# Contraction mapping theorem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A self-map on a nonempty complete metric space with one global contraction constant q<1 has a unique fixed point and convergent Picard iterates.

mko-contraction-mapping-theorem · theorem · v0.10.0



## Mathematical statement

The exact finite families qualify every metric axiom and a complete total self-map, then check d(Tx,Ty)≤q*d(x,y) for every ordered pair with fixed0≤q<1. From a declared start generate the entire requested Picard orbit and enumerate fixed points to check uniqueness. Compare d(x_n,x*) with q^n/(1−q)*d(x_1,x_0) exactly. A finite metric space is complete, but its certificate proves only that specified model. q=1, failure of the self-map premise, a noncomplete space or sampled-pair checks do not license the theorem.





## Representative formula (exact TeX)

C(X,T,q)=x





## Assumptions


- The complete finite rational metric matrix is checked for nonnegativity, separation, symmetry and every triangle inequality.
- The mapping is a complete total self-map and q is an exact rational.





## Symbol scope


- C — contraction-mapping construction
- X — complete metric space
- T — contraction self-map
- q — contraction constant
- x — unique fixed point





## Intuition, definition, and misconceptions

Full metric, self-map and contraction qualification precedes the theorem.

Check the Picard orbit and unique fixed point separately.

The geometric tail gives an exact error bound.

Sampled point pairs, q=1 or an omitted self-map/completeness premise do not establish a unique fixed point.





## Adopted derivation guide

[A complete contraction certificate on a finite metric](https://ocme.evemisslab.com/en/derivations/derivation-mko-contraction-mapping-theorem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Orbits, fixed points and periodic points](https://ocme.evemisslab.com/en/mko/mko-orbit-fixed-periodic/) — Use Picard orbits and fixed points.
- [Complete metric space](https://ocme.evemisslab.com/en/mko/mko-complete-metric-space/) — Completeness is a separate premise.
- [Convergence in a metric space](https://ocme.evemisslab.com/en/mko/mko-metric-convergence/) — Separate Cauchy behavior and convergence.
- [Infinite series](https://ocme.evemisslab.com/en/mko/mko-infinite-series/) — Use the geometric-tail bound.





## Verification boundary

Finite-model certificates do not replace the general complete-space proof. The sampled-pairs boundary gives a source scope description, not a supplied complete metric/map counterexample.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add applications such as Picard–Lindelof.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/contraction-mapping-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-contraction-mapping-theorem/index.md)