False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-29

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: the subtraction rule produces a unique member

Statement

In the subtraction surrogate, the member $z$ with $g z \equiv 0$ is unique.

Facts & Assumptions

Given: The category $A b$ .

[L1]

The category $A b$ is abelian (Abelian groups form an abelian category).

[L2]

The subtraction surrogate applies whenever $g x \equiv g y$ (The subtraction surrogate).

Refutation

technique · direct

1.1L1L2

In $A b$ , take $B = Z$ , $C = 0$ , $g = 0 : Z \to 0$ , and $x = y = 0_{Z, Z} : Z \to Z$ . Then $g x = g y = 0$ , so the hypotheses of [L2] are satisfied.

2.1L1step 1.1algebra

Both $z_{0} : = 0_{Z, Z}$ and $z_{1} : = 1_{Z}$ satisfy $g z_{i} = 0$ , hence $g z_{i} \equiv 0$ . Since $z_{0} \neq z_{1}$ , the asserted uniqueness fails in this instance.

3.1step 2.1∎

Therefore the statement is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Saunders Mac Lane, Categories for the Working Mathematician, Theorem VIII.4.3(vi) (standard reference, not scraped)