# Question 798: Audit All Vertex Permutations and Retain Every Valid Order

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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.

P1000-0798 · q-20261015-advance-20261116-r06-topological-ordering-dag-0048



## Original question and givens

Use unique vertices=[v0, v1, v2, v3, v4, v5] and loop-free duplicate-free directed edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v5]]. 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, v3], [v0, v5]]. 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).





## Definition and scope

For a finite digraph every valid order starts at a zero-indegree vertex; branching on that first vertex lists every order once.


- [mko-topological-ordering-dag](https://ocme.evemisslab.com/en/mko/mko-topological-ordering-dag/)





## Step-by-step derivation

S001 · impliesBefore

V=[v0, v1, v2, v3, v4, v5]

After

all directed edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v5]]

Conditions: none

This is a fully specified finite digraph, not necessarily a complete digraph.

Step identifiers and dependenciesRule: full_directed_graph · Uses: G001





S002 · impliesBefore

no vertices remain

After

one empty completion; C(∅)=1

Conditions: none

The empty remaining problem has one completion: place no further vertex.

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





S003 · calculationBefore

C(v5) := sum(1)

After

C(v5) = 1

Conditions: none

Each legal first vertex defines a disjoint branch; add the branch completion counts.

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





S004 · impliesBefore

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.

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





S005 · calculationBefore

C(v4) := sum(1)

After

C(v4) = 1

Conditions: none

Each legal first vertex defines a disjoint branch; add the branch completion counts.

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





S006 · impliesBefore

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.

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





S007 · calculationBefore

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.

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





S008 · impliesBefore

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.

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





S009 · calculationBefore

C(v3) := sum(1)

After

C(v3) = 1

Conditions: none

Each legal first vertex defines a disjoint branch; add the branch completion counts.

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





S010 · impliesBefore

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.

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





S011 · calculationBefore

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.

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





S012 · impliesBefore

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.

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





S013 · calculationBefore

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.

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





S014 · impliesBefore

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.

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





S015 · calculationBefore

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.

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





S016 · impliesBefore

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.

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





S017 · calculationBefore

C(v2) := sum(1)

After

C(v2) = 1

Conditions: none

Each legal first vertex defines a disjoint branch; add the branch completion counts.

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





S018 · impliesBefore

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.

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





S019 · calculationBefore

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.

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





S020 · impliesBefore

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.

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





S021 · calculationBefore

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.

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





S022 · impliesBefore

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.

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





S023 · calculationBefore

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.

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





S024 · impliesBefore

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.

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





S025 · calculationBefore

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.

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





S026 · impliesBefore

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.

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





S027 · calculationBefore

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.

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





S028 · impliesBefore

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.

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





S029 · calculationBefore

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.

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





S030 · impliesBefore

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.

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





S031 · calculationBefore

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.

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





S032 · impliesBefore

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.

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





S033 · calculationBefore

C(v1) := sum(1)

After

C(v1) = 1

Conditions: none

Each legal first vertex defines a disjoint branch; add the branch completion counts.

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





S034 · impliesBefore

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.

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





S035 · calculationBefore

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.

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





S036 · impliesBefore

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.

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





S037 · calculationBefore

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.

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





S038 · impliesBefore

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.

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





S039 · calculationBefore

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.

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





S040 · impliesBefore

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.

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





S041 · calculationBefore

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.

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





S042 · impliesBefore

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.

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





S043 · calculationBefore

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.

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





S044 · impliesBefore

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.

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





S045 · calculationBefore

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.

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





S046 · impliesBefore

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.

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





S047 · calculationBefore

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.

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





S048 · impliesBefore

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.

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





S049 · calculationBefore

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.

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





S050 · impliesBefore

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.

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





S051 · calculationBefore

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.

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





S052 · impliesBefore

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.

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





S053 · calculationBefore

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.

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





S054 · impliesBefore

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.

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





S055 · calculationBefore

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.

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





S056 · impliesBefore

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.

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





S057 · calculationBefore

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.

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





S058 · impliesBefore

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.

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





S059 · calculationBefore

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.

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





S060 · impliesBefore

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.

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





S061 · calculationBefore

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.

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





S062 · impliesBefore

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.

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





S063 · calculationBefore

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.

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





S064 · impliesBefore

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.

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





S065 · calculationBefore

C(v0,v1,v2,v3,v5) := sum(24)

After

C(v0,v1,v2,v3,v5) = 24

Conditions: none

Each legal first vertex defines a disjoint branch; add the branch completion counts.

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





S066 · impliesBefore

remaining=[v0, v1, v2, v3, v5]; eligible first vertices=[v0]

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.

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





S067 · calculationBefore

C(v0,v1,v2,v3,v4,v5) := sum(120, 24)

After

C(v0,v1,v2,v3,v4,v5) = 144

Conditions: none

Each legal first vertex defines a disjoint branch; add the branch completion counts.

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





S068 · impliesBefore

remaining=[v0, v1, v2, v3, v4, v5]; eligible first vertices=[v0, v4]

After

complete continuations=144

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.

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





S069 · impliesBefore

all root branches; count=144

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.

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





S070 · impliesBefore

valid order count=144

After

at least one exists: Yes

Conditions: none

An existing valid order certifies acyclicity of this finite digraph; multiple valid orders are compatible.

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









## Reverse checks


- 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 — Audit all vertices and edges; one failed order is not a cycle proof, and no legal branch may be omitted.
- 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,v3,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,v3,v5) 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.





## Result

([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v5]], [[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], [v4, v0, v1, v2, v3, v5], [v4, v0, v1, v2, v5, v3], [v4, v0, v1, v3, v2, v5], [v4, v0, v1, v3, v5, v2], [v4, v0, v1, v5, v2, v3], [v4, v0, v1, v5, v3, v2], [v4, v0, v2, v1, v3, v5], [v4, v0, v2, v1, v5, v3], [v4, v0, v2, v3, v1, v5], [v4, v0, v2, v3, v5, v1], [v4, v0, v2, v5, v1, v3], [v4, v0, v2, v5, v3, v1], [v4, v0, v3, v1, v2, v5], [v4, v0, v3, v1, v5, v2], [v4, v0, v3, v2, v1, v5], [v4, v0, v3, v2, v5, v1], [v4, v0, v3, v5, v1, v2], [v4, v0, v3, v5, v2, v1], [v4, v0, v5, v1, v2, v3], [v4, v0, v5, v1, v3, v2], [v4, v0, v5, v2, v1, v3], [v4, v0, v5, v2, v3, v1], [v4, v0, v5, v3, v1, v2], [v4, v0, v5, v3, v2, v1]], 144, Yes, all_vertex_permutations_are_audited)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.





## 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: 0ff28f77563f867d18f780e24c258666744b981beed0b3976dde01b7a2d5a35d
- candidate record SHA-256: de5c20fe18b63f4fed2221ffc80cff7571721176db9b09fc594bad17231acfe5
- semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
- question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261116-r06-topological-ordering-dag-0048.json)