ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An affine response map realizes the contraction and residual estimates

Example

This is a scalar fixed-point response model, not an occupied spectral projector. On $K = [0, 1]$ let $Φ (x) = x / 2 + 1 / 4$ . It is a $1 / 2$ -contraction with fixed point $x_{*} = 1 / 2$ , and its residual-to-error bound is exact.

Facts & Assumptions

Given: $K = [0, 1]$ with the usual distance, $Φ (x) = x / 2 + 1 / 4$ , and an initial value $x_{0} \in [0, 1]$ .

[F1]

Under its nonempty-complete-space and contraction hypotheses, the general theorem gives $d (x, x_{*}) \leq d (x, Φ (x)) / (1 - q)$ (A contractive self-consistent map converges and admits residual and inexact-solve bounds).

Verification

1.1givenF1algebra

For $x, y \in [0, 1]$ , $Φ (x) \in [1 / 4, 3 / 4] \subseteq [0, 1]$ and $∣ Φ (x) - Φ (y) ∣ = ∣ x - y ∣ / 2$ . Solving $x = x / 2 + 1 / 4$ gives the unique fixed point $x_{*} = 1 / 2$ .

2.1step 1.1algebrainduction

The exact iterates satisfy $x_{k + 1} - 1 / 2 = (x_{k} - 1 / 2) / 2$ , so induction gives $∣ x_{k} - x_{*} ∣ = 2^{- k} ∣ x_{0} - x_{*} ∣$ .

3.1givenstep 1.1F1algebra∎

For any $x \in [0, 1]$ , $∣ x - Φ (x) ∣ = ∣ x - 1 / 2 ∣ / 2$ , hence $∣ x - x_{*} ∣ = ∣ x - Φ (x) ∣ / (1 - 1 / 2)$ . This algebraic identity directly realizes the same residual-to-error formula stated conditionally in [F1], without invoking that theorem's hypotheses.

Depends on

A contractive self-consistent map converges and admits residual and inexact-solve bounds

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.