# Question 8422: Qualified Tree Nonedge and Its Unique Cycle Witness [Return to catalogue page 43](https://ocme.evemisslab.com/en/questions/page/43/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261017-r06-tree-foundations/#q-20261012-advance-20261017-r06-tree-0097) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0015-08422 · q-20261012-advance-20261017-r06-tree-0097 ## Original question and givens Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint. - G001 — Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-tree](https://ocme.evemisslab.com/en/mko/mko-tree/) ## Step-by-step derivation S001 · impliesBefore n=6;raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]] After canonical edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]] Conditions: none Validate every simple undirected edge for typed distinct endpoints.Sort endpoints and count reversed or repeated edges once;do not infer adjacency from a drawing. Step identifiers and dependenciesRule: simple_graph_edge_normalization · Uses: G001 S002 · impliesBefore phase=original;vertex=0;all edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]] After neighbors=[2, 4, 5];degree=3 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S001 S003 · impliesBefore phase=original;vertex=1;all edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]] After neighbors=[2];degree=1 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S002 S004 · impliesBefore phase=original;vertex=2;all edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]] After neighbors=[0, 1, 3];degree=3 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S003 S005 · impliesBefore phase=original;vertex=3;all edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]] After neighbors=[2];degree=1 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S004 S006 · impliesBefore phase=original;vertex=4;all edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]] After neighbors=[0];degree=1 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S005 S007 · impliesBefore phase=original;vertex=5;all edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]] After neighbors=[0];degree=1 Conditions: none Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels. Step identifiers and dependenciesRule: complete_graph_neighbor_inventory · Uses: G001, S006 S008 · impliesBefore phase=original;root=0;depth=0;frontier=[0] After next newly reached=[2, 4, 5];component seen=[0, 2, 4, 5] Conditions: none Expand along canonical edges only to unseen original labels.A component is complete only when its frontier empties;then start at another unseen vertex. Step identifiers and dependenciesRule: tree_complete_reachability_layer · Uses: G001, S007 S009 · impliesBefore phase=original;root=0;depth=1;frontier=[2, 4, 5] After next newly reached=[1, 3];component seen=[0, 1, 2, 3, 4, 5] Conditions: none Expand along canonical edges only to unseen original labels.A component is complete only when its frontier empties;then start at another unseen vertex. Step identifiers and dependenciesRule: tree_complete_reachability_layer · Uses: G001, S008 S010 · impliesBefore phase=original;root=0;depth=2;frontier=[1, 3] After next newly reached=[];component seen=[0, 1, 2, 3, 4, 5] Conditions: none Expand along canonical edges only to unseen original labels.A component is complete only when its frontier empties;then start at another unseen vertex. Step identifiers and dependenciesRule: tree_complete_reachability_layer · Uses: G001, S009 S011 · impliesBefore phase=original;spanning forest=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]];all edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]] After nonforest edges=[];cyclic=No Conditions: none First-discovery edges form a tree in each reached component.Any remaining edge closes an existing tree path into a cycle.Components were established first;this is not an edge-count-only tree test. Step identifiers and dependenciesRule: spanning_forest_cycle_witness · Uses: G001, S010 S012 · impliesBefore phase=original;vertices=[0, 1, 2, 3, 4, 5];components=[[0, 1, 2, 3, 4, 5]];cyclic=No After pairwise reachable=Yes;nonempty=Yes;tree=Yes Conditions: none A tree must be nonempty,have one component and be acyclic.Pairwise reachability is vacuously true on an empty vertex set,but nonemptiness fails.A singleton with no edges is a tree;its degree0 vertex is not a leaf. Step identifiers and dependenciesRule: nonempty_connected_acyclic_tree_gate · Uses: G001, S011 S013 · impliesBefore original tree=Yes;edge=[5, 2];typed=[Yes, Yes];distinct=Yes;already edge=No After new nonedge qualified=Yes Conditions: none Require an original tree,typed distinct endpoints and an absent original edge.Repeating an existing edge does not create a new cycle in this simple-graph model,and a loop is not permitted. Step identifiers and dependenciesRule: original_tree_new_nonedge_guard · Uses: G001, S012 S014 · impliesBefore phase=new nonedge;prefix=[2];next=0 After edge=[0, 2] present;new prefix=[2, 0] Conditions: none Each step follows an original edge to a nonrepeated vertex.The simple path between tree vertices is unique;equal endpoints need no edge. Step identifiers and dependenciesRule: qualified_tree_path_edge · Uses: G001, S013 S015 · impliesBefore phase=new nonedge;prefix=[2, 0];next=5 After edge=[0, 5] present;new prefix=[2, 0, 5] Conditions: none Each step follows an original edge to a nonrepeated vertex.The simple path between tree vertices is unique;equal endpoints need no edge. Step identifiers and dependenciesRule: qualified_tree_path_edge · Uses: G001, S014 S016 · impliesBefore phase=new nonedge;start=2;end=5 After unique path=[2, 0, 5] Conditions: none Path uniqueness relies on the earlier original-tree qualification;it is not asserted for a general cyclic graph. Step identifiers and dependenciesRule: qualified_tree_path_result · Uses: G001, S015 S017 · impliesBefore original path=[2, 0, 5];new edge=[2, 5] After closed cycle=[2, 0, 5, 2] Conditions: none Start the original unique path at the smaller new-edge endpoint,reach the larger endpoint,and close along the new edge.Do not rotate the witness to the globally smallest vertex on the cycle. Step identifiers and dependenciesRule: new_edge_closes_original_unique_path · Uses: G001, S016 ## Reverse checks - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2]) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: addd90893ebc391ee2b34212f052694c030be4383851ab829313b0b4e903e327 - candidate record SHA-256: 11c46468b027e5b7b9294d361aaf26de3cdaba779fa8cbf86dcb372678e0092c - semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18 - question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r06-tree-0097.json)