LemmaStatement: Literature-sourcedProof: 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.

The diagonal self-reference construction from s-m-n

Statement

For an acceptable numbering, there exists a total computable function $d : N \to N$ such that for every index $u$ and every input $x$ , $φ_{d (u)} (x) ≃ φ_{u} (⌜ u, x ⌝_{s e q}) .$

Facts & Assumptions

Given: An acceptable numbering $(φ_{e})_{e \in N}$ .

[L1]

The $s$ -m- $n$ theorem gives a total computable specialization function $s_{1}^{1}$ , by The s-m-n theorem.

Proof

technique · direct

1.1

Let $s_{1}^{1}$ be the specialization map from [L1]. Define $d (u) : = s_{1}^{1} (u, u) .$ Because $s_{1}^{1}$ is total computable, so is $d$ .

L1givenconstruct

2.1

By the defining property of $s_{1}^{1}$ , for every $u$ and $x$ one has $φ_{d (u)} (x) = φ_{s_{1}^{1} (u, u)} (x) ≃ φ_{u} (⌜ u, x ⌝_{s e q}) .$ That is exactly the required diagonal identity.

L1step 1.1

3.1

Therefore the desired total computable self-reference constructor exists.

step 2.1∎

Depends on

Used by

Kleene's second recursion theorem Theorem

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

Lawrence S. Moss, Invitation to Computability and Recursion, The Recursion Theorem (standard reference, not scraped)
Robert I. Soare, Turing Computability: Theory and Applications (standard reference, not scraped)