"""Direct mixed-state passage and capture for the original C9 preparation. Proof: theory/nodal/SINE_PREPARATION_PATH_CAPTURE.md. The source is rebuilt at tau=200 and continued through y=theta+z, z=gamma*x. No reference trajectory, nominal reset, phase-only replacement law or supplied endpoint is consumed. The two stages end at memberwise times; a common later deadline follows from full-state storage trapping, not from identifying their hitting times. """ from __future__ import annotations from dataclasses import dataclass from fractions import Fraction as Q from ..mathematics._rational_interval import I, cos, pi_interval, sin, sqrt from ._sine_preparation_path_selection import ( _assess_preparation_path_selection, _PreparationPathSelection, ) from .relational_sine_formed_classes import _GAP from .reversible_eigenmode_reference import _negative_exp_bounds _INITIAL_TIME = Q(200) _FIRST_DURATION = Q(8) _SECOND_DURATION = Q(128) _GAP_LOWER = Q(9, 10) _GAP_UPPER = Q(13, 10) _CLOSING_UPPER = Q(63, 100) _CLOSING_TARGET = Q(-7, 4) _PASSAGE_SPEED = Q(3, 10) _ACUTE_CAP = Q(7, 5) _INITIAL_DISTANCE = Q(15) _CONTRACTION = Q(1, 30) @dataclass(frozen=True) class _PreparationPathCapture: """Sufficient continuation certificate; failed margins remain unresolved. Bounds for stage two describe each member's actual passage time plus 128 slow units. Only after the strict capture margins pass does storage nonincrease carry that result to the common deadline and all later times. Candidate arithmetic remains visible when its sufficient premises fail. """ source: _PreparationPathSelection uniform_scaled_form_norm_upper_bound: Q mixed_field_error_upper_bound: Q initial_mixed_phase_error_norm_upper_bound: Q initial_ordinary_gap_bounds: tuple[I, ...] initial_closing_coordinate_bounds: I initial_closing_lower_margin: Q initial_closing_upper_margin: Q initial_ordinary_upper_margin: Q initial_coupled_lower_margin: Q stage_one_coupled_lower_bound: Q stage_one_ordinary_lower_bound: Q stage_one_ordinary_upper_bound: Q stage_one_sine_speed_lower_bounds: tuple[Q, Q, Q] stage_one_passage_speed_lower_bound: Q stage_one_passage_time_upper_bound: Q stage_two_initial_gap_bounds: I stage_two_actual_gap_absolute_upper_bound: Q stage_two_curvature_lower_bound: Q stage_two_initial_distance_squared_upper_bound: Q stage_two_decay_bounds: I stage_two_phase_distance_upper_bound: Q actual_phase_distance_at_capture_upper_bound: Q form_norm_upper_bound: Q form_storage_upper_bound: Q phase_storage_upper_bound: Q total_storage_upper_bound: Q sector_boundary_storage_lower_bound: Q capture_margin: Q sufficient_margins: tuple[tuple[str, Q], ...] source_policy_certified: bool passage_certified: bool capture_certified: bool captured_winding: int | None slow_time_deadline: Q scaled_time_deadline_bounds: I original_time_deadline_bounds: I status: str unresolved_conditions: tuple[str, ...] initial_scaled_time: Q = _INITIAL_TIME stage_one_duration: Q = _FIRST_DURATION stage_two_duration: Q = _SECOND_DURATION closing_target: Q = _CLOSING_TARGET acute_gap_cap: Q = _ACUTE_CAP contraction_rate: Q = _CONTRACTION scope: tuple[str, ...] = ( "same_original_Cayley_source_and_latent_errors_rebuilt_at_tau_200", "actual_mixed_state_y_equals_theta_plus_gamma_centered_form", "mixed_error_from_exact_initial_identity_without_a_second_fast_transient", "all_eighteen_original_coordinates_and_conserved_means_retained", "closing_edge_passage_is_smooth_flow_not_a_hybrid_event", "stage_two_begins_at_each_actual_first_hitting_time", "distance_bound_at_memberwise_capture_time_not_a_common_sampled_endpoint", "strict_full_state_storage_barrier_extends_capture_to_common_deadline", "acute_winding_one_capture_not_fixed_small_ball_entry", "failed_sufficient_margins_are_unresolved_not_nonformation", "no_reference_trajectory_parameter_sweep_or_frozen_producer", ) def _assess_preparation_path_capture( *, parameter_interval, form_error_bound, phase_error_bound ) -> _PreparationPathCapture: """Re-admit the original preparation and test the fixed two-stage proof. The path owner admits the exact interval and nonnegative finite residual budgets before any continuation arithmetic. Times, source, law, capacities and the proof constants are fixed; no incoming report replaces primitives. """ source = _assess_preparation_path_selection( parameter_interval=parameter_interval, scaled_time=_INITIAL_TIME, form_error_bound=form_error_bound, phase_error_bound=phase_error_bound, ) pi = pi_interval() eta, forcing = source.eta_bounds.hi, source.forcing_norm_bounds.hi z = max(source.scaled_form_radius_upper_bound, eta * forcing / _GAP) error = 2 * z # y(tau)=theta(0)+gamma*x(0)+eta*integral_0^tau f(theta). The # homogeneous form term cancels exactly here; adding Rz would charge it twice. initial_error = ( source.scaled_initial_error_norm_upper_bound + source.initial_phase_error_norm_upper_bound + eta * forcing * _INITIAL_TIME ) edge_error = I(-initial_error, initial_error) dipole = tuple(Q(int(j == 3) - int(j == 4)) for j in range(9)) ordinary = tuple( source.ramp_angle_bounds + source.lift_correction_bounds * (dipole[j + 1] - dipole[j]) + edge_error for j in range(8) ) closing = 8 * source.ramp_angle_bounds - 3 * pi + edge_error initial_margins = ( ("initial_closing_positive", closing.lo), ("initial_closing_below_declared_ceiling", _CLOSING_UPPER - closing.hi), ( "initial_ordinary_gaps_below_declared_ceiling", _GAP_UPPER - max(value.hi for value in ordinary), ), ( "initial_coupled_gap_lower_bound", min(value.lo for value in ordinary) - closing.hi / 2 - _GAP_LOWER, ), ) coupled_lower = _GAP_LOWER - Q(3, 2) * error * _FIRST_DURATION ordinary_lower = coupled_lower + _CLOSING_TARGET / 2 ordinary_upper = _GAP_UPPER + error * _FIRST_DURATION sine_speeds = ( (sin(I(Q(607, 500))) - sin(I(_CLOSING_UPPER))).lo, sin(I(Q(649, 1000))).lo, sin(I(Q(1, 2))).lo, ) speed = min(sine_speeds) - error passage_time = (_CLOSING_UPPER - _CLOSING_TARGET) / _PASSAGE_SPEED first_margins = initial_margins + ( ("stage_one_coupled_gap_floor", coupled_lower - Q(899, 1000)), ("stage_one_ordinary_positive_floor", ordinary_lower - Q(3, 125)), ("stage_one_ordinary_acute_ceiling", pi.lo / 2 - ordinary_upper), ("stage_one_strict_passage_speed", speed - _PASSAGE_SPEED), ("stage_one_strict_hitting_deadline", _FIRST_DURATION - passage_time), ) # At q=-7/4 the closing edge has the winding-one representative 7/4-pi. # Within the acute chart the gap extrema drift by at most error per unit # slow time. The actual phase adds at most z to every mixed edge gap. initial_gaps = I(Q(7, 4) - pi.hi, ordinary_upper) actual_cap = initial_gaps.abs_max + _SECOND_DURATION * error + z curvature = cos(I(_ACUTE_CAP)).lo initial_squared = (45 * (_ACUTE_CAP + 2 * pi / 9) ** 2).hi decay = I(*_negative_exp_bounds(_CONTRACTION * _SECOND_DURATION)) phase_distance = _INITIAL_DISTANCE * decay.hi + error / _CONTRACTION actual_distance = phase_distance + z inverse_gamma = 1023 * pi form_norm = inverse_gamma.hi * z / sqrt(I(2)).lo form_storage = 2 * form_norm**2 phase_storage = (9 * (1 - cos(2 * pi / 9))).hi + actual_distance**2 storage = form_storage + phase_storage barrier = source.sector_barrier_lower_bounds[source.candidate_windings.index(1)] capture_margin = barrier - storage all_margins = first_margins + ( ("stage_two_strict_acute_tube", _ACUTE_CAP - actual_cap), ("stage_two_cap_is_acute", pi.lo / 2 - _ACUTE_CAP), ("stage_two_uniform_phase_curvature", curvature - Q(1, 6)), ("stage_two_initial_distance", _INITIAL_DISTANCE**2 - initial_squared), ("strict_full_state_capture_barrier", capture_margin), ) source_ok = source.source_budget_certified and source.initial_zero_winding_certified passage = source_ok and all(margin > 0 for _, margin in first_margins) capture = source_ok and all(margin > 0 for _, margin in all_margins) unresolved = tuple(name for name, margin in all_margins if margin <= 0) + tuple( name for name, ok in ( ("original_source_storage_policy", source.source_budget_certified), ( "original_zero_winding_preparation", source.initial_zero_winding_certified, ), ) if not ok ) deadline = _FIRST_DURATION + _SECOND_DURATION # Use the exact model formula, never division by an outward eta interval. tau = _INITIAL_TIME + deadline * 1023**2 * pi**2 original_time = tau / Q(1023, 1024) return _PreparationPathCapture( source=source, uniform_scaled_form_norm_upper_bound=z, mixed_field_error_upper_bound=error, initial_mixed_phase_error_norm_upper_bound=initial_error, initial_ordinary_gap_bounds=ordinary, initial_closing_coordinate_bounds=closing, initial_closing_lower_margin=initial_margins[0][1], initial_closing_upper_margin=initial_margins[1][1], initial_ordinary_upper_margin=initial_margins[2][1], initial_coupled_lower_margin=initial_margins[3][1], stage_one_coupled_lower_bound=coupled_lower, stage_one_ordinary_lower_bound=ordinary_lower, stage_one_ordinary_upper_bound=ordinary_upper, stage_one_sine_speed_lower_bounds=sine_speeds, stage_one_passage_speed_lower_bound=speed, stage_one_passage_time_upper_bound=passage_time, stage_two_initial_gap_bounds=initial_gaps, stage_two_actual_gap_absolute_upper_bound=actual_cap, stage_two_curvature_lower_bound=curvature, stage_two_initial_distance_squared_upper_bound=initial_squared, stage_two_decay_bounds=decay, stage_two_phase_distance_upper_bound=phase_distance, actual_phase_distance_at_capture_upper_bound=actual_distance, form_norm_upper_bound=form_norm, form_storage_upper_bound=form_storage, phase_storage_upper_bound=phase_storage, total_storage_upper_bound=storage, sector_boundary_storage_lower_bound=barrier, capture_margin=capture_margin, sufficient_margins=all_margins, source_policy_certified=source_ok, passage_certified=passage, capture_certified=capture, captured_winding=1 if capture else None, slow_time_deadline=deadline, scaled_time_deadline_bounds=tau, original_time_deadline_bounds=original_time, status="certified_winding_one_capture" if capture else "unavailable", unresolved_conditions=unresolved, )