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from fractions import Fraction
from itertools import combinations, permutations
from math import factorial, prod, comb, sqrt
import numpy as np
from continuous_factorization import (
repair_factor, reference_frame, reference_factor, nearest_blind_projector,
wronskian_atlas, coordinate_atlas, select_reference_frame, jet_reference,
subspace_transport, subspace_holonomy,
)
def _det(matrix):
a = [[Fraction(x) for x in row] for row in matrix]
value = Fraction(1)
for k in range(len(a)):
pivot = next((j for j in range(k, len(a)) if a[j][k]), None)
if pivot is None:
return Fraction(0)
if pivot != k:
a[k], a[pivot] = a[pivot], a[k]
value = -value
z = a[k][k]
value *= z
for j in range(k+1, len(a)):
multiplier = a[j][k]/z
for l in range(k+1, len(a)):
a[j][l] -= multiplier*a[k][l]
return value
def _multiply(a, b):
result = [Fraction(0)]*(len(a)+len(b)-1)
for j, x in enumerate(a):
for k, y in enumerate(b):
result[j+k] += x*y
return result
def _derivative(a, k):
return [Fraction(a[j])*(factorial(j)//factorial(j-k))
for j in range(k, len(a))] or [Fraction(0)]
def _wronskian(columns):
r = len(columns)
degree = r*(len(columns[0])-1)-r*(r-1)//2
result = [Fraction(0)]*(degree+1)
for p in permutations(range(r)):
sign = (-1)**sum(p[j] > p[k] for j in range(r) for k in range(j+1, r))
term = [Fraction(1)]
for k in range(r):
term = _multiply(term, _derivative(columns[p[k]], k))
for k, x in enumerate(term):
result[k] += sign*x
while len(result) > 1 and result[-1] == 0:
result.pop()
return result
def _evaluate(a, t):
return sum(x*Fraction(t)**k for k, x in enumerate(a))
def _exact_wronski_checks():
monomials, evaluations = 0, 0
for n in range(2, 9):
for r in range(1, n):
d = r*(n-r)
for degrees in combinations(range(n), r):
leading = prod(degrees[j]-degrees[i]
for i in range(r) for j in range(i+1, r))
power = sum(degrees)-r*(r-1)//2
assert 0 <= power <= d and leading > 0
nonzero = False
for t in range(d+1):
rows = [[0 if k > j else factorial(j)//factorial(j-k)*t**(j-k)
for j in degrees] for k in range(r)]
observed = _det(rows)
expected = leading*Fraction(t)**power
assert observed == expected
nonzero |= observed != 0
evaluations += 1
assert nonzero
monomials += 1
rng = np.random.default_rng(30001004)
polynomials = 0
for _ in range(64):
n = int(rng.integers(2, 7)); r = int(rng.integers(1, min(n, 5)))
matrix = rng.integers(-3, 4, size=(n, r))
# An identity block gives an exact independence certificate.
matrix[:r, :] = np.eye(r, dtype=int)
columns = [[int(x) for x in matrix[:, j]] for j in range(r)]
coefficients = _wronskian(columns)
assert any(coefficients) and len(coefficients)-1 <= r*(n-r)
nonzero = False
for t in range(r*(n-r)+1):
exact = _det([[_evaluate(_derivative(col, k), t)
for col in columns] for k in range(r)])
assert exact == _evaluate(coefficients, t)
nonzero |= exact != 0
assert nonzero
polynomials += 1
return {"exact_monomial_subspaces": monomials,
"exact_monomial_jet_determinants": evaluations,
"exact_integer_polynomial_subspaces": polynomials}
def _random_frame(rng, n, r):
a = rng.normal(size=(n, r))+1j*rng.normal(size=(n, r))
return np.linalg.qr(a)[0][:, :r]
def _rotation(rng, n, scale):
h = rng.normal(size=(n, n))+1j*rng.normal(size=(n, n))
h = (h+h.conj().T)/2
values, vectors = np.linalg.eigh(h)
return (vectors*np.exp(1j*scale*values)) @ vectors.conj().T
def run():
rng = np.random.default_rng(30001004)
maxima = {k: 0. for k in ["repaired_gram_error", "reference_gram_error",
"blind_distance_error", "transport_error", "decoder_error",
"cauchy_binet_error"]}
counts = {k: 0 for k in ["repair_cases", "reference_cases", "blind_witnesses",
"inside_radius_cases", "wronskian_atlas_cases", "coordinate_atlas_cases",
"transport_cycles", "noisy_decoder_cases", "inverse_margin_cases",
"gap_boundary_rejections", "structured_blind_families"]}
observed_atlas_margin = 1.
for n in range(2, 9):
for r in range(1, n):
wronski, coordinates = wronskian_atlas(n, r), coordinate_atlas(n, r)
assert len(wronski) == r*(n-r)+1
assert len(coordinates) == comb(n, r)
for _ in range(8):
u, w = _random_frame(rng, n, r), _random_frame(rng, n, r)
p = u @ u.conj().T
values = rng.uniform(.5, 1.5, r)
g = (u*values) @ u.conj().T
out = _random_frame(rng, r+2, r)
c0 = (out*np.sqrt(values)) @ u.conj().T
perturbation = rng.normal(size=c0.shape)+1j*rng.normal(size=c0.shape)
perturbation *= .01/np.linalg.norm(perturbation, 2)
c = c0+perturbation
d, info = repair_factor(g, c)
error = float(np.linalg.norm(d.conj().T @ d-g, 2))
maxima["repaired_gram_error"] = max(maxima["repaired_gram_error"], error)
assert error < 2e-12 and d.shape == c.shape
assert np.linalg.norm(d-c@p, 2) <= info["support_correction_bound"]+2e-12
assert np.linalg.norm(d-c, 2) <= info["total_correction_bound"]+2e-12
counts["repair_cases"] += 1
frame, delta = reference_frame(p, w)
factor, _ = reference_factor(p, w, .7)
referr = float(np.linalg.norm(factor.conj().T@factor-.7*p, 2))
maxima["reference_gram_error"] = max(maxima["reference_gram_error"], referr)
assert referr < 2e-11
counts["reference_cases"] += 1
blind, margin = nearest_blind_projector(p, w)
assert np.linalg.norm(blind@blind-blind, 2) < 1e-10
assert abs(np.trace(blind).real-r) < 1e-10
assert np.linalg.svd(blind@w, compute_uv=False)[-1] < 1e-10
distance_error = abs(np.linalg.norm(p-blind, 2)-delta)
maxima["blind_distance_error"] = max(maxima["blind_distance_error"], float(distance_error))
assert distance_error < 1e-10 and abs(margin-delta) < 1e-10
counts["blind_witnesses"] += 1
rotation = _rotation(rng, n, delta/100)
q = rotation @ p @ rotation.conj().T
eta = np.linalg.norm(q-p, 2)
assert eta < delta
frame2, delta2 = reference_frame(q, w)
assert delta2 >= delta-eta-1e-12
bound = 2*np.linalg.norm((q-p)@w, 'fro')/(delta+delta2)
assert np.linalg.norm(frame2-frame, 'fro') <= bound+2e-11
counts["inside_radius_cases"] += 1
_, _, chart_margin = select_reference_frame(p, wronski)
assert chart_margin > 1e-12
observed_atlas_margin = min(observed_atlas_margin, chart_margin)
counts["wronskian_atlas_cases"] += 1
_, _, chart_margin = select_reference_frame(p, coordinates)
determinants = [abs(np.linalg.det(cw.conj().T @ u))**2 for cw in coordinates]
cb_error = abs(sum(determinants)-1)
maxima["cauchy_binet_error"] = max(maxima["cauchy_binet_error"], float(cb_error))
assert cb_error < 2e-12 and chart_margin+2e-12 >= 1/sqrt(comb(n,r))
counts["coordinate_atlas_cases"] += 1
# The noisy completion is checked on arbitrary complex inputs.
weight = .6
visible = np.eye(n)+(np.sqrt(1-weight)-1)*p
cf = np.sqrt(weight)*u.conj().T
noise = rng.normal(size=cf.shape)+1j*rng.normal(size=cf.shape)
noise *= .005/np.linalg.norm(noise, 2)
cf += noise
repaired, _ = repair_factor(weight*p, cf)
completion = np.vstack([visible, repaired])
x = rng.normal(size=n)+1j*rng.normal(size=n)
decoder_error = float(np.linalg.norm(completion.conj().T @ completion@x-x))
maxima["decoder_error"] = max(maxima["decoder_error"], decoder_error)
assert decoder_error < 2e-11
unrepaired = np.vstack([visible, cf])
eps = np.linalg.norm(cf.conj().T@cf-weight*p, 2)
decoder = np.linalg.solve(unrepaired.conj().T@unrepaired, unrepaired.conj().T)
assert np.linalg.norm(decoder, 2) <= 1/np.sqrt(1-eps)+2e-12
counts["noisy_decoder_cases"] += 1
qframe, sframe = _random_frame(rng, n, r), _random_frame(rng, n, r)
q, s = qframe@qframe.conj().T, sframe@sframe.conj().T
transport = subspace_transport(p, q)
te = max(np.linalg.norm(transport.conj().T@transport-q, 2),
np.linalg.norm(transport@transport.conj().T-p, 2))
maxima["transport_error"] = max(maxima["transport_error"], float(te))
assert te < 2e-10
hol = subspace_holonomy([p,q,s], u)
assert np.linalg.norm(hol.conj().T@hol-np.eye(r), 2) < 5e-10
gauge = _random_frame(rng, r, r)
assert np.allclose(subspace_holonomy([p,q,s], u@gauge),
gauge.conj().T@hol@gauge, atol=1e-10)
global_rotation = _random_frame(rng, n, n)
rotated = subspace_transport(global_rotation@p@global_rotation.conj().T,
global_rotation@q@global_rotation.conj().T)
assert np.allclose(rotated, global_rotation@transport@global_rotation.conj().T,
atol=1e-10)
counts["transport_cycles"] += 1
# Explicit inverse-margin family: one direction varies, r-1 remain fixed.
w = np.eye(n, dtype=complex)[:, :r]
v = np.eye(n, dtype=complex)[:, r]
for delta in [.0001, .003, .04, .3, .8]:
phase = .7
last0 = np.sqrt(1-delta**2)*v+delta*w[:, -1]
last1 = np.sqrt(1-delta**2)*v+delta*np.exp(1j*phase)*w[:, -1]
e0 = np.column_stack([w[:, :-1],last0]); e1 = np.column_stack([w[:, :-1],last1])
p, q = e0@e0.conj().T, e1@e1.conj().T
f0,_ = reference_frame(p,w); f1,_ = reference_frame(q,w)
ratio = np.linalg.norm(f0-f1,'fro')/np.linalg.norm(p-q,'fro')
assert abs(ratio*delta-1/np.sqrt(2)) < 1e-9
counts["inverse_margin_cases"] += 1
equal = w@w.conj().T
blind, delta = nearest_blind_projector(equal,w)
assert abs(delta-1)<1e-12 and abs(np.linalg.norm(equal-blind,2)-1)<1e-12
counts["blind_witnesses"] += 1
# Equality at the positive-gap threshold permits losing a support direction.
g = np.diag([1.]*r+[0.]*(n-r)).astype(complex)
c = np.eye(n, dtype=complex)[:r-1]
try:
repair_factor(g,c)
except ValueError:
counts["gap_boundary_rejections"] += 1
else:
raise AssertionError("gap equality must not be accepted")
# A fixed m<n scanner family has an explicit blind support direction.
m = n-1
h = np.eye(n,dtype=complex)[:m]
support = np.column_stack([np.eye(n)[:, -1],np.eye(n)[:, :r-1]])
p = support@support.conj().T
c = h@p
assert np.linalg.norm(c@np.eye(n)[:, -1]) == 0
assert abs(np.linalg.norm(c.conj().T@c-p,2)-1)<1e-12
counts["structured_blind_families"] += 1
zero,_ = repair_factor(np.zeros((3,3)),np.ones((0,3)))
assert zero.shape == (0,3)
# Four distinct real jet charts have a common blind two-plane in C^4.
a = (5+sqrt(73))/6; b = (-5+sqrt(73))/2
raw = np.array([[-a,0],[0,-b],[1,0],[0,1]],dtype=complex)
u = np.linalg.qr(raw)[0][:,:2]; p = u@u.conj().T
for t in [-2,-1,1,2]:
assert np.linalg.svd(p@jet_reference(4,2,t),compute_uv=False)[-1]<1e-12
assert np.linalg.svd(p@jet_reference(4,2,0),compute_uv=False)[-1]>.1
# Noncommuting U(2) cycle phases, in one common base frame.
base = np.eye(3,dtype=complex)[:,:2]; p0=base@base.conj().T
def graph(z):
frame=np.linalg.qr(np.vstack([np.eye(2),np.array(z)]))[0][:,:2]
return frame@frame.conj().T
pa,pb,pc = graph([[.6,0]]),graph([[0,.6]]),graph([[0,.6j]])
h1=subspace_holonomy([p0,pa,pb],base)
h2=subspace_holonomy([p0,pa,pc],base)
commutator=float(np.linalg.norm(h1@h2-h2@h1,2))
ha,hb=10*sqrt(34)/59,9/59
assert np.allclose(h1,[[ha,hb],[-hb,ha]],atol=1e-12)
assert np.allclose(h2,[[ha,1j*hb],[1j*hb,ha]],atol=1e-12)
assert Fraction(3400,3481)+Fraction(81,3481)==1
assert abs(commutator-162/3481)<1e-12
return {"status":"PASS", "seed":30001004, **_exact_wronski_checks(), **counts,
"maximum_errors":maxima,
"sampled_minimum_wronskian_atlas_margin":float(observed_atlas_margin),
"wronskian_margin_scope":"sample statistic, not a uniform certificate",
"four_reference_blind_counterexample":"Gr(2,4), nodes -2,-1,1,2",
"noncommuting_holonomy_commutator_norm":commutator,
"exact_holonomy_commutator_norm":"162/3481",
"topological_scope":"finite checks do not prove the global lower bounds"}
if __name__ == '__main__':
import json
print(json.dumps(run(),indent=2))
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