# Question 35420: Unique Cycle from a Genuinely New Tree Edge

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0069-35420 · q-20261007-advance-20261009-r02-tree-0095



## Original question and givens

The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 4,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.


- G001 — The finite simple undirected graph has V=[0, 1, 2, 3, 4] and E=[[2, 1], [1, 0], [3, 0], [0, 4]]. Edges are unordered, with repeated listings normalized and no loops or absent endpoints. The original nonempty graph is a tree, and the edge between distinct 4,3 is absent. Add it and list the unique cycle: start at the smaller endpoint, follow the original path to the larger, then close at the start.





## 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.


- [mko-tree](https://ocme.evemisslab.com/en/mko/mko-tree/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-tree-added-edge-cycle, [E, V, domain, operation, u, v]]

After

derived = [tree_added_edge_cycle, finite_simple_undirected]

Conditions: none

Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.

Step identifiers and dependenciesRule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001





S002 · impliesBefore

source roles = [TREE vertices ORIGINAL, 10, [0, 1, 2, 3, 4]]

After

derived = [[0, 1, 2, 3, 4]]

Conditions: none

Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits.

Step identifiers and dependenciesRule: Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort · Uses: G001, S001





S003 · impliesBefore

source roles = [TREE raw edges ORIGINAL, 0, [2, 1]]

After

derived = [[2, 1]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

Step identifiers and dependenciesRule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S002





S004 · impliesBefore

source roles = [TREE raw edges ORIGINAL, 1, [1, 0]]

After

derived = [[1, 0]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

Step identifiers and dependenciesRule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S003





S005 · impliesBefore

source roles = [TREE raw edges ORIGINAL, 2, [3, 0]]

After

derived = [[3, 0]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

Step identifiers and dependenciesRule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S004





S006 · impliesBefore

source roles = [TREE raw edges ORIGINAL, 3, [0, 4]]

After

derived = [[0, 4]]

Conditions: none

BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.

Step identifiers and dependenciesRule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S005





S007 · impliesBefore

source roles = [[0, 1, 2, 3, 4], [[0, 4], [1, 0], [2, 1], [3, 0]], Yes]

After

derived = [[[0, 1], [0, 3], [0, 4], [1, 2]], [[0, [1, 3, 4]], [1, [0, 2]], [2, [1]], [3, [0]], [4, [0]]]]

Conditions: none

Type the WHOLE raw edge list,then normalize undirected endpoints/deduplicate;reversed duplicates are allowed.INPUT edges have no loops/outside endpoints,while isolated vertices stay in V.TREE raw E does NOT acquire LDL’s200-entry limit.

Step identifiers and dependenciesRule: Graph_WHOLE_simple_undirected_edges_normalized_NO_loops_or_outside · Uses: G001, S006





S008 · impliesBefore

source roles = [0, 1, 0, 1]

After

derived = [Yes]

Conditions: none

Merge endpoint components ONLY when different.If already in the same component,this edge closes a cycle.Check EVERY edge across ALL components,not only one reached block.

Step identifiers and dependenciesRule: Tree_EACH_edge_union_components_cycle_audit · Uses: G001, S007





S009 · impliesBefore

source roles = [0, 3, 1, 3]

After

derived = [Yes]

Conditions: none

Merge endpoint components ONLY when different.If already in the same component,this edge closes a cycle.Check EVERY edge across ALL components,not only one reached block.

Step identifiers and dependenciesRule: Tree_EACH_edge_union_components_cycle_audit · Uses: G001, S008





S010 · impliesBefore

source roles = [0, 4, 3, 4]

After

derived = [Yes]

Conditions: none

Merge endpoint components ONLY when different.If already in the same component,this edge closes a cycle.Check EVERY edge across ALL components,not only one reached block.

Step identifiers and dependenciesRule: Tree_EACH_edge_union_components_cycle_audit · Uses: G001, S009





S011 · impliesBefore

source roles = [1, 2, 4, 2]

After

derived = [Yes]

Conditions: none

Merge endpoint components ONLY when different.If already in the same component,this edge closes a cycle.Check EVERY edge across ALL components,not only one reached block.

Step identifiers and dependenciesRule: Tree_EACH_edge_union_components_cycle_audit · Uses: G001, S010





S012 · impliesBefore

source roles = [0, [1, 3, 4]]

After

derived = [[0, 1, 3, 4], [0]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependenciesRule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S011





S013 · impliesBefore

source roles = [1, [0, 2]]

After

derived = [[0, 1, 3, 4, 2], [0, 1]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependenciesRule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S012





S014 · impliesBefore

source roles = [3, [0]]

After

derived = [[0, 1, 3, 4, 2], [0, 1, 3]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependenciesRule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S013





S015 · impliesBefore

source roles = [4, [0]]

After

derived = [[0, 1, 3, 4, 2], [0, 1, 3, 4]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependenciesRule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S014





S016 · impliesBefore

source roles = [2, [1]]

After

derived = [[0, 1, 3, 4, 2], [0, 1, 2, 3, 4]]

Conditions: none

Expand actual neighbors from the least label and retain reached vertices.Edge count|V|−1 ALONE does not replace connectivity.

Step identifiers and dependenciesRule: Tree_CONNECTIVITY_actual_reachability_from_min_label · Uses: G001, S015





S017 · impliesBefore

source roles = [[0, 1, 2, 3, 4], [[0, 1], [0, 3], [0, 4], [1, 2]]]

After

derived = [Yes, Yes, Yes, Yes]

Conditions: none

THIS entry defines a tree as NONEMPTY,connected and acyclic.Empty graph is No;an edgeless singleton is Yes.A triangle plus an isolated vertex is No EVEN when|E|=|V|−1.

Step identifiers and dependenciesRule: Tree_NONEMPTY_connected_acyclic_ALL_three · Uses: G001, S016





S018 · impliesBefore

source roles = [3, 0]

After

derived = [[3, 0]]

Conditions: none

Retain the parent on FIRST discovery.Label0 is a VALID vertex,not an end sentinel.

Step identifiers and dependenciesRule: Tree_PATH_EACH_parent_discovery_keep_label0 · Uses: G001, S017





S019 · impliesBefore

source roles = [0, 1]

After

derived = [[3, 0, 1]]

Conditions: none

Retain the parent on FIRST discovery.Label0 is a VALID vertex,not an end sentinel.

Step identifiers and dependenciesRule: Tree_PATH_EACH_parent_discovery_keep_label0 · Uses: G001, S018





S020 · impliesBefore

source roles = [0, 4]

After

derived = [[3, 0, 1, 4]]

Conditions: none

Retain the parent on FIRST discovery.Label0 is a VALID vertex,not an end sentinel.

Step identifiers and dependenciesRule: Tree_PATH_EACH_parent_discovery_keep_label0 · Uses: G001, S019





S021 · impliesBefore

source roles = [3, 4]

After

derived = [[3, 0, 4]]

Conditions: none

Backtrack parents and reverse to retain s→t ORDER.Verify EVERY adjacent edge and distinct vertices.TWO different simple paths would form a cycle,contradicting acyclicity.The zero-edge path for s=t is[s].

Step identifiers and dependenciesRule: Tree_PATH_ORDERED_edges_verified_unique_by_acyclicity · Uses: G001, S020





S022 · impliesBefore

source roles = [4, 3, [3, 0, 4]]

After

derived = [[3, 0, 4, 3]]

Conditions: none

Require DISTINCT endpoints in V and a GENUINELY absent original edge.Start at the SMALLER endpoint,follow the UNIQUE original tree path to the larger,THEN close by the new edge and repeat the start.Do not sort cycle vertices or accept a reversed existing edge.

Step identifiers and dependenciesRule: Tree_GENUINELY_NEW_edge_closes_UNIQUE_old_path_MIN_to_MAX · Uses: G001, S021









## Reverse checks


- S022 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S021 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S020 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S019 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S018 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S017 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S016 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S015 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S014 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S013 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S012 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S011 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S010 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S009 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S008 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S007 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S006 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S005 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S004 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S003 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S002 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
- S001 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.





## Result

[3, 0, 4, 3]

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: dbef03602dcc475e05e8611e4f642afacdd6fdb0ec2b7d6b974cf0111862b082
- candidate record SHA-256: aaa299e5068088de0b63f75dafd47e4cdcf5984b559ff8791a43ae82f9c890a6
- semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b
- question-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r02-tree-0095.json)