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R0015-08436 · q-20261012-advance-20261017-r06-tree-0011
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 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. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.
G001 — Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 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. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.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=5;raw edges=[[0, 1], [0, 3]]
After
canonical edges=[[0, 1], [0, 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, 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, S001
Before
phase=original;vertex=1;all edges=[[0, 1], [0, 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, S002
Before
phase=original;vertex=2;all edges=[[0, 1], [0, 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, S003
Before
phase=original;vertex=3;all edges=[[0, 1], [0, 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, S004
Before
phase=original;vertex=4;all edges=[[0, 1], [0, 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, S005
Before
phase=original;root=0;depth=0;frontier=[0]
After
next newly reached=[1, 3];component seen=[0, 1, 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=0;depth=1;frontier=[1, 3]
After
next newly reached=[];component seen=[0, 1, 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;root=2;depth=0;frontier=[2]
After
next newly reached=[];component seen=[2]
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=4;depth=0;frontier=[4]
After
next newly reached=[];component seen=[4]
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;spanning forest=[[0, 1], [0, 3]];all edges=[[0, 1], [0, 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, S010
Before
phase=original;vertices=[0, 1, 2, 3, 4];components=[[0, 1, 3], [2], [4]];cyclic=No
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, S011
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.(5, [[0, 1], [0, 3]], [[0, 1, 3], [2], [4]], No, No, Yes, No)
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: 67021cd0cfa3dbfd2af2a856c5f63bc61ebac715d3956fd8d0f291e90869ea4fcandidate record SHA-256: 00daaee974939aaceb1349858654b43650085124921865f367788a139c13c089semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED