ExampleConstruction: AI-generatedVerification: 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 cusp in a chosen exchange-correlation term violates the C2 hypothesis

Example

With $H = diag (1, - 1)$ on $C^{2}$ and $r = 1$ , consider the candidate term $E_{x c} (P) = ∣ tr (H P) ∣$ . Along the rank-one projectors $P_{θ} = u_{θ} u_{θ}^{*}$ , where $u_{θ} = (\cos θ, \sin θ)$ , its value is $∣ \cos (2 θ) ∣$ and has a cusp at $θ = π / 4$ . This candidate is not $C^{1}$ on a neighborhood of the projector domain and so is outside the $C^{2}$ finite-basis model hypothesis.

Facts & Assumptions

Given: $H = diag (1, - 1)$ , $r = 1$ , and the candidate functional $E_{x c} (P) = ∣ tr (H P) ∣$ .

[F1]

The model's exchange-correlation term must be real and $C^{2}$ on a neighborhood of $P_{r}$ (Finite-basis projector energy model).

[F2]

A rank-one projector is $u u^{*}$ for a unit vector $u$ (Rank-r orthogonal projector domain).

[F3]

The composition of totally differentiable maps is totally differentiable (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

Verification

1.1F2givenalgebra

For every $θ$ , $u_{θ}^{*} u_{θ} = 1$ , so [F2] gives $P_{θ} \in P_{1}$ . Multiplication yields $tr (H P_{θ}) = \cos^{2} θ - \sin^{2} θ = \cos (2 θ)$ .

2.1F1F3step 1.1algebra∎

Near $θ = π / 4$ , the left branch of $∣ \cos (2 θ) ∣$ has derivative $- 2$ at $π / 4$ , while the right branch has derivative $2$ . The restriction is not differentiable there; by [F3], a $C^{2}$ ambient functional composed with the smooth curve $P_{θ}$ would be differentiable. Hence the candidate cannot satisfy the $C^{2}$ requirement in [F1].

Depends on

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