# Question 18978: A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

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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.

R0036-18978 · q-20261014-advance-20261105-r04-eulerian-trail-0078



## Original question and givens

Use vertex carrier Fin(3)={0,…,2} and no edges, with n=3≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).


- G001 — Use vertex carrier Fin(3)={0,…,2} and no edges, with n=3≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).





## 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-eulerian-trail](https://ocme.evemisslab.com/en/mko/mko-eulerian-trail/)





## Step-by-step derivation

S001 · impliesBefore

source operation=euler-zero-edge-convention;original snapshot and mathematics retained

After

explicit ordered source output fields=[vertex_count, edge_count, length_zero_circuit, convention, boundary]

Conditions: none

Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ACTUAL nonempty original carrier Fin(3);NO edges

After

choose single-vertex walk[0],length0;all edges used exactly once vacuously;length_zero_circuit_on_nonempty_vertex_set

Conditions: none

Since n≥1 vertex0 is available.A one-vertex walk has length0 and is a circuit by the stated convention.It need not visit all isolated vertices,does not admit the empty-vertex graph,and gives no Hamiltonian claim.

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









## Reverse checks


- 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

(3, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

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: a6bc80b645df918492a660c478b58c53e6d916e11a9203b0803b1bfb35f99e98
- candidate record SHA-256: 0f308837bdcdcc859382930caabc407035bda47a75f1070da3d4372a9212f511
- semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301
- question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r04-eulerian-trail-0078.json)