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], [2, 4], [3, 4], [4, 0], [4, 1]]. (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], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]. (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).
Step-by-step derivation
S001 · implies
Before
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 dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL complete directed carrier Fin(5);ORIGINAL lexicographically canonical ORDERED arcs=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4], [4, 0], [4, 1]]
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 dependencies
Rule: digraph_ordered_pairs_not_undirected_endpoint_sorting · Uses: G001, S001
S003 · calculation
Before
digraph vertex0 exact outgoing count := sum(1, 1)
After
digraph vertex0 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
ORIGINAL vertex=0;ALL outgoing original-arc indices=[0, 1];ALL incoming indices=[7]
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 dependencies
Rule: digraph_every_vertex_separate_in_out_incidence · Uses: G001, S004
S006 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
digraph vertex1 exact incoming count := sum(1, 1)
After
digraph vertex1 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
ORIGINAL vertex=1;ALL outgoing original-arc indices=[2, 3];ALL incoming indices=[0, 8]
After
outdegree=2;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 dependencies
Rule: digraph_every_vertex_separate_in_out_incidence · Uses: G001, S007
S009 · calculation
Before
digraph vertex2 exact outgoing count := sum(1, 1)
After
digraph vertex2 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
digraph vertex2 exact incoming count := sum(1, 1)
After
digraph vertex2 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
ORIGINAL vertex=2;ALL outgoing original-arc indices=[4, 5];ALL incoming indices=[1, 2]
After
outdegree=2;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 dependencies
Rule: digraph_every_vertex_separate_in_out_incidence · Uses: G001, S010
S012 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · implies
Before
ORIGINAL vertex=3;ALL outgoing original-arc indices=[6];ALL incoming indices=[3, 4]
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 dependencies
Rule: digraph_every_vertex_separate_in_out_incidence · Uses: G001, S013
S015 · calculation
Before
digraph vertex4 exact outgoing count := sum(1, 1)
After
digraph vertex4 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · calculation
Before
digraph vertex4 exact incoming count := sum(1, 1)
After
digraph vertex4 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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · implies
Before
ORIGINAL vertex=4;ALL outgoing original-arc indices=[7, 8];ALL incoming indices=[5, 6]
After
outdegree=2;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 dependencies
Rule: digraph_every_vertex_separate_in_out_incidence · Uses: G001, S016
S018 · calculation
Before
digraph complete outgoing-degree sum := sum(2, 2, 2, 1, 2)
After
digraph complete outgoing-degree sum = 9
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S017
S019 · calculation
Before
digraph complete incoming-degree sum := sum(1, 2, 2, 2, 2)
After
digraph complete incoming-degree sum = 9
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · implies
Before
ALL vertex-order outdegrees=[2, 2, 2, 1, 2];indegrees=[1, 2, 2, 2, 2];complete directed arc count=9
After
each arc counted once at its source and once at its target;SUM(out)=9=|E| and SUM(in)=9=|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 dependencies
Rule: digraph_two_separate_degree_sum_certificates · Uses: G001, S019