ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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.

Free abelian groups have polynomial growth of the expected degree

Example

For the standard generating set of $Z^{n}$ , the growth function is equivalent to $m^{n}$ . Hence every free abelian group of rank $n$ has polynomial growth of degree $n$ .

Facts & Assumptions

Given: The standard generators $\pm e_{1}, \dots, \pm e_{n}$ of $Z^{n}$ .

[L1]

Polynomial growth means comparison with $m^{d}$ for some integer $d \geq 0$ (Polynomial, subexponential, exponential, and intermediate growth).

[L2]

Growth type is independent of the chosen finite generating set (Growth type is independent of the finite generating set).

Verification

technique · direct

1.1

In the standard word metric on $Z^{n}$ , the radius- $m$ ball is ${a \in Z^{n} : ∣ a_{1} ∣ + \dots + ∣ a_{n} ∣ \leq m}$ . It is contained in the cube ${- m, \dots, m}^{n}$ , so its cardinality is at most $(2 m + 1)^{n}$ .

givenalgebra

2.1

For each $k \in N$ , the cube ${- k, \dots, k}^{n}$ is contained in the radius- $n k$ ball, because $∣ a_{1} ∣ + \dots + ∣ a_{n} ∣ \leq n k$ there. So $β_{Z^{n}} (n k) \geq (2 k + 1)^{n}$ . Together with step 1.1, this shows the growth function is equivalent to $m^{n}$ .

step 1.1algebra

3.1

Step 2.1 gives polynomial growth of degree $n$ for the standard generators, and [L2] transports the same growth type to every finite generating set.

L1L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

C. Löh, Geometric Group Theory, Sections 5.1-5.3 (standard reference, not scraped)
C. Drutu and M. Kapovich, Lectures on Geometric Group Theory, Chapter 5 (standard reference, not scraped)