# Question 12440: Check the Whole Finite Vertex Map for Bijectivity

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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.

R0023-12440 · q-20261013-advance-20261026-r06-graph-isomorphism-0015



## Original question and givens

The complete map on Fin(6)={0,…,5} has images=[0, 1, 4, 3, 2, 5] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.


- G001 — The complete map on Fin(6)={0,…,5} has images=[0, 1, 4, 3, 2, 5] in source-label order, all in the target carrier of the same size. Give (n,mapping,bijective?). This family allows the unique empty map when n=0.





## 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-graph-isomorphism](https://ocme.evemisslab.com/en/mko/mko-graph-isomorphism/)





## Step-by-step derivation

S001 · impliesBefore

complete mapping=[0, 1, 4, 3, 2, 5];target label=0

After

source fiber=[0];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

Step identifiers and dependenciesRule: finite_vertex_map_each_target_fiber · Uses: G001





S002 · impliesBefore

complete mapping=[0, 1, 4, 3, 2, 5];target label=1

After

source fiber=[1];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

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





S003 · impliesBefore

complete mapping=[0, 1, 4, 3, 2, 5];target label=2

After

source fiber=[4];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

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





S004 · impliesBefore

complete mapping=[0, 1, 4, 3, 2, 5];target label=3

After

source fiber=[3];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

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





S005 · impliesBefore

complete mapping=[0, 1, 4, 3, 2, 5];target label=4

After

source fiber=[2];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

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





S006 · impliesBefore

complete mapping=[0, 1, 4, 3, 2, 5];target label=5

After

source fiber=[5];exactly one=Yes

Conditions: none

For a complete total map between equally sized finite carriers,every target needs exactly one preimage.Typedness alone is not bijectivity.

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





S007 · impliesBefore

n=6;all target fibers=[[0], [1], [4], [3], [2], [5]]

After

bijective=Yes;empty carrier uses unique empty map

Conditions: none

The unique empty map is bijective on empty carriers.This checks the mapping obligation only,not adjacency.

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









## Reverse checks


- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

(6, [0, 1, 4, 3, 2, 5], Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: aff01377d70614b09005cec5f681c7673da496c915d89398452f38651439894d
- candidate record SHA-256: fbb2641b11a54a3d9a52e6d54fab3a683cf7b82f597d25194d46078d649ca60d
- semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff
- question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261026-r06-graph-isomorphism-0015.json)