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.

Removing the trace preserves a stationary remainder

Statement

In ZFC, if $S \subseteq κ$ is stationary and $κ$ is regular uncountable, then $S ∖ Tr (S)$ is stationary.

Facts & Assumptions

[F1]

Cofinality strata, trace, and reflection: Trace membership requires uncountable cofinality and stationarity of the initial restriction.

[F2]

Limit points of an unbounded set form a club: Limit points of an unbounded subset form a club when the ambient cofinality is uncountable.

[F3]

Basic stationary-set calculus: A stationary set meets each club; intersecting it with a club preserves stationarity.

Proof

Given: The objects and hypotheses in the statement.

1.1

Suppose a club $C$ avoids $S ∖ Tr (S)$ . Since $acc (C)$ is club, let $α$ be the least point of $S \cap acc (C)$ . Closure of C gives $α \in C$ , so avoidance forces $α \in Tr (S)$ . In particular $cf (α) > ω$ .

F1F2F3

2.1

Now $C \cap α$ is unbounded in $α$ , so ${acc}_{α} (C \cap α) = {acc}_{κ} (C) \cap α$ is a club in $α$ . By minimality of $α$ it is disjoint from $S \cap α$ . This contradicts trace membership.

F1F2step 1.1∎

Depends on

Used by

Splitting a stationary set concentrated on regular cardinals Lemma

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

Lietz, Lemma 5.16, p.43 (standard reference, not scraped)