Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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FALSE: double-complex totalisation is unambiguous with infinite diagonals

Statement

FALSE: double-complex totalisation is unambiguous with infinite diagonals

Facts & Assumptions

Given: The zero-differential double complex Kp,q=Z when p+q=0 and Kp,q=0 otherwise, indexed over all p,q∈Z.

Refutation

technique · direct
1.1

Its degree-zero diagonal has one copy of Z for every p∈Z. The direct-sum totalisation has Tot⁡⊕0=⨁p∈ZZ, whose elements have finite support.

givenalgebra
2.1

The product totalisation instead has Tot⁡∏0=∏p∈ZZ, which contains the all-ones family and is strictly larger. Hence infinite diagonals leave a genuine sum/product choice; first-quadrant finite diagonals are what remove it.

step 1.1algebra∎

Depends on

Used by

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Sources