TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

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The bounded adaptive Runge–Kutta controller terminates

Statement

For finite-dimensional states and the finite-budget procedure in Finite-budget adaptive Runge–Kutta controller, the controller returns one of success, failure, minimum-step-rejected, or attempt-budget-exhausted after at most $N$ trial attempts. Each individual attempt uses finitely many stage evaluations and arithmetic operations.

Facts & Assumptions

Given: A controller input satisfying Finite-budget adaptive Runge–Kutta controller with finite stage count $s$ and attempt budget $N$ .

[F1]

Finite-budget adaptive Runge–Kutta controller: the procedure runs attempt indices $k = 1, \dots, N$ in order, advances to $k + 1$ only after a nonterminal attempt, and returns attempt-budget-exhausted after the $N$ th attempt unless an earlier terminal branch applies. Each trial uses the fixed finite tableau stage count and finite state dimension.

Proof

1.1givenF1algebra

Fix an attempt. Computing its trial size, its $s$ stages, its two weighted sums, and the finite-dimensional scaled maximum uses finitely many operations. Any stipulated undefined or nonfinite stage returns failure; an accepted step reaching $T$ returns success; and a rejection at the minimum step returns its stated status, in that order before the budget branch. A nonterminal attempt computes its next proposal and advances to the next loop index.

2.1F1step 1.1induction∎

The loop visits only the finite indices $1, \dots, N$ , with exactly one trial per visited index. If no earlier terminal branch returns, the $N$ th attempt returns attempt-budget-exhausted; in particular, an accepted step reaching $T$ on that attempt has already returned success. Thus termination occurs after at most $N$ attempts.

Depends on

Finite-budget adaptive Runge–Kutta controller

Used by

Dependency tree · two levels

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Sources

Ernst Hairer, course notes, Chapter III: Équations Différentielles Ordinaires, §III.2 (standard reference, not scraped)
Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Chapter 6 (standard reference, not scraped)