LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Splitting stationary sets of fixed smaller cofinality

Statement

In ZFC, if $κ$ is regular uncountable, $λ < κ$ is infinite regular, and $S \subseteq E_{λ}^{κ}$ is stationary, then $S$ has a partition into $κ$ stationary sets.

Facts & Assumptions

[F1]

Cofinality strata, trace, and reflection: Every point of the stratum has cofinality lambda.

[F2]

Intersections of fewer than the cofinality many clubs: Fewer than kappa clubs intersect to a club on regular uncountable kappa.

[F3]

Unboundedly many stationary fibres yield a partition: A regressive function with stationary tail domains at every threshold yields the required partition.

Proof

Given: The objects and hypotheses in the statement.

1.1

Use AC to choose for each $α \in S$ an increasing cofinal sequence $c^{α} : λ \to α$ . Suppose every coordinate $ξ < λ$ has a threshold $b_{ξ} < κ$ for which ${α \in S : c^{α} (ξ) \geq b_{ξ}}$ is nonstationary; choose an avoiding club $C_{ξ}$ .

2.1

The intersection $C = ⋂_{ξ < λ} C_{ξ}$ is club, and $b = \sup_{ξ < λ} b_{ξ} < κ$ by regularity. Take $α \in S \cap C$ above b (the intersection is unbounded, by testing it against additional tails). Then $c^{α} (ξ) < b_{ξ} \leq b$ for every $ξ$ , contradicting cofinality in $α > b$ .

F2step 1.1

3.1

Therefore some fixed coordinate $ξ$ has stationary tail domains at every threshold. The map $g (α) = c^{α} (ξ)$ is regressive on all S. The fibre-partition lemma applies.

F3step 2.1∎

Depends on

Used by

Solovay’s stationary partition theorem Theorem

Dependency tree · two levels

12 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

Lietz, Claim 5.17 and first splitting case, p.44 (standard reference, not scraped)
Vasey, Lemmas 15.5–15.7, pp.84–85 (standard reference, not scraped)