TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Sigma-one truth goes upward

Statement

For nonempty transitive $M \subseteq N$ , a literal existential block over a $Δ_{0}$ matrix transfers truth upward, and its universal dual transfers truth downward. For formulas classified only by ZF-provable equivalence, assume both structures satisfy ZF (or all axioms used in the equivalence proof).

Facts & Assumptions

[F1]

Bounded formulas are absolute for transitive sets: If $M \subseteq N$ are nonempty transitive sets, every $Δ_{0}$ formula is absolute between them on parameter tuples from $M$ . No internal set-theory axioms are required. The analogous assertion for definable transitive classes is a formula-by-formula scheme.

[F2]

Sigma-one formulas admit existential bounded matrices: Over ZF, every $Σ_{1}$ formula in the bounded-closure convention is equivalent to $\exists u_{1} \dots \exists u_{k} δ$ with $δ$ bounded. The equivalence is asserted over ZF, not over arbitrary transitive structures.

Proof

Given: Parameters in nonempty transitive $M \subseteq N$ , with the displayed syntax or equivalence axioms.

1.1

If $M ⊨ \exists \overset{ˉ}{u} δ (\overset{ˉ}{u}, \overset{ˉ}{a})$ , take its finite tuple of witnesses $\overset{ˉ}{b} \in M$ . The tuple remains in $N$ , and bounded absoluteness F1 gives $N ⊨ δ (\overset{ˉ}{b}, \overset{ˉ}{a})$ . Thus $N$ satisfies the existential formula. The empty block is exactly F1.

F1given

2.1

If a universal dual were true in $N$ and false in $M$ , its existential negation would be true in $M$ and hence in $N$ by step 1.1. This contradicts its truth in $N$ , proving downward transfer.

step 1.1algebra

3.1

When classification is modulo ZF, F2 supplies a ZF equivalence to the literal normal form. Each model satisfies that equivalence under the extra hypothesis. Translate to the normal form in the source model, apply steps 1.1 or 2.1, and translate back in the destination model.

F2step 1.1step 2.1∎

Depends on

Used by

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

Andrew Marks, Set Theory lecture notes — Proposition 18.13, upward and downward absoluteness, p78 (standard reference, not scraped)