CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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 embedded integers form an abstractly proper but topologically dense subgroup of Zp

Statement refuted

Every dense subgroup of $Z_{p}$ is all of $Z_{p}$ as an abstract subgroup.

Facts & Assumptions

Given: The embedded copy of $Z$ inside $Z_{p}$ .

[L1]

The image of $Z$ in $Z_{p}$ is dense (The canonical map from the integers into Zp is injective and has dense image).

[L2]

The additive group of $Z_{p}$ is not abstractly cyclic, even though the closure of $Z \cdot 1$ is all of $Z_{p}$ (The additive topological group of Zp is topologically generated by 1, although it is not abstractly cyclic).

Counterexample

technique · direct

1.1

By [L1], the embedded copy of $Z$ is dense in $Z_{p}$ .

L1given

2.1

The embedded copy of $Z$ is exactly $Z \cdot 1$ . By [L2], its closure is all of $Z_{p}$ but it is not equal to $Z_{p}$ as an abstract subgroup. Thus a dense subgroup can be proper.

L2step 1.1

3.1

Therefore the stated universal claim is false, and the embedded copy of $Z$ is the required counterexample.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Jordan Bell, The profinite completion of the integers, the p-adic integers, and Prufer p-groups (standard reference, not scraped)