| """Scientific summary of the proved rotation bound; plots are not certification.""" |
| from pathlib import Path |
| from fractions import Fraction |
| import json |
| import numpy as np |
| import matplotlib |
| matplotlib.use('Agg') |
| import matplotlib.pyplot as plt |
|
|
|
|
| def main(): |
| root=Path(__file__).parent |
| cert=json.loads((root/'ROTATION_BOUND_CERTIFICATE.json').read_text()) |
| pars=cert['parameters'] |
| r,t,d,g=[float(Fraction(pars[k])) for k in ('r','t','delta','kappa')] |
| a=1+r |
| w=np.sqrt(a*a-t*t) |
| C,S=np.cos(np.pi/8),np.sin(np.pi/8) |
| A,H=2*(t*S-w*C),2*(t*C+w*S) |
| eta0=(1-t)/2 |
| q=lambda z:A*A*z*z/(4*a*(A*A+(z+H)**2)) |
| reward=lambda eta:q(4*eta/3)+q(2*eta/3) |
| derivative=float(cert['packing_reward_derivative_at_eta0'][0]) |
| plt.rcParams.update({'font.size':10,'axes.spines.top':False, |
| 'axes.spines.right':False,'figure.facecolor':'white'}) |
| fig,axes=plt.subplots(1,3,figsize=(13.2,4.2),constrained_layout=True) |
| ax=axes[0] |
| angles=np.linspace(0,2*np.pi,1000) |
| ax.fill(np.cos(angles),np.sin(angles),color='#edf2f7') |
| for rotation,color,alpha in [(0,'#1591a2',.75),(np.pi/6,'#d48332',.55)]: |
| for k in range(4): |
| phi=np.pi/4+k*np.pi/2+rotation |
| u=np.array([np.cos(phi),np.sin(phi)]) |
| v=np.array([-np.sin(phi),np.cos(phi)]) |
| ss=np.linspace(t,1,100) |
| zz=np.sqrt(a*a-ss*ss) |
| pts=np.r_[ss[:,None]*u+zz[:,None]*v,ss[::-1,None]*u-zz[::-1,None]*v] |
| ax.fill(pts[:,0],pts[:,1],color=color,alpha=alpha) |
| ax.plot(np.cos(angles),np.sin(angles),color='#1e3a5f',lw=1.2) |
| ax.set(xlim=(-1.18,1.18),ylim=(-1.18,1.18),aspect='equal', |
| title='A. Average over all cap rotations') |
| ax.set_xticks([-1,0,1]);ax.set_yticks([-1,0,1]) |
| ax.text(.5,.53,r'Point membership $\leq 2\arccos t$',transform=ax.transAxes, |
| ha='center',fontsize=10) |
| ax.text(.5,.41,'Two example orientations shown',transform=ax.transAxes, |
| ha='center',fontsize=8,color='#526276') |
| ax.text(.5,-.1,'Cap regions; no candidate barrier shown',transform=ax.transAxes, |
| ha='center',fontsize=8,color='#526276') |
| ax=axes[1] |
| vals=[1.076457,d*1e6] |
| ax.bar([0,1],vals,color=['#94a3b8','#1591a2'],width=.58) |
| ax.set_xticks([0,1],['Published parameter\nchoice (2026)','Rotation certificate\nhere']) |
| ax.set(ylabel=r'Excess above $\pi$, in units of $10^{-6}$',ylim=(0,9.7), |
| title='B. Complete-barrier lower bound') |
| for x,y in enumerate(vals):ax.text(x,y+.14,f'{y:.7f}',ha='center',fontsize=9) |
| ax.text(.04,.96,f'Excess ratio: {vals[1]/vals[0]:.3f}'+r'$\times$', |
| transform=ax.transAxes,va='top') |
| ax=axes[2] |
| x=np.linspace(0,1,501) |
| U=np.pi*eta0*x |
| nonlinear=g*reward(eta0-U/np.pi)/d |
| linear=np.maximum(0,g*(reward(eta0)-derivative*U/np.pi)/d) |
| ax.plot(x,nonlinear,color='#1591a2',lw=2.5,label='Nonlinear robust gain') |
| ax.plot(x,linear,color='#d48332',lw=1.8,ls='--',label='Linear corollary (with zero)') |
| ax.set(xlabel=r'Missed-line measure $U/(\pi\eta_0)$', |
| ylabel=r'Gain over $L\geq\pi-U/2$, divided by $d$',ylim=(-.02,1.12), |
| title='C. Robustness to incomplete coverage') |
| ax.grid(alpha=.17);ax.legend(fontsize=8,loc='upper right') |
| fig.suptitle(r'Rotation-averaged disk opacity: $L>\pi+'+f'{d*1e6:.7f}'+r'\times10^{-6}$', |
| fontsize=16,fontweight='bold') |
| for suffix in ('png','pdf'): |
| fig.savefig(root/'figures'/f'disk_bound.{suffix}',dpi=180,bbox_inches='tight') |
| plt.close(fig) |
|
|
|
|
| if __name__=='__main__': |
| main() |
|
|
