LemmaStatement: AI-generatedProof: AI-generated

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.

Nesting of image filtration groups

PDF p. 3: Section 2, image filtration

Statement

For a decreasing filtration $F$ on a spectrum $E$ and integers $i$ and $a \geq b$ , the image of $π_{i} F^{a} \to π_{i} E$ is contained in the image of $π_{i} F^{b} \to π_{i} E$ .

Facts & Assumptions

Given: $F$ , $E$ , $i$ , and $a \geq b$ as stated.

Goal: Prove the containment.

[F1]

Decreasing spectral filtrations and associated graded gives a transition $F^{a} \to F^{b}$ compatible with the maps to $E$ .

[F2]

Images of filtration stages identifies these images as subgroups.

Proof

1.1F1F2given

Applying $π_{i}$ to the compatible maps in [F1] shows that $π_{i} F^{a} \to π_{i} E$ factors through $π_{i} F^{b} \to π_{i} E$ .

2.1F1F2givenstep 1.1∎

Thus the image of the former lies in the image of the latter, as subgroups by [F2]. This is the desired containment.

Depends on

Used by · 1

Motivic filtration and image groups

Dependency tree · 3 levels

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