Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12
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: a partial computable function has a unique program index

Statement

Every partial computable function has exactly one program index.

Facts & Assumptions

Given: An acceptable numbering (φe)e∈N.

[L1]

Every total computable program transformer has a fixed point, by Kleene's second recursion theorem.

Refutation

technique · direct
1.1

Consider the total computable transformer F(e):=e+1. It changes only the numerical index and is therefore a computable program transformer.

givenconstruct
2.1

By [L1], there exists an index e with φe=φF(e)=φe+1. Since e≠e+1, the same partial computable function has at least two distinct indices.

L1step 1.1
3.1

Therefore the statement is false: a partial computable function can have many distinct indices.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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