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R0015-08485 · q-20261012-advance-20261017-r06-tree-0035
Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.
G001 — Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.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, 2], [0, 3], [0, 4], [0, 5], [1, 3]]
After
canonical edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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, 2], [0, 3], [0, 4], [0, 5], [1, 3]]
After
neighbors=[2, 3, 4, 5];degree=4
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, 2], [0, 3], [0, 4], [0, 5], [1, 3]]
After
neighbors=[3];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, 2], [0, 3], [0, 4], [0, 5], [1, 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, S003
Before
phase=original;vertex=3;all edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]]
After
neighbors=[0, 1];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;vertex=4;all edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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, 2], [0, 3], [0, 4], [0, 5], [1, 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=[2, 3, 4, 5];component seen=[0, 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, S007
Before
phase=original;root=0;depth=1;frontier=[2, 3, 4, 5]
After
next newly reached=[1];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, S008
Before
phase=original;root=0;depth=2;frontier=[1]
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, S009
Before
phase=original;spanning forest=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]];all edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 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, 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, S011
Before
original tree=Yes;endpoints=[3, 0];typed=[Yes, Yes]
After
path lookup qualified=Yes
Conditions: none
A failed original-tree guard takes precedence.Only for an original tree do endpoint types permit path lookup.A local path's existence does not establish whole-graph treehood.
Rule: tree_path_whole_graph_and_endpoint_guard · Uses: G001, S012
Before
phase=query;prefix=[3];next=0
After
edge=[0, 3] present;new prefix=[3, 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, S013
Before
phase=query;start=3;end=0
After
unique path=[3, 0]
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, S014
Before
qualified path edge count := sub(2, 1)
After
qualified path edge count = 1
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S015
S016 · premise_and_result_audit — Recompute qualified path edge count with exact rational arithmetic, not rounded decimals.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, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [3, 0], 1)
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: d2709f0d0f57ff049f1de00c0fe60d44f0dc20fff79956ec9a5ba979747c72a5candidate record SHA-256: 9ca2c3def72e4f6a0d820b590efe902cf7f2dd36b95409b508422f1e7bd0883dsemantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED