PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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.

Termination and operation count of the periodic wave benchmark

Physical scope

A finite exact-arithmetic property of the one-dimensional scalar benchmark; it asserts no numerical-relativity solver performance or physical observation.

Statement

For every admissible integer $N \geq 3$ , time-step count $n_{\max} \geq 1$ , real step $k > 0$ , and pair of real periodic input arrays, the algorithm terminates after exactly $n_{\max} - 1$ update passes and outputs all levels through $u^{n_{\max}}$ . In the unit-cost exact-real-arithmetic model, where each scalar arithmetic operation and each array read or write has bounded cost independent of $N$ and $n_{\max}$ , its operation count is $O (N n_{\max})$ .

Statement references

A finite periodic scalar-wave benchmark

Facts & Assumptions

Given: The finite periodic scalar-wave algorithm with integer inputs $N \geq 3$ and $n_{\max} \geq 1$ .

[F1]

A finite periodic scalar-wave benchmark: one initialization computes $u^{1}$ by one finite grid pass; the update loop runs through the fixed integer range $1, \dots, n_{\max} - 1$ , and each pass updates exactly $N$ grid entries.

Proof

technique · direct

1.1givenF1algebra

The initialization uses a finite formula at each of the $N$ sites. The outer index takes each of the finitely many values $1$ through $n_{\max} - 1$ and has no data-dependent branch, so the procedure makes exactly $n_{\max} - 1$ update passes. Each pass writes the next $N$ -tuple, giving the output levels $u^{0}, u^{1}, \dots, u^{n_{\max}}$ .

2.1givenF1step 1.1algebra∎

The initialization costs at most $C_{0} N$ scalar and array operations for a fixed constant $C_{0}$ . Each update pass costs at most $C_{1} N$ such operations for a fixed $C_{1}$ , because every site uses the same fixed stencil. Thus the total is at most $C_{0} N + C_{1} N (n_{\max} - 1) \leq (C_{0} + C_{1}) N n_{\max}$ , proving the claimed bound.

Depends on

A finite periodic scalar-wave benchmark