# Complete orbits and their partition

[← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-finite-group-orbit/)

This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-finite-group-orbit-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation.

The same complete six-element/four-point action, using every g rather than bounded generator steps.





## Givens and symbols


- calc


- G1 — G={e,r,r2,s,rs,r2s}, encoded respectively (0,0),(1,0),(2,0),(0,1),(1,1),(2,1), with (i,j)(k,l)=(i+(-1)^j*k mod3,j+l mod2). Points X={0,1,2,3}; (i,j) acts on x<3 by i+(-1)^j*x mod3 and fixes 3.


- G — six-element group carrier
- r,s — labels with r^3=e,s^2=e,sr=r^2s
- G\cdot x — all group images of a specified point





## Rules and sources


- calc · prerequisite — Assume exact arithmetic modulo 3 and 2, extensional finite-set equality, and definitions of groups/actions; this operation is not pre-assumed to be a group.
- mko_definition · cited — Start with a fully qualified finite group action and enumerate G*x={g*x:g∈G} using every group element. The relation x~y exactly when some g sends x to y is an equivalence relation. Orbits are equal or disjoint and cover the carrier. One action image, finitely sampled generator steps and a stabilizer are not the whole orbit. · mko-finite-group-orbit





## Main derivation

S01 · impliesBefore

G=\{e,r,r^2,s,rs,r^2s\}

After

\begin{array}{c|cccccc}*&e&r&r^2&s&rs&r^2s\\\hline e&e&r&r^2&s&rs&r^2s\\r&r&r^2&e&rs&r^2s&s\\r^2&r^2&e&r&r^2s&s&rs\\s&s&r^2s&rs&e&r^2&r\\rs&rs&s&r^2s&r&e&r^2\\r^2s&r^2s&rs&s&r^2&r&e\\\end{array}

Conditions: none

Rule: calc · Uses: G1

Leaf basis: calc

Expand the pair formula for all 36 products; every output belongs to the same six-element carrier and no cell is missing.



S02 · impliesBefore

(i,j)(k,l)=(i+(-1)^jk,j+l)\quad(\bmod(3,2))

After

((i,j)(k,l))(m,n)=(i+(-1)^jk+(-1)^{j+l}m,j+l+n)=(i,j)((k,l)(m,n));\quad e=(0,0);\quad(i,j)^{-1}=(-(-1)^ji,j)

Conditions: none

Rule: calc · Uses: G1, S01

Leaf basis: calc

Both bracketings have identical coordinates, proving associativity for all 216 triples; (0,0) is a two-sided identity and the inverse formula multiplies back on both sides.



S03 · impliesBefore

X=\{0,1,2,3\}

After

\begin{array}{c|cccc}\cdot&0&1&2&3\\\hline e&0&1&2&3\\r&1&2&0&3\\r^2&2&0&1&3\\s&0&2&1&3\\rs&1&0&2&3\\r^2s&2&1&0&3\\\end{array}

Conditions: none

Rule: calc · Uses: G1, S02

Leaf basis: calc

Expand all 24 action values. For x<3, g*(h*x)=i+(-1)^j*k+(-1)^(j+l)*x=(gh)*x; both sides fix 3, and e fixes every point.



S04 · impliesBefore

g=e,r,r^2,s,rs,r^2s

After

g\cdot0=0,1,2,0,1,2\Rightarrow G\cdot0=\{0,1,2\}

Conditions: none

Rule: calc · Uses: G1, S03

Leaf basis: calc

Enumerate all six group elements in the 0-column and deduplicate to obtain the whole orbit.



S05 · impliesBefore

g\in G

After

g\cdot3=3\Rightarrow G\cdot3=\{3\}

Conditions: none

Rule: calc · Uses: G1, S03

Leaf basis: calc

Every entry of the complete 3-column fixes 3, so this orbit is a singleton.



S06 · impliesBefore

x\sim y\iff\exists g:\ g\cdot x=y

After

e\cdot x=x;\quad g\cdot x=y\Rightarrow g^{-1}\cdot y=x;\quad h\cdot y=z\Rightarrow(hg)\cdot x=z

Conditions: none

Rule: calc · Uses: G1, S02, S03

Leaf basis: calc

Identity, inverse and composition establish reflexivity, symmetry and transitivity, proving same-orbit equivalence.



S07 · impliesBefore

g\cdot1=1,2,0,2,0,1;\quad g\cdot2=2,0,1,1,2,0

After

G\cdot1=G\cdot2=\{0,1,2\}

Conditions: none

Rule: calc · Uses: G1, S03, S04, S06

Leaf basis: calc

Deduplicating the other two complete columns accounts for the orbits of all four points.



S08 · impliesBefore

O=\{0,1,2\},\quad P=\{3\}

After

O\cap P=\varnothing,\quad O\cup P=X,\quad X/G=\{O,P\}

Conditions: none

Rule: mko_definition · Uses: G1, S04, S05, S07

Leaf basis: mko_definition

All distinct orbits are nonempty, disjoint and cover X, directly certifying a partition.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S08 · premise_audit · G1, S04, S05, S07 — This partitions points and is not a collection of stabilizers or group cosets.
- S07 · premise_audit · G1, S03, S04, S06 — Different base points may have equal orbit sets, which must not be counted as distinct orbits.
- S06 · premise_audit · G1, S02, S03 — The combined witness is hg because g acts first and then h.
- S05 · premise_audit · G1, S03 — The orbit is nonempty since e*3=3; a fixed point still has an orbit.
- S04 · premise_audit · G1, S03 — Taking only e or the single image r*0=1 is not the whole orbit.
- S03 · premise_audit · G1, S02 — Every group-point pair has one value; the formula covers all 144 triples, not merely permutation rows.
- S02 · premise_audit · G1, S01 — The sign depends only on j+l modulo 2 since (-1)^2=1; the full inverse list is e,r^2,r,s,rs,r^2s.
- S01 · premise_audit · G1 — The six labels bind the six pairs; r^2 is not an additional seventh element.





## Error controls and negative controls

Same group tableBind every product, inverse and coset to this complete Cayley table.

Complete enumerationEvery universal finite property needs the full carrier or a universal formula, not a proper sample.


- r*0=1 is the orbit. → reject — The orbit is the full image set {0,1,2}, not point 1.
- Point 3 has no orbit because it is fixed. → reject — Its orbit is exactly {3}.
- The orbits of 0 and 1 are disjoint. → reject — Both are {0,1,2} and are equal.





## Result and limitations

The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation.







## Verification and source binding

0 declared checks for this guide; only the listed expressions are covered.


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: 15c74347343414557ec306e3bda093c9fb53df12da58f4a045f551f1a6c45b11
- original source SHA-256: 15c74347343414557ec306e3bda093c9fb53df12da58f4a045f551f1a6c45b11
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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