ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

Four-point Fourier collocation differentiates the Nyquist cosine at its nodes

Example

Take $N = 4$ and $p (x) = 1 + 2 \cos x + 3 \sin x - 4 \cos (2 x)$ on the grid $x_{j} = π j / 2$ . Its samples are $U = (- 1, 8, - 5, 2)$ . The interpolating space contains the Nyquist cosine $\cos (2 x)$ , while the derivative matrix omits that mode's derivative because it vanishes at every grid point. The returned derivative vector is $(3, - 2, - 3, 2)$ .

Facts & Assumptions

Given: The four-point periodic Fourier grid, the real interpolant convention, and the polynomial $p$ .

[F1]

Periodic Fourier collocation grid and even-grid Nyquist convention: for even $N = 4$ the interpolation space contains $1, \cos x, \sin x, \cos (2 x)$ and uses the cosine Nyquist mode only.

[F2]

Quarter-turn values and shifts by pi/2 and pi: the sine and cosine values at integer multiples of $π / 2$ are $0, \pm 1$ .

[F3]

The derivatives of sine and cosine are cosine and minus sine and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ : $(\cos (k x))^{'} = - k \sin (k x)$ and $(\sin (k x))^{'} = k \cos (k x)$ .

[F5]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : derivatives distribute over finite sums and scalar multiples.

[F4]

Periodic Fourier collocation differentiates its interpolant exactly: the matrix output on nodal data equals the derivative of their periodic trigonometric interpolant at every grid point, with the Nyquist derivative zero at the nodes.

Proof

1.1givenF1F2algebra

Substitution of $x_{0} = 0, x_{1} = π / 2, x_{2} = π, x_{3} = 3 π / 2$ into $p$ using [F2] gives respectively $- 1, 8, - 5, 2$ . The frequencies $0, 1, 2$ lie in [F1]'s interpolation space, so $p$ is the interpolant of these samples.

1.2givenF2F3F5algebra

By [F3] and [F5], $p^{'} (x) = - 2 \sin x + 3 \cos x + 8 \sin (2 x)$ . At the four nodes, the final term is zero and the first two terms give $3, - 2, - 3, 2$ .

2.1F1F4step 1.1step 1.2∎

Exactness [F4] therefore gives $D_{4} U = (3, - 2, - 3, 2)$ , including the vanishing nodal derivative of the Nyquist mode.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

30 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

Jie Shen and Tao Tang, Spectral and High-Order Methods with Applications, §2.2 (standard reference, not scraped)
Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Ch. 7 (standard reference, not scraped)