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R0015-08001 · q-20261012-advance-20261017-r05-graph-0001
Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. 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(0)=∅ with raw edges=[]. 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=0;raw edges=[]
After
canonical edges=[]
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
empty vertex list;degree list=[]
After
empty degree sum=0
Conditions: none
The empty sum is0;do not invent vertices or adjacency rows for the empty graph.
Rule: empty_graph_degree_sum · Uses: G001, S001
Before
twice distinct edge count := mul(2, 0)
After
twice distinct edge count = 0
Conditions: none
Substitute the source data exactly in the stated operand order;division requires a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
degrees=[];edge count=0
After
sum=0;2|E|=0
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, S003
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 — Recompute twice distinct edge count with exact rational arithmetic, not rounded decimals.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.(0, [], [], [], 0, 0)
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: e06cfc90fc9613da5566d9ed2c2b1d12da44100c6736449f306b88ffa2b9e191candidate record SHA-256: 6188a2809918350bd10eda77605d17c94c7f13a7cf16b6edf4f41e51f781e44bsemantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED