test_symplectic_substrate.py

"""Tests for the specified auxiliary symplectic substrate.

Module under test: physics/symplectic_substrate.py.
Verifies the specified ambient harmonic model: antisymmetric non-degenerate
closed 2-form, canonical
Poisson brackets, Jacobi identity, Liouville volume preservation, harmonic
Hamiltonian flow, and consistency with the canonical energy functional. These
identities do not establish that the graph's nodal dynamics produces this flow
or that its evolving coordinates remain realizable as graph fields.
"""

from __future__ import annotations

import math
import random

import networkx as nx
import numpy as np
import pytest

from tnfr.dynamics.dnfr import default_compute_delta_nfr
from tnfr.physics.conservation import compute_energy_functional
from tnfr.physics.symplectic_substrate import (
    BLOCK_COMPATIBLE_METRIC,
    BLOCK_COMPLEX_STRUCTURE,
    BLOCK_SYMPLECTIC_FORM,
    PhaseSpacePoint,
    background_potential,
    canonical_bracket_table,
    compatible_metric_matrix,
    complex_structure_matrix,
    diagonal_moment_map,
    evolve_substrate_flow,
    extract_phase_space_point,
    geometric_sector_energy,
    hamiltonian_vector_field,
    kahler_potential,
    liouville_divergence,
    loop_action_integral,
    noether_charges,
    polarization_density,
    polarization_vector,
    potential_sector_energy,
    reduced_symplectic_form_matrix,
    substrate_flow_matrix,
    substrate_hamiltonian,
    symplectic_form_matrix,
    symplectic_pullback_residual,
    to_action_angle,
    to_complex_coordinates,
    verify_adiabatic_invariance,
    verify_canonical_structure,
    verify_hermitian_structure,
    verify_integrability,
    verify_noether_conservation,
    verify_poincare_cartan,
    verify_polarization_symmetry,
    verify_substrate_geometry,
    verify_symplectic_reduction,
)


def _canonical_graph(n: int = 30, seed: int = 5) -> nx.Graph:
    rng = random.Random(seed)
    G = nx.watts_strogatz_graph(n, 4, 0.3, seed=seed)
    for node in G.nodes():
        G.nodes[node]["theta"] = rng.uniform(0.0, 2.0 * math.pi)
        G.nodes[node]["EPI"] = rng.uniform(0.2, 0.8)
        G.nodes[node]["nu_f"] = rng.uniform(0.5, 1.5)
    default_compute_delta_nfr(G)
    return G


@pytest.fixture(scope="module")
def adiabatic_report():
    return verify_adiabatic_invariance(ramp_times=(1.0, 5.0, 20.0, 80.0))


@pytest.fixture(scope="module")
def substrate_geometry_report():
    return verify_substrate_geometry(_canonical_graph(20))


@pytest.fixture(scope="module")
def unified_substrate_telemetry():
    from tnfr.physics.fields import compute_unified_telemetry

    return compute_unified_telemetry(_canonical_graph(20))


class TestBlockSymplecticForm:
    """The per-node canonical block J4 has the symplectic properties."""

    def test_shared_pullback_residual_for_harmonic_flow(self):
        for t in (0.0, 0.3, 1.7):
            matrix = substrate_flow_matrix(2, t)
            assert symplectic_pullback_residual(matrix, 2) < 1e-12

    def test_antisymmetric(self) -> None:
        assert np.allclose(BLOCK_SYMPLECTIC_FORM.T, -BLOCK_SYMPLECTIC_FORM)

    def test_nondegenerate_determinant_one(self) -> None:
        assert abs(float(np.linalg.det(BLOCK_SYMPLECTIC_FORM)) - 1.0) < 1e-12

    def test_square_is_minus_identity(self) -> None:
        # J^2 = -I is the defining property of the canonical complex
        # structure underlying the symplectic form.
        j2 = BLOCK_SYMPLECTIC_FORM @ BLOCK_SYMPLECTIC_FORM
        assert np.allclose(j2, -np.eye(4))


class TestSymplecticFormMatrix:
    """The full 4N x 4N form is block-diagonal and symplectic."""

    def test_dimension(self) -> None:
        omega = symplectic_form_matrix(7)
        assert omega.shape == (28, 28)

    def test_antisymmetric_and_nondegenerate(self) -> None:
        omega = symplectic_form_matrix(10)
        assert np.allclose(omega.T, -omega)
        # det of a 4N block-diagonal of det-1 blocks is 1.
        assert abs(float(np.linalg.det(omega)) - 1.0) < 1e-9

    def test_rejects_zero_nodes(self) -> None:
        import pytest

        with pytest.raises(ValueError):
            symplectic_form_matrix(0)


class TestCanonicalBrackets:
    """Poisson brackets are canonical: {q,p}=delta, others zero."""

    def test_conjugate_brackets_unity(self) -> None:
        table = canonical_bracket_table()
        assert abs(table["{qA,pA}"] - 1.0) < 1e-12
        assert abs(table["{qB,pB}"] - 1.0) < 1e-12

    def test_cross_and_self_brackets_vanish(self) -> None:
        table = canonical_bracket_table()
        assert abs(table["{qA,qB}"]) < 1e-12
        assert abs(table["{pA,pB}"]) < 1e-12
        assert abs(table["{qA,pB}"]) < 1e-12
        assert abs(table["{pA,qB}"]) < 1e-12


class TestPhaseSpaceExtraction:
    """The phase-space point is extracted from canonical fields."""

    def test_dimension_is_four_n(self) -> None:
        G = _canonical_graph(20)
        pt = extract_phase_space_point(G)
        assert pt.n_nodes == 20
        assert pt.dimension == 80
        assert pt.to_vector().shape == (80,)

    def test_vector_ordering(self) -> None:
        G = _canonical_graph(10)
        pt = extract_phase_space_point(G)
        z = pt.to_vector().reshape(pt.n_nodes, 4)
        # Per-node order is (K_phi, J_phi, Phi_s, J_dnfr).
        assert np.allclose(z[:, 0], pt.k_phi)
        assert np.allclose(z[:, 1], pt.j_phi)
        assert np.allclose(z[:, 2], pt.phi_s)
        assert np.allclose(z[:, 3], pt.j_dnfr)


class TestHamiltonianFlow:
    """The Hamiltonian flow is harmonic and volume-preserving."""

    def test_flow_is_harmonic(self) -> None:
        G = _canonical_graph(15)
        pt = extract_phase_space_point(G)
        z = pt.to_vector().reshape(pt.n_nodes, 4)
        zdot = hamiltonian_vector_field(pt).reshape(pt.n_nodes, 4)
        # q_dot = p, p_dot = -q per sector.
        expected = np.stack([z[:, 1], -z[:, 0], z[:, 3], -z[:, 2]], axis=1)
        assert np.allclose(zdot, expected)

    def test_liouville_divergence_zero(self) -> None:
        G = _canonical_graph(25)
        pt = extract_phase_space_point(G)
        assert abs(liouville_divergence(pt)) < 1e-9


class TestEnergyConsistency:
    """H_sub + background potential reconstructs the energy functional."""

    def test_decomposition_matches_energy_functional(self) -> None:
        G = _canonical_graph(30, seed=7)
        pt = extract_phase_space_point(G)
        h_sub = substrate_hamiltonian(pt)
        u_bg = background_potential(pt)
        energy = compute_energy_functional(G)
        assert abs((h_sub + u_bg) - energy) < 1e-9


class TestCanonicalCertificate:
    """The certificate checks the specified ambient canonical structure."""

    def test_valid_manifold(self) -> None:
        G = _canonical_graph(30)
        cert = verify_canonical_structure(G)
        assert cert.is_valid_symplectic_manifold
        assert cert.is_antisymmetric
        assert cert.is_nondegenerate
        assert cert.is_closed
        assert cert.brackets_canonical
        assert cert.jacobi_satisfied
        assert cert.flow_is_harmonic
        assert abs(cert.liouville_divergence) < 1e-9
        assert cert.dimension == 120

    def test_certificate_summary_string(self) -> None:
        G = _canonical_graph(12)
        cert = verify_canonical_structure(G)
        summary = cert.summary()
        assert "VALID" in summary
        assert "dim=48" in summary


class TestNoetherCharges:
    """Noether's theorem: substrate symmetries generate conserved charges."""

    def test_charges_split_total_energy(self) -> None:
        import pytest

        G = _canonical_graph(25)
        pt = extract_phase_space_point(G)
        charges = noether_charges(pt)
        # H_sub = E_geo + E_pot exactly.
        assert charges["time_translation"] == pytest.approx(
            charges["geometric_u1"] + charges["potential_u1"], abs=1e-12
        )
        assert charges["time_translation"] == pytest.approx(
            substrate_hamiltonian(pt), abs=1e-12
        )

    def test_sector_energies_nonnegative(self) -> None:
        G = _canonical_graph(20)
        pt = extract_phase_space_point(G)
        assert geometric_sector_energy(pt) >= 0.0
        assert potential_sector_energy(pt) >= 0.0

    def test_charges_conserved_under_flow(self) -> None:
        G = _canonical_graph(30)
        cert = verify_noether_conservation(G)
        assert cert.is_conserved
        assert cert.splits_exactly
        assert cert.max_hamiltonian_drift < 1e-9
        assert cert.max_geometric_drift < 1e-9
        assert cert.max_potential_drift < 1e-9

    def test_flow_preserves_hamiltonian(self) -> None:
        import pytest

        G = _canonical_graph(20)
        pt = extract_phase_space_point(G)
        h0 = substrate_hamiltonian(pt)
        for t in (0.5, 1.3, 2.7, 10.0):
            evolved = evolve_substrate_flow(pt, t)
            assert substrate_hamiltonian(evolved) == pytest.approx(h0, abs=1e-9)

    def test_flow_at_zero_is_identity(self) -> None:
        G = _canonical_graph(15)
        pt = extract_phase_space_point(G)
        evolved = evolve_substrate_flow(pt, 0.0)
        assert np.allclose(evolved.to_vector(), pt.to_vector())

    def test_certificate_summary(self) -> None:
        G = _canonical_graph(12)
        cert = verify_noether_conservation(G)
        summary = cert.summary()
        assert "CONSERVED" in summary
        assert "H_sub" in summary


class TestHermitianStructure:
    """The substrate carries a compatible Hermitian (flat Kähler) structure."""

    def test_complex_structure_squares_to_minus_id(self) -> None:
        j = BLOCK_COMPLEX_STRUCTURE
        assert np.allclose(j @ j, -np.eye(4))

    def test_complex_structure_is_minus_symplectic_form(self) -> None:
        assert np.allclose(BLOCK_COMPLEX_STRUCTURE, -BLOCK_SYMPLECTIC_FORM)

    def test_compatible_metric_is_identity(self) -> None:
        assert np.allclose(BLOCK_COMPATIBLE_METRIC, np.eye(4))

    def test_metric_is_positive_definite(self) -> None:
        eigvals = np.linalg.eigvalsh(BLOCK_COMPATIBLE_METRIC)
        assert np.all(eigvals > 0)

    def test_compatibility_omega_equals_jt_g(self) -> None:
        # ω(u,v) = g(Ju,v)  ⟺  Ω = Jᵀ g.
        j = BLOCK_COMPLEX_STRUCTURE
        g = BLOCK_COMPATIBLE_METRIC
        assert np.allclose(BLOCK_SYMPLECTIC_FORM, j.T @ g)

    def test_j_is_metric_orthogonal(self) -> None:
        j = BLOCK_COMPLEX_STRUCTURE
        g = BLOCK_COMPATIBLE_METRIC
        assert np.allclose(j.T @ g @ j, g)

    def test_block_matrices_dimension(self) -> None:
        assert complex_structure_matrix(5).shape == (20, 20)
        assert compatible_metric_matrix(5).shape == (20, 20)

    def test_complex_structure_acts_as_i(self) -> None:
        # J(q,p) = (−p, q) for ζ = q + i·p (multiplication by i).
        j = BLOCK_COMPLEX_STRUCTURE
        assert np.allclose(j @ np.array([1.0, 0, 0, 0]), [0, 1, 0, 0])
        assert np.allclose(j @ np.array([0, 1.0, 0, 0]), [-1, 0, 0, 0])

    def test_psi_is_geometric_complex_coordinate(self) -> None:
        from tnfr.physics.unified import compute_complex_geometric_field

        G = _canonical_graph(20)
        point = extract_phase_space_point(G)
        coords = to_complex_coordinates(point)
        psi = compute_complex_geometric_field(G)
        psi_arr = np.array([psi[n] for n in point.nodes], dtype=complex)
        assert np.allclose(coords["geometric"], psi_arr)

    def test_kahler_potential_equals_hamiltonian(self) -> None:
        import pytest

        G = _canonical_graph(25)
        point = extract_phase_space_point(G)
        assert kahler_potential(point) == pytest.approx(
            substrate_hamiltonian(point), abs=1e-9
        )

    def test_certificate_valid(self) -> None:
        G = _canonical_graph(30)
        cert = verify_hermitian_structure(G)
        assert cert.is_valid_hermitian_structure
        assert cert.psi_is_geometric_coordinate
        assert cert.kahler_potential_matches
        assert cert.complex_dimension == 60
        assert "VALID" in cert.summary()


class TestIntegrability:
    """The substrate flow is completely integrable (Liouville–Arnold)."""

    def test_action_count_equals_dof(self) -> None:
        G = _canonical_graph(24)
        cert = verify_integrability(G)
        # 2N action variables for 2N degrees of freedom.
        assert cert.degrees_of_freedom == 48
        assert cert.n_action_variables == 48
        assert cert.n_action_variables == cert.degrees_of_freedom

    def test_actions_are_half_modulus_squared(self) -> None:
        G = _canonical_graph(20)
        point = extract_phase_space_point(G)
        aa = to_action_angle(point)
        coords = to_complex_coordinates(point)
        assert np.allclose(
            aa["action_geometric"], 0.5 * np.abs(coords["geometric"]) ** 2
        )
        assert np.allclose(
            aa["action_potential"], 0.5 * np.abs(coords["potential"]) ** 2
        )

    def test_actions_conserved_along_flow(self) -> None:
        G = _canonical_graph(24)
        point = extract_phase_space_point(G)
        aa0 = to_action_angle(point)
        evolved = evolve_substrate_flow(point, 2.3)
        aa = to_action_angle(evolved)
        assert np.allclose(aa["action_geometric"], aa0["action_geometric"])
        assert np.allclose(aa["action_potential"], aa0["action_potential"])

    def test_angles_advance_linearly(self) -> None:
        G = _canonical_graph(22)
        point = extract_phase_space_point(G)
        aa0 = to_action_angle(point)
        t = 0.7
        evolved = evolve_substrate_flow(point, t)
        aa = to_action_angle(evolved)
        # θ(t) = θ(0) − t, compared on the circle.
        delta = np.angle(
            np.exp(1j * (aa["angle_geometric"] - (aa0["angle_geometric"] - t)))
        )
        assert np.allclose(delta, 0.0, atol=1e-9)

    def test_zero_action_pair_is_excluded_from_angle_check(self, monkeypatch) -> None:
        point = PhaseSpacePoint(
            (0,),
            np.array([0.0]),
            np.array([0.0]),
            np.array([1.0]),
            np.array([0.0]),
            np.array([0.0]),
        )
        monkeypatch.setattr(
            "tnfr.physics.symplectic_substrate.extract_phase_space_point",
            lambda graph: point,
        )

        cert = verify_integrability(object(), flow_times=(0.7,))

        assert cert.angles_advance_linearly
        assert cert.max_angle_error < 1e-12

    def test_actions_in_involution(self) -> None:
        G = _canonical_graph(24)
        cert = verify_integrability(G)
        assert cert.actions_in_involution
        assert cert.max_involution_bracket < 1e-12

    def test_sector_actions_match_noether_charges(self) -> None:
        import pytest

        G = _canonical_graph(24)
        point = extract_phase_space_point(G)
        aa = to_action_angle(point)
        assert float(np.sum(aa["action_geometric"])) == pytest.approx(
            geometric_sector_energy(point)
        )
        assert float(np.sum(aa["action_potential"])) == pytest.approx(
            potential_sector_energy(point)
        )

    def test_certificate_completely_integrable(self) -> None:
        G = _canonical_graph(30)
        cert = verify_integrability(G)
        assert cert.is_completely_integrable
        assert cert.actions_conserved
        assert cert.angles_advance_linearly
        assert cert.sector_actions_match_charges
        assert cert.degrees_of_freedom == 60
        assert "INTEGRABLE" in cert.summary()


class TestPoincareCartan:
    """The flow preserves the Poincaré–Cartan integral invariants."""

    def test_flow_matrix_is_symplectic(self) -> None:
        m = substrate_flow_matrix(4, 0.7)
        omega = symplectic_form_matrix(4)
        # 1st Poincaré invariant: M preserves ω.
        assert np.allclose(m.T @ omega @ m, omega)

    def test_flow_matrix_unit_determinant(self) -> None:
        m = substrate_flow_matrix(5, 1.3)
        # Top invariant ω^N = Liouville volume.
        assert abs(float(np.linalg.det(m)) - 1.0) < 1e-12

    def test_flow_matrix_dimension(self) -> None:
        assert substrate_flow_matrix(7, 0.4).shape == (28, 28)

    def test_flow_matrix_palindromic_spectrum(self) -> None:
        m = substrate_flow_matrix(3, 0.9)
        coeffs = np.poly(m)
        # Reciprocal symplectic spectrum → palindromic char poly.
        assert np.allclose(coeffs, coeffs[::-1], atol=1e-9)
        eig = np.linalg.eigvals(m)
        assert np.allclose(np.abs(eig), 1.0)

    def test_flow_matrix_invalid_nodes(self) -> None:
        import pytest

        with pytest.raises(ValueError):
            substrate_flow_matrix(0, 0.5)

    def test_loop_action_integral_equals_minus_2pi_I(self) -> None:
        # ∮ p dq = −2π·I (negative enclosed area) for the torus loop.
        val = loop_action_integral(2.0)
        assert abs(val - (-2.0 * np.pi * 2.0)) < 1e-3

    def test_relative_invariant_preserved(self) -> None:
        G = _canonical_graph(24)
        cert = verify_poincare_cartan(G)
        assert cert.relative_invariant_preserved
        assert cert.max_relative_drift < 1e-9

    def test_bohr_sommerfeld_holds(self) -> None:
        G = _canonical_graph(24)
        cert = verify_poincare_cartan(G)
        # |∮ p dq| = 2π I on the action torus.
        assert cert.bohr_sommerfeld_holds

    def test_certificate_all_invariants_hold(self) -> None:
        G = _canonical_graph(30)
        cert = verify_poincare_cartan(G)
        assert cert.all_invariants_hold
        assert cert.preserves_symplectic_form
        assert cert.volume_preserved
        assert cert.char_poly_palindromic
        assert cert.phase_space_dimension == 120
        assert "ALL HOLD" in cert.summary()


class TestMarsdenWeinstein:
    """The flow's diagonal U(1) admits a Marsden–Weinstein reduction."""

    def test_moment_map_equals_hamiltonian(self) -> None:
        import pytest

        G = _canonical_graph(20)
        point = extract_phase_space_point(G)
        j = diagonal_moment_map(point)
        assert j == pytest.approx(substrate_hamiltonian(point))

    def test_moment_map_conserved(self) -> None:
        G = _canonical_graph(24)
        cert = verify_symplectic_reduction(G)
        assert cert.moment_map_conserved
        assert cert.max_moment_drift < 1e-9

    def test_reduced_dimension_is_4n_minus_2(self) -> None:
        G = _canonical_graph(20)
        cert = verify_symplectic_reduction(G)
        assert cert.phase_space_dimension == 80
        assert cert.reduced_dimension == 78

    def test_reduced_form_shape_and_determinant(self) -> None:
        # Reduced form is (4N−2)×(4N−2), det = (2N)².
        red = reduced_symplectic_form_matrix(3)
        assert red.shape == (10, 10)
        assert abs(float(np.linalg.det(red)) - 36.0) < 1e-6

    def test_reduced_form_nondegenerate(self) -> None:
        G = _canonical_graph(24)
        cert = verify_symplectic_reduction(G)
        assert cert.reduced_form_nondegenerate
        assert cert.reduced_form_determinant == pytest.approx(1.0)

    def test_zero_level_is_a_point_not_a_regular_reduction(self) -> None:
        graph = nx.path_graph(2)
        for node in graph:
            graph.nodes[node].update(EPI=0.0, nu_f=0.0, theta=0.0, delta_nfr=0.0)
        cert = verify_symplectic_reduction(graph)
        assert cert.moment_map_value == 0.0
        assert cert.reduced_dimension == 0
        assert cert.is_regular_level is False
        assert cert.is_valid_reduction is False
        assert cert.reduction_status == "singular_zero_level"
        assert cert.relative_phases_invariant is None
        assert math.isnan(cert.reduced_form_determinant)
        assert "SINGULAR_ZERO_LEVEL" in cert.summary()

    @pytest.mark.parametrize("scale", [1e-12, 1.0, 1e12])
    def test_regular_level_with_zero_actions_uses_defined_phase_reference(
        self, monkeypatch, scale
    ):
        # The first pair has zero action, so its angle cannot be a reference.
        point = PhaseSpacePoint(
            (0,),
            np.array([0.0]),
            np.array([0.0]),
            np.array([scale]),
            np.array([0.0]),
            np.array([0.0]),
        )
        monkeypatch.setattr(
            "tnfr.physics.symplectic_substrate.extract_phase_space_point",
            lambda graph: point,
        )
        cert = verify_symplectic_reduction(nx.empty_graph(1))
        assert cert.is_regular_level
        assert cert.is_valid_reduction
        assert cert.reduced_dimension == 2
        assert cert.relative_phase_pairs == 0
        assert cert.reduced_form_determinant == pytest.approx(1.0)

    def test_relative_phase_comparison_respects_the_angle_wrap(self, monkeypatch):
        point = PhaseSpacePoint(
            (0,),
            np.array([1.0]),
            np.array([0.0]),
            np.array([-1.0]),
            np.array([1e-16]),
            np.array([0.0]),
        )
        monkeypatch.setattr(
            "tnfr.physics.symplectic_substrate.extract_phase_space_point",
            lambda graph: point,
        )
        cert = verify_symplectic_reduction(nx.empty_graph(1))
        assert cert.relative_phase_pairs == 1
        assert cert.relative_phases_invariant
        assert cert.is_valid_reduction

    def test_reduced_form_invalid_nodes(self) -> None:
        import pytest

        with pytest.raises(ValueError):
            reduced_symplectic_form_matrix(0)

    def test_relative_phases_invariant(self) -> None:
        G = _canonical_graph(24)
        cert = verify_symplectic_reduction(G)
        assert cert.relative_phases_invariant

    def test_certificate_valid_reduction(self) -> None:
        G = _canonical_graph(30)
        cert = verify_symplectic_reduction(G)
        assert cert.is_valid_reduction
        assert cert.moment_map_is_hamiltonian
        assert cert.reduced_dimension == 118
        assert "VALID" in cert.summary()


class TestPolarizationSymmetry:
    """The substrate carries a polarization symmetry (U(2)) with Stokes vector."""

    def test_polarization_vector_present(self) -> None:
        G = _canonical_graph(24)
        ch = polarization_vector(extract_phase_space_point(G))
        assert set(ch) == {"p_1", "p_2", "p_3", "magnitude_sq"}

    def test_p3_equals_sector_energy_difference(self) -> None:
        import pytest

        G = _canonical_graph(24)
        point = extract_phase_space_point(G)
        ch = polarization_vector(point)
        e_diff = geometric_sector_energy(point) - potential_sector_energy(point)
        assert ch["p_3"] == pytest.approx(e_diff)

    def test_magnitude_is_sum_of_squares(self) -> None:
        import pytest

        G = _canonical_graph(20)
        ch = polarization_vector(extract_phase_space_point(G))
        assert ch["magnitude_sq"] == pytest.approx(
            ch["p_1"] ** 2 + ch["p_2"] ** 2 + ch["p_3"] ** 2
        )

    def test_su2_algebra_closes(self) -> None:
        G = _canonical_graph(24)
        cert = verify_polarization_symmetry(G)
        assert cert.su2_algebra_closes
        assert cert.max_algebra_residual < 1e-6

    def test_rotation_is_symplectic(self) -> None:
        G = _canonical_graph(24)
        cert = verify_polarization_symmetry(G)
        assert cert.rotation_is_symplectic

    def test_charges_conserved_along_flow(self) -> None:
        G = _canonical_graph(24)
        cert = verify_polarization_symmetry(G)
        assert cert.charges_conserved
        assert cert.max_charge_drift < 1e-6

    def test_certificate_valid_polarization(self) -> None:
        G = _canonical_graph(30)
        cert = verify_polarization_symmetry(G)
        assert cert.is_valid_polarization_symmetry
        assert cert.p3_equals_energy_difference
        assert "VALID" in cert.summary()

    def test_full_polarization_per_node_radius_equals_energy(self) -> None:
        G = _canonical_graph(24)
        d = polarization_density(extract_phase_space_point(G))
        # |P_node| = e_node (the Poincaré sphere S²; fully polarized).
        assert np.allclose(d["radius"], d["energy"])

    def test_polarization_density_poincare_is_unit(self) -> None:
        G = _canonical_graph(20)
        d = polarization_density(extract_phase_space_point(G))
        norms = np.sqrt((d["poincare"] ** 2).sum(axis=0))
        assert np.allclose(norms, 1.0)

    def test_density_sums_to_global_charges(self) -> None:
        import pytest

        G = _canonical_graph(24)
        point = extract_phase_space_point(G)
        d = polarization_density(point)
        glob = polarization_vector(point)
        assert float(d["p_1"].sum()) == pytest.approx(glob["p_1"])
        assert float(d["p_2"].sum()) == pytest.approx(glob["p_2"])
        assert float(d["p_3"].sum()) == pytest.approx(glob["p_3"])

    def test_certificate_reports_full_polarization(self) -> None:
        G = _canonical_graph(30)
        cert = verify_polarization_symmetry(G)
        assert cert.full_polarization_holds
        assert cert.max_polarization_residual < 1e-9
        assert "fully polarized" in cert.summary()


class TestSubstrateGeometryReport:
    """The consolidated aggregator runs the whole tower in one call."""

    def test_bundles_seven_certificates(self, substrate_geometry_report) -> None:
        report = substrate_geometry_report
        assert report.all_structures_valid
        assert report.n_nodes == 20
        assert report.phase_space_dimension == 80
        assert report.marsden_weinstein.reduced_dimension == 78
        # Each sub-certificate is the same type the individual verify returns.
        assert report.canonical.is_valid_symplectic_manifold
        assert report.noether.is_conserved
        assert report.hermitian.is_valid_hermitian_structure
        assert report.integrability.is_completely_integrable
        assert report.poincare_cartan.all_invariants_hold
        assert report.marsden_weinstein.is_valid_reduction
        assert report.polarization.is_valid_polarization_symmetry

    def test_summary_lists_all_seven(self, substrate_geometry_report) -> None:
        report = substrate_geometry_report
        summary = report.summary()
        assert "ALL VALID" in summary
        # Seven numbered structures appear in the multi-line summary.
        for k in range(1, 8):
            assert f"  {k}." in summary


class TestSubstrateIntegration:
    """The auxiliary substrate is exposed across SDK, telemetry, and API."""

    def test_physics_package_exports(self) -> None:
        from tnfr.physics import extract_phase_space_point as _eps
        from tnfr.physics import substrate_hamiltonian as _sh
        from tnfr.physics import symplectic_form_matrix as _sfm
        from tnfr.physics import verify_canonical_structure as _vcs

        assert callable(_eps)
        assert callable(_vcs)
        assert callable(_sh)
        assert callable(_sfm)

    def test_unified_telemetry_includes_substrate(
        self, unified_substrate_telemetry
    ) -> None:

        telemetry = unified_substrate_telemetry
        assert "symplectic_substrate" in telemetry
        sub = telemetry["symplectic_substrate"]
        assert sub["phase_space_dimension"] == 80
        assert abs(sub["liouville_divergence"]) < 1e-9

    def test_unified_telemetry_includes_pulse(
        self, unified_substrate_telemetry
    ) -> None:
        # dual-face telemetry: the conservative pulse (the resonant rhythm)
        # alongside the dissipative canonical coherence read-out

        telemetry = unified_substrate_telemetry
        assert "pulse" in telemetry
        pulse = telemetry["pulse"]
        assert pulse["n_modes"] >= 1
        assert pulse["fundamental"] > 0.0
        assert pulse["vibration_energy"] > 0.0

    def test_unified_telemetry_includes_resonance(
        self, unified_substrate_telemetry
    ) -> None:
        # the per-NFR pulse / resonance face (the source the collective
        # rhythm emerges from) alongside the collective pulse

        telemetry = unified_substrate_telemetry
        assert "resonance" in telemetry
        res = telemetry["resonance"]
        assert 0.0 <= res["phase_coherence"] <= 1.0
        assert 0.0 <= res["mean_local_resonance"] <= 1.0
        assert res["n_nodes"] == 20

    def test_sdk_symplectic_substrate_method(self) -> None:
        from tnfr.sdk import TNFR, SymplecticReport

        net = TNFR.create(20).ring()
        report = net.symplectic_substrate()
        assert isinstance(report, SymplecticReport)
        assert report.is_valid_manifold
        assert report.phase_space_dimension == 80
        assert "VALID" in report.summary()

    def test_sdk_analyze_includes_substrate(self) -> None:
        from tnfr.sdk import TNFR, Network

        net = Network(_canonical_graph(15))
        analysis = TNFR.analyze(net)
        assert "symplectic_substrate" in analysis
        assert analysis["features"]["symplectic_substrate"] is True


class TestAdiabaticInvariance:
    """The auxiliary oscillator action is adiabatic under a slow omega ramp."""

    def test_slow_ramp_conserves_action(self, adiabatic_report) -> None:
        cert = adiabatic_report
        assert cert.is_adiabatic_invariant
        assert cert.slow_drift < 1e-2

    def test_drift_decreases_with_slowness(self, adiabatic_report) -> None:
        # the adiabatic signature: slower ramp -> smaller action drift
        cert = adiabatic_report
        assert cert.drift_decreases_with_slowness
        assert cert.slow_drift < cert.fast_drift

    def test_fast_ramp_breaks_invariance(self, adiabatic_report) -> None:
        # a sudden ramp (T=1) injects/extracts action: large drift
        cert = adiabatic_report
        assert cert.fast_drift > 0.1

    def test_drift_series_trends_down(self, adiabatic_report) -> None:
        cert = adiabatic_report
        drifts = cert.action_drifts
        # the slow end is far below the fast end (orders of magnitude)
        assert drifts[-1] < drifts[0] / 10.0

    def test_certificate_valid(self, adiabatic_report) -> None:
        from tnfr.physics.symplectic_substrate import AdiabaticInvarianceCertificate

        cert = adiabatic_report
        assert isinstance(cert, AdiabaticInvarianceCertificate)
        assert "VALID" in cert.summary()
        assert "auxiliary oscillator" in cert.summary()