# Question 20828: In-Degrees and Out-Degrees Each Sum to the Edge Count [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-0028) 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. R0040-20828 · q-20261014-advance-20261108-r07-finite-directed-graph-0028 ## Original question and givens Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [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. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts). - G001 — Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [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. Give (out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts). ## 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-in-out-degree;original snapshot and mathematics retained After explicit ordered source output fields=[out_degrees, in_degrees, total_out, total_in, edge_count, equal_counts] 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], [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 · calculationBefore digraph vertex0 exact outgoing count := sum(1) After digraph vertex0 exact outgoing count = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore digraph vertex0 exact incoming count := sum(1) After digraph vertex0 exact incoming count = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore ORIGINAL vertex=0;ALL outgoing original-arc indices=[0];ALL incoming indices=[5] After outdegree=1;indegree=1;a loop at this vertex contributes ONCE to EACH separate count Conditions: none Each arc has one source and one target;a loop contributes once to each.Do not import an undirected loop-degree rule or merge the two degree types.Retain both zeros for isolates. Step identifiers and dependenciesRule: digraph_every_vertex_separate_in_out_incidence · Uses: G001, S004 S006 · calculationBefore digraph vertex1 exact outgoing count := sum(1, 1) After digraph vertex1 exact outgoing count = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore digraph vertex1 exact incoming count := sum(1) After digraph vertex1 exact incoming count = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore ORIGINAL vertex=1;ALL outgoing original-arc indices=[1, 2];ALL incoming indices=[0] After outdegree=2;indegree=1;a loop at this vertex contributes ONCE to EACH separate count Conditions: none Each arc has one source and one target;a loop contributes once to each.Do not import an undirected loop-degree rule or merge the two degree types.Retain both zeros for isolates. Step identifiers and dependenciesRule: digraph_every_vertex_separate_in_out_incidence · Uses: G001, S007 S009 · calculationBefore digraph vertex2 exact outgoing count := sum(1) After digraph vertex2 exact outgoing count = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore digraph vertex2 exact incoming count := sum(1) After digraph vertex2 exact incoming count = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore ORIGINAL vertex=2;ALL outgoing original-arc indices=[3];ALL incoming indices=[1] After outdegree=1;indegree=1;a loop at this vertex contributes ONCE to EACH separate count Conditions: none Each arc has one source and one target;a loop contributes once to each.Do not import an undirected loop-degree rule or merge the two degree types.Retain both zeros for isolates. Step identifiers and dependenciesRule: digraph_every_vertex_separate_in_out_incidence · Uses: G001, S010 S012 · calculationBefore digraph vertex3 exact outgoing count := sum(1) After digraph vertex3 exact outgoing count = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore digraph vertex3 exact incoming count := sum(1, 1) After digraph vertex3 exact incoming count = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore ORIGINAL vertex=3;ALL outgoing original-arc indices=[4];ALL incoming indices=[2, 3] After outdegree=1;indegree=2;a loop at this vertex contributes ONCE to EACH separate count Conditions: none Each arc has one source and one target;a loop contributes once to each.Do not import an undirected loop-degree rule or merge the two degree types.Retain both zeros for isolates. Step identifiers and dependenciesRule: digraph_every_vertex_separate_in_out_incidence · Uses: G001, S013 S015 · calculationBefore digraph vertex4 exact outgoing count := sum(1) After digraph vertex4 exact outgoing count = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore digraph vertex4 exact incoming count := sum(1) After digraph vertex4 exact incoming count = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore ORIGINAL vertex=4;ALL outgoing original-arc indices=[5];ALL incoming indices=[4] After outdegree=1;indegree=1;a loop at this vertex contributes ONCE to EACH separate count Conditions: none Each arc has one source and one target;a loop contributes once to each.Do not import an undirected loop-degree rule or merge the two degree types.Retain both zeros for isolates. Step identifiers and dependenciesRule: digraph_every_vertex_separate_in_out_incidence · Uses: G001, S016 S018 · calculationBefore digraph complete outgoing-degree sum := sum(1, 2, 1, 1, 1) After digraph complete outgoing-degree sum = 6 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore digraph complete incoming-degree sum := sum(1, 1, 1, 2, 1) After digraph complete incoming-degree sum = 6 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore ALL vertex-order outdegrees=[1, 2, 1, 1, 1];indegrees=[1, 1, 1, 2, 1];complete directed arc count=6 After each arc counted once at its source and once at its target;SUM(out)=6=|E| and SUM(in)=6=|E|;the SEPARATE counting rule is |E|,not an imported undirected2|E| rule (numbers may coincide when E=0) Conditions: none Grouping all arcs by their sources or by their targets gives two separate totals,each |E|.This supplies no undirected-connectivity,DAG,strong-connectivity or Hamiltonian conclusion. Step identifiers and dependenciesRule: digraph_two_separate_degree_sum_certificates · Uses: G001, S019 ## Reverse checks - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute digraph complete incoming-degree sum with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute digraph complete outgoing-degree sum with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute digraph vertex4 exact incoming count with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute digraph vertex4 exact outgoing count with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute digraph vertex3 exact incoming count with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute digraph vertex3 exact outgoing count with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute digraph vertex2 exact incoming count with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute digraph vertex2 exact outgoing count with exact rational arithmetic, not rounded decimals. - 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 — Recompute digraph vertex1 exact incoming count with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute digraph vertex1 exact outgoing count with exact rational arithmetic, not rounded decimals. - 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 — Recompute digraph vertex0 exact incoming count with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute digraph vertex0 exact outgoing count with exact rational arithmetic, not rounded decimals. - 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 ([1, 2, 1, 1, 1], [1, 1, 1, 2, 1], 6, 6, 6, Yes) 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: b4abc97341bb17b33dce0c5da661225dbb66cefd45e16f3181e33affe54c317e - candidate record SHA-256: 1282bd09c007c940d7a3d4a8f588ef2878a5b2c6dba63c211c3578c7d9e6ed74 - 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-0028.json)