ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-01

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.

The integers sit densely but not closedly inside their profinite completion

Example

The canonical copy of $Z$ is dense but not closed in its profinite completion $\hat{Z}$ .

Facts & Assumptions

Given: The canonical map $ι : Z \to \hat{Z}$ .

[L1]

The image of the canonical map is dense in every profinite completion (The canonical map to the profinite completion has kernel equal to the finite residual and has dense image).

[L2]

The profinite completion of $Z$ is the inverse limit ${\underset{\leftarrow}{l i m}}_{n} Z / n Z$ (The profinite completion of the integers is the inverse limit of the rings Z mod n).

Verification

technique · direct

1.1

By [L1], $ι [Z]$ is dense in $\hat{Z}$ .

L1given

2.1

The space $\hat{Z}$ is compact and Hausdorff by [L2] together with the general inverse-limit theorem, so any closed dense subset would have to be the whole space. But $ι [Z]$ is countable, whereas $\hat{Z}$ contains the uncountable subset $\prod_{p} {0, 1}$ indexed by the primes inside its product model. Therefore $ι [Z]$ is a proper dense subset and cannot be closed.

L2step 1.1algebra

3.1

So $Z$ sits densely but not closedly inside its profinite completion.

step 1.1step 2.1∎

Depends on

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

Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)