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.

Factoring the Solovay collapse around a real parameter

Example

Let $t \in V [G_{ξ}]$ be a real and display both relevant factorizations.

Facts & Assumptions

Given: The Solovay collapse and $t \in V [G_{ξ}]$ .

[F1]

The inaccessible Lévy-collapse setup for Solovay's construction: restriction to $ξ \times ω$ is a complete projection with the corresponding initial-extension/quotient factorization.

[F2]

Absorption, factorization, and homogeneous truth in the Solovay collapse: small initial factors are absorbed over a real parameter and the remaining collapse is homogeneous.

Verification

1.1

The complete restriction map gives $V [G] = V [G_{ξ}] [G^{ξ}]$ , where $G^{ξ}$ is generic for the quotient supported on $[ξ, κ) \times ω$ . The actual initial generic, not merely $t$ , is present at this stage.

2.1

Absorption recodes $G_{ξ}$ together with the quotient into an $H$ generic for a fresh $Lv (κ)^{V [t]}$ , yielding $V [G] = V [t] [H]$ . A formula $φ (t, \vec{α})$ omitting $H$ is therefore decided by tail homogeneity; a formula mentioning $G_{ξ}$ need not be fixed.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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.