Contraction mapping theorem

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

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 — Use Picard orbits and fixed points.
• Complete metric space — Completeness is a separate premise.
• Convergence in a metric space — Separate Cauchy behavior and convergence.
• 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.