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Question 8401: Qualified Tree Nonedge and Its Unique Cycle Witness

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0015-08401 · q-20261012-advance-20261017-r06-tree-0076

Original question and givens

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. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

  • 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. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.

Definition and scope

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.

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: simple_graph_edge_normalization · Uses: G001

S002 · implies

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.

Step identifiers and dependencies

Rule: spanning_forest_cycle_witness · Uses: G001, S001

S003 · implies

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.

Step identifiers and dependencies

Rule: nonempty_connected_acyclic_tree_gate · Uses: G001, S002

S004 · implies

Before

original tree=No;edge=[0, 1];typed=[No, No];distinct=Yes;already edge=No

After

new nonedge qualified=No

Conditions: none

Require an original tree,typed distinct endpoints and an absent original edge.Repeating an existing edge does not create a new cycle in this simple-graph model,and a loop is not permitted.

Step identifiers and dependencies

Rule: original_tree_new_nonedge_guard · Uses: G001, S003

Reverse checks

  • 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.

Result

(0, [], No, [No, No], Yes, No, No, not-applicable, not-applicable)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 960034a044d7489e8043521e115f96232d06c0781a5946ab7fdf6cbcfadc80da
  • candidate record SHA-256: 845b1f6b37a70719a83b735ab8cea51ab9ca526223765dc33f63e23e3d01568d
  • semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18
  • question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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