"""Synthetic educational response. No material certification or remaining-life assessment."""
import math
BOUNDS={"T":(450,600),"sigma":(30,100),"time":(1,100000)}
def solve(T=550,sigma=80,time=10000):
    p={"T":T,"sigma":sigma,"time":time}
    for key,(lo,hi) in BOUNDS.items():
        if not isinstance(p[key],(int,float)) or not math.isfinite(p[key]) or not lo<=p[key]<=hi: raise ValueError(key)
    thermal=math.exp(-250000/8.314*(1/(T+273.15)-1/823.15))
    epsilon=.002*(sigma/80)**5.2*(time/10000)**.22*thermal
    return {"epsilon":epsilon,"rate_per_hour":.22*epsilon/time,"thermal":thermal,"relative":epsilon/.002}
if __name__=="__main__": print(solve())