# Question 18946: Isolated Vertices Do Not Disconnect the Edge Support

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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.

R0036-18946 · q-20261014-advance-20261105-r04-eulerian-trail-0071



## Original question and givens

Use the complete finite simple undirected graph on Fin(23)={0,…,22}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).


- G001 — Use the complete finite simple undirected graph on Fin(23)={0,…,22}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, 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-support-connectivity;original snapshot and mathematics retained

After

explicit ordered source output fields=[nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, 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

complete ORIGINAL carrier Fin(23);canonical ORIGINAL simple undirected edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21]]

After

edge indices are EXACT original list positions0..20;retain ALL vertices;no12-vertex or10-edge bound

Conditions: none

Keep edge indices at their original numeric positions,not string-sorted vertex pairs.This family permits up to60 vertices rather than importing earlier bounds;empty-edge graphs appear only where the operation allows them.

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





S003 · calculationBefore

Euler vertex0 exact incident edge count := sum(1)

After

Euler vertex0 exact incident edge count = 1

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S004 · impliesBefore

original vertex=0;ALL original incident edge indices=[0]

After

degree=1;nonisolated=Yes;odd=Yes

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

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





S005 · calculationBefore

Euler vertex1 exact incident edge count := sum(1, 1)

After

Euler vertex1 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S006 · impliesBefore

original vertex=1;ALL original incident edge indices=[0, 1]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

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





S007 · calculationBefore

Euler vertex2 exact incident edge count := sum(1, 1)

After

Euler vertex2 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S008 · impliesBefore

original vertex=2;ALL original incident edge indices=[1, 2]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

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





S009 · calculationBefore

Euler vertex3 exact incident edge count := sum(1, 1)

After

Euler vertex3 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S010 · impliesBefore

original vertex=3;ALL original incident edge indices=[2, 3]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

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





S011 · calculationBefore

Euler vertex4 exact incident edge count := sum(1, 1)

After

Euler vertex4 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S012 · impliesBefore

original vertex=4;ALL original incident edge indices=[3, 4]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

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





S013 · calculationBefore

Euler vertex5 exact incident edge count := sum(1, 1)

After

Euler vertex5 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S014 · impliesBefore

original vertex=5;ALL original incident edge indices=[4, 5]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

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





S015 · calculationBefore

Euler vertex6 exact incident edge count := sum(1, 1)

After

Euler vertex6 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S016 · impliesBefore

original vertex=6;ALL original incident edge indices=[5, 6]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

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





S017 · calculationBefore

Euler vertex7 exact incident edge count := sum(1, 1)

After

Euler vertex7 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S018 · impliesBefore

original vertex=7;ALL original incident edge indices=[6, 7]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

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





S019 · calculationBefore

Euler vertex8 exact incident edge count := sum(1, 1)

After

Euler vertex8 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S020 · impliesBefore

original vertex=8;ALL original incident edge indices=[7, 8]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

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





S021 · calculationBefore

Euler vertex9 exact incident edge count := sum(1, 1)

After

Euler vertex9 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S022 · impliesBefore

original vertex=9;ALL original incident edge indices=[8, 9]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

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





S023 · calculationBefore

Euler vertex10 exact incident edge count := sum(1, 1)

After

Euler vertex10 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · impliesBefore

original vertex=10;ALL original incident edge indices=[9, 10]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S023





S025 · calculationBefore

Euler vertex11 exact incident edge count := sum(1, 1)

After

Euler vertex11 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · impliesBefore

original vertex=11;ALL original incident edge indices=[10, 11]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S025





S027 · calculationBefore

Euler vertex12 exact incident edge count := sum(1, 1)

After

Euler vertex12 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · impliesBefore

original vertex=12;ALL original incident edge indices=[11, 12]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S027





S029 · calculationBefore

Euler vertex13 exact incident edge count := sum(1, 1)

After

Euler vertex13 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · impliesBefore

original vertex=13;ALL original incident edge indices=[12, 13]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S029





S031 · calculationBefore

Euler vertex14 exact incident edge count := sum(1, 1)

After

Euler vertex14 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · impliesBefore

original vertex=14;ALL original incident edge indices=[13, 14]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S031





S033 · calculationBefore

Euler vertex15 exact incident edge count := sum(1, 1)

After

Euler vertex15 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · impliesBefore

original vertex=15;ALL original incident edge indices=[14, 15]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S033





S035 · calculationBefore

Euler vertex16 exact incident edge count := sum(1, 1)

After

Euler vertex16 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · impliesBefore

original vertex=16;ALL original incident edge indices=[15, 16]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S035





S037 · calculationBefore

Euler vertex17 exact incident edge count := sum(1, 1)

After

Euler vertex17 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · impliesBefore

original vertex=17;ALL original incident edge indices=[16, 17]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S037





S039 · calculationBefore

Euler vertex18 exact incident edge count := sum(1, 1)

After

Euler vertex18 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · impliesBefore

original vertex=18;ALL original incident edge indices=[17, 18]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S039





S041 · calculationBefore

Euler vertex19 exact incident edge count := sum(1, 1)

After

Euler vertex19 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · impliesBefore

original vertex=19;ALL original incident edge indices=[18, 19]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S041





S043 · calculationBefore

Euler vertex20 exact incident edge count := sum(1, 1)

After

Euler vertex20 exact incident edge count = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · impliesBefore

original vertex=20;ALL original incident edge indices=[19, 20]

After

degree=2;nonisolated=Yes;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S043





S045 · calculationBefore

Euler vertex21 exact incident edge count := sum(1)

After

Euler vertex21 exact incident edge count = 1

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044





S046 · impliesBefore

original vertex=21;ALL original incident edge indices=[20]

After

degree=1;nonisolated=Yes;odd=Yes

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S045





S047 · calculationBefore

Euler vertex22 exact incident edge count := sum()

After

Euler vertex22 exact incident edge count = 0

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · impliesBefore

original vertex=22;ALL original incident edge indices=[]

After

degree=0;nonisolated=No;odd=No

Conditions: none

Count this vertex's incident edges in the complete original list.Retain degree0 vertices in the full carrier;neither vertex labels nor support size are degrees.

Step identifiers and dependenciesRule: euler_degree_actual_incidence_not_label_or_support_size · Uses: G001, S047





S049 · impliesBefore

nonisolated edge support:original edge index=0;edge=[0, 1];from=0;new vertex=1

After

original-edge path from0=[0, 1];reached=[0, 1]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S048





S050 · impliesBefore

nonisolated edge support:original edge index=1;edge=[1, 2];from=1;new vertex=2

After

original-edge path from0=[0, 1, 2];reached=[0, 1, 2]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S049





S051 · impliesBefore

nonisolated edge support:original edge index=2;edge=[2, 3];from=2;new vertex=3

After

original-edge path from0=[0, 1, 2, 3];reached=[0, 1, 2, 3]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S050





S052 · impliesBefore

nonisolated edge support:original edge index=3;edge=[3, 4];from=3;new vertex=4

After

original-edge path from0=[0, 1, 2, 3, 4];reached=[0, 1, 2, 3, 4]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S051





S053 · impliesBefore

nonisolated edge support:original edge index=4;edge=[4, 5];from=4;new vertex=5

After

original-edge path from0=[0, 1, 2, 3, 4, 5];reached=[0, 1, 2, 3, 4, 5]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S052





S054 · impliesBefore

nonisolated edge support:original edge index=5;edge=[5, 6];from=5;new vertex=6

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6];reached=[0, 1, 2, 3, 4, 5, 6]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S053





S055 · impliesBefore

nonisolated edge support:original edge index=6;edge=[6, 7];from=6;new vertex=7

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7];reached=[0, 1, 2, 3, 4, 5, 6, 7]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S054





S056 · impliesBefore

nonisolated edge support:original edge index=7;edge=[7, 8];from=7;new vertex=8

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S055





S057 · impliesBefore

nonisolated edge support:original edge index=8;edge=[8, 9];from=8;new vertex=9

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S056





S058 · impliesBefore

nonisolated edge support:original edge index=9;edge=[9, 10];from=9;new vertex=10

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S057





S059 · impliesBefore

nonisolated edge support:original edge index=10;edge=[10, 11];from=10;new vertex=11

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S058





S060 · impliesBefore

nonisolated edge support:original edge index=11;edge=[11, 12];from=11;new vertex=12

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S059





S061 · impliesBefore

nonisolated edge support:original edge index=12;edge=[12, 13];from=12;new vertex=13

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S060





S062 · impliesBefore

nonisolated edge support:original edge index=13;edge=[13, 14];from=13;new vertex=14

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S061





S063 · impliesBefore

nonisolated edge support:original edge index=14;edge=[14, 15];from=14;new vertex=15

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S062





S064 · impliesBefore

nonisolated edge support:original edge index=15;edge=[15, 16];from=15;new vertex=16

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S063





S065 · impliesBefore

nonisolated edge support:original edge index=16;edge=[16, 17];from=16;new vertex=17

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S064





S066 · impliesBefore

nonisolated edge support:original edge index=17;edge=[17, 18];from=17;new vertex=18

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S065





S067 · impliesBefore

nonisolated edge support:original edge index=18;edge=[18, 19];from=18;new vertex=19

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S066





S068 · impliesBefore

nonisolated edge support:original edge index=19;edge=[19, 20];from=19;new vertex=20

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S067





S069 · impliesBefore

nonisolated edge support:original edge index=20;edge=[20, 21];from=20;new vertex=21

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S068





S070 · impliesBefore

nonisolated edge support:complete permitted carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];unreached=[];ALL original edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21]]

After

no edge in carrier crosses reached/unreached cut;connected=Yes

Conditions: none

Exhausting BFS proves no edge crosses the cut,so an unreached vertex has no path from the start;otherwise path witnesses cover the carrier.Audit support and full carrier separately without silently deleting isolates to claim whole-graph connectivity.

Step identifiers and dependenciesRule: euler_complete_reachability_closed_cut_not_isolate_deletion · Uses: G001, S069





S071 · impliesBefore

whole original carrier:original edge index=0;edge=[0, 1];from=0;new vertex=1

After

original-edge path from0=[0, 1];reached=[0, 1]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S070





S072 · impliesBefore

whole original carrier:original edge index=1;edge=[1, 2];from=1;new vertex=2

After

original-edge path from0=[0, 1, 2];reached=[0, 1, 2]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S071





S073 · impliesBefore

whole original carrier:original edge index=2;edge=[2, 3];from=2;new vertex=3

After

original-edge path from0=[0, 1, 2, 3];reached=[0, 1, 2, 3]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S072





S074 · impliesBefore

whole original carrier:original edge index=3;edge=[3, 4];from=3;new vertex=4

After

original-edge path from0=[0, 1, 2, 3, 4];reached=[0, 1, 2, 3, 4]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S073





S075 · impliesBefore

whole original carrier:original edge index=4;edge=[4, 5];from=4;new vertex=5

After

original-edge path from0=[0, 1, 2, 3, 4, 5];reached=[0, 1, 2, 3, 4, 5]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S074





S076 · impliesBefore

whole original carrier:original edge index=5;edge=[5, 6];from=5;new vertex=6

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6];reached=[0, 1, 2, 3, 4, 5, 6]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S075





S077 · impliesBefore

whole original carrier:original edge index=6;edge=[6, 7];from=6;new vertex=7

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7];reached=[0, 1, 2, 3, 4, 5, 6, 7]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S076





S078 · impliesBefore

whole original carrier:original edge index=7;edge=[7, 8];from=7;new vertex=8

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S077





S079 · impliesBefore

whole original carrier:original edge index=8;edge=[8, 9];from=8;new vertex=9

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S078





S080 · impliesBefore

whole original carrier:original edge index=9;edge=[9, 10];from=9;new vertex=10

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S079





S081 · impliesBefore

whole original carrier:original edge index=10;edge=[10, 11];from=10;new vertex=11

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S080





S082 · impliesBefore

whole original carrier:original edge index=11;edge=[11, 12];from=11;new vertex=12

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S081





S083 · impliesBefore

whole original carrier:original edge index=12;edge=[12, 13];from=12;new vertex=13

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S082





S084 · impliesBefore

whole original carrier:original edge index=13;edge=[13, 14];from=13;new vertex=14

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S083





S085 · impliesBefore

whole original carrier:original edge index=14;edge=[14, 15];from=14;new vertex=15

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S084





S086 · impliesBefore

whole original carrier:original edge index=15;edge=[15, 16];from=15;new vertex=16

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S085





S087 · impliesBefore

whole original carrier:original edge index=16;edge=[16, 17];from=16;new vertex=17

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S086





S088 · impliesBefore

whole original carrier:original edge index=17;edge=[17, 18];from=17;new vertex=18

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S087





S089 · impliesBefore

whole original carrier:original edge index=18;edge=[18, 19];from=18;new vertex=19

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S088





S090 · impliesBefore

whole original carrier:original edge index=19;edge=[19, 20];from=19;new vertex=20

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S089





S091 · impliesBefore

whole original carrier:original edge index=20;edge=[20, 21];from=20;new vertex=21

After

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]

Conditions: none

Give an actual-edge path for every discovered vertex.Start at the first actual support vertex rather than assuming vertex0 has an edge.

Step identifiers and dependenciesRule: euler_connectivity_actual_path_witness · Uses: G001, S090





S092 · impliesBefore

whole original carrier:complete permitted carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];unreached=[22];ALL original edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21]]

After

no edge in carrier crosses reached/unreached cut;connected=No

Conditions: none

Exhausting BFS proves no edge crosses the cut,so an unreached vertex has no path from the start;otherwise path witnesses cover the carrier.Audit support and full carrier separately without silently deleting isolates to claim whole-graph connectivity.

Step identifiers and dependenciesRule: euler_complete_reachability_closed_cut_not_isolate_deletion · Uses: G001, S091





S093 · impliesBefore

complete degree list=[1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 0];ALL odd vertices=[0, 21];actual nonisolated support=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];support connected=Yes;whole graph connected=No

After

ANY full-edge trail joins all nonisolated vertices in one support component;each internal visit pairs incident edges,leaving only0 or2 odd endpoints

Conditions: none

Any all-edge trail puts every nonisolated vertex into its single walk component.Each internal entry/exit pairs incident edges;only distinct endpoints can leave one unmatched incidence,so there are0 or2 odd vertices.This is the universal necessity argument,not a guess from one tested walk.Isolates are outside edge support.

Step identifiers and dependenciesRule: euler_necessary_connected_support_and_degree_parity · Uses: G001, S092





S094 · impliesBefore

Hierholzer stack top=0;first unused incident ORIGINAL edge index=0;edge=[0, 1]

After

mark edge used ONCE;push vertex=1;current stack=[0, 1];used edge set=[0]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S093





S095 · impliesBefore

Hierholzer stack top=1;first unused incident ORIGINAL edge index=1;edge=[1, 2]

After

mark edge used ONCE;push vertex=2;current stack=[0, 1, 2];used edge set=[0, 1]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S094





S096 · impliesBefore

Hierholzer stack top=2;first unused incident ORIGINAL edge index=2;edge=[2, 3]

After

mark edge used ONCE;push vertex=3;current stack=[0, 1, 2, 3];used edge set=[0, 1, 2]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S095





S097 · impliesBefore

Hierholzer stack top=3;first unused incident ORIGINAL edge index=3;edge=[3, 4]

After

mark edge used ONCE;push vertex=4;current stack=[0, 1, 2, 3, 4];used edge set=[0, 1, 2, 3]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S096





S098 · impliesBefore

Hierholzer stack top=4;first unused incident ORIGINAL edge index=4;edge=[4, 5]

After

mark edge used ONCE;push vertex=5;current stack=[0, 1, 2, 3, 4, 5];used edge set=[0, 1, 2, 3, 4]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S097





S099 · impliesBefore

Hierholzer stack top=5;first unused incident ORIGINAL edge index=5;edge=[5, 6]

After

mark edge used ONCE;push vertex=6;current stack=[0, 1, 2, 3, 4, 5, 6];used edge set=[0, 1, 2, 3, 4, 5]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S098





S100 · impliesBefore

Hierholzer stack top=6;first unused incident ORIGINAL edge index=6;edge=[6, 7]

After

mark edge used ONCE;push vertex=7;current stack=[0, 1, 2, 3, 4, 5, 6, 7];used edge set=[0, 1, 2, 3, 4, 5, 6]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S099





S101 · impliesBefore

Hierholzer stack top=7;first unused incident ORIGINAL edge index=7;edge=[7, 8]

After

mark edge used ONCE;push vertex=8;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8];used edge set=[0, 1, 2, 3, 4, 5, 6, 7]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S100





S102 · impliesBefore

Hierholzer stack top=8;first unused incident ORIGINAL edge index=8;edge=[8, 9]

After

mark edge used ONCE;push vertex=9;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S101





S103 · impliesBefore

Hierholzer stack top=9;first unused incident ORIGINAL edge index=9;edge=[9, 10]

After

mark edge used ONCE;push vertex=10;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S102





S104 · impliesBefore

Hierholzer stack top=10;first unused incident ORIGINAL edge index=10;edge=[10, 11]

After

mark edge used ONCE;push vertex=11;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S103





S105 · impliesBefore

Hierholzer stack top=11;first unused incident ORIGINAL edge index=11;edge=[11, 12]

After

mark edge used ONCE;push vertex=12;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S104





S106 · impliesBefore

Hierholzer stack top=12;first unused incident ORIGINAL edge index=12;edge=[12, 13]

After

mark edge used ONCE;push vertex=13;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S105





S107 · impliesBefore

Hierholzer stack top=13;first unused incident ORIGINAL edge index=13;edge=[13, 14]

After

mark edge used ONCE;push vertex=14;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S106





S108 · impliesBefore

Hierholzer stack top=14;first unused incident ORIGINAL edge index=14;edge=[14, 15]

After

mark edge used ONCE;push vertex=15;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S107





S109 · impliesBefore

Hierholzer stack top=15;first unused incident ORIGINAL edge index=15;edge=[15, 16]

After

mark edge used ONCE;push vertex=16;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S108





S110 · impliesBefore

Hierholzer stack top=16;first unused incident ORIGINAL edge index=16;edge=[16, 17]

After

mark edge used ONCE;push vertex=17;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S109





S111 · impliesBefore

Hierholzer stack top=17;first unused incident ORIGINAL edge index=17;edge=[17, 18]

After

mark edge used ONCE;push vertex=18;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S110





S112 · impliesBefore

Hierholzer stack top=18;first unused incident ORIGINAL edge index=18;edge=[18, 19]

After

mark edge used ONCE;push vertex=19;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S111





S113 · impliesBefore

Hierholzer stack top=19;first unused incident ORIGINAL edge index=19;edge=[19, 20]

After

mark edge used ONCE;push vertex=20;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S112





S114 · impliesBefore

Hierholzer stack top=20;first unused incident ORIGINAL edge index=20;edge=[20, 21]

After

mark edge used ONCE;push vertex=21;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]

Conditions: none

Extend along an actual unused incident edge and mark it once.The finite used-edge count increases;vertices may recur but edges do not.

Step identifiers and dependenciesRule: euler_constructive_unused_edge_extension · Uses: G001, S113





S115 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];top vertex has NO unused incident edge

After

finalize top into reverse output=[21];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S114





S116 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S115





S117 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S116





S118 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S117





S119 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S118





S120 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S119





S121 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S120





S122 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S121





S123 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S122





S124 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S123





S125 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S124





S126 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S125





S127 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S126





S128 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S127





S129 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7];remaining stack=[0, 1, 2, 3, 4, 5, 6]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S128





S130 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6];remaining stack=[0, 1, 2, 3, 4, 5]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S129





S131 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4, 5];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5];remaining stack=[0, 1, 2, 3, 4]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S130





S132 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3, 4];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4];remaining stack=[0, 1, 2, 3]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S131





S133 · impliesBefore

Hierholzer current stack=[0, 1, 2, 3];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3];remaining stack=[0, 1, 2]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S132





S134 · impliesBefore

Hierholzer current stack=[0, 1, 2];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2];remaining stack=[0, 1]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S133





S135 · impliesBefore

Hierholzer current stack=[0, 1];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1];remaining stack=[0]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S134





S136 · impliesBefore

Hierholzer current stack=[0];top vertex has NO unused incident edge

After

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0];remaining stack=[]

Conditions: none

Backtrack and collect vertices in reverse only when no unused incident edge remains.This records the splicing of closed segments without traversing nonexistent edges.

Step identifiers and dependenciesRule: euler_constructive_backtracking_cycle_splice · Uses: G001, S135





S137 · impliesBefore

connected nonisolated support=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];odd vertices=[0, 21];finished reversed construction

After

ACTUAL Euler witness walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];actual ORIGINAL edge order=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];every original edge once;endpoints have required parity

Conditions: none

Map every step of the final walk back to original edges and verify length m+1,unique indices and complete coverage.This directly witnesses existence in this graph.With no odd vertices it closes;with two its endpoints are precisely those vertices.Parity alone is not the existence certificate,and no Hamiltonian property follows.

Step identifiers and dependenciesRule: euler_sufficiency_actual_full_edge_witness_not_parity_alone · Uses: G001, S136





S138 · impliesBefore

nonisolated support=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];support connected=Yes;WHOLE carrier Fin(23) connected=No

After

Euler type=open_euler_trail;isolated_vertices_do_not_break_edge_support_connectivity

Conditions: none

Euler's objective is using every edge,not visiting every vertex.Retain the two distinct connectivity fields without substituting one for the other.No Hamiltonian conclusion is supplied.

Step identifiers and dependenciesRule: euler_isolates_not_full_connectivity_or_Hamilton_claim · Uses: G001, S137









## Reverse checks


- S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S047 · premise_and_result_audit — Recompute Euler vertex22 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S045 · premise_and_result_audit — Recompute Euler vertex21 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S043 · premise_and_result_audit — Recompute Euler vertex20 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S041 · premise_and_result_audit — Recompute Euler vertex19 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S039 · premise_and_result_audit — Recompute Euler vertex18 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S037 · premise_and_result_audit — Recompute Euler vertex17 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S035 · premise_and_result_audit — Recompute Euler vertex16 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S033 · premise_and_result_audit — Recompute Euler vertex15 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S031 · premise_and_result_audit — Recompute Euler vertex14 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S029 · premise_and_result_audit — Recompute Euler vertex13 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S027 · premise_and_result_audit — Recompute Euler vertex12 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S025 · premise_and_result_audit — Recompute Euler vertex11 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S023 · premise_and_result_audit — Recompute Euler vertex10 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S021 · premise_and_result_audit — Recompute Euler vertex9 exact incident edge count with exact rational arithmetic, not rounded decimals.
- 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 Euler vertex8 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S017 · premise_and_result_audit — Recompute Euler vertex7 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S015 · premise_and_result_audit — Recompute Euler vertex6 exact incident edge 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 Euler vertex5 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S011 · premise_and_result_audit — Recompute Euler vertex4 exact incident edge count with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Recompute Euler vertex3 exact incident edge 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 Euler vertex2 exact incident edge count with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Euler vertex1 exact incident edge count with exact rational arithmetic, not rounded decimals.
- 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 — Recompute Euler vertex0 exact incident edge 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

([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], Yes, No, [0, 21], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

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: 97ea687be4de265dfa47f01af031fa04d115b48be4b2a060f574e9194be31764
- candidate record SHA-256: d8369250e9daee6a8a028ca2d1e26455a450d1b85e4e77498cf1a07fee54782e
- 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-0071.json)