Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12
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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: the recursion theorem requires source-code access

Statement

Kleene's recursion theorem can produce self-referential programs only when the program can inspect its own literal source file.

Facts & Assumptions

Given: The fixed-point theorem of Kleene's second recursion theorem.

[L1]

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

Refutation

technique · direct
1.1

Consider the total computable transformer F that sends an index e to a program which ignores its actual input and simply outputs the numeral e. This transformer manipulates indices only; it does not read any source file.

givenconstruct
2.1

By [L1], there exists q with φq=φF(q). But φF(q) is precisely the constant program that outputs q, so φq does the same. Thus a quine-style fixed point exists without any inspection of a literal source file.

L1step 1.1
3.1

Therefore the statement is false. The recursion theorem needs numbering, specialization, and computable index transformation, not filesystem access.

step 2.1∎

Depends on

Used by

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