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R0015-08019 · q-20261012-advance-20261017-r05-graph-0019
Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. 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. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).
G001 — Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. 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. Give (n, normalized edges, Boolean adjacency matrix, degree list, degree sum, twice the edge count).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], [0, 4]]
After
canonical edges=[[0, 1], [0, 3], [0, 4]]
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], [0, 4]]
After
neighbors=[1, 3, 4];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, 3], [0, 4]]
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], [0, 4]]
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], [0, 4]]
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], [0, 4]]
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
vertex=0;ordered targets=[0, 1, 2, 3, 4];neighbors=[1, 3, 4]
After
adjacency row=[No, Yes, No, Yes, Yes]
Conditions: none
Each Boolean entry records edge adjacency.The diagonal is false and the matrix is symmetric;this is not a reachability matrix.
Rule: complete_boolean_adjacency_row · Uses: G001, S006
Before
vertex=1;ordered targets=[0, 1, 2, 3, 4];neighbors=[0]
After
adjacency row=[Yes, No, No, No, No]
Conditions: none
Each Boolean entry records edge adjacency.The diagonal is false and the matrix is symmetric;this is not a reachability matrix.
Rule: complete_boolean_adjacency_row · Uses: G001, S007
Before
vertex=2;ordered targets=[0, 1, 2, 3, 4];neighbors=[]
After
adjacency row=[No, No, No, No, No]
Conditions: none
Each Boolean entry records edge adjacency.The diagonal is false and the matrix is symmetric;this is not a reachability matrix.
Rule: complete_boolean_adjacency_row · Uses: G001, S008
Before
vertex=3;ordered targets=[0, 1, 2, 3, 4];neighbors=[0]
After
adjacency row=[Yes, No, No, No, No]
Conditions: none
Each Boolean entry records edge adjacency.The diagonal is false and the matrix is symmetric;this is not a reachability matrix.
Rule: complete_boolean_adjacency_row · Uses: G001, S009
Before
vertex=4;ordered targets=[0, 1, 2, 3, 4];neighbors=[0]
After
adjacency row=[Yes, No, No, No, No]
Conditions: none
Each Boolean entry records edge adjacency.The diagonal is false and the matrix is symmetric;this is not a reachability matrix.
Rule: complete_boolean_adjacency_row · Uses: G001, S010
Before
degree sum := sum(3, 1, 0, 1, 1)
After
degree sum = 6
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
twice distinct edge count := mul(2, 3)
After
twice distinct edge count = 6
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
degrees=[3, 1, 0, 1, 1];edge count=3
After
sum=6;2|E|=6
Conditions: none
Each undirected nonloop edge contributes1 at each endpoint,so the degree sum is twice the canonical edge count.This does not test connectedness or treehood.
Rule: handshake_each_edge_two_ends · Uses: G001, S013
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 — Recompute twice distinct edge count with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute degree sum with exact rational arithmetic, not rounded decimals.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, 4]], [[No, Yes, No, Yes, Yes], [Yes, No, No, No, No], [No, No, No, No, No], [Yes, No, No, No, No], [Yes, No, No, No, No]], [3, 1, 0, 1, 1], 6, 6)
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: 71bfa61acf8ccf44d1fcd3d27a61d092f70e7911c459a880f9814ffbe7b67dc6candidate record SHA-256: 26783f69b942efe7650db972306a8a93266e5f6b66f2ec12e5c5c9ae32eea64bsemantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED