ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Two filtered abelian groups with the same associated graded

Example

For A=ℤ/4 filtered by $F_{0} A = {0, 2}$ and B=(ℤ/2)² filtered by $F_{0} B = (Z / 2) \times 0$ , take $F_{- 1} = 0$ and $F_{1}$ the whole group in each case. The two graded families are ℤ/2 at indices 0 and 1 and zero elsewhere, though the groups are not isomorphic.

Facts & Assumptions

Given: The two finite filtrations stated in the example, extended constantly outside -1≤p≤1.

[F1]

These filtrations have isomorphic graded families but different exponent behavior (Isomorphic associated graded objects need not give isomorphic filtered objects).

[F2]

The cyclic groups, their finite products, ordinary subgroup kernels, and coset quotients are legitimate abelian-group objects (Abelian-group model for spectral-sequence computations).

[F3]

Addition in $Z / n$ is addition of residue classes, with $- [a]_{n} = [- a]_{n}$ (For every natural $n$ , $(Z / n, +)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

[F4]

A direct product of groups has componentwise operation and homomorphic coordinate projections ( $G \times H$ is a group with identity $(e_{G}, e_{H})$ , coordinatewise inverses, and homomorphic coordinate projections).

Verification

technique · direct

1.1

At index zero, the isomorphisms from ℤ/2 are [a]₂↦[2a]₄ and [a]₂↦([a]₂,0). By [F3] and [F4] these preserve addition. Their respective images are {0,2} and {(0,0),(1,0)}, each with generator of order 2. These are the two zero-level graded quotients since $F_{- 1} = 0$ .

F1F2F3F4

2.1

At index one, the quotients map to ℤ/2 by [a]₄+{0,2}↦[a]₂ and $(a, b) + F_{0} B \mapsto b$ . The coset lists {0,2},{1,3}, together with [F3] and the homomorphic second projection of [F4], prove bijectivity and addition preservation. At every other index adjacent pieces coincide, giving zero. Finally 2[1]₄=[2]₄≠0 but 2(a,b)=(0,0), so no group isomorphism exists.

F1F2F3F4step 1.1∎

Source notes

Weibel, Chapter 5, §5.2 pp.122–125 and Definitions 5.4.2–4 pp.132–133; Sharifi, §4.1 pp.87–89 and Definition 4.2.1 p.90.

Depends on

Used by

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Dependency tree · two levels

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Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)