Question 22353: A Fiber Equals a Left Coset of the Same Point Stabilizer

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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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R0043-22353 · q-20261014-advance-20261110-r01-orbit-stabilizer-theorem-0053

Original question and givens

Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, point carrier Fin(4)={0,…,3}, total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and specified point=2. Row g/column h gives g*h, and action row g/column x gives g*x. Certify every group/action axiom, then enumerate all group elements with the same action and point binding. Free, faithful or transitive action is not required. Use orbit_point=1. Give (orbit_point, representative, fiber, left_coset, equal, certificate).

G001 — Use complete group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, point carrier Fin(4)={0,…,3}, total action_table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and specified point=2. Row g/column h gives g*h, and action row g/column x gives g*x. Certify every group/action axiom, then enumerate all group elements with the same action and point binding. Free, faithful or transitive action is not required. Use orbit_point=1. Give (orbit_point, representative, fiber, left_coset, equal, certificate).

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-orbit-stabilizer-theorem

Step-by-step derivation

S001 · implies

Before

source operation=orbitstab-coset-fiber-bijection;original snapshot and mathematics retained

After

explicit ordered source output fields=[orbit_point, representative, fiber, left_coset, equal, certificate]

Conditions: none

Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL complete group-label carrier Fin(4);declared identity label=0;Cayley table row g,column h gives g*h=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]]

After

ORIGINAL complete point carrier Fin(4);TOTAL action table row g,column x gives g*x=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]];all entries in range,so structural closure/totality checked;GROUP AXIOMS not assumed from shape

Conditions: none

The two tables have different column roles:group label h versus action point x.Complete in-range tables establish closure/totality only.A candidate group may fail its axioms;its table name does not certify them.

Step identifiers and dependencies

Rule: groupaction_two_table_roles_complete_candidate_not_group_assumption · Uses: G001, S001

S003 · implies

Before

ORIGINAL closed Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]];ACTUAL declared group identity=0

After

ALL [g,e*g,g*e,both_equal_g] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]];two-sided group identity=Yes

Conditions: none

Both e*g and g*e in the group table must equal g.This is not certified by an identity-action row on points.

Step identifiers and dependencies

Rule: groupaction_group_identity_both_sides_not_action_identity · Uses: G001, S002

S004 · implies

Before

ORIGINAL group pair(g,h)=[0, 0];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S003

S005 · implies

Before

ORIGINAL group pair(g,h)=[0, 1];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S004

S006 · implies

Before

ORIGINAL group pair(g,h)=[0, 2];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S005

S007 · implies

Before

ORIGINAL group pair(g,h)=[0, 3];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S006

S008 · implies

Before

ORIGINAL group pair(g,h)=[1, 0];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S007

S009 · implies

Before

ORIGINAL group pair(g,h)=[1, 1];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S008

S010 · implies

Before

ORIGINAL group pair(g,h)=[1, 2];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S009

S011 · implies

Before

ORIGINAL group pair(g,h)=[1, 3];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S010

S012 · implies

Before

ORIGINAL group pair(g,h)=[2, 0];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S011

S013 · implies

Before

ORIGINAL group pair(g,h)=[2, 1];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S012

S014 · implies

Before

ORIGINAL group pair(g,h)=[2, 2];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S013

S015 · implies

Before

ORIGINAL group pair(g,h)=[2, 3];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S014

S016 · implies

Before

ORIGINAL group pair(g,h)=[3, 0];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S015

S017 · implies

Before

ORIGINAL group pair(g,h)=[3, 1];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S016

S018 · implies

Before

ORIGINAL group pair(g,h)=[3, 2];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S017

S019 · implies

Before

ORIGINAL group pair(g,h)=[3, 3];check ALL k in Fin(4)

After

ordered [k,(g*h)*k,g*(h*k),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]]

Conditions: none

Compute both parenthesizations in the complete original table without assuming cyclicity,commutativity or replacement of multiplication by integer addition.

Step identifiers and dependencies

Rule: groupaction_complete_group_associativity_not_cyclic_assumption · Uses: G001, S018

S020 · implies

Before

ORIGINAL group element g=0;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 0, 0, Yes], [1, 1, 1, No], [2, 2, 2, No], [3, 3, 3, No]]

After

first actual two-sided inverse witness=0;exists=Yes

Conditions: none

For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels.

Step identifiers and dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S019

S021 · implies

Before

ORIGINAL group element g=1;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 1, 1, No], [1, 2, 2, No], [2, 3, 3, No], [3, 0, 0, Yes]]

After

first actual two-sided inverse witness=3;exists=Yes

Conditions: none

For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels.

Step identifiers and dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S020

S022 · implies

Before

ORIGINAL group element g=2;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 2, 2, No], [1, 3, 3, No], [2, 0, 0, Yes], [3, 1, 1, No]]

After

first actual two-sided inverse witness=2;exists=Yes

Conditions: none

For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels.

Step identifiers and dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S021

S023 · implies

Before

ORIGINAL group element g=3;DECLARED identity=0;ALL [h,g*h,h*g,two_sided_inverse] rows=[[0, 3, 3, No], [1, 0, 0, Yes], [2, 1, 1, No], [3, 2, 2, No]]

After

first actual two-sided inverse witness=1;exists=Yes

Conditions: none

For each g,the same h must satisfy both inverse identities.An unqualified candidate table may have no witness;retain None rather than guessing inverses from labels.

Step identifiers and dependencies

Rule: groupaction_each_element_actual_two_sided_inverse · Uses: G001, S022

S024 · implies

Before

structurally closed COMPLETE group candidate;two-sided identity=Yes;ALL-triple associativity=Yes;ALL-element inverses=Yes

After

group_valid=Yes is the CONJUNCTION;False remains a reported candidate result,not a silent repair

Conditions: none

Group qualification is the conjunction of these complete obligations.A permutation-looking action table cannot repair a bad group table;do not replace the identity or modify the table.

Step identifiers and dependencies

Rule: groupaction_group_axiom_conjunction_candidate_may_fail · Uses: G001, S023

S025 · implies

Before

ACTUAL declared identity label=0;ORIGINAL identity action row=[0, 1, 2, 3];complete point carrier Fin(4)

After

ALL ordered [point,identity_image,equality] rows=[[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 3, Yes]];identity-action axiom=Yes

Conditions: none

Check e*x=x at every action point,using the actual declared identity label rather than assuming0.This action axiom is distinct from the two-sided identity law in the group table.

Step identifiers and dependencies

Rule: groupaction_every_point_identity_action_not_group_identity · Uses: G001, S024

S026 · implies

Before

ORIGINAL ordered group pair(g,h)=[0, 0];actual group product=0;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]];earliest failure retained,not overwritten

Conditions: none

For the left side compute the group product before acting;for the right act by h and then g,without interchanging them.Check every point for every ordered pair and retain the first lexicographic g,h,x failure.

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S025

S027 · implies

Before

ORIGINAL ordered group pair(g,h)=[0, 1];actual group product=1;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S026

S028 · implies

Before

ORIGINAL ordered group pair(g,h)=[0, 2];actual group product=2;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S027

S029 · implies

Before

ORIGINAL ordered group pair(g,h)=[0, 3];actual group product=3;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S028

S030 · implies

Before

ORIGINAL ordered group pair(g,h)=[1, 0];actual group product=1;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S029

S031 · implies

Before

ORIGINAL ordered group pair(g,h)=[1, 1];actual group product=2;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S030

S032 · implies

Before

ORIGINAL ordered group pair(g,h)=[1, 2];actual group product=3;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S031

S033 · implies

Before

ORIGINAL ordered group pair(g,h)=[1, 3];actual group product=0;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S032

S034 · implies

Before

ORIGINAL ordered group pair(g,h)=[2, 0];actual group product=2;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S033

S035 · implies

Before

ORIGINAL ordered group pair(g,h)=[2, 1];actual group product=3;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S034

S036 · implies

Before

ORIGINAL ordered group pair(g,h)=[2, 2];actual group product=0;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S035

S037 · implies

Before

ORIGINAL ordered group pair(g,h)=[2, 3];actual group product=1;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S036

S038 · implies

Before

ORIGINAL ordered group pair(g,h)=[3, 0];actual group product=3;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 3, 3, Yes], [1, 0, 0, Yes], [2, 1, 1, Yes], [3, 2, 2, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S037

S039 · implies

Before

ORIGINAL ordered group pair(g,h)=[3, 1];actual group product=0;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 0, 0, Yes], [1, 1, 1, Yes], [2, 2, 2, Yes], [3, 3, 3, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S038

S040 · implies

Before

ORIGINAL ordered group pair(g,h)=[3, 2];actual group product=1;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 1, 1, Yes], [1, 2, 2, Yes], [2, 3, 3, Yes], [3, 0, 0, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S039

S041 · implies

Before

ORIGINAL ordered group pair(g,h)=[3, 3];actual group product=2;action table roles unchanged

After

ALL ordered [point,(g*h)*point,g*(h*point),equal] rows=[[0, 2, 2, Yes], [1, 3, 3, Yes], [2, 0, 0, Yes], [3, 1, 1, Yes]];earliest failure retained,not overwritten

Conditions: none

Step identifiers and dependencies

Rule: groupaction_all_ordered_compatibility_triples_not_samples · Uses: G001, S040

S042 · calculation

Before

group action ordered group-pair count := mul(4, 4)

After

group action ordered group-pair count = 16

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

group action complete triple-domain count := mul(16, 4)

After

group action complete triple-domain count = 64

Conditions: none

Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

COMPLETE triple domain order^2*point_count=64;this derivation ACTUALLY checked all64 triples

After

compatibility=Yes;FIRST source-order failure[g,h,x,left,right]=null;null means NO failure anywhere

Conditions: none

The triple_count is the complete quantified domain size,not a measurement of an early-stopping prototype.This derivation actually checks every point for every g,h pair,so null certifies full compatibility and a first failure retains exact source data/order.

Step identifiers and dependencies

Rule: groupaction_full_domain_count_first_failure_not_early_stop_measurement · Uses: G001, S043

S045 · implies

Before

ORIGINAL group-row label=0;ORIGINAL point-indexed action row=[0, 1, 2, 3]

After

sorted image MULTISET=[0, 1, 2, 3];equals full point carrier=Yes;original row NOT reordered

Conditions: none

Sort only to test whether the image multiset covers the full carrier exactly once;do not reorder the original point-indexed row.Even all-permutation rows alone do not establish an action without the group and both action axioms.

Step identifiers and dependencies

Rule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S044

S046 · implies

Before

ORIGINAL group-row label=1;ORIGINAL point-indexed action row=[1, 2, 3, 0]

After

sorted image MULTISET=[0, 1, 2, 3];equals full point carrier=Yes;original row NOT reordered

Conditions: none

Step identifiers and dependencies

Rule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S045

S047 · implies

Before

ORIGINAL group-row label=2;ORIGINAL point-indexed action row=[2, 3, 0, 1]

After

sorted image MULTISET=[0, 1, 2, 3];equals full point carrier=Yes;original row NOT reordered

Conditions: none

Step identifiers and dependencies

Rule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S046

S048 · implies

Before

ORIGINAL group-row label=3;ORIGINAL point-indexed action row=[3, 0, 1, 2]

After

sorted image MULTISET=[0, 1, 2, 3];equals full point carrier=Yes;original row NOT reordered

Conditions: none

Step identifiers and dependencies

Rule: groupaction_every_row_permutation_actual_images_not_validity_converse · Uses: G001, S047

S049 · implies

Before

CURRENT qualified group/action T=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]],identity=0,inverse list=[0, 3, 2, 1];ORIGINAL SAME point=2

After

ALL [g,g*point]=[[0, 2], [1, 3], [2, 0], [3, 1]];full orbit=[0, 1, 2, 3];SAME-point stabilizer=[0];ALL [target,fiber] rows=[[0, [2]], [1, [3]], [2, [0]], [3, [1]]];action map onto orbit,NOT necessarily whole carrier

Conditions: none

Qualify the actual group and both action axioms,then enumerate all g at the same point.Another point's stabilizer or a proper sample cannot replace these bindings.No free,faithful or transitive premise is needed.

Step identifiers and dependencies

Rule: orbitstab_qualified_SAME_action_point_full_fibers · Uses: G001, S048

S050 · implies

Before

ACTUAL SAME-point stabilizer=[0];identity=0

After

ALL [g,h,product]=[[0, 0, 0]];ALL [g,inverse]=[[0, 0]];identity/product/inverse stay in SAME stabilizer

Conditions: none

The identity fixes the point;products of fixers fix it,and acting back by an inverse also fixes it.Check every member in the actual table,so the left cosets use this genuine subgroup,not any equally sized list.

Step identifiers and dependencies

Rule: orbitstab_SAME_point_stabilizer_actual_subgroup · Uses: G001, S049

S051 · calculation

Before

Orbit-stabilizer fiber target0 complete size := sum(1)

After

Orbit-stabilizer fiber target0 complete size = 1

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · implies

Before

ORIGINAL orbit target=0;LEAST group-index representative g0=2;SAME stabilizer=[0];full fiber=[2]

After

ALL [h,g0*h,(g0*h)*point]=[[0, 2, 0]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[2, 0, 2]];LEFT coset sorted=[2] equals fiber;fiber size=1

Conditions: none

The map h->g0*h sends this point's stabilizer into the fibre;g->g0_inverse*g lands back in it by action compatibility.The compositions are identities,so the maps are inverse bijections.Do not substitute a right coset;the multiplication order works for nonabelian groups too.

Step identifiers and dependencies

Rule: orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps · Uses: G001, S051

S053 · calculation

Before

Orbit-stabilizer fiber target1 complete size := sum(1)

After

Orbit-stabilizer fiber target1 complete size = 1

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · implies

Before

ORIGINAL orbit target=1;LEAST group-index representative g0=3;SAME stabilizer=[0];full fiber=[3]

After

ALL [h,g0*h,(g0*h)*point]=[[0, 3, 1]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[3, 0, 3]];LEFT coset sorted=[3] equals fiber;fiber size=1

Conditions: none

Step identifiers and dependencies

Rule: orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps · Uses: G001, S053

S055 · calculation

Before

Orbit-stabilizer fiber target2 complete size := sum(1)

After

Orbit-stabilizer fiber target2 complete size = 1

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · implies

Before

ORIGINAL orbit target=2;LEAST group-index representative g0=0;SAME stabilizer=[0];full fiber=[0]

After

ALL [h,g0*h,(g0*h)*point]=[[0, 0, 2]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[0, 0, 0]];LEFT coset sorted=[0] equals fiber;fiber size=1

Conditions: none

Step identifiers and dependencies

Rule: orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps · Uses: G001, S055

S057 · calculation

Before

Orbit-stabilizer fiber target3 complete size := sum(1)

After

Orbit-stabilizer fiber target3 complete size = 1

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · implies

Before

ORIGINAL orbit target=3;LEAST group-index representative g0=1;SAME stabilizer=[0];full fiber=[1]

After

ALL [h,g0*h,(g0*h)*point]=[[0, 1, 3]];ALL [g,g0_inverse*g,g0*(g0_inverse*g)]=[[1, 0, 1]];LEFT coset sorted=[1] equals fiber;fiber size=1

Conditions: none

Step identifiers and dependencies

Rule: orbitstab_ACTUAL_left_coset_fiber_two_inverse_maps · Uses: G001, S057

S059 · calculation

Before

Orbit-stabilizer complete orbit count := sum(1, 1, 1, 1)

After

Orbit-stabilizer complete orbit count = 4

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · calculation

Before

Orbit-stabilizer SAME-point stabilizer count := sum(1)

After

Orbit-stabilizer SAME-point stabilizer count = 1

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · calculation

Before

Orbit-stabilizer SAME-bound orbit times stabilizer sizes := mul(4, 1)

After

Orbit-stabilizer SAME-bound orbit times stabilizer sizes = 4

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · implies

Before

ALL original group elements partitioned by actual action-map image;fibers=[[0, [2]], [1, [3]], [2, [0]], [3, [1]]]

After

each fiber bijective with SAME stabilizer of size1;orbit size=4;group order=4=4*1=4

Conditions: none

Each g has exactly one image,so these fibres are disjoint and cover G.The explicit bijections establish equal sizes;this is not merely a coincidental numerical product.Surjectivity onto the orbit does not imply surjectivity onto the entire point carrier.

Step identifiers and dependencies

Rule: orbitstab_partition_group_into_equal_size_fibers_count · Uses: G001, S061

Reverse checks

S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S061 · premise_and_result_audit — Recompute Orbit-stabilizer SAME-bound orbit times stabilizer sizes with exact rational arithmetic, not rounded decimals.
S060 · premise_and_result_audit — Recompute Orbit-stabilizer SAME-point stabilizer count with exact rational arithmetic, not rounded decimals.
S059 · premise_and_result_audit — Recompute Orbit-stabilizer complete orbit count with exact rational arithmetic, not rounded decimals.
S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S057 · premise_and_result_audit — Recompute Orbit-stabilizer fiber target3 complete size with exact rational arithmetic, not rounded decimals.
S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S055 · premise_and_result_audit — Recompute Orbit-stabilizer fiber target2 complete size with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S053 · premise_and_result_audit — Recompute Orbit-stabilizer fiber target1 complete size with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S051 · premise_and_result_audit — Recompute Orbit-stabilizer fiber target0 complete size with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S043 · premise_and_result_audit — Recompute group action complete triple-domain count with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute group action ordered group-pair count with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S035 · premise_and_result_audit — 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

(1, 3, [3], [3], Yes, fiber_equals_left_coset_of_same_point_stabilizer)

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: c0dcbb55b25fdcb4707bee7274d963cdf28b49530e736b4fabd64d39300ea3ce
candidate record SHA-256: bee87debd6f849f0399a1822dffdda8fa06f2163f3b741bc8e4de6b51eb847e6
semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON