CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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.

A quine obtained without inspecting its own source file

Statement refuted

The recursion theorem can yield a self-referential program only if the program reads its own literal source file.

Facts & Assumptions

Given: The total computable transformer $F$ that sends an index $e$ to the constant-output program for $e$ .

[L1]

The companion false statement is refuted by the recursion-theorem fixed point construction, by FALSE: the recursion theorem requires source-code access.

Counterexample

technique · direct

1.1

Let $F$ be the total computable transformer sending an index $e$ to the program that ignores its actual input and prints the numeral $e$ . No filesystem or source-file query appears in the description of $F$ .

givenconstruct

2.1

By the fixed-point theorem discussed in [L1], there exists an index $q$ with $φ_{q} = φ_{F (q)}$ . Therefore running program $q$ makes it print its own index $q$ , even though the construction used only numbering and effective transformation of indices.

L1step 1.1

3.1

This explicit quine is a counterexample to the statement.

step 2.1∎

Depends on

FALSE: the recursion theorem requires source-code access

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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.