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P1000-0786 · q-20261015-advance-20261116-r06-topological-ordering-dag-0036
Use unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free directed edges=[[v0, v1], [v0, v2], [v0, v4]]. Every edge u→v needs position(u)<position(v) in a complete vertex permutation; labels use source lexical order. Give (vertices, edges, all_valid_orders, order_count, order_exists, certificate).
G001 — Use unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free directed edges=[[v0, v1], [v0, v2], [v0, v4]]. Every edge u→v needs position(u)<position(v) in a complete vertex permutation; labels use source lexical order. Give (vertices, edges, all_valid_orders, order_count, order_exists, certificate).For a finite digraph every valid order starts at a zero-indegree vertex; branching on that first vertex lists every order once.
Before
V=[v0, v1, v2, v3, v4, v5]
After
all directed edges=[[v0, v1], [v0, v2], [v0, v4]]
Conditions: none
This is a fully specified finite digraph, not necessarily a complete digraph.
Rule: full_directed_graph · Uses: G001
Before
no vertices remain
After
one empty completion; C(∅)=1
Conditions: none
The empty remaining problem has one completion: place no further vertex.
Rule: empty_order_base · Uses: G001, S001
Before
C(v5) := sum(1)
After
C(v5) = 1
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
remaining=[v5]; eligible first vertices=[v5]
After
complete continuations=1
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S003
Before
C(v4) := sum(1)
After
C(v4) = 1
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
remaining=[v4]; eligible first vertices=[v4]
After
complete continuations=1
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S005
Before
C(v4,v5) := sum(1, 1)
After
C(v4,v5) = 2
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
remaining=[v4, v5]; eligible first vertices=[v4, v5]
After
complete continuations=2
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S007
Before
C(v3) := sum(1)
After
C(v3) = 1
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
remaining=[v3]; eligible first vertices=[v3]
After
complete continuations=1
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S009
Before
C(v3,v5) := sum(1, 1)
After
C(v3,v5) = 2
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
remaining=[v3, v5]; eligible first vertices=[v3, v5]
After
complete continuations=2
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S011
Before
C(v3,v4) := sum(1, 1)
After
C(v3,v4) = 2
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
remaining=[v3, v4]; eligible first vertices=[v3, v4]
After
complete continuations=2
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S013
Before
C(v3,v4,v5) := sum(2, 2, 2)
After
C(v3,v4,v5) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
remaining=[v3, v4, v5]; eligible first vertices=[v3, v4, v5]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S015
Before
C(v2) := sum(1)
After
C(v2) = 1
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
remaining=[v2]; eligible first vertices=[v2]
After
complete continuations=1
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S017
Before
C(v2,v5) := sum(1, 1)
After
C(v2,v5) = 2
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
remaining=[v2, v5]; eligible first vertices=[v2, v5]
After
complete continuations=2
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S019
Before
C(v2,v4) := sum(1, 1)
After
C(v2,v4) = 2
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
remaining=[v2, v4]; eligible first vertices=[v2, v4]
After
complete continuations=2
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S021
Before
C(v2,v4,v5) := sum(2, 2, 2)
After
C(v2,v4,v5) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
remaining=[v2, v4, v5]; eligible first vertices=[v2, v4, v5]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S023
Before
C(v2,v3) := sum(1, 1)
After
C(v2,v3) = 2
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
remaining=[v2, v3]; eligible first vertices=[v2, v3]
After
complete continuations=2
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S025
Before
C(v2,v3,v5) := sum(2, 2, 2)
After
C(v2,v3,v5) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
remaining=[v2, v3, v5]; eligible first vertices=[v2, v3, v5]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S027
Before
C(v2,v3,v4) := sum(2, 2, 2)
After
C(v2,v3,v4) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
remaining=[v2, v3, v4]; eligible first vertices=[v2, v3, v4]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S029
Before
C(v2,v3,v4,v5) := sum(6, 6, 6, 6)
After
C(v2,v3,v4,v5) = 24
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
remaining=[v2, v3, v4, v5]; eligible first vertices=[v2, v3, v4, v5]
After
complete continuations=24
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S031
Before
C(v1) := sum(1)
After
C(v1) = 1
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
remaining=[v1]; eligible first vertices=[v1]
After
complete continuations=1
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S033
Before
C(v1,v5) := sum(1, 1)
After
C(v1,v5) = 2
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
remaining=[v1, v5]; eligible first vertices=[v1, v5]
After
complete continuations=2
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S035
Before
C(v1,v4) := sum(1, 1)
After
C(v1,v4) = 2
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
remaining=[v1, v4]; eligible first vertices=[v1, v4]
After
complete continuations=2
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S037
Before
C(v1,v4,v5) := sum(2, 2, 2)
After
C(v1,v4,v5) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
remaining=[v1, v4, v5]; eligible first vertices=[v1, v4, v5]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S039
Before
C(v1,v3) := sum(1, 1)
After
C(v1,v3) = 2
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
remaining=[v1, v3]; eligible first vertices=[v1, v3]
After
complete continuations=2
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S041
Before
C(v1,v3,v5) := sum(2, 2, 2)
After
C(v1,v3,v5) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
remaining=[v1, v3, v5]; eligible first vertices=[v1, v3, v5]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S043
Before
C(v1,v3,v4) := sum(2, 2, 2)
After
C(v1,v3,v4) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
remaining=[v1, v3, v4]; eligible first vertices=[v1, v3, v4]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S045
Before
C(v1,v3,v4,v5) := sum(6, 6, 6, 6)
After
C(v1,v3,v4,v5) = 24
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
remaining=[v1, v3, v4, v5]; eligible first vertices=[v1, v3, v4, v5]
After
complete continuations=24
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S047
Before
C(v1,v2) := sum(1, 1)
After
C(v1,v2) = 2
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
remaining=[v1, v2]; eligible first vertices=[v1, v2]
After
complete continuations=2
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S049
Before
C(v1,v2,v5) := sum(2, 2, 2)
After
C(v1,v2,v5) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
remaining=[v1, v2, v5]; eligible first vertices=[v1, v2, v5]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S051
Before
C(v1,v2,v4) := sum(2, 2, 2)
After
C(v1,v2,v4) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
remaining=[v1, v2, v4]; eligible first vertices=[v1, v2, v4]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S053
Before
C(v1,v2,v4,v5) := sum(6, 6, 6, 6)
After
C(v1,v2,v4,v5) = 24
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
remaining=[v1, v2, v4, v5]; eligible first vertices=[v1, v2, v4, v5]
After
complete continuations=24
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S055
Before
C(v1,v2,v3) := sum(2, 2, 2)
After
C(v1,v2,v3) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
remaining=[v1, v2, v3]; eligible first vertices=[v1, v2, v3]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S057
Before
C(v1,v2,v3,v5) := sum(6, 6, 6, 6)
After
C(v1,v2,v3,v5) = 24
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
remaining=[v1, v2, v3, v5]; eligible first vertices=[v1, v2, v3, v5]
After
complete continuations=24
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S059
Before
C(v1,v2,v3,v4) := sum(6, 6, 6, 6)
After
C(v1,v2,v3,v4) = 24
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
remaining=[v1, v2, v3, v4]; eligible first vertices=[v1, v2, v3, v4]
After
complete continuations=24
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S061
Before
C(v1,v2,v3,v4,v5) := sum(24, 24, 24, 24, 24)
After
C(v1,v2,v3,v4,v5) = 120
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
remaining=[v1, v2, v3, v4, v5]; eligible first vertices=[v1, v2, v3, v4, v5]
After
complete continuations=120
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S063
Before
C(v0,v1,v2,v4) := sum(6)
After
C(v0,v1,v2,v4) = 6
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
remaining=[v0, v1, v2, v4]; eligible first vertices=[v0]
After
complete continuations=6
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S065
Before
C(v0,v1,v2,v4,v5) := sum(24, 6)
After
C(v0,v1,v2,v4,v5) = 30
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
remaining=[v0, v1, v2, v4, v5]; eligible first vertices=[v0, v5]
After
complete continuations=30
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S067
Before
C(v0,v1,v2,v3,v4) := sum(24, 6)
After
C(v0,v1,v2,v3,v4) = 30
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S068
Before
remaining=[v0, v1, v2, v3, v4]; eligible first vertices=[v0, v3]
After
complete continuations=30
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S069
Before
C(v0,v1,v2,v3,v4,v5) := sum(120, 30, 30)
After
C(v0,v1,v2,v3,v4,v5) = 180
Conditions: none
Each legal first vertex defines a disjoint branch; add the branch completion counts.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
remaining=[v0, v1, v2, v3, v4, v5]; eligible first vertices=[v0, v3, v5]
After
complete continuations=180
Conditions: none
The first vertex of every legal order has zero remaining indegree. Conversely any such source followed by a legal tail gives an order, so these branches are exhaustive.
Rule: all_source_branches · Uses: G001, S071
Before
all root branches; count=180
After
enumeration contains each legal permutation exactly once
Conditions: none
Distinct first-vertex branches do not overlap and their recursive tails do not duplicate, so the answer retains every and only valid order.
Rule: enumeration_completeness · Uses: G001, S072
Before
valid order count=180
After
at least one exists: Yes
Conditions: none
An existing valid order certifies acyclicity of this finite digraph; multiple valid orders are compatible.
Rule: order_existence · Uses: G001, S073
S074 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S073 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S072 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S071 · premise_and_result_audit — Recompute C(v0,v1,v2,v3,v4,v5) with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S069 · premise_and_result_audit — Recompute C(v0,v1,v2,v3,v4) with exact rational arithmetic, not rounded decimals.S068 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S067 · premise_and_result_audit — Recompute C(v0,v1,v2,v4,v5) with exact rational arithmetic, not rounded decimals.S066 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S065 · premise_and_result_audit — Recompute C(v0,v1,v2,v4) with exact rational arithmetic, not rounded decimals.S064 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S063 · premise_and_result_audit — Recompute C(v1,v2,v3,v4,v5) with exact rational arithmetic, not rounded decimals.S062 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S061 · premise_and_result_audit — Recompute C(v1,v2,v3,v4) with exact rational arithmetic, not rounded decimals.S060 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S059 · premise_and_result_audit — Recompute C(v1,v2,v3,v5) with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S057 · premise_and_result_audit — Recompute C(v1,v2,v3) with exact rational arithmetic, not rounded decimals.S056 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S055 · premise_and_result_audit — Recompute C(v1,v2,v4,v5) with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S053 · premise_and_result_audit — Recompute C(v1,v2,v4) with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S051 · premise_and_result_audit — Recompute C(v1,v2,v5) with exact rational arithmetic, not rounded decimals.S050 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S049 · premise_and_result_audit — Recompute C(v1,v2) with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S047 · premise_and_result_audit — Recompute C(v1,v3,v4,v5) with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S045 · premise_and_result_audit — Recompute C(v1,v3,v4) with exact rational arithmetic, not rounded decimals.S044 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S043 · premise_and_result_audit — Recompute C(v1,v3,v5) with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S041 · premise_and_result_audit — Recompute C(v1,v3) with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S039 · premise_and_result_audit — Recompute C(v1,v4,v5) with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S037 · premise_and_result_audit — Recompute C(v1,v4) with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S035 · premise_and_result_audit — Recompute C(v1,v5) with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S033 · premise_and_result_audit — Recompute C(v1) with exact rational arithmetic, not rounded decimals.S032 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S031 · premise_and_result_audit — Recompute C(v2,v3,v4,v5) with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S029 · premise_and_result_audit — Recompute C(v2,v3,v4) with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S027 · premise_and_result_audit — Recompute C(v2,v3,v5) with exact rational arithmetic, not rounded decimals.S026 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S025 · premise_and_result_audit — Recompute C(v2,v3) with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S023 · premise_and_result_audit — Recompute C(v2,v4,v5) with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S021 · premise_and_result_audit — Recompute C(v2,v4) with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S019 · premise_and_result_audit — Recompute C(v2,v5) with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S017 · premise_and_result_audit — Recompute C(v2) with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S015 · premise_and_result_audit — Recompute C(v3,v4,v5) with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S013 · premise_and_result_audit — Recompute C(v3,v4) with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S011 · premise_and_result_audit — Recompute C(v3,v5) with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S009 · premise_and_result_audit — Recompute C(v3) with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S007 · premise_and_result_audit — Recompute C(v4,v5) with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S005 · premise_and_result_audit — Recompute C(v4) with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S003 · premise_and_result_audit — Recompute C(v5) with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.S001 · premise_and_result_audit — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v4]], [[v0, v1, v2, v3, v4, v5], [v0, v1, v2, v3, v5, v4], [v0, v1, v2, v4, v3, v5], [v0, v1, v2, v4, v5, v3], [v0, v1, v2, v5, v3, v4], [v0, v1, v2, v5, v4, v3], [v0, v1, v3, v2, v4, v5], [v0, v1, v3, v2, v5, v4], [v0, v1, v3, v4, v2, v5], [v0, v1, v3, v4, v5, v2], [v0, v1, v3, v5, v2, v4], [v0, v1, v3, v5, v4, v2], [v0, v1, v4, v2, v3, v5], [v0, v1, v4, v2, v5, v3], [v0, v1, v4, v3, v2, v5], [v0, v1, v4, v3, v5, v2], [v0, v1, v4, v5, v2, v3], [v0, v1, v4, v5, v3, v2], [v0, v1, v5, v2, v3, v4], [v0, v1, v5, v2, v4, v3], [v0, v1, v5, v3, v2, v4], [v0, v1, v5, v3, v4, v2], [v0, v1, v5, v4, v2, v3], [v0, v1, v5, v4, v3, v2], [v0, v2, v1, v3, v4, v5], [v0, v2, v1, v3, v5, v4], [v0, v2, v1, v4, v3, v5], [v0, v2, v1, v4, v5, v3], [v0, v2, v1, v5, v3, v4], [v0, v2, v1, v5, v4, v3], [v0, v2, v3, v1, v4, v5], [v0, v2, v3, v1, v5, v4], [v0, v2, v3, v4, v1, v5], [v0, v2, v3, v4, v5, v1], [v0, v2, v3, v5, v1, v4], [v0, v2, v3, v5, v4, v1], [v0, v2, v4, v1, v3, v5], [v0, v2, v4, v1, v5, v3], [v0, v2, v4, v3, v1, v5], [v0, v2, v4, v3, v5, v1], [v0, v2, v4, v5, v1, v3], [v0, v2, v4, v5, v3, v1], [v0, v2, v5, v1, v3, v4], [v0, v2, v5, v1, v4, v3], [v0, v2, v5, v3, v1, v4], [v0, v2, v5, v3, v4, v1], [v0, v2, v5, v4, v1, v3], [v0, v2, v5, v4, v3, v1], [v0, v3, v1, v2, v4, v5], [v0, v3, v1, v2, v5, v4], [v0, v3, v1, v4, v2, v5], [v0, v3, v1, v4, v5, v2], [v0, v3, v1, v5, v2, v4], [v0, v3, v1, v5, v4, v2], [v0, v3, v2, v1, v4, v5], [v0, v3, v2, v1, v5, v4], [v0, v3, v2, v4, v1, v5], [v0, v3, v2, v4, v5, v1], [v0, v3, v2, v5, v1, v4], [v0, v3, v2, v5, v4, v1], [v0, v3, v4, v1, v2, v5], [v0, v3, v4, v1, v5, v2], [v0, v3, v4, v2, v1, v5], [v0, v3, v4, v2, v5, v1], [v0, v3, v4, v5, v1, v2], [v0, v3, v4, v5, v2, v1], [v0, v3, v5, v1, v2, v4], [v0, v3, v5, v1, v4, v2], [v0, v3, v5, v2, v1, v4], [v0, v3, v5, v2, v4, v1], [v0, v3, v5, v4, v1, v2], [v0, v3, v5, v4, v2, v1], [v0, v4, v1, v2, v3, v5], [v0, v4, v1, v2, v5, v3], [v0, v4, v1, v3, v2, v5], [v0, v4, v1, v3, v5, v2], [v0, v4, v1, v5, v2, v3], [v0, v4, v1, v5, v3, v2], [v0, v4, v2, v1, v3, v5], [v0, v4, v2, v1, v5, v3], [v0, v4, v2, v3, v1, v5], [v0, v4, v2, v3, v5, v1], [v0, v4, v2, v5, v1, v3], [v0, v4, v2, v5, v3, v1], [v0, v4, v3, v1, v2, v5], [v0, v4, v3, v1, v5, v2], [v0, v4, v3, v2, v1, v5], [v0, v4, v3, v2, v5, v1], [v0, v4, v3, v5, v1, v2], [v0, v4, v3, v5, v2, v1], [v0, v4, v5, v1, v2, v3], [v0, v4, v5, v1, v3, v2], [v0, v4, v5, v2, v1, v3], [v0, v4, v5, v2, v3, v1], [v0, v4, v5, v3, v1, v2], [v0, v4, v5, v3, v2, v1], [v0, v5, v1, v2, v3, v4], [v0, v5, v1, v2, v4, v3], [v0, v5, v1, v3, v2, v4], [v0, v5, v1, v3, v4, v2], [v0, v5, v1, v4, v2, v3], [v0, v5, v1, v4, v3, v2], [v0, v5, v2, v1, v3, v4], [v0, v5, v2, v1, v4, v3], [v0, v5, v2, v3, v1, v4], [v0, v5, v2, v3, v4, v1], [v0, v5, v2, v4, v1, v3], [v0, v5, v2, v4, v3, v1], [v0, v5, v3, v1, v2, v4], [v0, v5, v3, v1, v4, v2], [v0, v5, v3, v2, v1, v4], [v0, v5, v3, v2, v4, v1], [v0, v5, v3, v4, v1, v2], [v0, v5, v3, v4, v2, v1], [v0, v5, v4, v1, v2, v3], [v0, v5, v4, v1, v3, v2], [v0, v5, v4, v2, v1, v3], [v0, v5, v4, v2, v3, v1], [v0, v5, v4, v3, v1, v2], [v0, v5, v4, v3, v2, v1], [v3, v0, v1, v2, v4, v5], [v3, v0, v1, v2, v5, v4], [v3, v0, v1, v4, v2, v5], [v3, v0, v1, v4, v5, v2], [v3, v0, v1, v5, v2, v4], [v3, v0, v1, v5, v4, v2], [v3, v0, v2, v1, v4, v5], [v3, v0, v2, v1, v5, v4], [v3, v0, v2, v4, v1, v5], [v3, v0, v2, v4, v5, v1], [v3, v0, v2, v5, v1, v4], [v3, v0, v2, v5, v4, v1], [v3, v0, v4, v1, v2, v5], [v3, v0, v4, v1, v5, v2], [v3, v0, v4, v2, v1, v5], [v3, v0, v4, v2, v5, v1], [v3, v0, v4, v5, v1, v2], [v3, v0, v4, v5, v2, v1], [v3, v0, v5, v1, v2, v4], [v3, v0, v5, v1, v4, v2], [v3, v0, v5, v2, v1, v4], [v3, v0, v5, v2, v4, v1], [v3, v0, v5, v4, v1, v2], [v3, v0, v5, v4, v2, v1], [v3, v5, v0, v1, v2, v4], [v3, v5, v0, v1, v4, v2], [v3, v5, v0, v2, v1, v4], [v3, v5, v0, v2, v4, v1], [v3, v5, v0, v4, v1, v2], [v3, v5, v0, v4, v2, v1], [v5, v0, v1, v2, v3, v4], [v5, v0, v1, v2, v4, v3], [v5, v0, v1, v3, v2, v4], [v5, v0, v1, v3, v4, v2], [v5, v0, v1, v4, v2, v3], [v5, v0, v1, v4, v3, v2], [v5, v0, v2, v1, v3, v4], [v5, v0, v2, v1, v4, v3], [v5, v0, v2, v3, v1, v4], [v5, v0, v2, v3, v4, v1], [v5, v0, v2, v4, v1, v3], [v5, v0, v2, v4, v3, v1], [v5, v0, v3, v1, v2, v4], [v5, v0, v3, v1, v4, v2], [v5, v0, v3, v2, v1, v4], [v5, v0, v3, v2, v4, v1], [v5, v0, v3, v4, v1, v2], [v5, v0, v3, v4, v2, v1], [v5, v0, v4, v1, v2, v3], [v5, v0, v4, v1, v3, v2], [v5, v0, v4, v2, v1, v3], [v5, v0, v4, v2, v3, v1], [v5, v0, v4, v3, v1, v2], [v5, v0, v4, v3, v2, v1], [v5, v3, v0, v1, v2, v4], [v5, v3, v0, v1, v4, v2], [v5, v3, v0, v2, v1, v4], [v5, v3, v0, v2, v4, v1], [v5, v3, v0, v4, v1, v2], [v5, v3, v0, v4, v2, v1]], 180, Yes, all_vertex_permutations_are_audited)
The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: ed2cff32a1c5bb47daf7c7f7e72ff93e40036dc8706a1f658e49df5258e0f7ffcandidate record SHA-256: c53c0abca1d1c35844aa42a90fefdf9995c2080290b0e76939a0a5168fe64e18semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612bLean: NOT_RUN · human review: NOT_PERFORMED