A complete contraction certificate on a finite metric

Goal and scope

Audit completeness, a unique fixed point and the error bound for a constant self-map on a three-point discrete metric.

Only the complete finite model, not a replacement for the general Banach theorem.

Givens and symbols

• algebra
• sets
• epsilon

• G1 — X={a,b,c}; d(x,x)=0 and d=1 on distinct points; T(a)=T(b)=T(c)=a, q=1/2, x_0=c.

• D — complete metric matrix
• T — total self-map
• q — strict contraction factor

Rules and sources

• mko_definition · cited — 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. · mko-contraction-mapping-theorem
• algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
• sets · prerequisite — Compute unions, intersections, complements, inclusion and preimages element by element.
• epsilon · prerequisite — Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals.

Main derivation

D=\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}

d\ge0,\quad d(x,y)=0\iff x=y,\quad d(x,y)=d(y,x)

x=z\text{ or }x\ne z

d(x,z)\le d(x,y)+d(y,z)

(z_n)\text{ is }d\text{-Cauchy},\quad\varepsilon=1/2

\exists N\ \forall m,n\ge N:\ d(z_m,z_n)<1/2\Longrightarrow z_m=z_n

T(a)=T(b)=T(c)=a\in X

T:X\to X\text{ is a total self-map}

x,y\in X

d(Tx,Ty)=0\le\tfrac12d(x,y),\quad0\le q<1

Tx=x

x=a\text{ is the only fixed point}

x_{n+1}=Tx_n,\quad x_0=c

x_1=a,\quad x_n=a\ (n\ge1),\quad x_n\to a

\frac{q^n}{1-q}d(x_1,x_0)=2^{1-n}

d(x_0,a)=1\le2;\quad d(x_n,a)=0\le2^{1-n}\ (n\ge1)

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S06, S07 — The bound certifies this instance without claiming a general complete-space proof.
• S07 · premise_audit · G1, S04, S06 — Convergence is not claimed merely from a finite Picard trace.
• S06 · premise_audit · G1, S04 — Uniqueness follows from all-point auditing and also from d≤q*d with q<1.
• S05 · premise_audit · G1, S04 — q=1/2 is strictly below one, not merely nonexpansive q=1.
• S04 · premise_audit · G1 — A self-map, not a map into another space, is required.
• S03 · premise_audit · S01, S02 — The minimum positive distance controls every Cauchy sequence, not selected samples.
• S02 · premise_audit · G1, S01 — The triangle argument covers every triple, not numerical samples.
• S01 · premise_audit · G1 — Distances are supplied for every pair, not sampled pairs.

Error controls and negative controls

Complete metric

Qualify every metric axiom and the total map before contraction.

Whole orbit

Inductively prove it stays fixed forever, separately from a finite trace.

• The identity with q=1 also has a unique fixed point. → reject — All three points are fixed; strict contraction fails.
• Checking only (a,b) proves contraction on every pair. → reject — One success cannot replace the all-pair quantifier.
• A map leaving X still qualifies for the self-map theorem. → reject — Iteration and the fixed-point obligation require the same X.

Result and limitations

This finite metric is complete, T has unique fixed point a, and the orbit from c reaches it in one step with every error bound satisfied.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 9495e80b14cbe0e32e555d76b8ebed0ff8eb48f593477974ab06adb8325a82a6
• original source SHA-256: 9495e80b14cbe0e32e555d76b8ebed0ff8eb48f593477974ab06adb8325a82a6
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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