Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-04
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 duals of an object are isomorphic

Statement

Left and right duals of an object are isomorphic.

Facts & Assumptions

Given: The rigid monoidal category and object constructed in the cited counterexample.

[L1]

The category R contains an object whose left and right duals are not isomorphic (Left and right duals, and double duals, need not collapse).

Refutation

technique · direct
1.1

The counterexample Left and right duals, and double duals, need not collapse constructs the composition-closed rigid monoidal subcategory R generated by an adjoint chain, containing an object D whose left dual is L(n)=⌈n/2⌉ and whose right dual is R(n)=⌊n/2⌋.

givenL1
2.1

In that category, endofunctors are isomorphic only when they agree pointwise, but L(1)=1 and R(1)=0. So L≇R.

step 1.1L1
3.1

Hence there exists an object whose left and right duals are not isomorphic, so the statement is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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