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R0015-08416 · q-20261012-advance-20261017-r06-tree-0091
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [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, 1], [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.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.
Before
n=6;raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]
After
canonical edges=[[0, 1], [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.
Rule: simple_graph_edge_normalization · Uses: G001
Before
phase=original;vertex=0;all edges=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]
After
neighbors=[1, 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.
Rule: complete_graph_neighbor_inventory · Uses: G001, S001
Before
phase=original;vertex=1;all edges=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]
After
neighbors=[0, 2];degree=2
Conditions: none
Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels.
Rule: complete_graph_neighbor_inventory · Uses: G001, S002
Before
phase=original;vertex=2;all edges=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]
After
neighbors=[1, 3];degree=2
Conditions: none
Scan all canonical edges.An incident edge contributes its other endpoint as a distinct neighbor.Keep original vertex labels.
Rule: complete_graph_neighbor_inventory · Uses: G001, S003
Before
phase=original;vertex=3;all edges=[[0, 1], [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.
Rule: complete_graph_neighbor_inventory · Uses: G001, S004
Before
phase=original;vertex=4;all edges=[[0, 1], [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.
Rule: complete_graph_neighbor_inventory · Uses: G001, S005
Before
phase=original;vertex=5;all edges=[[0, 1], [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.
Rule: complete_graph_neighbor_inventory · Uses: G001, S006
Before
phase=original;root=0;depth=0;frontier=[0]
After
next newly reached=[1, 4, 5];component seen=[0, 1, 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.
Rule: tree_complete_reachability_layer · Uses: G001, S007
Before
phase=original;root=0;depth=1;frontier=[1, 4, 5]
After
next newly reached=[2];component seen=[0, 1, 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.
Rule: tree_complete_reachability_layer · Uses: G001, S008
Before
phase=original;root=0;depth=2;frontier=[2]
After
next newly reached=[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.
Rule: tree_complete_reachability_layer · Uses: G001, S009
Before
phase=original;root=0;depth=3;frontier=[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.
Rule: tree_complete_reachability_layer · Uses: G001, S010
Before
phase=original;spanning forest=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]];all edges=[[0, 1], [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.
Rule: spanning_forest_cycle_witness · Uses: G001, S011
Before
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.
Rule: nonempty_connected_acyclic_tree_gate · Uses: G001, S012
Before
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.
Rule: original_tree_new_nonedge_guard · Uses: G001, S013
Before
phase=new nonedge;prefix=[2];next=1
After
edge=[1, 2] present;new prefix=[2, 1]
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.
Rule: qualified_tree_path_edge · Uses: G001, S014
Before
phase=new nonedge;prefix=[2, 1];next=0
After
edge=[0, 1] present;new prefix=[2, 1, 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.
Rule: qualified_tree_path_edge · Uses: G001, S015
Before
phase=new nonedge;prefix=[2, 1, 0];next=5
After
edge=[0, 5] present;new prefix=[2, 1, 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.
Rule: qualified_tree_path_edge · Uses: G001, S016
Before
phase=new nonedge;start=2;end=5
After
unique path=[2, 1, 0, 5]
Conditions: none
Path uniqueness relies on the earlier original-tree qualification;it is not asserted for a general cyclic graph.
Rule: qualified_tree_path_result · Uses: G001, S017
Before
original path=[2, 1, 0, 5];new edge=[2, 5]
After
closed cycle=[2, 1, 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.
Rule: new_edge_closes_original_unique_path · Uses: G001, S018
S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 397c934debb095d6ea82bf5f429eb69d633c7188e07308555805e723fe8c810ecandidate record SHA-256: e665e9165f5a977d6ff1ad015e0bff3da60ae762577709867176e62e48fe771dsemantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED