Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21
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.

False: Henstock–Kurzweil integrability implies Riemann integrability

Statement

False claim: Every Henstock–Kurzweil integrable function is Riemann integrable.

Facts & Assumptions

Given: The false implication.

[L1]

The indicator of the irrationals is Henstock–Kurzweil integrable with integral 1 and is not Riemann integrable (The indicator of the irrationals is Henstock–Kurzweil integrable with integral 1).

Refutation

technique · contradiction
1.1

Suppose the implication were true; [L1] gives a function satisfying its hypothesis but not its conclusion.

assume-contraL1
2.1

This contradiction refutes the implication, so the converse of the Riemann-to-HK theorem is false.

step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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