File size: 3,584 Bytes
3b80460 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | """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()
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