Datasets:

PureOne
/

controlled-spectral-reconstruction

#!/usr/bin/env python3
"""Independent exact and numerical tests of constructive spectral recovery.
Exact: graph cycle matrices and recovery from a magnetic ground-state Hessian;
finite-rank determinant normalization/interpolation; critical block inertia.
Numerical: independently assembled magnetic, gauge-covariant linear finite
elements for Kirchhoff metric graphs, using NumPy/SciPy only in that section.
Run: python3 validation_v2/constructive_verification.py
The exact checks are not formal machine proofs. FEM checks are floating-point
consistency tests, not certified eigenvalue enclosures or evidence of novelty.
"""

from __future__ import annotations

from collections import Counter, deque
from fractions import Fraction as Q
from itertools import permutations
from math import isqrt
from pathlib import Path
import argparse
import json
import random


SEED = 20261005


def transpose(a):
    return [list(row) for row in zip(*a)]


def mm(a, b):
    bt = transpose(b)
    return [[sum((x * y for x, y in zip(row, col)), Q(0)) for col in bt] for row in a]


def inverse(a):
    n = len(a)
    aug = [[Q(x) for x in row] + [Q(i == j) for j in range(n)] for i, row in enumerate(a)]
    for j in range(n):
        pivot = next((k for k in range(j, n) if aug[k][j]), None)
        if pivot is None:
            raise ValueError("singular exact matrix")
        aug[j], aug[pivot] = aug[pivot], aug[j]
        q = aug[j][j]
        aug[j] = [x / q for x in aug[j]]
        for i in range(n):
            if i != j and aug[i][j]:
                q = aug[i][j]
                aug[i] = [x - q * y for x, y in zip(aug[i], aug[j])]
    return [row[n:] for row in aug]


def rank(a):
    a = [[Q(x) for x in row] for row in a]
    r = 0
    for j in range(len(a[0]) if a else 0):
        pivot = next((i for i in range(r, len(a)) if a[i][j]), None)
        if pivot is None:
            continue
        a[r], a[pivot] = a[pivot], a[r]
        p = a[r][j]
        a[r] = [x / p for x in a[r]]
        for i in range(len(a)):
            if i != r and a[i][j]:
                p = a[i][j]
                a[i] = [x - p * y for x, y in zip(a[i], a[r])]
        r += 1
        if r == len(a):
            break
    return r


def solve_full_column_rank(a, b):
    rows, n = len(a), len(a[0])
    aug = [[Q(x) for x in row] + [Q(value)] for row, value in zip(a, b)]
    r = 0
    pivots = []
    for j in range(n):
        pivot = next((i for i in range(r, rows) if aug[i][j]), None)
        if pivot is None:
            continue
        aug[r], aug[pivot] = aug[pivot], aug[r]
        p = aug[r][j]
        aug[r] = [x / p for x in aug[r]]
        for i in range(rows):
            if i != r and aug[i][j]:
                p = aug[i][j]
                aug[i] = [x - p * y for x, y in zip(aug[i], aug[r])]
        pivots.append(j)
        r += 1
    if len(pivots) != n:
        raise ValueError("edge parameters not uniquely identifiable")
    if any(not any(row[:n]) and row[n] for row in aug):
        raise ValueError("inconsistent exact linear system")
    result = [Q(0)] * n
    for i, p in enumerate(pivots):
        result[p] = aug[i][-1]
    return result


def determinant(a):
    a = [[Q(x) for x in row] for row in a]
    result = Q(1)
    for j in range(len(a)):
        pivot = next((i for i in range(j, len(a)) if a[i][j]), None)
        if pivot is None:
            return Q(0)
        if pivot != j:
            a[j], a[pivot] = a[pivot], a[j]
            result = -result
        p = a[j][j]
        result *= p
        for i in range(j + 1, len(a)):
            q = a[i][j] / p
            for k in range(j + 1, len(a)):
                a[i][k] -= q * a[j][k]
    return result


def fundamental_cycles(nvertices, edges, *, reverse=False):
    """Cycle rows: positive chord u->v, followed by tree path v->u."""
    adjacency = [[] for _ in range(nvertices)]
    for e, (u, v) in enumerate(edges):
        adjacency[u].append((v, e))
        adjacency[v].append((u, e))
    for row in adjacency:
        row.sort(reverse=reverse)
    root = nvertices - 1 if reverse else 0
    seen, tree = {root}, set()
    queue = deque([root])
    while queue:
        u = queue.popleft()
        for v, e in adjacency[u]:
            if v not in seen:
                seen.add(v)
                tree.add(e)
                queue.append(v)
    if len(seen) != nvertices:
        raise ValueError("disconnected graph")
    tree_adjacency = [[] for _ in range(nvertices)]
    for e in tree:
        u, v = edges[e]
        tree_adjacency[u].append((v, e, 1))
        tree_adjacency[v].append((u, e, -1))
    chords = [e for e in range(len(edges)) if e not in tree]
    cycles = []
    for chord in chords:
        u, v = edges[chord]
        predecessor = {v: None}
        queue = deque([v])
        while u not in predecessor:
            node = queue.popleft()
            for next_node, edge, sign in tree_adjacency[node]:
                if next_node not in predecessor:
                    predecessor[next_node] = (node, edge, sign)
                    queue.append(next_node)
        row = [Q(0)] * len(edges)
        row[chord] = 1
        node = u
        while node != v:
            prev, edge, sign = predecessor[node]
            row[edge] += sign
            node = prev
        cycles.append(row)
    return cycles, chords


def edge_design(cycles):
    return [[cycles[i][e] * cycles[j][e] for e in range(len(cycles[0]))]
            for i in range(len(cycles)) for j in range(i, len(cycles))]


def exact_square_root(q):
    a, b = isqrt(q.numerator), isqrt(q.denominator)
    if a * a != q.numerator or b * b != q.denominator:
        raise ValueError("rational quantity is not an exact rational square")
    return Q(a, b)


def polynomial_multiply(a, b):
    result = [Q(0)] * (len(a) + len(b) - 1)
    for i, x in enumerate(a):
        for j, y in enumerate(b):
            result[i + j] += x * y
    return result


def spectral_polynomial(a):
    """Exact coefficients of det(a+zI), independently by permutations."""
    n = len(a)
    result = [Q(0)] * (n + 1)
    for permutation in permutations(range(n)):
        inversions = sum(permutation[i] > permutation[j] for i in range(n) for j in range(i + 1, n))
        product = [Q((-1) ** inversions)]
        for i, j in enumerate(permutation):
            product = polynomial_multiply(product, [Q(a[i][j]), Q(1)] if i == j else [Q(a[i][j])])
        for i, value in enumerate(product):
            result[i] += value
    return result


def poly_eval(p, z):
    result = Q(0)
    for c in reversed(p):
        result = result * z + c
    return result


def inertia_without_zero_pivots(a):
    """Exact congruence elimination for the nondegenerate cases below."""
    a = [[Q(x) for x in row] for row in a]
    negative = 0
    for j in range(len(a)):
        p = a[j][j]
        if p == 0:
            raise ValueError("zero pivot requires a 2x2 inertia block")
        negative += int(p < 0)
        for i in range(j + 1, len(a)):
            for k in range(j + 1, len(a)):
                a[i][k] -= a[i][j] * a[j][k] / p
    return negative


class Verification:
    def __init__(self):
        self.assertions = Counter()

    def check(self, category, condition, detail=None):
        if not condition:
            raise AssertionError(f"{category}: {detail!r}")
        self.assertions[category] += 1


def graphs():
    return {
        "K4": (4, [(u, v) for u in range(4) for v in range(u + 1, 4)]),
        "K3_3": (6, [(u, v) for u in range(3) for v in range(3, 6)]),
        "triangular_prism": (6, [(0, 1), (1, 2), (0, 2), (3, 4), (4, 5), (3, 5),
                                  (0, 3), (1, 4), (2, 5)]),
        "wheel_W6": (6, [(0, i) for i in range(1, 6)]
                          + [(1, 2), (2, 3), (3, 4), (4, 5), (1, 5)]),
        "cube": (8, [(u, v) for u in range(8) for v in range(u + 1, 8)
                     if (u ^ v).bit_count() == 1])
    }


def vertex_connected_after_removal(n, edges, removed):
    available = set(range(n)) - set(removed)
    seen = {next(iter(available))}
    stack = list(seen)
    while stack:
        u = stack.pop()
        for a, b in edges:
            v = b if a == u else a if b == u else None
            if v is not None and v in available and v not in seen:
                seen.add(v)
                stack.append(v)
    return seen == available


def exact_graph_checks(v, rng):
    reports, fem_cases = [], []
    for name, (nv, edges) in graphs().items():
        c, chords = fundamental_cycles(nv, edges)
        incidence = [[Q((head == node) - (tail == node)) for tail, head in edges]
                     for node in range(nv)]
        v.check("cycle_boundary_zero", mm(c, transpose(incidence))
                == [[Q(0)] * nv for _ in c])
        v.check("cycle_rank_and_chord_coordinates", len(c) == len(edges) - nv + 1
                and [[row[e] for e in chords] for row in c]
                == [[Q(i == j) for j in range(len(c))] for i in range(len(c))])
        for a in range(nv):
            for b in range(a + 1, nv):
                v.check("graph_three_vertex_connectivity", vertex_connected_after_removal(nv, edges, (a, b)))
        design = edge_design(c)
        v.check("edge_metric_linear_identifiability", rank(design) == len(edges))
        c2, _ = fundamental_cycles(nv, edges, reverse=True)
        u = [[row[e] for e in chords] for row in c2]
        v.check("cycle_basis_unimodular", abs(determinant(u)) == 1 and mm(u, c) == c2)
        for case in range(12):
            lengths = [Q(rng.randint(1, 10), rng.randint(2, 12)) for _ in edges]
            total = sum(lengths, Q(0))
            weighted = [[value * ell for value, ell in zip(row, lengths)] for row in c]
            gram = mm(weighted, transpose(c))
            hessian = [[2 * value / total for value in row] for row in inverse(gram)]
            observed = [[2 * value for value in row] for row in inverse(hessian)]
            rhs = [observed[i][j] for i in range(len(c)) for j in range(i, len(c))]
            weights = solve_full_column_rank(design, rhs)
            recovered_total = exact_square_root(sum(weights, Q(0)))
            recovered_lengths = [weight / recovered_total for weight in weights]
            v.check("hessian_total_length_reconstruction", recovered_total == total)
            v.check("hessian_all_edge_lengths_reconstruction", recovered_lengths == lengths)
            v.check("hessian_observation_is_LQ", observed
                    == [[total * value for value in row] for row in gram])
            gram2 = mm([[value * ell for value, ell in zip(row, lengths)] for row in c2], transpose(c2))
            v.check("cycle_gram_coordinate_covariance", gram2 == mm(mm(u, gram), transpose(u)))
            inv_u = inverse(u)
            hessian2 = [[2 * value / total for value in row] for row in inverse(gram2)]
            v.check("hessian_coordinate_covariance", hessian2 == mm(mm(transpose(inv_u), hessian), inv_u))
            if case == 0:
                reports.append({"graph": name, "vertices": nv, "edges": len(edges),
                                "cycle_rank": len(c), "design_rank": rank(design),
                                "lengths": [str(ell) for ell in lengths], "total_length": str(total),
                                "cycle_matrix": [[int(x) for x in row] for row in c],
                                "exact_recovered_lengths": [str(ell) for ell in recovered_lengths]})
                if name in ("K4", "K3_3", "cube"):
                    fem_cases.append((name, nv, edges, lengths, chords, gram, total))
    cycle_edges = [(0, 1), (1, 2), (2, 3), (0, 3)]
    cc, _ = fundamental_cycles(4, cycle_edges)
    v.check("nonidentifiable_cycle_negative_control", rank(edge_design(cc)) == 1 < len(cycle_edges))
    return reports, fem_cases


def magnetic_fem_ground(nv, edges, lengths, chords, theta, subdivisions):
    import numpy as np
    from scipy.linalg import eigh
    nodes = nv + len(edges) * (subdivisions - 1)
    stiffness = np.zeros((nodes, nodes), dtype=np.complex128)
    mass = np.zeros_like(stiffness)
    chord_flux = dict(zip(chords, theta))
    cursor = nv
    for edge, ((tail, head), ell) in enumerate(zip(edges, lengths)):
        interiors = list(range(cursor, cursor + subdivisions - 1))
        cursor += subdivisions - 1
        sequence = [tail] + interiors + [head]
        h = float(ell) / subdivisions
        phase = np.exp(-1j * chord_flux.get(edge, 0.0) / subdivisions)
        local_k = np.array([[1, -phase], [-phase.conjugate(), 1]], dtype=np.complex128) / h
        local_m = h / 6 * np.array([[2, phase], [phase.conjugate(), 2]], dtype=np.complex128)
        for a, b in zip(sequence, sequence[1:]):
            idx = np.ix_([a, b], [a, b])
            stiffness[idx] += local_k
            mass[idx] += local_m
    return float(eigh(stiffness, mass, subset_by_index=[0, 0], driver="gvx",
                      check_finite=True)[0][0])


def fem_checks(v, fem_cases):
    results = []
    for name, nv, edges, lengths, chords, gram, total in fem_cases:
        beta = len(chords)
        inv_gram = inverse(gram)
        vectors = [[int(i == j) for i in range(beta)] for j in range(min(2, beta))]
        vectors.append([1 if i % 2 == 0 else -1 for i in range(beta)])
        for subdivisions in (4, 8):
            baseline = magnetic_fem_ground(nv, edges, lengths, chords, [0.0] * beta, subdivisions)
            for vector in vectors:
                predicted = sum((Q(vector[i]) * inv_gram[i][j] * vector[j]
                                 for i in range(beta) for j in range(beta)), Q(0)) / total
                errors = []
                for epsilon in (0.01, 0.005, 0.0025):
                    positive = magnetic_fem_ground(nv, edges, lengths, chords,
                                                   [epsilon * a for a in vector], subdivisions)
                    negative = magnetic_fem_ground(nv, edges, lengths, chords,
                                                   [-epsilon * a for a in vector], subdivisions)
                    curvature_half = ((positive + negative) / 2 - baseline) / epsilon ** 2
                    relative = abs(curvature_half - float(predicted)) / float(predicted)
                    v.check("fem_hessian_normalization", relative < 0.002,
                            (name, subdivisions, vector, epsilon, relative))
                    v.check("fem_time_reversal_symmetry", abs(positive - negative) < 1e-9)
                    errors.append(relative)
                    results.append({"graph": name, "subdivisions_per_edge": subdivisions,
                                    "flux_direction": vector, "epsilon_radians": epsilon,
                                    "baseline_numerical_eigenvalue": baseline,
                                    "predicted_lambda_over_epsilon_squared": float(predicted),
                                    "observed_lambda_over_epsilon_squared": curvature_half,
                                    "relative_error": relative})
                v.check("fem_small_flux_consistency", max(errors) < 0.002)
    return results


def critical_coupling_checks(v, rng):
    examples = []
    n = 4
    t = [Q(8), Q(11), Q(15), Q(20)]
    base = [[t[i] if i == j else Q(0) for j in range(n)] for i in range(n)]
    expected_poly = spectral_polynomial(base)
    expected = [x / expected_poly[0] for x in expected_poly]
    for r in (1, 2, 3):
        for case in range(15):
            factor = [[Q(rng.randint(-4, 4), 50) for _ in range(r)] for _ in range(n)]
            interaction = mm(factor, transpose(factor))
            v.check("coupling_rank_bound", rank(interaction) <= r)
            settings = [Q((i + 1) ** 2) for i in range(r + 1)]
            polys, dets = [], []
            for setting in settings:
                e = [[base[i][j] - setting * interaction[i][j] for j in range(n)] for i in range(n)]
                v.check("coupled_matrix_positive_gershgorin", all(e[i][i] > sum(abs(e[i][j])
                                                for j in range(n) if j != i) for i in range(n)))
                p = spectral_polynomial(e)
                v.check("permutation_determinant_independent_check", p[0] == determinant(e))
                polys.append([a / p[0] for a in p])
                dets.append(p[0])
            alpha = [det / dets[0] for det in dets]
            weights = [Q(1)] * len(settings)
            for i, setting in enumerate(settings):
                for j, other in enumerate(settings):
                    if i != j:
                        weights[i] *= -other / (setting - other)
            denominator = sum((w * a for w, a in zip(weights, alpha)), Q(0))
            restored = [sum((weights[i] * alpha[i] * polys[i][j] for i in range(r + 1)), Q(0))
                        / denominator for j in range(n + 1)]
            v.check("determinant_normalization_denominator_positive", denominator > 0)
            v.check("finite_rank_spectral_polynomial_recovery", restored == expected)
            naive = [sum((weights[i] * polys[i][j] for i in range(r + 1)), Q(0)) for j in range(n + 1)]
            if any(any(row) for row in interaction):
                v.check("uncalibrated_products_negative_control", naive != expected)
            # Positive-axis ratios independently approach the determinant
            # calibrations. Record exact finite-s ratios rather than assume
            # eigenvectors or simultaneously diagonalizable matrices.
            if case == 0:
                ratios = [[poly_eval(polys[0], s) / poly_eval(polys[i], s) for i in range(r + 1)]
                          for s in (Q(100), Q(1000), Q(10000))]
                v.check("positive_axis_calibration_convergence", all(
                    abs(ratios[-1][i] - alpha[i]) <= abs(ratios[0][i] - alpha[i])
                    for i in range(r + 1)))
                examples.append({"rank_bound": r, "settings": [str(u) for u in settings],
                                 "T_diagonal": [str(a) for a in t],
                                 "interaction_commutes_with_T": mm(base, interaction) == mm(interaction, base),
                                 "alphas": [str(a) for a in alpha],
                                 "weights": [str(w) for w in weights],
                                 "reconstructed_D_T_coefficients": [str(a) for a in restored],
                                 "positive_axis_ratios": [[str(a) for a in row] for row in ratios]})

    # Exact critical-coupling cutoff counts using full block inertia, which is
    # independent of the determinant-polynomial interpolation test above.
    hard = [Q(3), Q(5)]
    w = [[Q(1, 5), Q(0)], [Q(1, 7), Q(1, 8)], [Q(0), Q(1, 6)], [Q(1, 9), Q(-1, 10)]]
    cutoff = Q(1)
    correction = mm([[a / (f - cutoff) for a, f in zip(row, hard)] for row in w], transpose(w))
    inertia_rows = []
    for setting in (Q(1), Q(4), Q(9)):
        effective = [[base[i][j] - setting * correction[i][j] for j in range(n)] for i in range(n)]
        for a in (Q(1, 4), Q(1, 9), Q(1, 16), Q(1, 25), Q(1, 100)):
            root = exact_square_root(a * setting)
            # Order hard block first so exact congruence elimination begins
            # with two strictly positive pivots.
            physical_minus_cutoff = [[Q(0)] * (n + 2) for _ in range(n + 2)]
            for i in range(2):
                physical_minus_cutoff[i][i] = hard[i] - cutoff
            for i in range(n):
                for j in range(n):
                    physical_minus_cutoff[i + 2][j + 2] = a * base[i][j] - (cutoff if i == j else 0)
                for j in range(2):
                    physical_minus_cutoff[j][i + 2] = physical_minus_cutoff[i + 2][j] = root * w[i][j]
            shifted_effective = [[effective[i][j] - (cutoff / a if i == j else 0)
                                  for j in range(n)] for i in range(n)]
            full_count = inertia_without_zero_pivots(physical_minus_cutoff)
            effective_count = inertia_without_zero_pivots(shifted_effective)
            v.check("critical_low_cutoff_exact_inertia_identity", full_count == effective_count)
            inertia_rows.append({"a": str(a), "u": str(setting),
                                 "physical_count_below_cutoff": full_count,
                                 "effective_count_below_cutoff_over_a": effective_count})

    # The manuscript's rank-two ambiguity, and arbitrary-finite-setting
    # parity construction, independently tested as multisets.
    ta, ca, tb, cb = [Q(11), Q(12)], [Q(2), Q(2)], [Q(10), Q(13)], [Q(1), Q(3)]
    for setting in (Q(1), Q(2)):
        v.check("rank_two_two_scan_nonuniqueness", sorted(t - setting * c for t, c in zip(ta, ca))
                == sorted(t - setting * c for t, c in zip(tb, cb)))
    v.check("rank_two_unperturbed_spectra_differ", sorted(ta) != sorted(tb))
    for k in range(1, 6):
        settings = list(range(1, k + 1))
        displacements = [10 ** i for i in range(k)]
        t0 = 10 + k * (1 + sum(displacements))
        even, odd = [], []
        for mask in range(2 ** k):
            point = (Q(t0 + sum(settings[i] * displacements[i] for i in range(k) if mask >> i & 1)),
                     Q(1 + sum(displacements[i] for i in range(k) if mask >> i & 1)))
            (even if mask.bit_count() % 2 == 0 else odd).append(point)
        for setting in settings:
            observed_a = sorted(t - setting * c for t, c in even)
            observed_b = sorted(t - setting * c for t, c in odd)
            v.check("arbitrary_setting_parity_nonuniqueness", observed_a == observed_b and min(observed_a) > 0)
        v.check("parity_unperturbed_spectra_differ", sorted(t for t, c in even) != sorted(t for t, c in odd))
    return {"determinant_examples": examples, "critical_cutoff_inertia_examples": inertia_rows}


def run(skip_fem=False):
    rng, v = random.Random(SEED), Verification()
    graph_reports, fem_cases = exact_graph_checks(v, rng)
    coupling = critical_coupling_checks(v, rng)
    exact_assertions = sum(v.assertions.values())
    fem_reports = [] if skip_fem else fem_checks(v, fem_cases)
    return {"status": "PASS", "seed": SEED,
            "exact_assertions": exact_assertions,
            "numerical_assertions": sum(v.assertions.values()) - exact_assertions,
            "assertions_total": sum(v.assertions.values()),
            "assertions_by_category": dict(sorted(v.assertions.items())),
            "flux_convention": "Cycle holonomies in radians; positive chord entry 1; tree-edge flux 0",
            "hessian_formula": "lambda0(theta)=theta^T Q^-1 theta/L+O(|theta|^4), H=2Q^-1/L",
            "inverse_metric_algorithm": "S=2H^-1=LQ; solve S=sum_e w_e c_e c_e^T; L=sqrt(sum_e w_e); ell_e=w_e/L",
            "graphs": graph_reports, "finite_element_normalization": fem_reports,
            "critical_coupling": coupling,
            "limitations": ["Exact rational finite tests are not formal proofs for all graphs or infinite operators.",
                            "The graph reconstruction assumes the labeled graph and a known fundamental cycle-coordinate basis.",
                            "Graph topology recovery from an unmarked period lattice requires separate theory and is not validated here.",
                            "FEM checks use floating-point eigenvalues and are not rigorous spectral enclosures.",
                            "No originality, priority or 10/10 importance certification is provided."]}


def main():
    parser = argparse.ArgumentParser(description=__doc__)
    parser.add_argument("--output", type=Path,
                        default=Path(__file__).resolve().parent / "constructive_verification_results.json")
    parser.add_argument("--skip-fem", action="store_true")
    args = parser.parse_args()
    result = run(args.skip_fem)
    args.output.parent.mkdir(parents=True, exist_ok=True)
    args.output.write_text(json.dumps(result, indent=2) + "\n", encoding="utf-8")
    print(json.dumps({"status": result["status"], "exact_assertions": result["exact_assertions"],
                      "numerical_assertions": result["numerical_assertions"],
                      "assertions_total": result["assertions_total"],
                      "worst_fem_relative_error": max((r["relative_error"] for r in result["finite_element_normalization"]),
                                                       default=None),
                      "assertions_by_category": result["assertions_by_category"],
                      "output": str(args.output.resolve())}, indent=2))


if __name__ == "__main__":
    main()

	#!/usr/bin/env python3
	"""Independent exact and numerical tests of constructive spectral recovery.

	Exact: graph cycle matrices and recovery from a magnetic ground-state Hessian;
	finite-rank determinant normalization/interpolation; critical block inertia.
	Numerical: independently assembled magnetic, gauge-covariant linear finite
	elements for Kirchhoff metric graphs, using NumPy/SciPy only in that section.

	Run: python3 validation_v2/constructive_verification.py
	The exact checks are not formal machine proofs. FEM checks are floating-point
	consistency tests, not certified eigenvalue enclosures or evidence of novelty.
	"""

	from __future__ import annotations

	from collections import Counter, deque
	from fractions import Fraction as Q
	from itertools import permutations
	from math import isqrt
	from pathlib import Path
	import argparse
	import json
	import random


	SEED = 20261005


	def transpose(a):
	return [list(row) for row in zip(*a)]


	def mm(a, b):
	bt = transpose(b)
	return [[sum((x * y for x, y in zip(row, col)), Q(0)) for col in bt] for row in a]


	def inverse(a):
	n = len(a)
	aug = [[Q(x) for x in row] + [Q(i == j) for j in range(n)] for i, row in enumerate(a)]
	for j in range(n):
	pivot = next((k for k in range(j, n) if aug[k][j]), None)
	if pivot is None:
	raise ValueError("singular exact matrix")
	aug[j], aug[pivot] = aug[pivot], aug[j]
	q = aug[j][j]
	aug[j] = [x / q for x in aug[j]]
	for i in range(n):
	if i != j and aug[i][j]:
	q = aug[i][j]
	aug[i] = [x - q * y for x, y in zip(aug[i], aug[j])]
	return [row[n:] for row in aug]


	def rank(a):
	a = [[Q(x) for x in row] for row in a]
	r = 0
	for j in range(len(a[0]) if a else 0):
	pivot = next((i for i in range(r, len(a)) if a[i][j]), None)
	if pivot is None:
	continue
	a[r], a[pivot] = a[pivot], a[r]
	p = a[r][j]
	a[r] = [x / p for x in a[r]]
	for i in range(len(a)):
	if i != r and a[i][j]:
	p = a[i][j]
	a[i] = [x - p * y for x, y in zip(a[i], a[r])]
	r += 1
	if r == len(a):
	break
	return r


	def solve_full_column_rank(a, b):
	rows, n = len(a), len(a[0])
	aug = [[Q(x) for x in row] + [Q(value)] for row, value in zip(a, b)]
	r = 0
	pivots = []
	for j in range(n):
	pivot = next((i for i in range(r, rows) if aug[i][j]), None)
	if pivot is None:
	continue
	aug[r], aug[pivot] = aug[pivot], aug[r]
	p = aug[r][j]
	aug[r] = [x / p for x in aug[r]]
	for i in range(rows):
	if i != r and aug[i][j]:
	p = aug[i][j]
	aug[i] = [x - p * y for x, y in zip(aug[i], aug[r])]
	pivots.append(j)
	r += 1
	if len(pivots) != n:
	raise ValueError("edge parameters not uniquely identifiable")
	if any(not any(row[:n]) and row[n] for row in aug):
	raise ValueError("inconsistent exact linear system")
	result = [Q(0)] * n
	for i, p in enumerate(pivots):
	result[p] = aug[i][-1]
	return result


	def determinant(a):
	a = [[Q(x) for x in row] for row in a]
	result = Q(1)
	for j in range(len(a)):
	pivot = next((i for i in range(j, len(a)) if a[i][j]), None)
	if pivot is None:
	return Q(0)
	if pivot != j:
	a[j], a[pivot] = a[pivot], a[j]
	result = -result
	p = a[j][j]
	result *= p
	for i in range(j + 1, len(a)):
	q = a[i][j] / p
	for k in range(j + 1, len(a)):
	a[i][k] -= q * a[j][k]
	return result


	def fundamental_cycles(nvertices, edges, *, reverse=False):
	"""Cycle rows: positive chord u->v, followed by tree path v->u."""
	adjacency = [[] for _ in range(nvertices)]
	for e, (u, v) in enumerate(edges):
	adjacency[u].append((v, e))
	adjacency[v].append((u, e))
	for row in adjacency:
	row.sort(reverse=reverse)
	root = nvertices - 1 if reverse else 0
	seen, tree = {root}, set()
	queue = deque([root])
	while queue:
	u = queue.popleft()
	for v, e in adjacency[u]:
	if v not in seen:
	seen.add(v)
	tree.add(e)
	queue.append(v)
	if len(seen) != nvertices:
	raise ValueError("disconnected graph")
	tree_adjacency = [[] for _ in range(nvertices)]
	for e in tree:
	u, v = edges[e]
	tree_adjacency[u].append((v, e, 1))
	tree_adjacency[v].append((u, e, -1))
	chords = [e for e in range(len(edges)) if e not in tree]
	cycles = []
	for chord in chords:
	u, v = edges[chord]
	predecessor = {v: None}
	queue = deque([v])
	while u not in predecessor:
	node = queue.popleft()
	for next_node, edge, sign in tree_adjacency[node]:
	if next_node not in predecessor:
	predecessor[next_node] = (node, edge, sign)
	queue.append(next_node)
	row = [Q(0)] * len(edges)
	row[chord] = 1
	node = u
	while node != v:
	prev, edge, sign = predecessor[node]
	row[edge] += sign
	node = prev
	cycles.append(row)
	return cycles, chords


	def edge_design(cycles):
	return [[cycles[i][e] * cycles[j][e] for e in range(len(cycles[0]))]
	for i in range(len(cycles)) for j in range(i, len(cycles))]


	def exact_square_root(q):
	a, b = isqrt(q.numerator), isqrt(q.denominator)
	if a * a != q.numerator or b * b != q.denominator:
	raise ValueError("rational quantity is not an exact rational square")
	return Q(a, b)


	def polynomial_multiply(a, b):
	result = [Q(0)] * (len(a) + len(b) - 1)
	for i, x in enumerate(a):
	for j, y in enumerate(b):
	result[i + j] += x * y
	return result


	def spectral_polynomial(a):
	"""Exact coefficients of det(a+zI), independently by permutations."""
	n = len(a)
	result = [Q(0)] * (n + 1)
	for permutation in permutations(range(n)):
	inversions = sum(permutation[i] > permutation[j] for i in range(n) for j in range(i + 1, n))
	product = [Q((-1) ** inversions)]
	for i, j in enumerate(permutation):
	product = polynomial_multiply(product, [Q(a[i][j]), Q(1)] if i == j else [Q(a[i][j])])
	for i, value in enumerate(product):
	result[i] += value
	return result


	def poly_eval(p, z):
	result = Q(0)
	for c in reversed(p):
	result = result * z + c
	return result


	def inertia_without_zero_pivots(a):
	"""Exact congruence elimination for the nondegenerate cases below."""
	a = [[Q(x) for x in row] for row in a]
	negative = 0
	for j in range(len(a)):
	p = a[j][j]
	if p == 0:
	raise ValueError("zero pivot requires a 2x2 inertia block")
	negative += int(p < 0)
	for i in range(j + 1, len(a)):
	for k in range(j + 1, len(a)):
	a[i][k] -= a[i][j] * a[j][k] / p
	return negative


	class Verification:
	def __init__(self):
	self.assertions = Counter()

	def check(self, category, condition, detail=None):
	if not condition:
	raise AssertionError(f"{category}: {detail!r}")
	self.assertions[category] += 1


	def graphs():
	return {
	"K4": (4, [(u, v) for u in range(4) for v in range(u + 1, 4)]),
	"K3_3": (6, [(u, v) for u in range(3) for v in range(3, 6)]),
	"triangular_prism": (6, [(0, 1), (1, 2), (0, 2), (3, 4), (4, 5), (3, 5),
	(0, 3), (1, 4), (2, 5)]),
	"wheel_W6": (6, [(0, i) for i in range(1, 6)]
	+ [(1, 2), (2, 3), (3, 4), (4, 5), (1, 5)]),
	"cube": (8, [(u, v) for u in range(8) for v in range(u + 1, 8)
	if (u ^ v).bit_count() == 1])
	}


	def vertex_connected_after_removal(n, edges, removed):
	available = set(range(n)) - set(removed)
	seen = {next(iter(available))}
	stack = list(seen)
	while stack:
	u = stack.pop()
	for a, b in edges:
	v = b if a == u else a if b == u else None
	if v is not None and v in available and v not in seen:
	seen.add(v)
	stack.append(v)
	return seen == available


	def exact_graph_checks(v, rng):
	reports, fem_cases = [], []
	for name, (nv, edges) in graphs().items():
	c, chords = fundamental_cycles(nv, edges)
	incidence = [[Q((head == node) - (tail == node)) for tail, head in edges]
	for node in range(nv)]
	v.check("cycle_boundary_zero", mm(c, transpose(incidence))
	== [[Q(0)] * nv for _ in c])
	v.check("cycle_rank_and_chord_coordinates", len(c) == len(edges) - nv + 1
	and [[row[e] for e in chords] for row in c]
	== [[Q(i == j) for j in range(len(c))] for i in range(len(c))])
	for a in range(nv):
	for b in range(a + 1, nv):
	v.check("graph_three_vertex_connectivity", vertex_connected_after_removal(nv, edges, (a, b)))
	design = edge_design(c)
	v.check("edge_metric_linear_identifiability", rank(design) == len(edges))
	c2, _ = fundamental_cycles(nv, edges, reverse=True)
	u = [[row[e] for e in chords] for row in c2]
	v.check("cycle_basis_unimodular", abs(determinant(u)) == 1 and mm(u, c) == c2)
	for case in range(12):
	lengths = [Q(rng.randint(1, 10), rng.randint(2, 12)) for _ in edges]
	total = sum(lengths, Q(0))
	weighted = [[value * ell for value, ell in zip(row, lengths)] for row in c]
	gram = mm(weighted, transpose(c))
	hessian = [[2 * value / total for value in row] for row in inverse(gram)]
	observed = [[2 * value for value in row] for row in inverse(hessian)]
	rhs = [observed[i][j] for i in range(len(c)) for j in range(i, len(c))]
	weights = solve_full_column_rank(design, rhs)
	recovered_total = exact_square_root(sum(weights, Q(0)))
	recovered_lengths = [weight / recovered_total for weight in weights]
	v.check("hessian_total_length_reconstruction", recovered_total == total)
	v.check("hessian_all_edge_lengths_reconstruction", recovered_lengths == lengths)
	v.check("hessian_observation_is_LQ", observed
	== [[total * value for value in row] for row in gram])
	gram2 = mm([[value * ell for value, ell in zip(row, lengths)] for row in c2], transpose(c2))
	v.check("cycle_gram_coordinate_covariance", gram2 == mm(mm(u, gram), transpose(u)))
	inv_u = inverse(u)
	hessian2 = [[2 * value / total for value in row] for row in inverse(gram2)]
	v.check("hessian_coordinate_covariance", hessian2 == mm(mm(transpose(inv_u), hessian), inv_u))
	if case == 0:
	reports.append({"graph": name, "vertices": nv, "edges": len(edges),
	"cycle_rank": len(c), "design_rank": rank(design),
	"lengths": [str(ell) for ell in lengths], "total_length": str(total),
	"cycle_matrix": [[int(x) for x in row] for row in c],
	"exact_recovered_lengths": [str(ell) for ell in recovered_lengths]})
	if name in ("K4", "K3_3", "cube"):
	fem_cases.append((name, nv, edges, lengths, chords, gram, total))
	cycle_edges = [(0, 1), (1, 2), (2, 3), (0, 3)]
	cc, _ = fundamental_cycles(4, cycle_edges)
	v.check("nonidentifiable_cycle_negative_control", rank(edge_design(cc)) == 1 < len(cycle_edges))
	return reports, fem_cases


	def magnetic_fem_ground(nv, edges, lengths, chords, theta, subdivisions):
	import numpy as np
	from scipy.linalg import eigh
	nodes = nv + len(edges) * (subdivisions - 1)
	stiffness = np.zeros((nodes, nodes), dtype=np.complex128)
	mass = np.zeros_like(stiffness)
	chord_flux = dict(zip(chords, theta))
	cursor = nv
	for edge, ((tail, head), ell) in enumerate(zip(edges, lengths)):
	interiors = list(range(cursor, cursor + subdivisions - 1))
	cursor += subdivisions - 1
	sequence = [tail] + interiors + [head]
	h = float(ell) / subdivisions
	phase = np.exp(-1j * chord_flux.get(edge, 0.0) / subdivisions)
	local_k = np.array([[1, -phase], [-phase.conjugate(), 1]], dtype=np.complex128) / h
	local_m = h / 6 * np.array([[2, phase], [phase.conjugate(), 2]], dtype=np.complex128)
	for a, b in zip(sequence, sequence[1:]):
	idx = np.ix_([a, b], [a, b])
	stiffness[idx] += local_k
	mass[idx] += local_m
	return float(eigh(stiffness, mass, subset_by_index=[0, 0], driver="gvx",
	check_finite=True)[0][0])


	def fem_checks(v, fem_cases):
	results = []
	for name, nv, edges, lengths, chords, gram, total in fem_cases:
	beta = len(chords)
	inv_gram = inverse(gram)
	vectors = [[int(i == j) for i in range(beta)] for j in range(min(2, beta))]
	vectors.append([1 if i % 2 == 0 else -1 for i in range(beta)])
	for subdivisions in (4, 8):
	baseline = magnetic_fem_ground(nv, edges, lengths, chords, [0.0] * beta, subdivisions)
	for vector in vectors:
	predicted = sum((Q(vector[i]) * inv_gram[i][j] * vector[j]
	for i in range(beta) for j in range(beta)), Q(0)) / total
	errors = []
	for epsilon in (0.01, 0.005, 0.0025):
	positive = magnetic_fem_ground(nv, edges, lengths, chords,
	[epsilon * a for a in vector], subdivisions)
	negative = magnetic_fem_ground(nv, edges, lengths, chords,
	[-epsilon * a for a in vector], subdivisions)
	curvature_half = ((positive + negative) / 2 - baseline) / epsilon ** 2
	relative = abs(curvature_half - float(predicted)) / float(predicted)
	v.check("fem_hessian_normalization", relative < 0.002,
	(name, subdivisions, vector, epsilon, relative))
	v.check("fem_time_reversal_symmetry", abs(positive - negative) < 1e-9)
	errors.append(relative)
	results.append({"graph": name, "subdivisions_per_edge": subdivisions,
	"flux_direction": vector, "epsilon_radians": epsilon,
	"baseline_numerical_eigenvalue": baseline,
	"predicted_lambda_over_epsilon_squared": float(predicted),
	"observed_lambda_over_epsilon_squared": curvature_half,
	"relative_error": relative})
	v.check("fem_small_flux_consistency", max(errors) < 0.002)
	return results


	def critical_coupling_checks(v, rng):
	examples = []
	n = 4
	t = [Q(8), Q(11), Q(15), Q(20)]
	base = [[t[i] if i == j else Q(0) for j in range(n)] for i in range(n)]
	expected_poly = spectral_polynomial(base)
	expected = [x / expected_poly[0] for x in expected_poly]
	for r in (1, 2, 3):
	for case in range(15):
	factor = [[Q(rng.randint(-4, 4), 50) for _ in range(r)] for _ in range(n)]
	interaction = mm(factor, transpose(factor))
	v.check("coupling_rank_bound", rank(interaction) <= r)
	settings = [Q((i + 1) ** 2) for i in range(r + 1)]
	polys, dets = [], []
	for setting in settings:
	e = [[base[i][j] - setting * interaction[i][j] for j in range(n)] for i in range(n)]
	v.check("coupled_matrix_positive_gershgorin", all(e[i][i] > sum(abs(e[i][j])
	for j in range(n) if j != i) for i in range(n)))
	p = spectral_polynomial(e)
	v.check("permutation_determinant_independent_check", p[0] == determinant(e))
	polys.append([a / p[0] for a in p])
	dets.append(p[0])
	alpha = [det / dets[0] for det in dets]
	weights = [Q(1)] * len(settings)
	for i, setting in enumerate(settings):
	for j, other in enumerate(settings):
	if i != j:
	weights[i] *= -other / (setting - other)
	denominator = sum((w * a for w, a in zip(weights, alpha)), Q(0))
	restored = [sum((weights[i] * alpha[i] * polys[i][j] for i in range(r + 1)), Q(0))
	/ denominator for j in range(n + 1)]
	v.check("determinant_normalization_denominator_positive", denominator > 0)
	v.check("finite_rank_spectral_polynomial_recovery", restored == expected)
	naive = [sum((weights[i] * polys[i][j] for i in range(r + 1)), Q(0)) for j in range(n + 1)]
	if any(any(row) for row in interaction):
	v.check("uncalibrated_products_negative_control", naive != expected)
	# Positive-axis ratios independently approach the determinant
	# calibrations. Record exact finite-s ratios rather than assume
	# eigenvectors or simultaneously diagonalizable matrices.
	if case == 0:
	ratios = [[poly_eval(polys[0], s) / poly_eval(polys[i], s) for i in range(r + 1)]
	for s in (Q(100), Q(1000), Q(10000))]
	v.check("positive_axis_calibration_convergence", all(
	abs(ratios[-1][i] - alpha[i]) <= abs(ratios[0][i] - alpha[i])
	for i in range(r + 1)))
	examples.append({"rank_bound": r, "settings": [str(u) for u in settings],
	"T_diagonal": [str(a) for a in t],
	"interaction_commutes_with_T": mm(base, interaction) == mm(interaction, base),
	"alphas": [str(a) for a in alpha],
	"weights": [str(w) for w in weights],
	"reconstructed_D_T_coefficients": [str(a) for a in restored],
	"positive_axis_ratios": [[str(a) for a in row] for row in ratios]})

	# Exact critical-coupling cutoff counts using full block inertia, which is
	# independent of the determinant-polynomial interpolation test above.
	hard = [Q(3), Q(5)]
	w = [[Q(1, 5), Q(0)], [Q(1, 7), Q(1, 8)], [Q(0), Q(1, 6)], [Q(1, 9), Q(-1, 10)]]
	cutoff = Q(1)
	correction = mm([[a / (f - cutoff) for a, f in zip(row, hard)] for row in w], transpose(w))
	inertia_rows = []
	for setting in (Q(1), Q(4), Q(9)):
	effective = [[base[i][j] - setting * correction[i][j] for j in range(n)] for i in range(n)]
	for a in (Q(1, 4), Q(1, 9), Q(1, 16), Q(1, 25), Q(1, 100)):
	root = exact_square_root(a * setting)
	# Order hard block first so exact congruence elimination begins
	# with two strictly positive pivots.
	physical_minus_cutoff = [[Q(0)] * (n + 2) for _ in range(n + 2)]
	for i in range(2):
	physical_minus_cutoff[i][i] = hard[i] - cutoff
	for i in range(n):
	for j in range(n):
	physical_minus_cutoff[i + 2][j + 2] = a * base[i][j] - (cutoff if i == j else 0)
	for j in range(2):
	physical_minus_cutoff[j][i + 2] = physical_minus_cutoff[i + 2][j] = root * w[i][j]
	shifted_effective = [[effective[i][j] - (cutoff / a if i == j else 0)
	for j in range(n)] for i in range(n)]
	full_count = inertia_without_zero_pivots(physical_minus_cutoff)
	effective_count = inertia_without_zero_pivots(shifted_effective)
	v.check("critical_low_cutoff_exact_inertia_identity", full_count == effective_count)
	inertia_rows.append({"a": str(a), "u": str(setting),
	"physical_count_below_cutoff": full_count,
	"effective_count_below_cutoff_over_a": effective_count})

	# The manuscript's rank-two ambiguity, and arbitrary-finite-setting
	# parity construction, independently tested as multisets.
	ta, ca, tb, cb = [Q(11), Q(12)], [Q(2), Q(2)], [Q(10), Q(13)], [Q(1), Q(3)]
	for setting in (Q(1), Q(2)):
	v.check("rank_two_two_scan_nonuniqueness", sorted(t - setting * c for t, c in zip(ta, ca))
	== sorted(t - setting * c for t, c in zip(tb, cb)))
	v.check("rank_two_unperturbed_spectra_differ", sorted(ta) != sorted(tb))
	for k in range(1, 6):
	settings = list(range(1, k + 1))
	displacements = [10 ** i for i in range(k)]
	t0 = 10 + k * (1 + sum(displacements))
	even, odd = [], []
	for mask in range(2 ** k):
	point = (Q(t0 + sum(settings[i] * displacements[i] for i in range(k) if mask >> i & 1)),
	Q(1 + sum(displacements[i] for i in range(k) if mask >> i & 1)))
	(even if mask.bit_count() % 2 == 0 else odd).append(point)
	for setting in settings:
	observed_a = sorted(t - setting * c for t, c in even)
	observed_b = sorted(t - setting * c for t, c in odd)
	v.check("arbitrary_setting_parity_nonuniqueness", observed_a == observed_b and min(observed_a) > 0)
	v.check("parity_unperturbed_spectra_differ", sorted(t for t, c in even) != sorted(t for t, c in odd))
	return {"determinant_examples": examples, "critical_cutoff_inertia_examples": inertia_rows}


	def run(skip_fem=False):
	rng, v = random.Random(SEED), Verification()
	graph_reports, fem_cases = exact_graph_checks(v, rng)
	coupling = critical_coupling_checks(v, rng)
	exact_assertions = sum(v.assertions.values())
	fem_reports = [] if skip_fem else fem_checks(v, fem_cases)
	return {"status": "PASS", "seed": SEED,
	"exact_assertions": exact_assertions,
	"numerical_assertions": sum(v.assertions.values()) - exact_assertions,
	"assertions_total": sum(v.assertions.values()),
	"assertions_by_category": dict(sorted(v.assertions.items())),
	"flux_convention": "Cycle holonomies in radians; positive chord entry 1; tree-edge flux 0",
	"hessian_formula": "lambda0(theta)=theta^T Q^-1 theta/L+O(\|theta\|^4), H=2Q^-1/L",
	"inverse_metric_algorithm": "S=2H^-1=LQ; solve S=sum_e w_e c_e c_e^T; L=sqrt(sum_e w_e); ell_e=w_e/L",
	"graphs": graph_reports, "finite_element_normalization": fem_reports,
	"critical_coupling": coupling,
	"limitations": ["Exact rational finite tests are not formal proofs for all graphs or infinite operators.",
	"The graph reconstruction assumes the labeled graph and a known fundamental cycle-coordinate basis.",
	"Graph topology recovery from an unmarked period lattice requires separate theory and is not validated here.",
	"FEM checks use floating-point eigenvalues and are not rigorous spectral enclosures.",
	"No originality, priority or 10/10 importance certification is provided."]}


	def main():
	parser = argparse.ArgumentParser(description=__doc__)
	parser.add_argument("--output", type=Path,
	default=Path(__file__).resolve().parent / "constructive_verification_results.json")
	parser.add_argument("--skip-fem", action="store_true")
	args = parser.parse_args()
	result = run(args.skip_fem)
	args.output.parent.mkdir(parents=True, exist_ok=True)
	args.output.write_text(json.dumps(result, indent=2) + "\n", encoding="utf-8")
	print(json.dumps({"status": result["status"], "exact_assertions": result["exact_assertions"],
	"numerical_assertions": result["numerical_assertions"],
	"assertions_total": result["assertions_total"],
	"worst_fem_relative_error": max((r["relative_error"] for r in result["finite_element_normalization"]),
	default=None),
	"assertions_by_category": result["assertions_by_category"],
	"output": str(args.output.resolve())}, indent=2))


	if __name__ == "__main__":
	main()