ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)verified 2026-09-10 (Codex)

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.

Tor detects n-torsion

Example

Assume AC (The Axiom of Choice) for balanced Tor. For $M = Z / 12$ and $n = 8$ , ${Tor}_{1}^{Z} (Z / 8, M) ≅ Z / 4$ . The calculation from the explicit multiplication-by- $8$ resolution is choice-free; AC is inherited only for comparison with arbitrary supplied resolutions in Tor one of a cyclic abelian group detects n-torsion.

Verification

Given: $M = Z / 12$ , $n = 8$ , and AC for the balanced comparison clause.

[L1] The repaired cyclic-Tor theorem supplies the explicit two-term resolution, its multiplication kernel, and the balanced identification under AC (Tor one of a cyclic abelian group detects n-torsion).

1.1givenL1

Apply [L1] to the explicit right resolution of $Z / 8$ and $M = Z / 12$ . Its tensor homology is the kernel of multiplication by $8$ ; AC is used only for the comparison giving balanced Tor. A left resolution, if used, is also explicitly supplied by the same multiplication construction for $12$ .

2.1step 1.1algebra

The kernel residues satisfy $12 ∣ 8 x$ , equivalently $3 ∣ 2 x$ . Since $x = 2 (2 x) - 3 x$ , this is equivalent to $3 ∣ x$ . Modulo $12$ the solutions are exactly $0, 3, 6, 9$ .

3.1step 1.1step 2.1algebra∎

The map $Z / 4 \to Z / 12$ , $[t]_{4} \mapsto [3 t]_{12}$ , is a well-defined homomorphism, with image those four residues and kernel zero since $12 ∣ 3 t$ iff $4 ∣ t$ . It identifies $Z / 4$ with the kernel in step 2.1, proving the claimed Tor calculation. The arithmetic uses no choice.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)