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.

The precise source stage bound and its homotopy interpretation

PDF p. 4: Motivic graded spectral sequence; PDF p. 9: Eventual filtration vanishing

Statement

Assume the Axiom of Choice. For every $X$ in the preceding full Noetherian equicharacteristic scope, there exists an integer $B (X) \geq 0$ such that $π_{q} F_{m o t}^{r} K (X) = 0 whenever q < r - B (X) .$ This is a bound on the actual stages, not just on the image filtration or the associated graded groups.

Facts & Assumptions

Given: The Axiom of Choice and the actual source filtration and its full-scope finite-valuative-dimension hypothesis.

Goal: Translate the source’s cohomological boundedness convention with the correct sign.

[F1]

General equicharacteristic scope and the standing filtration restriction checks the source hypothesis. External actual motivic stagewise boundedness imports actual stagewise boundedness in the source’s §2.2 convention.

[F2]

Stable spectra and exact triangles indexes homotopy so that cohomological degree $m$ corresponds to homotopical degree $- m$ .

[F3]

The assumed The Axiom of Choice discharges the inherited Choice premises of both suppliers in [F1] for their selected actual filtration and perfect-complex $K$ -theory realization. The sign conversion in [F2] requires no additional choice.

Proof

1.1F1F3given

Under the Choice premise in [F3], apply [F1] to obtain one finite $B (X)$ with each $F^{r}$ supported in cohomological degrees $\leq B (X) - r$ . The input is the explicit boundedness theorem, not the misprinted numerical homological estimates in its proof.

2.1F1F2step 1.1algebra∎

If $q < r - B (X)$ then $- q > B (X) - r$ , so the degree $- q$ cohomology group vanishes. By [F2] it is $π_{q} F^{r}$ , giving the displayed formula. This is a stagewise conclusion and does not follow merely by knowing anything about total $K_{q} (X)$ .

Depends on

Used by · 2

Dependency tree · 3 levels

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