"""Static constitutive comparison of locally identical phase-storage laws. The supplied smooth cutoff law agrees with sine on a complete open chamber but raises the antipodal edge cost. This owner certifies a local-flow equality and a global winding obstruction; it installs no runtime or trajectory law. """ from __future__ import annotations from dataclasses import dataclass from fractions import Fraction as Q from .._exact_time import exact_or_represented_real from ..mathematics._rational_interval import INTERVAL_METHOD, I, pi_interval from .relational_sine_comparison import ( SineExchangeComparison, _sine_phase_storage, _validate_comparison_labels, ) from .relational_sine_corridor import SineSaddlePreparation, prepare_sine_saddle_state from .relational_sine_partition import _private_leaf_support from .relational_sine_regional import _branches __all__ = ("SaddleStorageDiscriminator", "assess_saddle_storage_discriminator") @dataclass(frozen=True) class SaddleStorageDiscriminator: """A supplied alternative on the same freshly constructed initial state. The full independent coordinate cube inherits local equality of the two complete flows and, when certified, the alternative winding obstruction. Signed-family sine corridor gates belong only to the nominal preparation. The discriminator proves a global difference for that same nominal state; it supplies no new sine passage certificate for every point of the cube. """ source: SineExchangeComparison preparation: SineSaddlePreparation prepared_state: SineExchangeComparison cycle: tuple[object, ...] cycle_indices: tuple[int, ...] edge_indices: tuple[tuple[int, int], ...] epsilon: Q initial_coordinate_radius: Q initial_box: tuple[I, ...] cutoff_phase_bounds: I initial_edge_principal_gap_bounds: tuple[I, ...] | None initial_agreement_margin_lower_bound: Q | None initial_cycle_principal_gap_bounds: tuple[I, ...] | None initial_edge_turn_offsets: tuple[int, ...] | None initial_cycle_turn_offsets: tuple[int, ...] | None initial_winding: int | None initial_form_storage_bounds: I initial_sine_phase_storage_bounds: I sine_initial_storage_bounds: I alternative_initial_storage_bounds: I | None antipodal_storage: Q energy_margin_lower_bound: Q | None local_scaled_duration: Q local_phase_radius_bound: Q local_agreement_margin_lower_bound: Q initial_agreement_certified: bool whole_local_flow_agreement_certified: bool nominal_sine_outer_gate_certified: bool nominal_sine_inner_gate_certified: bool alternative_winding_invariant_both_time_directions_certified: bool alternative_zero_winding_passage_excluded: bool discriminator_certified: bool status: str reasons: tuple[str, ...] sine_full_cube_passage_certified: bool = False law: str = "smooth_cutoff_phase_storage_reciprocal_exchange" regularity: str = "C_infinity_but_not_real_analytic_at_the_cutoff" clock: str = "original structural t; local proof tau=t/pi" potential_formula: str = ( "U(delta)=1-cos(delta)+2*h(y); " "y=(-cos(delta)-c)/(1-c); c=sqrt(2)/2; " "h(y)=0 for y<=0 and exp(1-1/y) for y>0" ) current_formula: str = ( "j(delta)=sin(delta) for y<=0; " "j(delta)=sin(delta)*(1+2*h(y)/((1-c)*y^2)) for y>0" ) complete_rows: tuple[str, ...] = ( "dx_i/dt=sum_j j(theta_j-theta_i)/(pi*d_i)", "dtheta_i/dt=sum_j(x_i-x_j)/(pi*d_i)", "fixed complete support; held unit capacities; zero inputs and no events", ) storage_identity: str = ( "H=sum_edges((x_j-x_i)^2/2+U(theta_j-theta_i)); " "dH/dtau=-(Lx)^T*K*grad(V)+grad(V)^T*K*Lx=0" ) arithmetic_method: str = INTERVAL_METHOD scope: tuple[str, ...] = ( "source_anchors_reference_support_and_law_not_the_generated_preparation", "prepared_state_retains_a_sine_reference_declaration_not_an_installed_alternative", "smooth_even_periodic_nonnegative_potential_is_supplied_not_uniquely_derived", "global_first_resultant_sufficiency_and_native_pressure_alignment_do_not_hold", "cutoff_and_antipodal_cost_are_counterexample_choices_not_physical_constants", "all_full_state_coordinates_contacts_and_form_edge_storage_are_retained", "initial_box_outwardly_encloses_the_declared_exact_independent_coordinate_cube", "alternative_initial_storage_is_evaluated_only_inside_the_exact_zero_bump_chamber", "whole_local_flow_equality_uses_a_uniform_bound_and_smooth_uniqueness_in_both_time_directions", "independent_cube_perturbations_do_not_inherit_nominal_signed_family_corridor_gates", "discriminator_certified_compares_nominal_global_passage_not_full_cube_sine_passage", "strict_full_storage_below_antipodal_cost_prevents_any_cycle_winding_change", "winding_invariance_is_not_acute_identity_retention_or_attraction", "no_alternative_trajectory_solver_law_installation_or_physical_identification", "no_retained_response_is_authenticated_recomputed_or_rewritten", ) def to_dict(self): from ..sdk.relational_reports import _project, _validate_label_groups _validate_comparison_labels(self.source) _validate_comparison_labels(self.prepared_state) _validate_label_groups(self.cycle) self.preparation.to_dict() return { "schema": "tnfr.saddle-storage-discriminator.v1", "report": _project(self), } def assess_saddle_storage_discriminator( source, *, cycle, initial_coordinate_radius, epsilon=Q(1, 2**32) ) -> SaddleStorageDiscriminator: """Compare a rebuilt sine preparation with one explicit smooth alternative. No prepared state or cached verdict is accepted from the caller. The source only anchors the existing conservative unit C5/private-leaf law; the shared preparation owner constructs the actual rational center. Radius is a nonnegative exact or represented real error at every form and phase coordinate. Capacity, support and clock stay fixed. The alternative has zero added storage/current for principal edge gaps at most3*pi/4. Its complete storage is therefore exactly the rebuilt sine storage on that chamber. Outside it this static reader leaves alternative storage unavailable instead of evaluating an unimplemented runtime. The exact antipodal value U(pi)=4 supplies the global boundary proof. """ delta = exact_or_represented_real( initial_coordinate_radius, "initial_coordinate_radius" ) if delta < 0: raise ValueError("initial_coordinate_radius must be nonnegative") source, edges, indices, pairs, _ = _private_leaf_support(source, cycle) labels = tuple(source.nodes[index] for index in indices) preparation = prepare_sine_saddle_state(source, cycle=labels, epsilon=epsilon) prepared = preparation.prepared_state initial_box = tuple( I(value - delta, value + delta) for value in prepared.epi + prepared.phase ) count = len(prepared.nodes) forms, phases = initial_box[:count], initial_box[count:] pi = pi_interval() cutoff = 3 * pi / 4 edge_offsets, principal, _, _ = _branches( tuple(phases[j] - phases[i] for i, j in edges) ) agreement_margin = ( None if principal is None else min(cutoff.lo - gap.abs_max for gap in principal) ) initial_agreement = agreement_margin is not None and agreement_margin >= 0 offsets, cycle_gaps, winding, _ = _branches( tuple(phases[j] - phases[i] for i, j in pairs) ) form_storage = sum(((forms[j] - forms[i]) ** 2 / 2 for i, j in edges), I(0)) phase_storage = _sine_phase_storage(phases, edges) sine_storage = form_storage + phase_storage alternative_storage = sine_storage if initial_agreement else None energy_margin = ( None if alternative_storage is None else Q(4) - alternative_storage.hi ) local_radius = 729 * ( 3 * preparation.epsilon + preparation.preparation_error_bound + delta ) # Saddle gaps are bounded by2pi/3. Two nodal errors affect an edge, so # the extra gap allowance before the3pi/4 cutoff is only pi/12. local_margin = pi.lo / 12 - 2 * local_radius local_agreement = ( preparation.formation.correlated_preparation_certified and initial_agreement and local_margin > 0 ) invariant = ( initial_agreement and winding is not None and energy_margin is not None and energy_margin > 0 ) excludes_zero = invariant and winding != 0 outer = preparation.outer_corridor_gate_certified inner = preparation.inner_corridor_gate_certified certified = local_agreement and outer and inner and excludes_zero reasons = tuple( reason for passed, reason in ( ( initial_agreement, "initial_full_cube_not_certified_inside_agreement_chamber", ), (local_agreement, "whole_local_flow_agreement_not_certified"), (winding is not None, "initial_cycle_branch_not_certified"), ( energy_margin is not None and energy_margin > 0, "alternative_storage_below_seam_not_certified", ), (outer and inner, "nominal_sine_corridor_gates_not_certified"), (excludes_zero, "alternative_zero_winding_exclusion_not_certified"), ) if not passed ) return SaddleStorageDiscriminator( source=source, preparation=preparation, prepared_state=prepared, cycle=labels, cycle_indices=indices, edge_indices=edges, epsilon=preparation.epsilon, initial_coordinate_radius=delta, initial_box=initial_box, cutoff_phase_bounds=cutoff, initial_edge_principal_gap_bounds=principal, initial_agreement_margin_lower_bound=agreement_margin, initial_cycle_principal_gap_bounds=cycle_gaps, initial_edge_turn_offsets=edge_offsets, initial_cycle_turn_offsets=offsets, initial_winding=winding, initial_form_storage_bounds=form_storage, initial_sine_phase_storage_bounds=phase_storage, sine_initial_storage_bounds=sine_storage, alternative_initial_storage_bounds=alternative_storage, antipodal_storage=Q(4), energy_margin_lower_bound=energy_margin, local_scaled_duration=Q(3), local_phase_radius_bound=local_radius, local_agreement_margin_lower_bound=local_margin, initial_agreement_certified=initial_agreement, whole_local_flow_agreement_certified=local_agreement, nominal_sine_outer_gate_certified=outer, nominal_sine_inner_gate_certified=inner, alternative_winding_invariant_both_time_directions_certified=invariant, alternative_zero_winding_passage_excluded=excludes_zero, discriminator_certified=certified, status="certified" if certified else "unavailable", reasons=reasons, )