verify_continuous.py

"""Independent exact and numerical checks of the arbitrary-rank extension."""
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))