ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

Condensation of singularities

Example

Assume DC. Let $X$ be Banach, let $Y$ be normed, and let $F \subseteq B (X, Y)$ be not uniformly bounded. Then singular vectors, those $x$ with $\sup_{T \in F} ∥ T x ∥ = \infty$ , form a dense $G_{δ}$ set.

Facts & Assumptions

Given: DC, $X, Y, F$ as displayed, and failure of uniform boundedness.

Verification

technique · direct

1.1

The second alternative of Baire dichotomy for a pointwise-defined family of bounded linear operators applies directly.

given

2.1

Therefore every nonempty open ball contains a singular vector, which is the condensation assertion.

step 1.1∎

Depends on

Baire dichotomy for a pointwise-defined family of bounded linear operators

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Teschl, Topics in Real and Functional Analysis, Theorem 4.3 (standard reference, not scraped)