Original question and givens
Use the complete finite simple undirected graph on Fin(13)={0,…,12}, with canonical sorted edges=[[0, 1], [0, 12], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).
G001 — Use the complete finite simple undirected graph on Fin(13)={0,…,12}, with canonical sorted edges=[[0, 1], [0, 12], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).
Step-by-step derivation
S001 · implies
Before
source operation=euler-trail-audit;original snapshot and mathematics retained
After
explicit ordered source output fields=[vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary]
Conditions: none
Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
complete ORIGINAL carrier Fin(13);canonical ORIGINAL simple undirected edges=[[0, 1], [0, 12], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12]]
After
edge indices are EXACT original list positions0..12;retain ALL vertices;no12-vertex or10-edge bound
Conditions: none
Keep edge indices at their original numeric positions,not string-sorted vertex pairs.This family permits up to60 vertices rather than importing earlier bounds;empty-edge graphs appear only where the operation allows them.
Step identifiers and dependencies
Rule: euler_original_simple_graph_indices_full_carrier · Uses: G001, S001
S003 · implies
Before
original walk step=0;consecutive vertices=[0, 1];undirected candidate edge=[0, 1]
After
matching original edge index=0;recorded edge prefix=[0];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S002
S004 · implies
Before
original walk step=1;consecutive vertices=[1, 2];undirected candidate edge=[1, 2]
After
matching original edge index=2;recorded edge prefix=[0, 2];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S003
S005 · implies
Before
original walk step=2;consecutive vertices=[2, 3];undirected candidate edge=[2, 3]
After
matching original edge index=3;recorded edge prefix=[0, 2, 3];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S004
S006 · implies
Before
original walk step=3;consecutive vertices=[3, 4];undirected candidate edge=[3, 4]
After
matching original edge index=4;recorded edge prefix=[0, 2, 3, 4];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S005
S007 · implies
Before
original walk step=4;consecutive vertices=[4, 5];undirected candidate edge=[4, 5]
After
matching original edge index=5;recorded edge prefix=[0, 2, 3, 4, 5];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S006
S008 · implies
Before
original walk step=5;consecutive vertices=[5, 6];undirected candidate edge=[5, 6]
After
matching original edge index=6;recorded edge prefix=[0, 2, 3, 4, 5, 6];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S007
S009 · implies
Before
original walk step=6;consecutive vertices=[6, 7];undirected candidate edge=[6, 7]
After
matching original edge index=7;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S008
S010 · implies
Before
original walk step=7;consecutive vertices=[7, 8];undirected candidate edge=[7, 8]
After
matching original edge index=8;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S009
S011 · implies
Before
original walk step=8;consecutive vertices=[8, 9];undirected candidate edge=[8, 9]
After
matching original edge index=9;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S010
S012 · implies
Before
original walk step=9;consecutive vertices=[9, 10];undirected candidate edge=[9, 10]
After
matching original edge index=10;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S011
S013 · implies
Before
original walk step=10;consecutive vertices=[10, 11];undirected candidate edge=[10, 11]
After
matching original edge index=11;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S012
S014 · implies
Before
original walk step=11;consecutive vertices=[11, 12];undirected candidate edge=[11, 12]
After
matching original edge index=12;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S013
S015 · implies
Before
original walk step=12;consecutive vertices=[12, 0];undirected candidate edge=[0, 12]
After
matching original edge index=1;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1];length-and-adjacency status=Yes
Conditions: none
Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity.
Step identifiers and dependencies
Rule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S014
S016 · implies
Before
original complete walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0];required vertex-list length=14;actual length=14;found ORIGINAL edge indices=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1]
After
length-and-all-adjacency=Yes;recorded indices unique=Yes;recorded complete single-use coverage=Yes;vertex revisited=Yes;VALID Euler trail=Yes
Conditions: none
An Euler trail may revisit vertices but not edges.Final validity conjoins length/all-adjacency with complete single-use edge coverage.Extra missing steps can leave a recorded list that covers every edge while invalidating the walk.One bad supplied walk does not disprove existence of another Euler walk.
Step identifiers and dependencies
Rule: euler_trail_full_conjunction_repeated_vertices_permitted · Uses: G001, S015
Result
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.