Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

The jump is well-defined on degrees

Statement

If A≡TB, then A′≡TB′. Thus [A]T↦[A′]T is a well-defined operation on Turing degrees.

Facts & Assumptions

Given: total oracle deciders witnessing A≤TB and B≤TA.

Proof

technique · direct
1.1

Compile an A-oracle program e into a B-oracle program q(e) that answers each simulated A-query using the fixed B-oracle decider for A; q is computable and preserves halting on the same input.

givenconstruct
2.1

By parameter specialization, compute an index r(e) for the program that on its own input simulates q(e) on e. Then e∈A′ iff r(e)∈B′, so A′≤TB′.

step 1.1construct
3.1

Repeat the construction of step 2.1 using B≤TA to obtain B′≤TA′. Therefore A′≡TB′.

step 2.1∎

Depends on

Used by

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