Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01
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 degreewise split sequence with nonzero connecting map

Example

Take the identity map 1Z[0]:Z[0]→Z[0]. Its canonical cone sequence 0→Z[0]→Cone⁡(1Z[0])→Z[1]→0 is degreewise split short exact, but its connecting morphism H1(Z[1])→H0(Z[0]) is nonzero.

Facts & Assumptions

Given: The identity map on the stalk complex Z[0].

[L1]

Every canonical cone sequence is degreewise split short exact (The canonical mapping-cone sequence is degreewise split short exact).

[L2]

For the identity map, the connecting morphism agrees with the shifted identity up to sign (The cone connecting map agrees with the shifted identity up to the declared sign).

Verification

technique · direct
1.1L1given

The displayed cone sequence is degreewise split by [L1].

2.1L2step 1.1algebra∎

The source H1(Z[1]) and target H0(Z[0]) are both isomorphic to Z, and [L2] identifies the connecting map with ±1 on that group. Hence it is nonzero.

Depends on

Used by

Dependency tree · two levels

9 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