Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)↗ rests on later material
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 limit approximation computed from the halting oracle

Example

Use the fixed effective numbering ΦeA from the definition of the Turing jump. For 0′={e:Φe∅(e)↓}, put h(e,s)=1 if Φe∅(e) halts within s steps and put h(e,s)=0 otherwise.

Facts & Assumptions

Given: a program index e.

[F1]

The fixed effective numbering defines 0′ by diagonal halting (The Turing jump ↗).

Verification

technique · direct
1.1F1givenconstruct

The numbering and decoder are effective, so bounded simulation makes (e,s)↦h(e,s) computable.

2.1F1step 1.1∎

If Φe∅(e) never halts every value is 0; if it halts at stage r, every value from r onward is 1. Thus lim⁡sh(e,s)=χ0′(e), without supplying a computable stage at which this happens.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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