ExampleConstruction: AI-adaptedVerification: 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.

A strict tent-map itinerary has exponential finite-time sensitivity growth

Example

Let $T (x) = 2 x$ for $x < 1 / 2$ and $T (x) = 2 - 2 x$ for $x > 1 / 2$ . Fix a finite integer $N \geq 1$ , take $x_{0} (θ) = θ$ near $θ_{*} = 1 / 3$ , and set the terminal objective to $J (θ) = x_{N} (θ)$ .

Facts & Assumptions

Given: The two affine branches, a finite horizon $N$ , and the finite state-step adjoint rule.

[F1]

On a neighborhood with a fixed smooth branch at every step, the reverse recursion gives the exact derivative of the finite graph (The reverse recursion is the exact gradient of a finite state-step objective).

Verification

1.1givenconstructalgebra

At $θ_{*} = 1 / 3$ , the orbit is $x_{0} = 1 / 3$ and $x_{n} = 2 / 3$ for every $n \geq 1$ . If $∣ θ - 1 / 3 ∣ < 1 / (6 \cdot 2^{N})$ , then the first state remains below $1 / 2$ and each later state remains within $1 / 6$ of $2 / 3$ through time $N$ ; hence this whole open neighborhood has the same strict branch itinerary.

2.1F1step 1.1algebra∎

On that neighborhood the first derivative is $T^{'} (x_{0}) = 2$ , and each of the next $N - 1$ derivatives is $- 2$ . The chain rule gives $J^{'} (1 / 3) = 2 (- 2)^{N - 1}$ , whose magnitude is $2^{N}$ . By [F1], reverse accumulation returns this same finite-horizon derivative. No infinite-time exponent or orbit-shadowing claim is involved.

Depends on

The reverse recursion is the exact gradient of a finite state-step objective

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.

Sources

Predrag Cvitanović et al., ChaosBook, Chapter 6: Lyapunov exponents, §6.2 (standard reference, not scraped)
Michael B. Giles and Niles A. Pierce, An Introduction to the Adjoint Approach to Design, §2.4 (standard reference, not scraped)