code/independent_verify.py

"""Independent exact arithmetic checks for the exterior kernel-feedback candidate.

These finite checks are an implementation audit; the accompanying prose review
contains the universal arguments. SymPy performs all calculations over Q.
"""
from itertools import combinations
from random import Random
import json
import sympy as s

rng = Random(202610071)


def dot(x, y):
    return (x.T * y)[0]


def norm2(x):
    return dot(x, x)


def wedge(x, a):
    return x * a.T - a * x.T


def matrix_columns(cols, d):
    return s.Matrix.hstack(*cols) if cols else s.zeros(d, 0)


def split(A, x):
    d = len(x)
    K = matrix_columns(A.nullspace(), d)
    P = K * (K.T * K).inv() * K.T if K.cols else s.zeros(d)
    w = P * x
    u = x - w
    h = (A + P).inv() * u
    return K, P, u, w, h


def pf(A):
    if A.rows == 0:
        return s.Integer(1)
    if A.rows == 2:
        return A[0, 1]
    out = s.Integer(0)
    for j in range(1, A.rows):
        ii = [i for i in range(1, A.rows) if i != j]
        out += (-1) ** (j + 1) * A[0, j] * pf(A.extract(ii, ii))
    return out


def one_step(A, x, a):
    d = len(x)
    assert A + A.T == s.zeros(d)
    assert A * a == s.zeros(d, 1)
    K, P, u, w, h = split(A, x)
    B, y = A + wedge(x, a), x + a
    L, Q, v, z, hh = split(B, y)
    independent = matrix_columns([w, a], d).rank() == 2
    if independent:
        pred = matrix_columns(A.col_join(w.T).col_join(a.T).nullspace(), d)
        expected_rho = norm2(u) + norm2(w + a)
        assert B.rank() == A.rank() + 2
        assert z == s.zeros(d, 1)
    else:
        pred = K - h * (a.T * K)
        expected_rho = norm2(u) + norm2(h) * dot(a, w + a) ** 2 / (1 + norm2(h) * norm2(a))
        assert B.rank() == A.rank()
    assert B * pred == s.zeros(d, pred.cols)
    assert pred.rank() == L.cols
    assert norm2(v) == s.cancel(expected_rho)
    assert norm2(v) >= norm2(u)
    for k in range(1, d // 2 + 1):
        old = [pf(A.extract(ii, ii)) for ii in combinations(range(d), 2 * k)]
        new = [pf(B.extract(ii, ii)) for ii in combinations(range(d), 2 * k)]
        cross = sum(p * (q - p) for p, q in zip(old, new))
        assert cross == 0
        assert sum(q * q for q in new) == sum(p * p for p in old) + sum((q - p) ** 2 for p, q in zip(old, new))
    C, t = A + wedge(x, -w), u
    KK, PP, uu, ww, hhh = split(C, t)
    assert ww == s.zeros(d, 1)
    assert norm2(uu) == norm2(u)
    return B, y, independent


def random_state(d, rank):
    while True:
        C = s.Matrix(d, rank, lambda i, j: rng.randint(-2, 2))
        if C.rank() == rank:
            break
    J = s.zeros(rank)
    for i in range(0, rank, 2):
        J[i, i + 1], J[i + 1, i] = 1, -1
    A = C * J * C.T
    x = s.Matrix([rng.randint(-2, 2) for i in range(d)])
    return A, x


results = {"rational_one_step_cases": 0, "independent_cases": 0, "dependent_cases": 0, "maxrank_paths": 0}
for d in range(1, 8):
    for rank in range(0, d + 1, 2):
        for repeat in range(3):
            A, x = random_state(d, rank)
            K, P, u, w, h = split(A, x)
            controls = [s.zeros(d, 1), -w]
            if K.cols:
                controls.append(K * s.Matrix([rng.randint(-2, 2) for _ in range(K.cols)]))
            for a in controls:
                B, y, independent = one_step(A, x, a)
                results["rational_one_step_cases"] += 1
                results["independent_cases" if independent else "dependent_cases"] += 1
            # The construction also covers states with w forced to zero.
            for start_x in (x, u):
                AA, xx = A, start_x
                KK, PP, uu, ww, hh = split(AA, xx)
                m = KK.cols // 2
                expected = 0 if m == 0 else 2 * m - (0 if ww == s.zeros(d, 1) else 1)
                count = 0
                while AA.rank() < 2 * (d // 2):
                    KK, PP, uu, ww, hh = split(AA, xx)
                    if ww == s.zeros(d, 1):
                        a = KK[:, 0]
                    else:
                        a = next(KK[:, i] for i in range(KK.cols) if matrix_columns([ww, KK[:, i]], d).rank() == 2)
                    AA, xx, independent = one_step(AA, xx, a)
                    count += 1
                assert count == expected
                results["maxrank_paths"] += 1

# Separation family, including exact signs of lambda.
for lam in map(s.Rational, [-5, -2, -1, 1, 2, 5]):
    x = s.Matrix([1, 1, 0])
    A = s.Matrix([[0, lam, 0], [-lam, 0, 0], [0, 0, 0]])
    a = s.Matrix([0, 0, 1])
    B, y, independent = one_step(A, x, a)
    assert B * s.Matrix([1, -1, lam]) == s.zeros(3, 1)
    K, P, u, w, h = split(B, y)
    assert norm2(u) == 2 + 2 / (lam * lam + 2)
results["separation_lambdas"] = 6

# Dormant block replacement: run the same legal exact words in both systems.
for weight in (1, 2, 3):
    J = s.Matrix([[0, 1], [-1, 0]])
    A = s.diag(J, weight * J, s.zeros(3))
    B = s.diag(J, (weight + 5) * J, s.zeros(3))
    x = s.Matrix([1, 2, 0, 0, 1, -1, 0])
    y = x.copy()
    for step in range(5):
        K, P, u, w, h = split(A, x)
        L, Q, v, z, hh = split(B, y)
        assert P == Q
        assert x == y
        a = K * s.Matrix([rng.randint(-2, 2) for _ in range(K.cols)]) if K.cols else s.zeros(7, 1)
        assert a[2] == a[3] == 0
        A, x = A + wedge(x, a), x + a
        B, y = B + wedge(y, a), y + a
        assert A[2:4, 2:4] == weight * J
        assert B[2:4, 2:4] == (weight + 5) * J
results["dormant_replacement_paths"] = 3

# General three-dimensional accessibility: no projection preparation needed.
bb, cc, zz, pp, qq, rr = s.symbols("b c z p q r", real=True)
xx, kk = s.Matrix([bb, 0, zz]), s.Matrix([0, 0, cc])
for tt in (pp, qq, rr):
    xx, kk = s.expand(xx + tt * kk), s.expand(kk + tt * xx.cross(kk))
expected_endpoint = s.Matrix([
    bb + bb * cc**2 * pp**2 * qq * rr + bb * cc * zz * pp * qq * rr,
    -bb * cc * (pp * qq + pp * rr + qq * rr),
    zz + cc * (pp + qq + rr - bb**2 * pp * qq * rr),
])
assert s.simplify(xx - expected_endpoint) == s.zeros(3, 1)
jacobian_at_point = xx.jacobian((pp, qq, rr)).subs({pp: 1, qq: 1, rr: 2})
assert s.factor(jacobian_at_point.det()) == -2 * bb**2 * cc**4 * (bb**2 + 1)
results["symbolic_general_accessibility_checks"] = 1
print(json.dumps(results, indent=2))