ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Coding countably many Solovay definition parameters

Example

Explicitly combine countably many countable ordinal parameters into one member of $S$ .

Facts & Assumptions

Given: $s_{n} : ω \to α_{n}$ for $n < ω$ and a fixed bijection $π : ω^{2} \to ω$ .

[F1]

The hereditarily ordinal-sequence-definable Solovay model: $S$ consists of countable ordinal sequences and $M = H O D (S)$ permits an $S$ -parameter.

Verification

1.1

Let $β = \sup_{n} (α_{n} + 1)$ and define $s (π (n, k)) = s_{n} (k)$ . Then $s : ω \to β$ is in $S$ . The fixed inverse of $π$ recovers $s_{n} (k) = s (π (n, k))$ uniformly.

Given

2.1

Encode the finite formula number and finite ordinal tuple for the $n$ th definition in slots $π (n, 0), π (n, 1), \dots$ , shifting the values of $s_{n}$ to later tagged slots. Finite tags are ordinals below a common bound. Thus one sequence recovers every formula, ordinal tuple and $S$ -parameter, exactly as permitted by F1. Empty tuples use their length tag zero. This is an explicit coding calculation, not an application of the separate omega-closure theorem.

F1step 1.1∎

Depends on

The hereditarily ordinal-sequence-definable Solovay model

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.