research/src/make_figures.py

"""Rebuild numerical figures and their CSV data (no external data required)."""
from pathlib import Path
import sys,json
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import mpmath as mp
from event_calculus import Event,compile_score,observation_matrix
ROOT=Path(__file__).resolve().parents[1]; F=ROOT/'figures';D=ROOT/'data'
F.mkdir(exist_ok=True)

def save(name,fig,data=None,header=''):
    fig.tight_layout()
    fig.savefig(F/(name+'.pdf'),bbox_inches='tight')
    fig.savefig(F/(name+'.png'),dpi=170,bbox_inches='tight')
    plt.close(fig)
    if data is not None:np.savetxt(D/(name+'.csv'),data,delimiter=',',header=header,comments='')

def axes(xlabel,ylabel):
    fig,ax=plt.subplots(figsize=(6.7,4.0));ax.set_xlabel(xlabel);ax.set_ylabel(ylabel);ax.grid(True,alpha=.22)
    return fig,ax

s=compile_score([Event('0',2),Event('.18',1),Event('.33',2)],'.5',precision=100)
t=np.linspace(0,.8,901);h=np.array([float(s.helicity(str(x))) for x in t]);e=np.array([float(s.energy(str(x))) for x in t])
fig,ax=axes('Operational time','Normalized helicity')
ax.plot(t,h);ax.set_yscale('symlog',linthresh=.05)
for x in [0,.18,.33]:ax.axvline(x,ls=':',lw=.8)
ax.set_title('Two contacts and one crossing, compiled exactly')
save('01_event_divisor',fig,np.c_[t,h,e],'time,helicity,energy')
fig,ax=axes('Carrier wavenumber','Signed helicity coefficient')
cs=np.array(list(map(float,s.coefficients)));ax.stem(s.wavenumbers,cs);ax.set_title('Every carrier is present; handedness alternates')
save('02_scissor_carriers',fig,np.c_[s.wavenumbers,cs],'wavenumber,coefficient')

fig,ax=axes('Time','Third-order response coefficient')
data=[t]
for nu in [.2,1,3]:
    c=np.exp(-6*nu*t)*(np.expm1(4*nu*t)-4*nu*t)/(8*nu)
    ax.plot(t,c,label=f'viscosity {nu:g}');data.append(c)
ax.legend();ax.set_title('The closed triad creates a positive response kernel')
save('03_triad_kernel',fig,np.array(data).T,'time,nu_0p2,nu_1,nu_3')
phi=np.linspace(-np.pi,np.pi,501); val=np.cos(phi)
fig,ax=axes('Closed-loop phase (radians)','Initial helicity acceleration / (2 nu a b c)')
ax.plot(phi,val);ax.axhline(0,ls=':',lw=.8);ax.set_title('Phase reversal and exact quadrature extinction')
save('04_phase_gate',fig,np.c_[phi,val],'phase,normalized_acceleration')

nd=json.loads((D/'nonlinear_demo.json').read_text());rs=nd['results']
eps=np.array([r['epsilon'] for r in rs]);corr=np.array([r['amplitude_correction_norm'] for r in rs])
fig,ax=axes('Transverse perturbation amplitude','Norm of axial amplitude correction')
ax.loglog(eps,corr,'o-',label='Galerkin continuation')
ax.loglog(eps,corr[0]*(eps/eps[0])**2,'--',label='Quadratic reference')
ax.legend();ax.set_title('Retuning begins at second order')
save('05_quadratic_retuning',fig,np.c_[eps,corr],'epsilon,amplitude_correction_norm')
fig,ax=axes('Transverse perturbation amplitude','Relative diagnostic (normalized at first point)')
nl=np.array([r['nonlinear_l2'] for r in rs]);st=np.array([r['stretching_l2'] for r in rs])
ax.loglog(eps,nl/nl[0],'o-',label='Projected advection')
ax.loglog(eps,corr/corr[0],'s-',label='Axial retuning')
ax.legend();ax.set_title('First-order interaction; second-order scalar correction')
save('06_interaction_orders',fig,np.c_[eps,nl,st,corr],'epsilon,advection,stretching,retuning')
base=compile_score([Event('0',1),Event('.12',1)],'.25');tt=np.array(rs[-1]['times'])
fig,ax=axes('Time','Helicity')
ax.plot(tt,[.01*float(base.helicity(str(x))) for x in tt],ls='--',label='Exact shear scaffold')
ax.plot(tt,rs[-1]['helicity'],label='Nonlinear retuned flow, K=3')
for x in [0,.12,.25]:ax.axvline(x,ls=':',lw=.7)
ax.legend();ax.set_title('Zeros retained; the intermediate curve is not prescribed')
save('07_nonlinear_calendar',fig,np.c_[tt,rs[-1]['helicity']],'time,helicity')
fig,ax=axes('Time','Kinetic energy')
for r in rs:ax.plot(r['times'],r['energy'],label=f"epsilon={r['epsilon']:g}")
ax.legend();ax.set_title('Energy decreases throughout every demonstrated run')
save('08_energy_decay',fig)
fig,ax=axes('Kinetic energy','Helicity')
ax.plot(rs[-1]['energy'],rs[-1]['helicity']);ax.invert_xaxis();ax.set_title('An event curve without timestamps')
save('09_energy_clock',fig,np.c_[rs[-1]['energy'],rs[-1]['helicity']],'energy,helicity')

lab=np.linspace(0,1.1,1101)
# A precisely specified, externally imposed common clock.
T=lab-np.clip(lab-.15,0,.18)-np.clip(lab-.65,0,.2)
H=np.array([float(base.helicity(str(x))) for x in T]); E=np.array([float(base.energy(str(x))) for x in T])
fig,ax=axes('Laboratory time','Helicity / anchor helicity')
ax.plot(lab,H,label='Common-gated equation')
ax.plot(lab,[float(base.helicity(str(x))) for x in lab],ls='--',label='Ungated equation')
ax.legend();ax.set_title('Pauses are clock control, not ordinary fixed-viscosity dynamics')
save('10_common_clock',fig,np.c_[lab,T,E,H],'laboratory_time,operational_time,energy,helicity')
fig,ax=axes('Energy','Helicity')
ss=np.linspace(0,.72,700)
ax.plot([float(base.energy(str(x))) for x in ss],[float(base.helicity(str(x))) for x in ss],label='Ungated geometric curve')
ax.plot(E[::40],H[::40],'o',ms=3,label='Samples under the paused clock')
ax.invert_xaxis();ax.legend();ax.set_title('The energy-coordinate event multiplicity survives retiming')
save('11_clock_quotient',fig)

lp=json.loads((D/'optimal_score.json').read_text())
fig,ax=axes('Wavenumber in the finite library','Signed helicity coefficient')
ax.stem(lp['wavenumbers'],lp['coefficients']);ax.set_title('Energy optimum uses the minimum three carriers')
save('12_energy_optimizer',fig,np.c_[lp['wavenumbers'],lp['coefficients']],'wavenumber,coefficient')
ks=np.array(lp['wavenumbers']);A=np.array(observation_matrix([Event(0,1),Event(.12,1)],.25,1,ks).tolist(),dtype=float)
q=np.abs(A.T@np.array(lp['dual']));w=1/(2*ks)
fig,ax=axes('Wavenumber','Dual magnitude / energy weight')
ax.plot(ks,q,'o-',label='Absolute dual evaluation');ax.plot(ks,w,'s--',label='Allowed envelope 1/(2k)')
ax.legend();ax.set_title('A dual certificate supports the spectral energy optimum')
save('13_dual_certificate',fig,np.c_[ks,q,w],'wavenumber,dual_absolute,weight')

orders=np.arange(1,11);conds=[]
for N in orders:
    A=np.array(observation_matrix([Event(0,int(N))],.2,1,list(range(1,int(N)+2))).tolist(),dtype=float)
    A=A/np.max(np.abs(A),axis=1)[:,None]
    conds.append(np.linalg.cond(A))
fig,ax=axes('Initial vanishing order','Row-scaled observation condition number')
ax.semilogy(orders,conds,'o-');ax.set_title('Exact solvability is not numerical robustness')
save('14_conditioning',fig,np.c_[orders,conds],'order,row_scaled_condition_number')
eta=np.logspace(-12,-2,250)
fig,ax=axes('Uniform observation error','Characteristic root displacement')
for m in [1,2,4,8]:ax.loglog(eta,eta**(1/m),label=f'order {m}')
ax.legend();ax.set_title('High-order silence has a sharp noise penalty')
save('15_root_sensitivity',fig,np.c_[eta,*[eta**(1/m) for m in [1,2,4,8]]],'noise,order1,order2,order4,order8')

# The exact rational four-carrier example, with coefficient c0=1.
r=np.array([1,4,9,16]); cc=np.array([27141,-227464,331779,-131456])/27141
te=np.linspace(0,2.2,701);he=np.sum(cc[None,:]*np.exp(-te[:,None]*r),axis=1)
fig,ax=axes('Time (2 nu = 1)','Helicity with first coefficient normalized to 1')
ax.plot(te,he);ax.axvline(np.log(2),ls=':',lw=.8);ax.axhline(0,ls=':',lw=.8)
ax.set_title('An exact double zero at 0 and a simple zero at log(2)')
save('16_exact_example',fig,np.c_[te,he],'time,helicity')
print('Built',len(list(F.glob('*.pdf'))),'figures.')