ExampleConstruction: 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.

A multiple residue root need not lift uniquely

Example

Over the $2$ -adic integers $Z_{2}$ , the polynomial $f (T) = T^{2} - 1$ has the multiple residue root $\overline{1} \in F_{2}$ , but that root does not lift uniquely.

Facts & Assumptions

Given: The polynomial $f (T) = T^{2} - 1$ over $Z_{2}$ .

[L1]

Simple roots lift uniquely in Henselian local rings; the derivative hypothesis is therefore the load-bearing condition (Factor lifting implies simple-root lifting).

Verification

technique · exhibit two distinct lifts of the same multiple residue root

1.1

Modulo $2$ , the polynomial becomes $\overline{f} (T) = T^{2} - 1 = (T - 1)^{2},$ so the residue root $\overline{1}$ has multiplicity $2$ . Equivalently, ${\overline{f}}^{'} (\overline{1}) = 2 \overline{1} = 0$ .

givenalgebra

2.1

In $Z_{2}$ , both $1$ and $- 1$ satisfy $f (T) = 0$ , and both reduce to $\overline{1}$ modulo $2$ . Hence the residue root $\overline{1}$ has at least two lifts.

step 1.1given

3.1

Therefore the derivative-unit hypothesis in [L1] cannot be dropped: a multiple residue root need not lift uniquely even in a complete local ring.

L1step 2.1∎

Depends on

Factor lifting implies simple-root lifting

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

The Stacks Project, Section 10.153: Henselian local rings (standard reference, not scraped)