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18: The QICLean contributors. Complete-positivity scope of the peripheral cyclicity theorem. Paper-gap note wolf_thm6_6_kraus_scope, QICLean, 2026. https://sirui-lu.com/QICLean/paper-gaps/wolf_thm6_6_kraus_scope.pdf.
19: The QICLean contributors. Normalization of the tensor in CPSV16 example 4.12. Paper-gap note cpsv16_example_4_12_normalization, QICLean, 2026. https://sirui-lu.com/QICLean/paper-gaps/cpsv16_example_4_12_normalization.pdf.
20: The QICLean contributors. The mutual-information bound for matrix product density operators. Paper-gap note cpgsv17_mpdo_mutual_information_bound, QICLean, 2026. https://sirui-lu.com/QICLean/paper-gaps/cpgsv17_mpdo_mutual_information_bound.pdf.
21: The QICLean contributors. The nonzero-coefficient convention for CPSV16 canonical forms. Paper-gap note cpsv16_bnt_uniqueness_zero_coefficient, QICLean, 2026. https://sirui-lu.com/QICLean/paper-gaps/cpsv16_bnt_uniqueness_zero_coefficient.pdf.
22: The QICLean contributors. The structure theorem for finite-dimensional \(c^\ast \)-algebras: matrix-realized single-block scope. Paper-gap note sw04_csform_matrix_scope, QICLean, 2026. https://sirui-lu.com/QICLean/paper-gaps/sw04_csform_matrix_scope.pdf.
23: The QICLean contributors. The CPSV16 unit-weight convention and renormalization-fixed-point scale. Paper-gap note cpsv16_unit_weight_rfp_scale_tension, QICLean, 2026. https://sirui-lu.com/QICLean/paper-gaps/cpsv16_unit_weight_rfp_scale_tension.pdf.
24: The QICLean contributors. The Perron–Frobenius rank-one step in Cirac–Perez-Garcia–Schuch–Verstraete 2017 appendix C.2. Paper-gap note cpgsv17_pf_rank_one, QICLean, 2026. https://sirui-lu.com/QICLean/paper-gaps/cpgsv17_pf_rank_one.pdf.
25: The QICLean contributors. Zero correlation length uses the physical-trace transfer in Cirac–Perez-Garcia–Schuch–Verstraete 2017. Paper-gap note cpsv16_zcl_canonical_form_normalization, QICLean, 2026. https://sirui-lu.com/QICLean/paper-gaps/cpsv16_zcl_canonical_form_normalization.pdf.
26: The TNLean contributors. A corrected approximating state for a tensor with repeated blocks. Paper-gap note mswc24_repeated_block_corrected_state, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_repeated_block_corrected_state.pdf.
27: The TNLean contributors. A gapless short-range parent Hamiltonian of a canonical MPS. Paper-gap note cpgsv21_short_range_parent_gap, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv21_short_range_parent_gap.pdf.
28: The TNLean contributors. A mixed-prefix repair of the two-site sector comparison. Paper-gap note cpsv16_two_site_sector_unitary_gauge_gap, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_two_site_sector_unitary_gauge_gap.pdf.
29: The TNLean contributors. A two-site counterexample to cyclic MPS bond uniformity. Paper-gap note peps_cyclic_mps_two_site_bond_uniformity, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/peps_cyclic_mps_two_site_bond_uniformity.pdf.
30: The TNLean contributors. Active physical support compression in the CPSV16 commuting-form argument. Paper-gap note cpsv16_active_physical_support_compression, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_active_physical_support_compression.pdf.
31: The TNLean contributors. Adjacent regions obstruct the unrestricted CID forward implication in Cirac–Perez-Garcia–Schuch–Verstraete 2017. Paper-gap note cpsv16_pure_zcl_adjacent_gap_cid_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_pure_zcl_adjacent_gap_cid_scope.pdf.
32: The TNLean contributors. Asymptotic local channel conversion without classical feedforward. Paper-gap note psc21_local_channel_phase_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/psc21_local_channel_phase_scope.pdf.
33: The TNLean contributors. Basis-of-representatives scope restriction in the fundamental theorem formalization. Paper-gap note cpsv16_ft_one_copy_scope_restriction, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_ft_one_copy_scope_restriction.pdf.
34: The TNLean contributors. Blocking and the parent-hamiltonian range claim in CPSV16. Paper-gap note cpsv16_parent_hamiltonian_range_short_ring, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_parent_hamiltonian_range_short_ring.pdf.
35: The TNLean contributors. Bond dimensions of the successive decompositions: upper bounds only. Paper-gap note ssvcw05_sequential_bond_upper_bound_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/ssvcw05_sequential_bond_upper_bound_scope.pdf.
36: The TNLean contributors. Czx defect domains from the source diagrams. Paper-gap note fbc25_czx_defect_domains, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_czx_defect_domains.pdf.
37: The TNLean contributors. Canonical-form weight normalization and the proportional MPV comparison. Paper-gap note cpsv16_cf_normalization_and_proportional_comparison, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_cf_normalization_and_proportional_comparison.pdf.
38: The TNLean contributors. Circle and nonzero-complex coefficients in degree-two cohomology. Paper-gap note fbc25_circle_complex_units_cohomology, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_circle_complex_units_cohomology.pdf.
39: The TNLean contributors. Circuits between states in the same phase: the normal case. Paper-gap note mswc24_same_phase_circuit_normal_case, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_same_phase_circuit_normal_case.pdf.
40: The TNLean contributors. Common identity coefficients in the block ground-space intersection. Paper-gap note pgvwc07_common_identity_coefficients, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgvwc07_common_identity_coefficients.pdf.
41: The TNLean contributors. Commuting parent hamiltonians of \(G\)-isometric tensors: the projector and the scope. Paper-gap note scp10_g_isometric_commuting_parent_hamiltonian, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/scp10_g_isometric_commuting_parent_hamiltonian.pdf.
42: The TNLean contributors. Compatibility of czx fusion and action coordinates. Paper-gap note fbc25_czx_fusion_coordinate_compatibility, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_czx_fusion_coordinate_compatibility.pdf.
43: The TNLean contributors. Critical parameters and boundary localization in the one-species pvbs chain. Paper-gap note cpgsv21_pvbs_threshold, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv21_pvbs_threshold.pdf.
44: The TNLean contributors. Decaying correlations: windows of separations and the connected correlator. Paper-gap note mswc24_decaying_correlations_windowed_connected, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_decaying_correlations_windowed_connected.pdf.
45: The TNLean contributors. Defect covariance in state-level gauging. Paper-gap note fbc25_state_level_gauging_covariance, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_state_level_gauging_covariance.pdf.
46: The TNLean contributors. Deferred result: the translation-invariant bond-dimension lower bound for the W state. Paper-gap note rmp_w_state_ti_bound, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_w_state_ti_bound.pdf.
47: The TNLean contributors. Domain walls: nonzero tensors, phases and blocked local actions. Paper-gap note gs24_domain_wall_nondegenerate, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/gs24_domain_wall_nondegenerate.pdf.
48: The TNLean contributors. Endpoint-coordinate scope of truncated-symmetry unitarity. Paper-gap note fbc25_truncated_symmetry_endpoint_unitarity_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_truncated_symmetry_endpoint_unitarity_scope.pdf.
49: The TNLean contributors. Exact preparation at a zero subleading transfer spectrum. Paper-gap note mswc24_zero_subleading_spectrum_endpoint, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_zero_subleading_spectrum_endpoint.pdf.
50: The TNLean contributors. Example parent hamiltonians in cirac–pérez-garcía–schuch–verstraete 2021. Paper-gap note rmp_example_parent_hamiltonian_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_example_parent_hamiltonian_scope.pdf.
51: The TNLean contributors. Exchange of domain-wall endpoints on glued half-chains. Paper-gap note gs24_domain_wall_half_string_gluing, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/gs24_domain_wall_half_string_gluing.pdf.
52: The TNLean contributors. Fibonacci string-net: entries of the two operator blocks. Paper-gap note bmwshv17_fibonacci_block_entries_provenance, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/bmwshv17_fibonacci_block_entries_provenance.pdf.
53: The TNLean contributors. Fibonacci string-net: the projector identity at positive length. Paper-gap note bmwshv17_fibonacci_projector_positive_length, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/bmwshv17_fibonacci_projector_positive_length.pdf.
54: The TNLean contributors. Finite adaptive local channels and source protocol blocks. Paper-gap note psc21_adaptive_channel_round_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/psc21_adaptive_channel_round_scope.pdf.
55: The TNLean contributors. Finite-group commutants: the unital qualification. Paper-gap note scp10_finite_group_commutant_unital, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/scp10_finite_group_commutant_unital.pdf.
56: The TNLean contributors. Finite-range Knabe coefficients for cyclic and open chains. Paper-gap note knabe88_finite_range_coefficient, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/knabe88_finite_range_coefficient.pdf.
57: The TNLean contributors. Fixed virtual bond dimension in mpu tensor equivalence. Paper-gap note mpu_equivalence_fixed_bond, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mpu_equivalence_fixed_bond.pdf.
58: The TNLean contributors. Full support in MPU normality and theorem III.8. Paper-gap note mpu_canonical_form_full_support, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mpu_canonical_form_full_support.pdf.
59: The TNLean contributors. Full-matrix scope for the GNVW support-algebra formalization. Paper-gap note gnvw12_support_algebra_full_matrix_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/gnvw12_support_algebra_full_matrix_scope.pdf.
60: The TNLean contributors. Global phase freedom in the PGVWC07 quadratic reconstruction system. Paper-gap note pgvwc07_quadratic_reconstruction_phase, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgvwc07_quadratic_reconstruction_phase.pdf.
61: The TNLean contributors. Ground-space subtraction in the infinite-volume gap argument. Paper-gap note nachtergaele96_infinite_volume_ground_projection, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/nachtergaele96_infinite_volume_ground_projection.pdf.
62: The TNLean contributors. Guide properties in the angular separation of dyadic repaintings. Paper-gap note openai26_dyadic_geometry_guide_properties, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/openai26_dyadic_geometry_guide_properties.pdf.
63: The TNLean contributors. Half-chain spectrum: canonical orientation, normalization, and decay. Paper-gap note pgvwc07_half_chain_spectrum_conventions, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgvwc07_half_chain_spectrum_conventions.pdf.
64: The TNLean contributors. Homogeneous-chain scope for the GNVW support algebras, adjacent generation, and index. Paper-gap note gnvw12_site_dependent_adjacent_generation_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/gnvw12_site_dependent_adjacent_generation_scope.pdf.
65: The TNLean contributors. Identity-ancilla transport of mpu symmetries. Paper-gap note mpu_symmetry_ancilla_transport, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mpu_symmetry_ancilla_transport.pdf.
66: The TNLean contributors. Inhomogeneous matrix product states: block length. Paper-gap note mswc24_inhomogeneous_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_inhomogeneous_scope.pdf.
67: The TNLean contributors. Injective PEPS fundamental theorem: section 3 route. Paper-gap note peps_injective_ft_section3_route, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/peps_injective_ft_section3_route.pdf.
68: The TNLean contributors. Injective PEPS parent interactions: the physical reconstruction condition. Paper-gap note cpgsv21_injective_parent_reconstruction, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv21_injective_parent_reconstruction.pdf.
69: The TNLean contributors. Ising string-net operators: the normalization of the σ tensor. Paper-gap note bmwshv17_ising_boundary_tensor_normalization, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/bmwshv17_ising_boundary_tensor_normalization.pdf.
70: The TNLean contributors. Kitaev chain: the bosonic contraction of the fixed-point matrices. Paper-gap note rmp_kitaev_chain_bosonic_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_kitaev_chain_bosonic_scope.pdf.
71: The TNLean contributors. Klein three-cocycles: circle and nonzero-complex coefficients. Paper-gap note glm23_klein_h3_circle_coefficients, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_klein_h3_circle_coefficients.pdf.
72: The TNLean contributors. Long-range matrix product states with measurements: scope of the formal statement. Paper-gap note mswc24_measurement_preparation_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_measurement_preparation_scope.pdf.
73: The TNLean contributors. Matrix product operator symmetric states: the periodic-boundary layer. Paper-gap note glm23_mpo_symmetric_mps_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_mpo_symmetric_mps_scope.pdf.
74: The TNLean contributors. Nilpotency length one after blocking: a terminology offset. Paper-gap note mgsc18_nilpotency_length_one_terminology, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mgsc18_nilpotency_length_one_terminology.pdf.
75: The TNLean contributors. Nondegeneracy in the alternating-network proportionality lemma. Paper-gap note fbc25_alternating_network_nondegeneracy, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_alternating_network_nondegeneracy.pdf.
76: The TNLean contributors. Nondegenerate rectangles and triangles in area-law templates. Paper-gap note arealaw2d_nondegenerate_templates, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/arealaw2d_nondegenerate_templates.pdf.
77: The TNLean contributors. Nonzero weights in the MPU canonical form. Paper-gap note mpu_canonical_form_nonzero_weights, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mpu_canonical_form_nonzero_weights.pdf.
78: The TNLean contributors. Normal PEPS fundamental theorem: section 3 route. Paper-gap note peps_normal_ft_section3_route, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/peps_normal_ft_section3_route.pdf.
79: The TNLean contributors. One left shift per site in the periodic group operators. Paper-gap note glm23_pbc_group_mpo_single_shift, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_pbc_group_mpo_single_shift.pdf.
80: The TNLean contributors. One-dimensional scope of Schumacher–Werner support-algebra covariance. Paper-gap note sw04_support_covariance_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/sw04_support_covariance_scope.pdf.
81: The TNLean contributors. One-sided swap transformation of mpu symmetries. Paper-gap note mpu_swap_symmetry_one_sided, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mpu_swap_symmetry_one_sided.pdf.
82: The TNLean contributors. Outer-factor basis correction in the support-algebra commutation lemma. Paper-gap note gnvw12_support_algebra_outer_factor_basis, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/gnvw12_support_algebra_outer_factor_basis.pdf.
83: The TNLean contributors. Party-network bounds in the dyadic routing lemma. Paper-gap note polypeps_routing_network_bounds, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/polypeps_routing_network_bounds.pdf.
84: The TNLean contributors. Per-pair support in the CPSV16 figure 11 decomposition. Paper-gap note cpsv16_figure11_per_pair_support, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_figure11_per_pair_support.pdf.
85: The TNLean contributors. Per-block RFP isometry form and the source isometry condition in CPSV16. Paper-gap note cpsv16_rfp_isometry_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_rfp_isometry_scope.pdf.
86: The TNLean contributors. Peripheral roots after blocking a periodic tensor. Paper-gap note dccsp17_blocking_peripheral_roots, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/dccsp17_blocking_peripheral_roots.pdf.
87: The TNLean contributors. Physical bond separation and the remaining regular PEPS blocking step. Paper-gap note scp10_general_group_physical_blocking, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/scp10_general_group_physical_blocking.pdf.
88: The TNLean contributors. Physical-port simulation of bounded register channels. Paper-gap note psc21_physical_port_simulation_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/psc21_physical_port_simulation_scope.pdf.
89: The TNLean contributors. Polynomial approximation accuracy at positive correlation length. Paper-gap note mswc24_polynomial_accuracy_uniform_blocks, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_polynomial_accuracy_uniform_blocks.pdf.
90: The TNLean contributors. Preparation in logarithmic depth: chain lengths divisible by the block length. Paper-gap note mswc24_depth_upper_bound_divisible_length, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_depth_upper_bound_divisible_length.pdf.
91: The TNLean contributors. Preparation in logarithmic depth: the periodic state must not vanish. Paper-gap note mswc24_depth_upper_bound_nonzero_state, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_depth_upper_bound_nonzero_state.pdf.
92: The TNLean contributors. Product-reference estimates and a column on the open square. Paper-gap note polypeps_product_reference_column, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/polypeps_product_reference_column.pdf.
93: The TNLean contributors. Proof-route alignment for the periodic overlap dichotomy (de las Cuevas–Cirac–Schuch–Pérez-García). Paper-gap note dccsp17_periodic_overlap_route_alignment, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/dccsp17_periodic_overlap_route_alignment.pdf.
94: The TNLean contributors. Pure-exponential correlator expansion and Jordan blocks in CPGSV21. Paper-gap note cpgsv21_correlator_diagonalizable_expansion, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv21_correlator_diagonalizable_expansion.pdf.
95: The TNLean contributors. Quantum-double hamiltonian terms: normalization and local scope. Paper-gap note scp10_quantum_double_local_hamiltonian, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/scp10_quantum_double_local_hamiltonian.pdf.
96: The TNLean contributors. Quantum-double PEPS: normalization, representations and scope. Paper-gap note rmp_peps_quantum_double_g_isometry, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_peps_quantum_double_g_isometry.pdf.
97: The TNLean contributors. Reciprocal factors in the block-independence argument. Paper-gap note fbc25_block_independence_reciprocal_errors, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_block_independence_reciprocal_errors.pdf.
98: The TNLean contributors. Regular PEPS blocking with native closures and coherent sectors. Paper-gap note scp10_twisted_two_by_two_regular_fixed_point, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/scp10_twisted_two_by_two_regular_fixed_point.pdf.
99: The TNLean contributors. Repeated overlapping blocks with a small weight sum. Paper-gap note mswc24_repeated_overlap_small_weight_sum, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_repeated_overlap_small_weight_sum.pdf.
100: The TNLean contributors. Scalar phases in the physical string-order criterion. Paper-gap note pgwsvc08_string_order_scalar_phase, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgwsvc08_string_order_scalar_phase.pdf.
101: The TNLean contributors. Schuch–pérez-garcía–cirac SPT interpolation: upper-range correction. Paper-gap note spc11_spt_interpolation_upper_range, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/spc11_spt_interpolation_upper_range.pdf.
102: The TNLean contributors. Scope of the formal block-separation lemmas relative to PGVWC07 uniqueness. Paper-gap note pgvwc07_ti_uniqueness_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgvwc07_ti_uniqueness_scope.pdf.
103: The TNLean contributors. Scope of the formal canonical-form reductions relative to PGVWC07. Paper-gap note pgvwc07_ti_canonical_form_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgvwc07_ti_canonical_form_scope.pdf.
104: The TNLean contributors. Scope of the physical-deformation ground-space correspondence relative to CPGSV21. Paper-gap note cpgsv21_physical_deformation_general_graph, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv21_physical_deformation_general_graph.pdf.
105: The TNLean contributors. Sequential factorization of the blocked isometry: positive block length. Paper-gap note mswc24_sequential_factorization_positive_block_length, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_sequential_factorization_positive_block_length.pdf.
106: The TNLean contributors. Sequential generation with an ancilla: nonempty chains and ancillas. Paper-gap note pgvwc07_sequential_with_ancilla_positive_length, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgvwc07_sequential_with_ancilla_positive_length.pdf.
107: The TNLean contributors. Sequential generation without an ancilla needs two sites. Paper-gap note pgvwc07_sequential_no_ancilla_two_sites, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgvwc07_sequential_no_ancilla_two_sites.pdf.
108: The TNLean contributors. Sharp finite-length block separation for canonical-form tensors. Paper-gap note cpgsv17_bicf_block_separation, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv17_bicf_block_separation.pdf.
109: The TNLean contributors. Simultaneous left inverse of injective operator blocks. Paper-gap note bmwshv17_joint_block_left_inverse, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/bmwshv17_joint_block_left_inverse.pdf.
110: The TNLean contributors. Single-block scope for the schumacher–werner commuting-algebra proposition. Paper-gap note gnvw12_cscom_single_block_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/gnvw12_cscom_single_block_scope.pdf.
111: The TNLean contributors. Source boundary for mpu action tensors and stabilizer data. Paper-gap note fbc25_mpu_action_stabilizer_source_audit, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_mpu_action_stabilizer_source_audit.pdf.
112: The TNLean contributors. Source-notation gap: the Majumdar–Ghosh example tensor. Paper-gap note rmp_majumdar_ghosh_tensor_gap, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_majumdar_ghosh_tensor_gap.pdf.
113: The TNLean contributors. Star-closure correction for the GNVW support-algebra lemma. Paper-gap note gnvw12_support_algebra_star_closure, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/gnvw12_support_algebra_star_closure.pdf.
114: The TNLean contributors. Supplied two-site decompositions of the shift matrix product unitaries. Paper-gap note mpu_shift_supplied_standard_forms, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mpu_shift_supplied_standard_forms.pdf.
115: The TNLean contributors. The anomaly class of the two-qubit \(\mathbb {Z}\_ 2\times \mathbb {Z}\_ 2\) tensors: from detector values to the class. Paper-gap note glm23_z2z2_anomaly_detector_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_z2z2_anomaly_detector_scope.pdf.
116: The TNLean contributors. The arbitrary-chain topological projector recursion. Paper-gap note cpsv16_topological_projector_recursion, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_topological_projector_recursion.pdf.
117: The TNLean contributors. The block index in the positive generalized-cocycle corollary. Paper-gap note fbc25_positive_generalized_cocycle_block_index, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_positive_generalized_cocycle_block_index.pdf.
118: The TNLean contributors. The commuting-form-to-SAL implication for simple MPDOs. Paper-gap note cpsv16_commuting_form_to_sal, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_commuting_form_to_sal.pdf.
119: The TNLean contributors. The constant in the depth lower bound. Paper-gap note mswc24_depth_lower_bound_constant, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_depth_lower_bound_constant.pdf.
120: The TNLean contributors. The depth lower bound in the gauge of a normal tensor. Paper-gap note mswc24_depth_lower_bound_gauge, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_depth_lower_bound_gauge.pdf.
121: The TNLean contributors. The direct-sum input from the older MPS representation paper. Paper-gap note pgvwc07_direct_sum_input, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgvwc07_direct_sum_input.pdf.
122: The TNLean contributors. The domain boundary of the topological Gibbs decomposition. Paper-gap note cpsv16_topological_gibbs_length_one, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_topological_gibbs_length_one.pdf.
123: The TNLean contributors. The empty interior word in the printed czx action pair. Paper-gap note fbc25_czx_action_empty_word, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_czx_action_empty_word.pdf.
124: The TNLean contributors. The empty-word summation error in the MPS residual expansion. Paper-gap note mgsc18_residual_expansion_empty_word_error, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mgsc18_residual_expansion_empty_word_error.pdf.
125: The TNLean contributors. The figure 11 fusion map is a coisometry onto the active sectors. Paper-gap note cpsv16_figure11_fusion_coisometry, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_figure11_fusion_coisometry.pdf.
126: The TNLean contributors. The final index in the block-dependent two-cochain coboundary. Paper-gap note fbc25_two_cochain_coboundary_index_typo, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_two_cochain_coboundary_index_typo.pdf.
127: The TNLean contributors. The finite periodic one-species PVBS gap. Paper-gap note bn12_pvbs_periodic_gap, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/bn12_pvbs_periodic_gap.pdf.
128: The TNLean contributors. The fixed-point state of a tensor with repeated blocks. Paper-gap note mswc24_multiplicity_fixed_point, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_multiplicity_fixed_point.pdf.
129: The TNLean contributors. The full-input pseudoinverse obstruction and its support correction. Paper-gap note mswc24_mixed_polar_injectivity_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_mixed_polar_injectivity_scope.pdf.
130: The TNLean contributors. The fusion gauge of the stabilizer reconstruction formula. Paper-gap note glm23_eq20_fusion_gauge, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_eq20_fusion_gauge.pdf.
131: The TNLean contributors. The interaction range in the block-injective parent-Hamiltonian gap. Paper-gap note cpgsv21_block_parent_interaction_range, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv21_block_parent_interaction_range.pdf.
132: The TNLean contributors. The intertwiner count in the quadratic reconstruction argument. Paper-gap note pgvwc07_intertwiner_chain_off_by_one, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgvwc07_intertwiner_chain_off_by_one.pdf.
133: The TNLean contributors. The left action vectors of the CZX product states. Paper-gap note gs24_czx_action_left_vectors, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/gs24_czx_action_left_vectors.pdf.
134: The TNLean contributors. The lower endpoint of Nachtergaele’s theorem 2.1(i). Paper-gap note nachtergaele96_theorem_2_1_lower_endpoint, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/nachtergaele96_theorem_2_1_lower_endpoint.pdf.
135: The TNLean contributors. The martingale principal-angle estimate for MPS parent Hamiltonians. Paper-gap note cpgsv21_martingale_overlap, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv21_martingale_overlap.pdf.
136: The TNLean contributors. The missing algebraic specification of the PEPS boundary-RFP diagrams. Paper-gap note cpsv16_peps_boundary_rfp_specification, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_peps_boundary_rfp_specification.pdf.
137: The TNLean contributors. The missing tilde in the inverse-compatible gate argument. Paper-gap note fbc25_inverse_compatible_tilde_omission, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_inverse_compatible_tilde_omission.pdf.
138: The TNLean contributors. The one-step subspace in the quantum Wielandt inequality. Paper-gap note spwc10_wielandt_one_step_subspace, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/spwc10_wielandt_one_step_subspace.pdf.
139: The TNLean contributors. The parent commuting hamiltonian condition in Cirac–Pérez-García–Schuch–Verstraete 2016. Paper-gap note cpsv16_parent_commuting_hamiltonian_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_parent_commuting_hamiltonian_scope.pdf.
140: The TNLean contributors. The physical complement in the topological Gibbs decomposition. Paper-gap note cpsv16_topological_gibbs_physical_complement, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_topological_gibbs_physical_complement.pdf.
141: The TNLean contributors. The polynomial count of the small-patch rewrite. Paper-gap note polypeps_small_rewrite_monomials, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/polypeps_small_rewrite_monomials.pdf.
142: The TNLean contributors. The positive part of a blocked tensor with overlapping blocks. Paper-gap note mswc24_block_form_mixed_overlap, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_block_form_mixed_overlap.pdf.
143: The TNLean contributors. The positive primitive in the generalized-cocycle corollary. Paper-gap note fbc25_positive_generalized_cocycle_determinant_sign, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_positive_generalized_cocycle_determinant_sign.pdf.
144: The TNLean contributors. The positive-gap boundary for distance- and length-independent MPDO correlations. Paper-gap note cpsv16_mpdo_zcl_correlation_length_boundary, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_mpdo_zcl_correlation_length_boundary.pdf.
145: The TNLean contributors. The printed one-qubit \(\mathbb {Z}\_ 2\times \mathbb {Z}\_ 2\) symmetry: the anomaly sign at \(ab\). Paper-gap note glm23_klein_printed_anomaly_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_klein_printed_anomaly_scope.pdf.
146: The TNLean contributors. The proportionality scalar in the MPS reduction theorem. Paper-gap note mgsc18_reduction_proportionality_scalar, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mgsc18_reduction_proportionality_scalar.pdf.
147: The TNLean contributors. The repeated case in the scalar three-cocycle normalization. Paper-gap note fbc25_three_cocycle_normalization_typo, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_three_cocycle_normalization_typo.pdf.
148: The TNLean contributors. The repeated determinant factor in the generalized-cocycle lemma. Paper-gap note fbc25_generalized_cocycle_determinant_typo, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_generalized_cocycle_determinant_typo.pdf.
149: The TNLean contributors. The representative index in the first stabilizer factor. Paper-gap note fbc25_stabilizer_transition_index_typo, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/fbc25_stabilizer_transition_index_typo.pdf.
150: The TNLean contributors. The root-channel kraus-rank step in theorem 4.1. Paper-gap note dccsp17_root_kraus_rank_thm41, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/dccsp17_root_kraus_rank_thm41.pdf.
151: The TNLean contributors. The sign of \(\sigma \_ y\) in the Pauli form of the AKLT tensor. Paper-gap note rmp_aklt_pauli_basis_sign, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_aklt_pauli_basis_sign.pdf.
152: The TNLean contributors. The three-object ising twist: channels and formal scope. Paper-gap note tnlean_ising_three_object_twist_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/tnlean_ising_three_object_twist_scope.pdf.
153: The TNLean contributors. The trace-ratio exponent in the blocked vertical-operator formula. Paper-gap note cpsv16_blocked_operator_trace_ratio_exponent, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_blocked_operator_trace_ratio_exponent.pdf.
154: The TNLean contributors. The tree circuit as a multiscale network: two-site injectivity and chain length. Paper-gap note mswc24_tree_mera_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_tree_mera_scope.pdf.
155: The TNLean contributors. The tree circuit with measurements: rounds, registers, and chain length. Paper-gap note mswc24_tree_measurement_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_tree_measurement_scope.pdf.
156: The TNLean contributors. The truncated-symmetry detector depends on the endpoints. Paper-gap note gs24_truncation_detector_endpoint_dependence, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/gs24_truncation_detector_endpoint_dependence.pdf.
157: The TNLean contributors. The unblocked parity reading of the fundamental theorem of matrix product unitaries. Paper-gap note mpu_standard_form_parity_gap, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mpu_standard_form_parity_gap.pdf.
158: The TNLean contributors. The unbroken subgroup is the stabilizer of a block. Paper-gap note gs24_unbroken_subgroup_stabilizer, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/gs24_unbroken_subgroup_stabilizer.pdf.
159: The TNLean contributors. The vertical canonical form retains only the nonzero physical sectors. Paper-gap note cpgsv17_vertical_isometry_zero_sector, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv17_vertical_isometry_zero_sector.pdf.
160: The TNLean contributors. The virtual unitary in the symmetric isometric deformation. Paper-gap note spc11_isometric_symmetry_unitary_virtual, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/spc11_isometric_symmetry_unitary_virtual.pdf.
161: The TNLean contributors. The \(\mathbb Z\_ 3\) phase of \(\mathrm{Rep}(S\_ 3)\): the printed action of ψ. Paper-gap note glm23_reps3_z3_table, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_reps3_z3_table.pdf.
162: The TNLean contributors. The \(\mathrm{Rep}(S\_ 3)\) and \(su(2)\_ 4\) phases: completeness of the lists. Paper-gap note glm23_reps3_su24_module_list, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_reps3_su24_module_list.pdf.
163: The TNLean contributors. The CZX tensor: orientation of the bond labels. Paper-gap note rmp_peps_czx_bond_orientation, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_peps_czx_bond_orientation.pdf.
164: The TNLean contributors. The CZX PEPS: actual rectangle boundaries and their symmetry. Paper-gap note rmp_peps_czx_boundary_chain, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_peps_czx_boundary_chain.pdf.
165: The TNLean contributors. The Fibonacci module: direct-sum classification. Paper-gap note glm23_fibonacci_module_rank_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_fibonacci_module_rank_scope.pdf.
166: The TNLean contributors. The L-symbol relation imposed by time reversal. Paper-gap note glm23_time_reversal_l_symbol, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_time_reversal_l_symbol.pdf.
167: The TNLean contributors. The NNCPH ground-state clause in Cirac–Pérez-García–Schuch–Verstraete 2016. Paper-gap note cpsv16_nncph_ground_state_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_nncph_ground_state_scope.pdf.
168: The TNLean contributors. The RVB tensor: the singlet matrix on the vacuum. Paper-gap note rmp_peps_rvb_bond_vacuum, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_peps_rvb_bond_vacuum.pdf.
169: The TNLean contributors. The SAL to eta-local structure step in Cirac–Perez-Garcia–Schuch–Verstraete 2017 appendix C.2. Paper-gap note cpgsv17_mpdo_sal_zcl_eta_local_structure, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv17_mpdo_sal_zcl_eta_local_structure.pdf.
170: The TNLean contributors. The SAL–ZCL blocking-channel conclusion of CPSV16 proposition C.15 is false. Paper-gap note cpsv16_prop_c15_blocking_channel_false, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpsv16_prop_c15_blocking_channel_false.pdf.
171: The TNLean contributors. Trace preservation in the forward direction of theorem 4.1 (de las Cuevas–Cirac–Schuch–Pérez-García). Paper-gap note dccsp17_thm41_forward_trace_preservation, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/dccsp17_thm41_forward_trace_preservation.pdf.
172: The TNLean contributors. Translation-invariant MPS and the trivial phase under circuits with measurements: one direction. Paper-gap note psc21_mps_classification_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/psc21_mps_classification_scope.pdf.
173: The TNLean contributors. Two-dimensional PEPS examples: tori of width two. Paper-gap note rmp_peps_examples_small_torus, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_peps_examples_small_torus.pdf.
174: The TNLean contributors. Two-by-two regular PEPS renormalization with the original tensor. Paper-gap note scp10_two_by_two_regular_fixed_point, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/scp10_two_by_two_regular_fixed_point.pdf.
175: The TNLean contributors. Typed multiplicity indices for the coupled pentagon. Paper-gap note glm23_multiplicity_l_indices, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_multiplicity_l_indices.pdf.
176: The TNLean contributors. Uniform parent-hamiltonian gaps along injective paths only. Paper-gap note spc11_uniform_gap_injective_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/spc11_uniform_gap_injective_scope.pdf.
177: The TNLean contributors. Vertical canonical-form grouping in proposition 4.13. Paper-gap note cpgsv17_vertical_cf_grouping, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/cpgsv17_vertical_cf_grouping.pdf.
178: The TNLean contributors. Weak-hopf averaging and the canonical parent complement. Paper-gap note glm23_wha_parent_completion, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/glm23_wha_parent_completion.pdf.
179: The TNLean contributors. Wraparound in the orbit-phase telescoping product. Paper-gap note dccsp17_orbit_phase_wraparound, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/dccsp17_orbit_phase_wraparound.pdf.
180: The TNLean contributors. CZX: normalization and the physical Z gates. Paper-gap note mpu_czx_tensor_normalization, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mpu_czx_tensor_normalization.pdf.
181: The TNLean contributors. CZY fusion: associativity on the physical image. Paper-gap note, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/lmsvkl25_czy_associator.pdf.
182: The TNLean contributors. GHZ form of the fixed-point state: local orthogonality as a hypothesis. Paper-gap note mswc24_ghz_form_local_orthogonality, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_ghz_form_local_orthogonality.pdf.
183: The TNLean contributors. PGWSVC08 physical string order versus virtual-boundary nondecay. Paper-gap note pgwsvc08_string_order_virtual_boundary, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/pgwsvc08_string_order_virtual_boundary.pdf.
184: The TNLean contributors. SPT fixed points from a supplied projective representation. Paper-gap note rmp_spt_fixed_point_supplied_representation, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_spt_fixed_point_supplied_representation.pdf.
185: The TNLean contributors. SPT fixed points with time reversal and reflection. Paper-gap note rmp_spt_fixed_point_onsite_scope, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_spt_fixed_point_onsite_scope.pdf.
186: The TNLean contributors. SPT fixed points with a nontrivial character. Paper-gap note rmp_spt_fixed_point_trivial_character, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_spt_fixed_point_trivial_character.pdf.
187: The TNLean contributors. SPT fixed points: the printed tensor is not normal. Paper-gap note rmp_spt_fixed_point_tensor, TNLean, 2026. https://sirui-lu.com/TNLean/paper-gaps/rmp_spt_fixed_point_tensor.pdf.
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189: TNLean contributors. Compatible coherent seeds and non-normal state conversion. Technical Report, TNLean project, 2026. https://sirui-lu.com/TNLean/paper-gaps/mswc24_same_phase_compatible_seed.pdf.
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