Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Freeness of chain groups cannot simply be dropped from the classical Kunneth statement

Statement

The classical free-complex Kunneth argument cannot simply omit freeness of chain groups: it uses freeness to make cycle and boundary modules flat and to control the Tor edge.

Refutation

Given: the complexes C=D=Z/2 concentrated in degree zero, viewed as complexes of nonfree abelian groups.

1.1given

Their ordinary tensor complex is concentrated in degree zero, so H1(C⊗ZD)=0. On the other hand, H0C=H0D=Z/2 and Tor⁡1Z(H0C,H0D)=Z/2.

2.1step 1.1∎

If the classical free-complex Kunneth short exact sequence were asserted unchanged after simply deleting freeness, then in total degree one it would surject from the zero group H1(C⊗D) onto the nonzero Tor group from step 1.1. That is impossible. Hence freeness cannot simply be dropped without replacing ordinary tensor by a derived construction or adding suitable flatness hypotheses.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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