmup500测量窗3500代。最长周期T1257代覆盖2.8个周期——够。更长的周期需要更多代数。跟踪区仅Ω0.005一个格点支撑。更慢周期Ω0.005未测需要更长模拟窗口。单位点模型。多位点、上位效应、连锁不平衡待后续。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188天 第一篇 让环境开始轮回——周期峰的低通滤波响应 PID: TC-188A-V3.3.26.0 V3.3.26.0 · 2026-10-07 接续187-1定向移动趋势分量→ 周期移动季节分量 187-1: x*(t)x*_0v·t单调趋势种群响应常数滞后 188-1: x*(t)x*_0A·sin(Ω·t)周期驱动种群响应低通滤波 理论AR(1)均值动力学 μ(t1)(1-c)·μ(t)c·x*(t)cβ·h²·K z域传递函数 H(z)c/(z-(1-c)) 令ze^(jΩ)离散频率响应 |H(Ω)| c/√(1−2(1−c)cos(Ω)(1−c)²) φ(Ω) −atan2(sin(Ω), cos(Ω)−1c) 连续极限Ω→0, c1 |H(Ω)| → c/√(c²Ω²) φ(Ω) → −arctan(Ω/c) 临界周期 T*2π/c≈523代半功率点 预测 慢周期TT*A≈1, φ≈0种群紧跟峰 快周期TT*A→0种群均值躺平峰在动种群不动 T*≈523代是数字生命的时间尺度——季节比这短就感知不到 数值扫Ω正弦拟合提取幅值衰减相位滞后理论vs数值对比 importsysimportmathimportnumpyasnpifhasattr(sys.stdout,reconfigure):sys.stdout.reconfigure(encodingutf-8)PIDTC-188A-V3.3.26.0TARGET0.5X_STAR_00.8AMPLITUDE0.3FITNESS_WIDTH0.10BETA0.3SIGMA_E0.02SIGMA_V0.02N_POP200N_GENERATIONS4000WARMUP_STATIC500N_SEEDS20TOL1e-12MAX_ITER5000OMEGAS[0.005,0.01,0.012,0.02,0.03,0.05,0.1,0.2]defbar(title):print(*72)print( title)print(*72)print()defsub(title):print(【title)print(-*72)deffitness(x,x_star):returnnp.exp(-(x-x_star)**2/(2*FITNESS_WIDTH**2))defsolve_self_consistent(sigma_v):var_gsigma_v**2/(1-BETA**2)foritinrange(MAX_ITER):sigma_x2var_gSIGMA_E**2h2var_g/sigma_x2 v_x_selsigma_x2*FITNESS_WIDTH**2/(sigma_x2FITNESS_WIDTH**2)var_g_selh2**4*v_x_selvar_g*(1-h2**2)var_g_newBETA**2*var_g_selsigma_v**2ifabs(var_g_new-var_g)TOL:returnvar_g_new,it1var_gvar_g_newreturnvar_g,MAX_ITERdeftheory_response(omega,track_rate):ctrack_rate denom1.0-2.0*(1.0-c)*math.cos(omega)(1.0-c)**2gainc/math.sqrt(denom)ifdenom0elsefloat(inf)phase-math.atan2(math.sin(omega),math.cos(omega)-1.0c)returngain,phasedeftheory_response_cont(omega,track_rate):ctrack_rate gainc/math.sqrt(c**2omega**2)phase-math.atan2(omega,c)returngain,phasedefrun_one(omega,seed):rngnp.random.RandomState(seed)genesrng.normal(TARGET,0.01,N_POP)mu_trace[]xstar_trace[]fit_trace[]forgeninrange(N_GENERATIONS):ifgenWARMUP_STATIC:x_starX_STAR_0else:tgen-WARMUP_STATIC x_starX_STAR_0AMPLITUDE*math.sin(omega*t)phenosgenesrng.normal(0,SIGMA_E,N_POP)fitsfitness(phenos,x_star)fit_sumfloat(fits.sum())iffit_sum1e-10:forg2inrange(gen,N_GENERATIONS):ifg2WARMUP_STATIC:fit_trace.append(0.0)breakprobsfits/fit_sum sel_idxrng.choice(N_POP,sizeN_POP,pprobs)parent_genesgenes[sel_idx]mufloat(np.mean(genes))genesmuBETA*(parent_genes-mu)rng.normal(0,SIGMA_V,N_POP)ifgenWARMUP_STATIC:mu_trace.append(float(np.mean(genes)))xstar_trace.append(x_star)fit_trace.append(float(np.mean(fits)))returnnp.array(mu_trace),np.array(xstar_trace),np.array(fit_trace)defsine_fit(mu_trace,xstar_trace,omega):tnp.arange(len(mu_trace))1residualmu_trace-X_STAR_0 cos_basisnp.cos(omega*t)sin_basisnp.sin(omega*t)A_matnp.array([[np.sum(cos_basis*cos_basis),np.sum(cos_basis*sin_basis)],[np.sum(sin_basis*cos_basis),np.sum(sin_basis*sin_basis)]])b_vecnp.array([np.sum(residual*cos_basis),np.sum(residual*sin_basis)])try:coeffsnp.linalg.solve(A_mat,b_vec)exceptnp.linalg.LinAlgError:return0.0,0.0a,bcoeffs ampmath.sqrt(a**2b**2)phasemath.atan2(-a,b)returnamp,phasedefmain():bar(f{PID}让环境开始轮回——周期峰的低通滤波响应)print(f模型: gμβ(g_sel-μ)v_mut回归到种群均值μ)print(ffitness峰: x*(t)x*_0A·sin(Ω·t), x*_0{X_STAR_0}, A{AMPLITUDE})print(f参数: β{BETA}, σ_e{SIGMA_E}, σ_v{SIGMA_V}, N{N_POP}, seeds{N_SEEDS})print(f理论: AR(1)低通滤波 |H(Ω)|c/√(1−2(1−c)cos(Ω)(1−c)²))print(f 连续极限 |H|c/√(c²Ω²), φ−arctan(Ω/c))print(f 临界周期 T*2π/c)print(f数值: gen{N_GENERATIONS}, warmup{WARMUP_STATIC}, 正弦拟合提取幅值相位)print()var_g_star,_solve_self_consistent(SIGMA_V)sigma_x2var_g_starSIGMA_E**2h2var_g_star/sigma_x2 Ksigma_x2/(sigma_x2FITNESS_WIDTH**2)track_rateBETA*h2*K T_star2*math.pi/track_rate sub(理论参数σ_v0.02自洽)print(f h²* {h2:.4f})print(f K {K:.4f})print(f c β·h²·K {track_rate:.6f}跟踪速率)print(f T* 2π/c {T_star:.1f}代临界周期)print()sub(频率响应理论 vs 数值)print(f{Ω:8s}{T(代):8s}{|H|离散:8s}{|H|连续:8s}{|H|数值:8s}{φ离散:8s}{φ数值:8s}{平均fit:10s}{状态:6s})print(-*90)results[]foromegainOMEGAS:T2*math.pi/omega gain_discrete,phase_discretetheory_response(omega,track_rate)gain_cont,phase_conttheory_response_cont(omega,track_rate)gains_seed[]phases_seed[]fits_seed[]forseedinrange(N_SEEDS):mu_trace,xstar_trace,fit_tracerun_one(omega,seed)amp,phasesine_fit(mu_trace,xstar_trace,omega)gains_seed.append(amp/AMPLITUDE)phases_seed.append(phase)fits_seed.append(float(np.mean(fit_trace))iflen(fit_trace)0else0.0)gain_numfloat(np.mean(gains_seed))phase_numfloat(np.mean(phases_seed))mean_fitfloat(np.mean(fits_seed))ifgain_num0.9:status跟踪elifgain_num0.1:status衰减else:status躺平results.append({omega:omega,T:T,gain_discrete:gain_discrete,gain_cont:gain_cont,gain_num:gain_num,phase_discrete:phase_discrete,phase_num:phase_num,mean_fit:mean_fit,status:status,})print(f{omega:8.4f}{T:8.1f}{gain_discrete:8.4f}{gain_cont:8.4f}{gain_num:8.4f}{phase_discrete:8.4f}{phase_num:8.4f}{mean_fit:10.6f}{status:6s})print()sub(理论 vs 数值对比)forrinresults:ratior[gain_num]/r[gain_discrete]ifr[gain_discrete]1e-6elsefloat(inf)print(f Ω{r[omega]:.4f}(T{r[T]:.0f}代): |H|离散{r[gain_discrete]:.4f}vs |H|数值{r[gain_num]:.4f}比值{ratio:.2f}φ离散{r[phase_discrete]:.4f}vs φ数值{r[phase_num]:.4f})print()sub(结论)print(f187-1定向移动常数滞后lag*v·(1−c)/c)print(f188-1周期驱动低通滤波响应|H(Ω)|φ(Ω))print(f 临界周期 T*2π/c{T_star:.0f}代——数字生命的时间尺度)track_vs[rforrinresultsifr[status]跟踪]decay_vs[rforrinresultsifr[status]衰减]flat_vs[rforrinresultsifr[status]躺平]iftrack_vs:print(f 跟踪区T≥{track_vs[-1][T]:.0f}代Ω≤{track_vs[-1][omega]:.4f}|H|≈1)ifdecay_vs:print(f 衰减区T∈[{decay_vs[-1][T]:.0f},{decay_vs[0][T]:.0f}]代0|H|1)ifflat_vs:print(f 躺平区T≤{flat_vs[0][T]:.0f}代Ω≥{flat_vs[0][omega]:.4f}|H|→0)print()print(f 弧签名动作187定向追踪 → 188周期滤波)print(f 主线意义T*≈{T_star:.0f}代是数字生命感知环境变化的时间尺度下限)if__name____main__:main()天赐范式 V3.3.26.0 · 2026-10-07