ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

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

An interior stationary point adds a Fresnel-scale term

Example

For $f (x) = 1$ , $g (x) = x^{2} / 2$ on $[- 1, 1]$ , and $ω \geq 1$ , the integral has an interior contribution of order $ω^{- 1 / 2}$ in addition to its order- $ω^{- 1}$ endpoint term: $I_{ω} (1, g; [- 1, 1]) = \sqrt{\frac{2 π}{ω}} e^{i π / 4} + \frac{2 e^{i ω / 2}}{i ω} + R_{ω}, ∣ R_{ω} ∣ \leq C ω^{- 3 / 2} .$ The nonstationary expansion cannot be applied on the full interval because $g^{'} (0) = 0$ .

Facts & Assumptions

Given: $f (x) = 1$ , $g (x) = x^{2} / 2$ on $[- 1, 1]$ , and $ω \geq 1$ .

[F1]

Stationary-phase expansion at one simple interior critical point: a single simple interior stationary point contributes $f (x_{0}) e^{i ω g (x_{0})} \sqrt{2 π / ∣ ω g^{''} (x_{0}) ∣} e^{i sgn (ω g^{''} (x_{0})) π / 4}$ , with the endpoint term and an $O (∣ ω ∣^{- 3 / 2})$ remainder.

[F2]

Repeated integration by parts for a nonstationary oscillatory integral: the nonstationary branch requires $∣ g^{'} ∣ \geq c > 0$ on the whole interval to which it is applied.

[F3]

The complex modulus is subadditive, $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Verification

1.1givenF1algebra

Here $g^{'} (x) = x$ , $g^{''} (0) = 1$ , and $g (\pm 1) = 1 / 2$ . Applying [F1] at $x_{0} = 0$ gives the stationary term $\sqrt{2 π / ω} e^{i π / 4}$ , while the endpoint bracket is $\frac{1}{i ω} (\frac{e^{i ω / 2}}{1} - \frac{e^{i ω / 2}}{- 1}) = \frac{2 e^{i ω / 2}}{i ω} .$ The remainder bound is the one in [F1].

2.1givenF1F2F3step 1.1algebra∎

Since $g^{'} (0) = 0$ , no $c > 0$ satisfies [F2] on $[- 1, 1]$ . By [F3], the endpoint term and remainder together are bounded by $2 / ω + C ω^{- 3 / 2}$ , both smaller than the stationary term's magnitude $\sqrt{2 π / ω}$ by a factor tending to zero. Thus the leading scale is $ω^{- 1 / 2}$ , not the nonstationary integer-power expansion.

Depends on

Used by

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Sources

NIST Digital Library of Mathematical Functions, §2.3(iv) (standard reference, not scraped)
A. Iserles, S. P. Nørsett, and S. Olver, Highly oscillatory quadrature: The story so far, §4 (standard reference, not scraped)