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R0015-08426 · q-20261012-advance-20261017-r06-tree-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. 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(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. 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=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
phase=original;spanning forest=[];all edges=[]
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, S001
Before
phase=original;vertices=[];components=[];cyclic=No
After
pairwise reachable=Yes;nonempty=No;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, S002
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.(0, [], [], Yes, No, No, 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: 814ca3989f5c9ba871a98f2c6d1855c6e2bdd8b5decb73af716b2c4bd4b3b5cbcandidate record SHA-256: 625cfeabb86f3a41e2a1fa46ccfb10146789c449a3ecec4e362129688b98efe0semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED