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.

FALSE: left and right global dimensions are equal by definition

Statement

FALSE: left and right global dimensions are equal by definition

Facts & Assumptions

Given: An arbitrary ring R and the definitions of left and right global dimension.

Refutation

technique · direct
1.1

Left global dimension is sup⁡{pd⁡RM:M a left R-module}, while right global dimension is the corresponding supremum over right R-modules, equivalently left Rop-modules.

givenalgebra
2.1

These are distinct definitions on distinct module categories; no equality is built into either one. Any theorem equating them must add hypotheses or prove a comparison, so equality cannot hold “by definition.”

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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