# Fibers as left cosets of the stabilizer

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derivation-mko-orbit-stabilizer-theorem-01 · ocme-new142-reader-adoption-20261021



## Goal and scope

At point 0 the orbit has size 3 and stabilizer size 2; all three fibers are left cosets, giving 6=3*2.

The same complete finite action and point zero, with a two-way fiber/coset proof and translation bijections.





## 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
- \phi_0 — action map at point zero
- H — stabilizer at the same 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 — Fix the same fully certified action of G on X and point x. The action map phi_x(g)=g*x is surjective onto the orbit. If g0*x=y, its fiber over y is exactly the left coset g0*Stab_G(x); hence |G|=|G*x|*|Stab_G(x)|. The action need not be free, faithful or transitive. A proper group sample, a different point binding or a coincidentally equal numerical product is insufficient. · mko-orbit-stabilizer-theorem





## 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

\phi_0:G\to X,\quad\phi_0(g)=g\cdot0

After

\operatorname{im}\phi_0=O=\{0,1,2\},\quad H=\phi_0^{-1}(\{0\})=\{e,s\}

Conditions: none

Rule: calc · Uses: G1, S03

Leaf basis: calc

The full 0-column computes orbit and stabilizer, both bound to point zero.



S05 · impliesBefore

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

After

\phi_0^{-1}(0)=\{e,s\},\quad\phi_0^{-1}(1)=\{r,rs\},\quad\phi_0^{-1}(2)=\{r^2,r^2s\}

Conditions: none

Rule: calc · Uses: G1, S04

Leaf basis: calc

All six group elements are partitioned among three fibers without omissions or overlap.



S06 · impliesBefore

H=\{e,s\}

After

eH=\{e,s\},\quad rH=\{r,rs\},\quad r^2H=\{r^2,r^2s\}

Conditions: none

Rule: calc · Uses: G1, S01, S05

Leaf basis: calc

Compute all three left cosets from the full table; they exactly match the fibers.



S07 · equivalentBefore

g_0\cdot0=y

After

g\cdot0=y\iff(g_0^{-1}g)\cdot0=0\iff g_0^{-1}g\in H\iff g\in g_0H

Conditions: none

Rule: calc · Uses: G1, S02, S03, S04, S06

Leaf basis: calc

Action laws and inverses give the two-way fiber/left-coset proof, not merely coincident lists.



S08 · impliesBefore

L_{g_0}:H\to g_0H,\quad h\mapsto g_0h

After

L_{g_0}^{-1}(k)=g_0^{-1}k\Rightarrow|g_0H|=|H|=2

Conditions: none

Rule: calc · Uses: G1, S02, S06, S07

Leaf basis: calc

Left translation has an explicit inverse, so every fiber has the same size two.



S09 · impliesBefore

G=\phi_0^{-1}(0)\sqcup\phi_0^{-1}(1)\sqcup\phi_0^{-1}(2)

After

|G|=3\cdot2=|G\cdot0|\,|\operatorname{Stab}_G(0)|=6

Conditions: none

Rule: mko_definition · Uses: G1, S04, S05, S08

Leaf basis: mko_definition

Disjoint complete fibers and equal-size bijections prove the counting identity.







## Substeps and lemmas

No substeps in this guide.





## Reverse audits


- S09 · premise_audit · G1, S04, S05, S08 — Neither freeness nor transitivity is needed: s fixes 0 and point 3 is outside its orbit.
- S08 · premise_audit · G1, S02, S06, S07 — Three observed pairs alone would not explain the mechanism; the inverse map supplies the bijection.
- S07 · inverse_equivalence · G1, S02, S03, S04, S06 — Conversely g=g0*h gives g*0=g0*(h*0)=g0*0, with h in the same H.
- S06 · premise_audit · G1, S01, S05 — These are left cosets g0*H; changing to H*g0 would change the order.
- S05 · premise_audit · G1, S04 — Every orbit point has a preimage, so phi0 is onto its orbit.
- S04 · premise_audit · G1, S03 — Point 3 also lies in X, but its stabilizer G cannot be substituted.
- 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.


- Use Stab(3)=G with orbit 0 in the product. → reject — Mixing points gives 3*6=18, not 6.
- The theorem requires a free action. → reject — Nonidentity s fixes 0, yet the fiber certificate and 6=3*2 hold.
- Numerical 6=3*2 alone certifies any action. → reject — The same action/point needs full fibers and coset bijections, which arbitrary numbers do not supply.





## Result and limitations

At point 0 the orbit has size 3 and stabilizer size 2; all three fibers are left cosets, giving 6=3*2.







## Verification and source binding

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


- source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
- adopted source SHA-256: f78007176051f8883326c27cdfa614a2deb1ac3401d22f708d72381b178efd39
- original source SHA-256: f78007176051f8883326c27cdfa614a2deb1ac3401d22f708d72381b178efd39
- Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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