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R0015-08407 · q-20261012-advance-20261017-r06-tree-0082
Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [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=[0, 1]. 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(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [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=[0, 1]. 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=4;raw edges=[[1, 2], [1, 3], [2, 3]]
After
canonical edges=[[1, 2], [1, 3], [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=[[1, 2], [1, 3], [2, 3]]
After
neighbors=[];degree=0
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=[[1, 2], [1, 3], [2, 3]]
After
neighbors=[2, 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, S002
Before
phase=original;vertex=2;all edges=[[1, 2], [1, 3], [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=[[1, 2], [1, 3], [2, 3]]
After
neighbors=[1, 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, S004
Before
phase=original;root=0;depth=0;frontier=[0]
After
next newly reached=[];component seen=[0]
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, S005
Before
phase=original;root=1;depth=0;frontier=[1]
After
next newly reached=[2, 3];component seen=[1, 2, 3]
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, S006
Before
phase=original;root=1;depth=1;frontier=[2, 3]
After
next newly reached=[];component seen=[1, 2, 3]
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;spanning forest=[[1, 2], [1, 3]];all edges=[[1, 2], [1, 3], [2, 3]]
After
nonforest edges=[[2, 3]];cyclic=Yes
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, S008
Before
phase=original;vertices=[0, 1, 2, 3];components=[[0], [1, 2, 3]];cyclic=Yes
After
pairwise reachable=No;nonempty=Yes;tree=No
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, S009
Before
original tree=No;edge=[0, 1];typed=[Yes, Yes];distinct=Yes;already edge=No
After
new nonedge qualified=No
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, S010
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.(4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], Yes, No, No, not-applicable, not-applicable)
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: e1df92246a478d439713820fc93a2e1a94fecd4e7e5d0d8c69150e0b243c6a70candidate record SHA-256: 1b468014364f77c34294006cce939da6fbae428cf8e6deba3eac62a0a7c5d0d4semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED