"""Reciprocity and conditional relaxation under the supplied law theta'=g. The phase metric represents a vector field; it does not install a phase law or select the undetermined joint potential. The final budget requires a global regular path and a uniform positive metric floor. No trajectory, pressure refresh, fitted gain or production evolution rule is constructed. """ from fractions import Fraction as Q import pytest from tests.physics._internal_mode_fixture import LIFT, NODES, _graph from tests.physics._internal_mode_fixture import ( _prepared_phase_metric as _prepared_metric, ) from tests.physics._internal_mode_fixture import _unit_regional_budget as _budget from tnfr.physics.support_transport import observe_support_transport def test_epi_independent_psi_cannot_remove_the_prepared_reciprocal_velocity(): s, _, _, metric, source, jacobian = _prepared_metric() direction = s.Matrix((1, -1, 0) * 2) first = s.ones(6, 1) / 2 + direction / 16 second = first + direction / 16 weight = s.Rational(1, 4) degree = 3 * s.eye(6) # At identical phase/capacity/support, every EPI-independent Psi has the # same phase gradient at both states; no particular Psi is selected. psi_gradient = s.Matrix(s.symbols("psi_theta0:6", real=True)) required = tuple( metric.inv() * (weight * jacobian.T * degree * epi - psi_gradient) for epi in (first, second) ) difference = s.simplify(required[1] - required[0]) expected = -direction / (64 * s.pi * (1 + s.sqrt(3))) assert s.simplify(difference - expected) == s.zeros(6, 1) assert difference != s.zeros(6, 1) residuals = tuple(metric * (source - value) for value in required) assert s.simplify(residuals[1] - residuals[0]) == 3 * direction / (64 * s.pi) # The proposed x-independent theta'=g is identical at these two states. # If Psi cancels its residual at the first, it cannot do so at the second. def test_phase_only_descent_does_not_erase_the_joint_reciprocal_work(): s, graph, phases, metric, source, jacobian = _prepared_metric() variables = s.symbols("theta0:6", real=True) positions = {node: index for index, node in enumerate(NODES)} potential = sum( 1 - s.cos(variables[positions[left]] - variables[positions[right]]) for left, right in graph.edges ) substitution = dict(zip(variables, phases, strict=True)) gradient = s.Matrix( [s.diff(potential, value).subs(substitution) for value in variables] ) assert s.simplify(gradient + metric * source) == s.zeros(6, 1) assert s.simplify(potential.subs(substitution)) == 5 - 2 * s.sqrt(3) phase_work = s.simplify((gradient.T * source)[0]) assert phase_work == -(1 + s.sqrt(3)) / 3 < 0 # Audit the illustrative choice Psi=V_phi without promoting it to a law. epi = s.ones(6, 1) / 2 + s.Matrix((1, -1, 0) * 2) / 16 joint_gradient = gradient - s.Rational(1, 4) * jacobian.T * (3 * epi) joint_phase_work = s.simplify((joint_gradient.T * source)[0]) assert s.simplify(joint_phase_work - phase_work) == 1 / (32 * s.pi) > 0 # This is a nonzero missing cross term, not a claim that the complete # joint energy increases or that a reciprocal gradient law was executed. def test_regional_owner_keeps_the_exact_young_slack_and_disagreement_loss(): e, weight = Q(1, 2), Q(1, 4) phase_source = (Q(1, 6), Q(-1, 6), Q(0)) * 2 source_squared = sum(value**2 for value in phase_source) rates = [] for sign in (1, -1): snapshot = observe_support_transport(_graph((Q(1, 16), 0, sign * Q(1, 16), 0))) internal = tuple(value - Q(1, 2) for value in snapshot.epi) difference = tuple( a - b for a, b in zip(internal[:3], internal[3:], strict=True) ) disagreement = sum(value**2 for value in difference) forcing = tuple(weight * value for value in phase_source) squared, rate, work = _budget(snapshot, forcing, e=e) assert squared == sum(value**2 for value in internal) == Q(1, 64) slack = e * sum( (y - weight * g / e) ** 2 for y, g in zip(internal, phase_source, strict=True) ) assert slack == e * squared + weight**2 * source_squared / e - work assert ( rate + e * squared - weight**2 * source_squared / e == (-slack - 2 * e * disagreement / 3) < 0 ) rates.append(rate) assert rates == [Q(1, 192), Q(-5, 192)] # These are exact declared source vectors at snapshots. Positive initial # source work is compatible with the later conditional decay estimate. def test_combined_budget_has_nonnegative_slack_under_a_uniform_phase_metric_floor(): s = pytest.importorskip("sympy") e, floor = s.symbols("e h_min", positive=True) weight, metric_excess = s.symbols("w metric_excess", nonnegative=True) coefficients = s.Matrix(s.symbols("u0 v0 u1 v1", real=True)) internal = s.Matrix(LIFT) * coefficients source = s.Matrix(s.symbols("g0:6", real=True)) squared = (internal.T * internal)[0] difference = internal[:3, :] - internal[3:, :] disagreement = (difference.T * difference)[0] source_squared = (source.T * source)[0] work = 2 * weight * (internal.T * source)[0] rate = -2 * e * squared - 2 * e * disagreement / 3 + work young_slack = ( e * ((internal - weight * source / e).T * (internal - weight * source / e))[0] ) # IF theta'=g and H_i>=h_min along the entire regular path, V' has # this form with metric_excess=sum_i(H_i-h_min)*g_i^2 >=0. phase_energy_rate = -floor * source_squared - metric_excess combined_rate = rate + weight**2 * phase_energy_rate / (e * floor) remainder = combined_rate + e * squared expected = ( -young_slack - 2 * e * disagreement / 3 - weight**2 * metric_excess / (e * floor) ) assert s.expand(remainder - expected) == 0 assert all( value.is_nonnegative for value in ( e, floor, weight, metric_excess, 2 * e / 3, weight**2 / (e * floor), ) ) # Z=S+w^2*V/(e*h_min)>=0 and Z'<=-e*S give integral S<=Z(0)/e. # V'<=-h_min*||g||^2 also gives an L1 forcing input. Convergence of S # additionally uses S'<=-e*S+(w^2/e)||g||^2 and the stable scalar # convolution estimate; finite integral S alone would not suffice.