# Question 20804: Every Successive Step Must Follow Its Own Directed Edge [Return to catalogue page 105](https://ocme.evemisslab.com/en/questions/page/105/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations/#q-20261014-advance-20261108-r07-finite-directed-graph-0054) 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. R0040-20804 · q-20261014-advance-20261108-r07-finite-directed-graph-0054 ## Original question and givens Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, boundary). - G001 — Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Supplied vertex list walk=[0, 1, 2, 3, 4, 0]. Give (walk, step_checks, all_oriented_steps_present, 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-finite-directed-graph](https://ocme.evemisslab.com/en/mko/mko-finite-directed-graph/) ## Step-by-step derivation S001 · impliesBefore source operation=digraph-directed-walk-audit;original snapshot and mathematics retained After explicit ordered source output fields=[walk, step_checks, all_oriented_steps_present, boundary] Conditions: none Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL complete directed carrier Fin(5);ORIGINAL lexicographically canonical ORDERED arcs=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4], [4, 0]] After all endpoints in range and same-direction arcs unique;actual loops=[];main loop-free directed-set convention qualified Conditions: none Check outer from/to order without sorting endpoints inside each arc.(u,v) and(v,u)are distinct and may coexist.The main convention forbids loops;the boundary temporarily accepts them to audit that convention,not to deny other digraph models. Step identifiers and dependenciesRule: digraph_ordered_pairs_not_undirected_endpoint_sorting · Uses: G001, S001 S003 · impliesBefore ORIGINAL consecutive positions=[0, 1];ORIGINAL directed step=[0, 1] After EXACT oriented arc(0,1) present=Yes;reverse arc is a DIFFERENT lookup Conditions: none Every step must follow its listed orientation;(b,a)cannot replace(a,b).Do not delete repeated vertices or arcs:a walk is not an additional edge-distinct trail requirement. Step identifiers and dependenciesRule: digraph_every_step_exact_orientation_not_reverse_edge · Uses: G001, S002 S004 · impliesBefore ORIGINAL consecutive positions=[1, 2];ORIGINAL directed step=[1, 2] After EXACT oriented arc(1,2) present=Yes;reverse arc is a DIFFERENT lookup Conditions: none Every step must follow its listed orientation;(b,a)cannot replace(a,b).Do not delete repeated vertices or arcs:a walk is not an additional edge-distinct trail requirement. Step identifiers and dependenciesRule: digraph_every_step_exact_orientation_not_reverse_edge · Uses: G001, S003 S005 · impliesBefore ORIGINAL consecutive positions=[2, 3];ORIGINAL directed step=[2, 3] After EXACT oriented arc(2,3) present=Yes;reverse arc is a DIFFERENT lookup Conditions: none Every step must follow its listed orientation;(b,a)cannot replace(a,b).Do not delete repeated vertices or arcs:a walk is not an additional edge-distinct trail requirement. Step identifiers and dependenciesRule: digraph_every_step_exact_orientation_not_reverse_edge · Uses: G001, S004 S006 · impliesBefore ORIGINAL consecutive positions=[3, 4];ORIGINAL directed step=[3, 4] After EXACT oriented arc(3,4) present=Yes;reverse arc is a DIFFERENT lookup Conditions: none Every step must follow its listed orientation;(b,a)cannot replace(a,b).Do not delete repeated vertices or arcs:a walk is not an additional edge-distinct trail requirement. Step identifiers and dependenciesRule: digraph_every_step_exact_orientation_not_reverse_edge · Uses: G001, S005 S007 · impliesBefore ORIGINAL consecutive positions=[4, 5];ORIGINAL directed step=[4, 0] After EXACT oriented arc(4,0) present=Yes;reverse arc is a DIFFERENT lookup Conditions: none Every step must follow its listed orientation;(b,a)cannot replace(a,b).Do not delete repeated vertices or arcs:a walk is not an additional edge-distinct trail requirement. Step identifiers and dependenciesRule: digraph_every_step_exact_orientation_not_reverse_edge · Uses: G001, S006 S008 · impliesBefore ORIGINAL vertex list=[0, 1, 2, 3, 4, 0];COMPLETE successive-step audit rows=[[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]] After ALL oriented steps present=Yes;empty/singleton list has no step to refute;NOT an extra nonempty-walk,simple-path or trail certificate Conditions: none The flag is only the conjunction of successive-step checks.Empty/singleton lists are vacuous and do not certify a separately required nonempty-walk convention.Neither full-edge coverage nor no-revisit conditions are imposed. Step identifiers and dependenciesRule: digraph_vacuous_step_conjunction_not_nonempty_walk_claim · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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, 1, 2, 3, 4, 0], [[0, 1, Yes], [1, 2, Yes], [2, 3, Yes], [3, 4, Yes], [4, 0, Yes]], Yes, edge_orientation_must_match) 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: 1cfc5225852300877ea7759b9e9ecc3783562b89cacf528f7a2c86180ea2a19a - candidate record SHA-256: 972c28fda4d7d5a770a522c5f5e3e10642275c50feef4c44ee4ccd57a3960bcb - semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d - question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261108-r07-finite-directed-graph-0054.json)