Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A triangle is distinguished whenever the three composites vanish

Statement

A triangle is distinguished if its three consecutive composites are zero.

Refutation

Given: The displayed data.

1.1

Distinguished cone triangles do have zero consecutive composites, but this is only a necessary condition.

given
2.1

In K(Z-Mod), consider S0Z⟶0⟶S0Z[1]⟶S0Z[1] with all three maps zero. Its consecutive composites vanish.

step 1.1given
3.1

If this triangle were distinguished, it would be isomorphic to the standard cone triangle of 0:S0Z→0. The final map of that cone triangle is the identity of S0Z[1]; commutativity of the final square would then force the third component of any purported triangle isomorphism to be zero, so it could not be an isomorphism. Thus the displayed triangle is not distinguished.

step 2.1given∎

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Dependency tree · two levels

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Sources