import json,math
w0=.01
exact=lambda t:1/(1+(1/w0-1)*math.exp(-2*t))
frozen=lambda t:1-(1-w0)*math.exp(-2*w0*t)
assert abs(exact(0)-w0)<1e-14 and abs(frozen(0)-w0)<1e-14
h=.001;w=w0
for _ in range(5000):
 f=lambda z:2*z*(1-z)
 k1=f(w);k2=f(w+h*k1/2);k3=f(w+h*k2/2);k4=f(w+h*k3)
 w+=h*(k1+2*k2+2*k3+k4)/6
assert abs(w-exact(5))<1e-10
for t in [0,1,2,3,4,5]:
 d=1e-5;assert abs((exact(t+d)-exact(t-d))/(2*d)-2*exact(t)*(1-exact(t)))<1e-8
a=b=.1;da=db=.9
assert abs((a+da)*(b+db)-(a*b+b*da+a*db)-da*db)<1e-12
ts=[0,1,2,3,4,5];ts2=[0,1,2,5,10]
figures=[{"id":"balanced-vs-frozen","kind":"analytical_curve","caption":"Author-constructed scalar model w0=.01: exact balanced gradient flow versus matched-initialization frozen tangent. Dimensionless; no empirical benchmark.","x":{"name":"Dimensionless time","values":ts},"series":[{"name":"Factorized dynamics","values":[exact(t)for t in ts]},{"name":"Frozen tangent","values":[frozen(t)for t in ts]}]},{"id":"spectral-residual","kind":"analytical_curve","caption":"Author-constructed fixed-kernel residual modes with eigenvalues 1 and .01, both initialized at one. No trained-network measurement.","x":{"name":"Dimensionless time","values":ts2},"series":[{"name":"Eigenvalue 1","values":[math.exp(-t)for t in ts2]},{"name":"Eigenvalue .01","values":[math.exp(-.01*t)for t in ts2]}]}]
print(json.dumps({"status":"PASS","checks":["Initial conditions","RK4 vs logistic closed form","ODE derivative at six times","Exact bilinear Taylor remainder"],"rk4_error":abs(w-exact(5)),"empirical_benchmark":False,"source_proofs_verified":False,"figures":figures}))