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.

Tor does not take two left modules over an arbitrary ring without extra bimodule structure

Statement

False claim: for every noncommutative ring R, Tor⁡iR(M,N) is defined for two left R-modules M,N.

Refutation

Given: the ring R and its left modules.

1.1

The tensor construction requires its first input to be a right R-module so that (xr)⊗y=x⊗(ry) is meaningful.

given
2.1

For two merely left modules, xr is not part of the supplied structure, so the balancing relation is not typed.

step 1.1algebra
3.1

Hence the displayed Tor expression is not even defined without additional bimodule or opposite-ring data, refuting the universal claim.

step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

3 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