Question 12305: Emit Classes Only after All Three Axioms Qualify

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0023-12305 · q-20261013-advance-20261026-r07-equivalence-relation-0080

Original question and givens

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [2, 0], [2, 2], [3, 1], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

G001 — Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [2, 0], [2, 2], [3, 1], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

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-equivalence-relation

Step-by-step derivation

S001 · implies

Before

raw relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [2, 0], [2, 2], [3, 1], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]];carrier Fin(6)

After

lexicographically sorted distinct ordered pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [2, 0], [2, 2], [3, 1], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]]

Conditions: none

List distinct directed pairs lexicographically.Duplicates are invalid;do not silently add reverses,diagonals or a transitive closure.

Step identifiers and dependencies

Rule: complete_relation_no_duplicate_or_implicit_repair · Uses: G001

S002 · implies

Before

carrier label=0;required diagonal=[0, 0]

After

present=Yes

Conditions: none

Every carrier element needs its diagonal pair,not only labels appearing in the relation.

Step identifiers and dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S001

S003 · implies

Before

carrier label=1;required diagonal=[1, 1]

After

present=Yes

Conditions: none

Every carrier element needs its diagonal pair,not only labels appearing in the relation.

Step identifiers and dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S002

S004 · implies

Before

carrier label=2;required diagonal=[2, 2]

After

present=Yes

Conditions: none

Every carrier element needs its diagonal pair,not only labels appearing in the relation.

Step identifiers and dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S003

S005 · implies

Before

carrier label=3;required diagonal=[3, 3]

After

present=Yes

Conditions: none

Every carrier element needs its diagonal pair,not only labels appearing in the relation.

Step identifiers and dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S004

S006 · implies

Before

carrier label=4;required diagonal=[4, 4]

After

present=Yes

Conditions: none

Every carrier element needs its diagonal pair,not only labels appearing in the relation.

Step identifiers and dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S005

S007 · implies

Before

carrier label=5;required diagonal=[5, 5]

After

present=Yes

Conditions: none

Every carrier element needs its diagonal pair,not only labels appearing in the relation.

Step identifiers and dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S006

S008 · implies

Before

original pair=[0, 0];required reverse=[0, 0]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S007

S009 · implies

Before

original pair=[0, 2];required reverse=[2, 0]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S008

S010 · implies

Before

original pair=[1, 1];required reverse=[1, 1]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S009

S011 · implies

Before

original pair=[1, 3];required reverse=[3, 1]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S010

S012 · implies

Before

original pair=[2, 0];required reverse=[0, 2]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S011

S013 · implies

Before

original pair=[2, 2];required reverse=[2, 2]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S012

S014 · implies

Before

original pair=[3, 1];required reverse=[1, 3]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S013

S015 · implies

Before

original pair=[3, 3];required reverse=[3, 3]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S014

S016 · implies

Before

original pair=[4, 4];required reverse=[4, 4]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S015

S017 · implies

Before

original pair=[4, 5];required reverse=[5, 4]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S016

S018 · implies

Before

original pair=[5, 4];required reverse=[4, 5]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S017

S019 · implies

Before

original pair=[5, 5];required reverse=[5, 5]

After

reverse present=Yes

Conditions: none

Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.

Step identifiers and dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S018

S020 · implies

Before

first original pair=[0, 0];all composable second endpoints=[0, 2]

After

required transitivity rows [a,b,d,aRd]=[[0, 0, 0, Yes], [0, 0, 2, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S019

S021 · implies

Before

first original pair=[0, 2];all composable second endpoints=[0, 2]

After

required transitivity rows [a,b,d,aRd]=[[0, 2, 0, Yes], [0, 2, 2, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S020

S022 · implies

Before

first original pair=[1, 1];all composable second endpoints=[1, 3]

After

required transitivity rows [a,b,d,aRd]=[[1, 1, 1, Yes], [1, 1, 3, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S021

S023 · implies

Before

first original pair=[1, 3];all composable second endpoints=[1, 3]

After

required transitivity rows [a,b,d,aRd]=[[1, 3, 1, Yes], [1, 3, 3, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S022

S024 · implies

Before

first original pair=[2, 0];all composable second endpoints=[0, 2]

After

required transitivity rows [a,b,d,aRd]=[[2, 0, 0, Yes], [2, 0, 2, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S023

S025 · implies

Before

first original pair=[2, 2];all composable second endpoints=[0, 2]

After

required transitivity rows [a,b,d,aRd]=[[2, 2, 0, Yes], [2, 2, 2, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S024

S026 · implies

Before

first original pair=[3, 1];all composable second endpoints=[1, 3]

After

required transitivity rows [a,b,d,aRd]=[[3, 1, 1, Yes], [3, 1, 3, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S025

S027 · implies

Before

first original pair=[3, 3];all composable second endpoints=[1, 3]

After

required transitivity rows [a,b,d,aRd]=[[3, 3, 1, Yes], [3, 3, 3, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S026

S028 · implies

Before

first original pair=[4, 4];all composable second endpoints=[4, 5]

After

required transitivity rows [a,b,d,aRd]=[[4, 4, 4, Yes], [4, 4, 5, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S027

S029 · implies

Before

first original pair=[4, 5];all composable second endpoints=[4, 5]

After

required transitivity rows [a,b,d,aRd]=[[4, 5, 4, Yes], [4, 5, 5, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S028

S030 · implies

Before

first original pair=[5, 4];all composable second endpoints=[4, 5]

After

required transitivity rows [a,b,d,aRd]=[[5, 4, 4, Yes], [5, 4, 5, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S029

S031 · implies

Before

first original pair=[5, 5];all composable second endpoints=[4, 5]

After

required transitivity rows [a,b,d,aRd]=[[5, 5, 4, Yes], [5, 5, 5, Yes]]

Conditions: none

List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.

Step identifiers and dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S030

S032 · implies

Before

least unseen representative=0;complete relation=[[0, 0], [0, 2], [1, 1], [1, 3], [2, 0], [2, 2], [3, 1], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]]

After

class=[0, 2];seen=[0, 2]

Conditions: none

Only after every axiom passes,derive classes from the original relation.Choose the least unseen representative and increasing class labels.The axioms ensure disjoint-or-equal classes covering the carrier.

Step identifiers and dependencies

Rule: equivalence_class_after_all_axioms · Uses: G001, S031

S033 · implies

Before

least unseen representative=1;complete relation=[[0, 0], [0, 2], [1, 1], [1, 3], [2, 0], [2, 2], [3, 1], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]]

After

class=[1, 3];seen=[0, 1, 2, 3]

Conditions: none

Step identifiers and dependencies

Rule: equivalence_class_after_all_axioms · Uses: G001, S032

S034 · implies

Before

least unseen representative=4;complete relation=[[0, 0], [0, 2], [1, 1], [1, 3], [2, 0], [2, 2], [3, 1], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]]

After

class=[4, 5];seen=[0, 1, 2, 3, 4, 5]

Conditions: none

Step identifiers and dependencies

Rule: equivalence_class_after_all_axioms · Uses: G001, S033

S035 · implies

Before

missing reflexive=[];missing reverses=[];missing transitive triples=[]

After

equivalence=Yes;classes=[[0, 2], [1, 3], [4, 5]]

Conditions: none

Emit classes only when all three obligations pass.On any failure leave classes empty,without repairing the relation by connected grouping.

Step identifiers and dependencies

Rule: equivalence_three_obligations_no_silent_closure · Uses: G001, S034

Reverse checks

S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — 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, [], [], [], Yes, [[0, 2], [1, 3], [4, 5]])

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: 8d7a41f715d40b1ea9a4f5296afb4289e151297d39a764325406ba7bd5d02cd9
candidate record SHA-256: ae128afc545446a479d54b0034a40ff07be8aed9b0ed6f729a2498a06be79687
semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff
question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18
Lean: NOT_RUN · human review: NOT_PERFORMED

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