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9755170 | 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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 | """Reproducible adversarial checks. Run: python code/verify.py
These finite tests corroborate the proofs; they are neither formal
verification nor a novelty/priority audit.
"""
from __future__ import annotations
import json, math, platform, sys
from pathlib import Path
from dataclasses import asdict
import numpy as np
import scipy
import sympy as sp
from scipy.integrate import quad
from floquet import (fixed_period, free_period, monodromy, log_spectral_radius,
cell_gain, transfer, solve_eta, defect_integrand)
from exact_inverse import (layer_product, moments, recover_exponential_polynomial,
peel_first_row)
def lifted_angle(psi, w):
n = math.floor((psi + math.pi/2)/math.pi)
p = psi - n*math.pi
return math.atan(w*math.tan(p)) + n*math.pi
def phase_segments(us, ts, M):
vals, vecs = np.linalg.eig(M)
j = int(np.argmax(np.abs(vals)))
x = np.real(vecs[:, j]).astype(float)
theta = math.atan2(-x[1], x[0])
initial = theta
out = []
for u, duration in zip(us, ts):
w = math.sqrt(u)
psi = math.atan2(-x[1]/w, x[0])
nxt = theta + lifted_angle(psi+w*duration, w) - lifted_angle(psi, w)
out.append((theta, nxt, float(u)))
x = transfer(float(u), float(duration)) @ x
x /= np.linalg.norm(x)
theta = nxt
n = round((theta-initial)/math.pi)
assert abs(theta-initial-n*math.pi) < 2e-8
return out, n
def integrate_defect(segments, eta, R):
value = 0.0
for left, right, u in segments:
points = []
for base in [math.pi/2, -math.atan(eta)]:
for j in range(math.floor((left-base)/math.pi)-1,
math.ceil((right-base)/math.pi)+2):
z = base+j*math.pi
if left < z < right:
points.append(z)
value += quad(lambda th: defect_integrand(th,u,eta,R), left, right,
points=sorted(set(points)), epsabs=2e-10, epsrel=2e-10,
limit=250)[0]
return value
def main():
rng = np.random.default_rng(240925)
random_count = 0
hyperbolic_count = 0
worst_excess = -math.inf
defect_cases = []
for R in [1.1, 1.5, 2.0, 3.0, 5.0]:
for S in np.linspace(0.3,9.0,22):
optimum, candidates = fixed_period(R, float(S))
for j in range(60):
r = int(rng.integers(1,17))
us = (rng.choice([1.,R*R],size=r) if j%2 == 0
else rng.uniform(1.,R*R,size=r))
ts = rng.dirichlet(np.ones(r))*S
M = monodromy(us,ts)
L = log_spectral_radius(M)
random_count += 1
worst_excess = max(worst_excess,L-optimum)
assert L <= optimum+1e-9, (R,S,us,ts,L,optimum)
if L > 1e-5:
hyperbolic_count += 1
if len(defect_cases) < 60:
defect_cases.append((R,float(S),us,ts,M,L))
defect_error = 0.0
for R,S,us,ts,M,L in defect_cases:
segments,n = phase_segments(us,ts,M)
eta = solve_eta(R,S/n)
exact = n*cell_gain(R,eta)-L
computed = integrate_defect(segments,eta,R)
defect_error = max(defect_error,abs(exact-computed))
assert abs(exact-computed) < 1e-7
canonical_count = 0
canonical_error = 0.0
for R in [1.1,1.5,2.,3.,5.]:
for S in np.linspace(0.25,6,40):
bound,candidates = fixed_period(R,float(S))
for c in candidates:
M=monodromy([R*R,1.]*c.winding,[c.high_time,c.low_time]*c.winding)
err=abs(log_spectral_radius(M)-c.log_gain)
canonical_error=max(canonical_error,err)
assert err<1e-9
canonical_count += 1
free = {str(R): asdict(free_period(R)) for R in [1.1,2.,3.,5.,10.]}
sharp = []
R=2.; S=3*math.pi/4
for s in [0.02,0.01,0.005,0.0025]:
M=monodromy([4.,1.],[math.pi/4+s,math.pi/2-s])
deficit=math.log(2)-log_spectral_radius(M)
ratio=deficit/s**2
sharp.append({'s':s,'deficit':deficit,'deficit_over_s_squared':ratio})
assert abs(ratio-1.5)<0.002
# Exact signed-frequency identity for the all-discriminant counterexample.
A=[(sp.Rational(4),sp.Rational(1,2)),(sp.Rational(2),sp.Rational(1,2))]
B=[(sp.Rational(5,2),sp.Rational(4,5)),(sp.Rational(5),sp.Rational(1,5))]
from exact_inverse import add, scaled
HA,HB=layer_product(A),layer_product(B)
assert scaled(add(HA[0][0],HA[1][1]),sp.Rational(1,2)) == \
scaled(add(HB[0][0],HB[1][1]),sp.Rational(1,2))
assert HA[0] != HB[0]
# Includes collisions of signed optical-duration sums.
datasets = [
[(sp.Rational(2),sp.Rational(3,7))],
A, B,
[(sp.Rational(2),sp.Rational(1,5)),(sp.Rational(3),sp.Rational(2,7)),
(sp.Rational(5),sp.Rational(3,11))],
[(sp.Rational(2),sp.Rational(1,2)),(sp.Rational(3),sp.Rational(1,3)),
(sp.Rational(4),sp.Rational(1,4)),(sp.Rational(5),sp.Rational(1,5))]
]
inverse_results=[]
for layers in datasets:
r=len(layers); N=2**r
H=layer_product(layers)
f=recover_exponential_polynomial(moments(H[0][0],2*N),N)
g=recover_exponential_polynomial(moments(H[0][1],2*N),N)
assert f == H[0][0] and g == H[0][1]
result=peel_first_row(f,g,r)
assert result == layers,(layers,result)
inverse_results.append({'r':r,'first_row_terms':[len(f),len(g)],
'recovered_layers':[[str(c),str(t)] for c,t in result],
'exact':True})
report={
'seed':240925,'random_profiles_tested':random_count,
'hyperbolic_random_profiles':hyperbolic_count,
'maximum_random_bound_violation':max(0.,worst_excess),
'canonical_profiles_tested':canonical_count,
'maximum_canonical_gain_error':canonical_error,
'defect_identities_tested':len(defect_cases),
'maximum_defect_identity_error':defect_error,
'free_period_examples':free,'sharpness_examples':sharp,
'exact_inverse_roundtrips':inverse_results,
'exact_discriminant_counterexample_passed':True,
'all_assertions_passed':True,
'python':platform.python_version(),'numpy':np.__version__,
'scipy':scipy.__version__,'sympy':sp.__version__,
'scope':'Finite computational tests; not formal verification or peer review.'
}
path=Path(__file__).resolve().parents[1]/'results'/'verification.json'
path.parent.mkdir(parents=True,exist_ok=True)
path.write_text(json.dumps(report,indent=2)+'\n')
print(json.dumps(report,indent=2))
if __name__ == '__main__':
main()
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