ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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 cyclic presentation and its five-term sequence

Example

For $m \geq 1$ , the presentation $F = Z ↠ C_{m} = Z / m Z$ has kernel $R = m Z$ , and its five-term sequence is $0 \to 0 \to m Z ↪ Z ↠ Z / m Z \to 0$ . Thus $H_{2} (C_{m}; Z) = 0$ .

Facts & Assumptions

Given: m is a positive integer; F is the free group on one generator, written additively; use the inherited DC and supplied-resolution convention.

[F1]

A free presentation gives an exact sequence with injection from H2 into R/[F,R] and the inclusion and quotient maps on abelianizations. (The free-presentation homology five-term sequence).

Verification

1.1

Every element of the free group on one generator has a unique integer exponent, so identify F with $Z$ . Reduction modulo m is onto and has kernel $m Z$ . All elements of F commute, hence $[F, R] = 0$ , $F_{a b} = Z$ , and $(C_{m})_{a b} = C_{m}$ . F1 therefore gives $0 \to H_{2} (C_{m}; Z) \overset{j}{\to} m Z \overset{ι}{\to} Z \to Z / m Z \to 0$ , where $ι$ is the actual subgroup inclusion.

F1algebra

2.1

The inclusion has zero kernel. Exactness gives $im j = 0$ , while j is injective, so $H_{2} (C_{m}; Z) = 0$ . Under the isomorphism $Z \to m Z$ , $a \mapsto m a$ , the middle map becomes multiplication by m; its image is exactly the kernel of reduction modulo m. For m=1 this map is identity and the final quotient is zero.

step 1.1algebra∎

Depends on

The free-presentation homology five-term sequence

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

Loh, Group Cohomology, Definitions 1.7.12–13 and Theorem 1.7.15 pp.63–64; Theorem 3.2.18 pp.129–132 (standard reference, not scraped)
Weibel, An Introduction to Homological Algebra, Chapter 6, Sections 6.4–6.8 (standard reference, not scraped)