Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedjudge pass (gpt-5.6-terra)audited 2026-09-06
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 absolutely convergent Fourier series that is not twice continuously differentiable

Example

Assume the Axiom of Countable Choice. The uniformly convergent series f(x)=∑k≠0∣k∣−2ek(x) defines a member of A(T) which is not C2(T).

Facts & Assumptions

Given: The Axiom of Countable Choice and the displayed Fourier series.

[L1]

Fourier coefficients of an integrable function tend to zero at infinity (Riemann-Lebesgue lemma for Fourier coefficients).

Verification

1.1

Since ∑k≠0∣k∣−2<∞, the coefficient sequence is in ℓ1, so the displayed function belongs to A(T).

givenalgebra
1.2

If f were C2, two integrations by parts would give f′′^(k)=(2πik)2f^(k)=−4π2 for every k≠0.

givenalgebra
2.1

This contradicts [L1] for the continuous, hence integrable, function f′′; therefore f∉C2(T).

L1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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