Result and reversal ledger

Detailed: established-in-model

The Jordan/TKK route reaches the Spin(2,3) hinge

For the declared four-real-dimensional carrier, Jordan translations, grading and a correctly typed continuous Furey channel rotation close the ten-dimensional real Lie algebra \(\mathfrak{sp}(4,\mathbb R)\), hence \(\mathrm{Spin}_0(2,3)\).

Boundary. This is a finite-dimensional model theorem, not evidence that nature selects the carrier or its interpretation.

• 1058_a_three_generator_jordan_furey_recursion_closes_all_of_sp4_but_requires_the_furey_channel_rotation_as_a_continuous_flow.md
• 1059_the_linear_furey_product_FCp_exponentiates_exactly_to_the_missing_continuous_sp4_channel_rotation_but_its_action_term_is_open.md
• 47_global_group_isomorphism_sp4r_spin023.md

Detailed: established-in-model

A concrete lower-rank Jordan entry point exists

The Gaussian residual selector is represented by the \(H_2(\mathbb R)\) logarithmic barrier and its Peirce–Schur recursion. Its conformal/TKK algebra supplies an explicit local rung into \(\mathfrak{sp}(4,\mathbb R)\).

Boundary. \(H_2(\mathbb R)\) is not the exceptional Albert algebra; the later \(J_3(\mathbb O)\), \(E_6\) and \(E_8\) continuation remains a separate bridge.

• 1023_the_gaussian_gram_selector_is_the_h2r_jordan_logdet_barrier_and_a_peirce_slice_of_the_rank_three_cubic.md
• 986_the_diagonal_cubic_barrier_is_flat_in_log_coordinates_and_cannot_generate_the_carrier_shear_without_tkk_prolongation.md
• 202_peirce_schur_triple_recursion_to_spin23.md

Detailed: established-in-model

The odd-cycle intuition has a projective double-cover realization

The three-relation cycle supports an odd-loop obstruction, a connected six-cycle cover and a coherent central-sign lift. Nested six- and twelve-step transports make the twist and return data explicit.

Boundary. Central −I holonomy is not automatically chirality, a spacetime braid, a Pfaffian reversal or a generation count.

• 994_the_three_placement_triangle_has_a_z2_sideband_frustration_and_requires_path_dependence_or_a_double_cover.md
• 995_the_unique_connected_double_cover_of_the_placement_triangle_is_the_six_cycle_with_sideband_as_deck_parity.md
• 996_the_graded_sideband_antiunitary_lifts_the_six_cycle_to_a_nested_c12_to_c6_to_c3_transport.md
• 1128_three_is_the_minimal_Mobius_gear_cycle_and_the_third_relation_of_the_typed_twin_wheels.md

Detailed: closed-identification

The central sign is not the missing orientation datum

The canonical projective lift returns the carrier frame by −I while preserving determinant orientation. The direct identification of Wilson-loop sign with response-Pfaffian sheet therefore fails.

Boundary. A separate reflection, oriented area or other odd datum could still supply a typed orientation bridge.

• 1066_the_Wilson_pi_flux_is_the_nontrivial_RP1_line_holonomy_but_its_canonical_frame_lift_is_central_minus_I_not_a_Pfaffian_sheet_flip.md
• 1046_the_oriented_area_is_exactly_the_pfaffian_line_of_the_response_root_but_its_microscopic_weak_higgs_identification_remains_open.md
• 1047_fixing_the_phase_symplectic_orientation_reduces_the_gram_root_gauge_to_SO2_and_turns_the_pfaffian_sign_into_a_rank_one_separated_Z2_sheet.md

Detailed: constructed-conditional

The Furey channel rotation participates in exact sp(4) closure

On the declared response carrier, the continuous Furey product rotation supplies the generator that enlarges the compact-turn/split-dilation subalgebra to \(\mathfrak{sp}(4,\mathbb R)\).

Boundary. The response operator is an effective typed construction, not a microscopic Standard-Model Hamiltonian or a theorem about Furey's complete programme.

• 1059_the_linear_furey_product_FCp_exponentiates_exactly_to_the_missing_continuous_sp4_channel_rotation_but_its_action_term_is_open.md
• 1060_a_positive_first_order_response_pulse_generates_the_full_furey_channel_flow_without_bulk_degeneracy_but_needs_microscopic_charge_completion.md
• 69_furey_weak_jordan_ladder_spin23.md

Detailed: established-obstruction

One S₁ mediator cannot realize the desired pure Furey pulse

The common mediator producing the two crossed Standard-Model operators also produces a third Wilson edge. One copy is rank-obstructed; common-orientation positive copies do not remove it.

Boundary. A split-copy involution, source-incidence rule, boundary realization or physical reading of the closed triangle remains possible only with additional structure.

• 1061_the_minimal_sm_field_completion_of_the_furey_channel_pulse_is_a_pair_of_dimension_six_S1_exchange_operators_with_an_underived_relative_phase.md
• 1062_one_S1_mediator_cannot_generate_a_pure_FCp_pulse_but_two_orthogonal_copies_realize_the_exact_oriented_star.md
• 1074_positive_Spin10_Clebsch_preserving_copies_can_never_remove_the_third_edge_but_a_twisted_two_copy_involution_can.md

Detailed: established-obstruction

Proton safety is a source-selection problem

The complete unified triplet contains the desired leptoquark arms together with four-quark and proton-decay channels. B−L and matter parity do not isolate the safe subset.

Boundary. This excludes the naive complete-multiplet reading, not every Pati–Salam, finite-geometry or multi-mediator completion.

• 1075_the_unified_vector_triplet_gives_two_FCp_arms_one_safe_extra_and_three_proton_decay_channels_all_with_BminusL_zero.md
• 1076_the_proton_safe_current_projector_exists_only_after_breaking_the_pati_salam_colour_lepton_roots.md
• 1078_no_minimal_family_universal_anomaly_free_U1_can_protect_the_proton_safe_S1_source_mask.md

Detailed: constructed-conditional

Pati–Salam finite geometry supplies a concrete source split

The retained Model C reconstruction separates leptoquark and diquark sources and reproduces opposite crossed coefficients; rank-one vacuum geometry protects selected mixing channels through the audited orders.

Boundary. Broken-phase counterterms, global anomalies and the third-edge burden remain open.

• 1079_NCG_Pati_Salam_Model_C_realizes_the_safe_source_split_but_its_minimal_left_right_pair_still_closes_the_Wilson_triangle.md
• 1084_the_minimal_spectral_graph_has_an_all_orders_rank_one_bridge_normal_form_that_forbids_holomorphic_cross_six_mixing.md
• 1085_the_rank_one_spectral_zero_is_the_unbroken_colour_lepton_reflection_Ward_sector_but_three_generation_weak_instantons_make_it_nonperturbatively_anomalous.md

Detailed: established-boundary

The simplest continuous and binary protections fail

Family-universal anomaly-free U(1) charges reduce to familiar directions and cannot distinguish the safe mask. A universal internal Z₂ is weak-anomalous for an odd family count, while connected flavor removes the two-family sign exit.

Boundary. Larger cyclic, flavor-dependent, non-Abelian, inflow and finite-geometry mechanisms remain live.

• 1078_no_minimal_family_universal_anomaly_free_U1_can_protect_the_proton_safe_S1_source_mask.md
• 1086_for_an_odd_number_of_families_any_universal_Z2_that_splits_leptoquark_from_diquark_triplets_has_the_same_mod_two_weak_anomaly.md
• 1087_connected_CKM_and_PMNS_Yukawa_graphs_force_any_exact_diagonal_family_Z2_back_to_the_universal_anomalous_class.md

Detailed: constructed-conditional

Baryon triality is the first surviving cyclic protection

The first family-universal cyclic witness appears at Z₉; quotienting its gauge cube leaves baryon triality. Three families cancel the audited low-energy residues.

Boundary. The full Pati–Salam bordism class, heavy-threshold descent and exact finite-spectral-triple normalizer remain open.

• 1088_Z9_is_the_minimal_family_universal_cyclic_lift_that_protects_triplet_companion_mixing_and_its_three_family_quotient_is_baryon_triality.md
• 1089_the_Z9_quark_number_lift_factors_exactly_into_a_chiral_module_phase_times_Pati_Salam_BminusL_and_its_gauge_quotient_is_baryon_triality.md
• 1092_the_Z9_extension_not_its_naive_mod3_reduction_realizes_baryon_triality_and_matches_the_Dai_Freed_mod9_anomaly.md
• 1094_the_Majorana_vacuum_stabilizer_is_the_Z6_quotiented_standard_model_times_a_split_baryon_triality_with_the_Z9_lift_cubing_to_colour_centre.md

Detailed: conditional

The exceptional continuation remains a separate bridge

The local \(H_2(\mathbb R)\)/TKK result strengthens the route into Spin(2,3), but does not derive \(J_3(\mathbb O)\), \(E_6\), \(E_8\), Golay, Leech or Monster structure.

Boundary. Shared dimensions, labels and visual resemblance are not carrier maps; each continuation needs an explicit map, invariant and observable.

• 33_octonions_leech_lattice_jordan_branch.md
• 40_grand_synthesis_spin23_pole.md
• 1217_leech_pair_gram_geometry_reaches_spin23_but_needs_an_orientation_twist_and_affine_theta_shift.md

Detailed: open-physical

Mathematical closure is not physical closure

The programme contains reproducible algebraic closure and microscopic candidates, but no selected action, vacuum, measure, recovered Standard-Model spectrum or target-blind prediction follows from closure alone.

Boundary. This is the principal firewall against presenting the programme as a completed theory of nature.

• 30_research_frontier.md
• 49_multi_branch_research_registry.md
• 1203_terminal_reflexive_closure_audit_conditional_eft_closure_and_sharp_indeterminacy_of_absolute_flavour_selection.md

Detailed: constructed-conditional

Charge-three Higgsing has a precise topological limit

With the declared normalization, a compact charge-three Abelian Higgs sector reduces in the strict infrared to level-three BF theory. A gapped unit-charge reader may retain sector memory without becoming a topological degree of freedom.

Boundary. The charge-three phase does not create the required wall: its apparent vacua are gauge-equivalent. A separate defect origin and orientation history are required.

• 1233_charge_three_abelian_higgs_reduces_conditionally_to_level_three_bf_and_a_gapped_unit_charge_reader_can_coexist_only_as_non_topological_memory.md

Detailed: constructed-conditional

A neutral kink supplies the minimal reader wall

Adding one neutral \(\mathbb Z_2\) order parameter produces an explicit kink and localizes one even, gapped, charge-one bosonic mode without mirror doubling.

Boundary. This is a minimal completion, not a derivation from the original action. Compact boundary conditions and the physical origin of the neutral field remain external.

• 1292_a_neutral_z2_kink_is_the_minimal_defect_origin_and_one_even_charge_one_boson_localizes_without_mirror_doubling.md

Detailed: established-boundary

Healthy bosonic poles do not settle the chiral fold

A frozen tree-level benchmark has positive kinetic eigenvalues, nonnegative masses and positive residues for every defined bosonic sector.

Boundary. Reflection exchanges four-dimensional chirality. The undoubled chiral Standard Model has no declared same-representation partner, self-adjoint fold domain or boundary BRST cohomology.

• 1290_s_v1_contains_a_fully_healthy_quadratic_benchmark_but_one_completion_cannot_discharge_the_global_a4_gate.md
• 1451_the_maximal_a4_action_freezes_all_bulk_poles_but_the_chiral_fold_has_no_declared_physical_state_domain.md

Detailed: internally-exhausted

The selector is missing a physical phase reference

Endpoint-blind Standard-Model data cannot supply the exchange-odd portal needed to choose one non-fixed orientation. CP, CPT, \(\mathbb Z_3\), boundary normal, index and known flavour data do not canonically fix the remaining phase circle.

Boundary. A physical charge-minus-two complex reference and a local lock to the charged carrier would reopen the problem; merely renaming an existing phase does not.

• 1395_no_declared_cp_boundary_z3_flavor_or_index_datum_fixes_the_a5_sterile_phase_reference.md

Detailed: established-boundary

The canonical completion carries index three, but its minimal 6D parent is anomalous

The selected-projector completion canonically induces O(3) on \(\mathbb{CP}^{1}\), and the twisted Dirac operator has index three for every spectator Standard-Model representation.

Boundary. The simplest same-six-dimensional-chirality local parent has irreducible gravitational, gauge and global SU(2) anomalies. A vectorlike repair returns mirror zero modes.

• 1397_the_completion_cp1_canonically_inherits_o3_and_index_three_but_no_current_action_makes_it_the_fermion_family_dirac_space.md
• 1450_the_o3_twisted_cp1_operator_has_exactly_three_universal_chiral_zero_modes_but_its_minimal_local_six_dimensional_parent_is_anomalous.md

Detailed: internally-exhausted

The traceless spectral threshold survives stress improvement

Within the displayed scalar improvement family, the traceless stress vertex and its pair-production threshold are independent of the curvature coupling xi.

Boundary. The full trace-mixed third response direction is not universal and can be tuned away. Curved microscopic matching and an operational port definition are still needed.

• 1393_the_full_scalar_stress_preserves_a_traceless_holonomy_threshold_but_not_a_scheme_invariant_third_matrix_direction.md
• 1396_no_current_principle_predicts_the_reader_curvature_coupling_so_only_the_traceless_threshold_is_scheme_robust.md

Detailed: simulation-ready

A calibrated pole-and-band experiment can be specified

The A11 packet includes finite-volume, disorder, sector-confusion, calibration and statistical-power gates. Its synthetic fixture has an effective resolution floor near 0.630 MHz and resolves an injected 1 MHz shift.

Boundary. Sensitivity is not a predicted signal. The programme still lacks a microscopic sector shift, spectral residue and physical port operator.

• 1452_the_scheme_invariant_a11_experiment_is_a_calibrated_spectral_pole_and_band_contrast_with_a_device_prediction_gate.md

Detailed: established-no-go

The softened QWZ curve is not the response of its stated Lindbladian

The published scalar-broadening quadrature and doubled cusp reproduce, but the stated unital L=\(\sigma_z\) dynamics generically drive each momentum sector to I/2. Exact stationary Kubo forcing therefore vanishes.

Boundary. The curve remains a phenomenological template. A non-unital bath, reservoir or finite-time preparation would define a different model and must be solved independently.

• 1454_the_zenodo_chern_quadrature_reproduces_but_is_not_the_exact_sigma_z_lindblad_hall_response.md

Detailed: corrected-conditional

The two-branch flow has an exact open-bosonic realization

After replacing the incorrect A-bar-B invariant by AB and common-phase symmetry by opposite-phase U(1), the two-branch flow closes on an SO(2,1) cone and is reproduced exactly by a two-mode squeezing Hamiltonian with completely positive loss.

Boundary. Its Lie closure is \(\mathfrak{sp}(2,\mathbb R)\), not \(\mathfrak{sp}(4,\mathbb R)\), and the octonionic bracket-to-bosonic-operator map remains assumed.

• 1455_the_corrected_octonionic_two_branch_flow_has_an_exact_cp_lindblad_completion_and_a_three_coordinate_lorentz_cone_but_not_full_spin23.md

Detailed: terminal-current-input

Seven gates now have explicit stopping and reopening rules

A5 and A7 are internally exhausted; A2 and A9 are conditionally constructed; A4, A6 and A11 are externally open. No whole gate is globally failed.

Boundary. Physical admission remains false. Exhaustion means that recombining the same inputs is no longer frontier work, not that every possible completion is impossible.

• 1453_genuine_exhaustion_leaves_no_untyped_internal_work_but_seven_named_external_or_conditional_frontiers.md
• 1456_the_two_zenodo_preprints_strengthen_response_controls_but_do_not_change_the_terminal_gate_partition.md

Detailed: established-boundary

The local gravity route has one healthy critical torsion sector but no closed quantum parent

The reopened mass-gap and action programme isolates a healthy classical/EFT zero-minus torsion sector and a complete quadratic shell, while the local quantum flow leaves the critical surface through a Weyl-square term.

Boundary. Positive enlarged completions remain possible, but they import new functional, metric, vacuum or matching data and do not select the axial portal.

• 1624_the_reopened_mass_gap_action_and_shared_scale_programme_reaches_five_irreducible_gates.md

Detailed: terminal-current-input

Every finite connection-parent escape now ends at new model construction

Ordinary Lorentz super-Yang--Mills is kinetically indefinite; self-duality requires complex reality constraints; nonlinear Lorentz realization gives a massive rotation-vector triplet rather than the target pseudoscalar.

Boundary. This is operational exhaustion under current inputs, not a theorem against future models. Reopening requires a complete action, vacuum, constraint solution, positive spectrum and normalized fermion matching.

• 1788_all_finite_escapes_from_the_minimal_connection_parent_now_end_at_new_model_construction.md

Detailed: established-in-model

The mixed admissibility interface is an A8 transfer lattice

For the declared 16-node, 27-relation quotient-scar graph, the constructive and guardrail incidence images intersect on the zero-sum integral lattice of a nine-node mixed core, hence A8 with determinant nine.

Boundary. The lattice is a corpus-derived transfer coordinate. It is not a physical A8 symmetry, octonion carrier, Leech lattice or exceptional-algebra identification; laboratory meaning requires an explicit cross-carrier map.

• frontier_wave7_39_mixed_interface_resolution.md
• frontier_wave7_44_refinement_stability_of_the_mixed_interface.md

Detailed: internally-exhausted

The pinned Claimipedia operator layers contain sl2 plus sl3, not sp4

The full relationship-path algebra contributes sl2, the reciprocal programme core contributes exact A2/sl3, and the evidence and provenance layers contribute no further semisimple Lie factor.

Boundary. This is an exhaustive result for the fixed captured schema, not a physical symmetry claim. Reopening requires a new evidence-bound cross-operator on one shared carrier satisfying the C2 Cartan, Serre and nondegenerate symplectic-form tests.

• frontier_wave7_39_programme_a2_double_centralizer_and_c2_no_go.md
• frontier_wave7_40_claimipedia_operator_exhaustion_and_sp4_gate.md

Detailed: constructed-conditional

Causal hydrodynamic relaxation admits positive stationary Gaussian fluctuations

The AZ-103 subluminal relaxation wedge admits positive stationary covariances and positive-semidefinite auxiliary noise satisfying the Lyapunov fluctuation–dissipation identity and explicit Onsager–Casimir reciprocity across the declared momentum scan.

Boundary. This is an effective classical Gaussian construction, not a unique physical noise model or a proof of exact quantum-frequency or nonlinear dynamical KMS. Stress Ward identities, contacts, physical normalization and microscopic matching remain open.

• admissibility_103_causal_stable_hydrodynamic_relaxation_wedge.md
• admissibility_104_relaxation_auxiliary_gaussian_kms_completion.md