TheoremStatement: Literature-sourcedProof: Literature-sourced

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.

External actual motivic stagewise boundedness

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

Statement

Assume the Axiom of Choice. For a qcqs scheme $X$ over a prime field $F = Q$ or $F_{p}$ with finite valuative dimension, there exists an integer $B (X) \geq 0$ such that the actual EM filtration satisfies $F_{m o t}^{r} K (X) \in {S p}_{\geq r - B (X)}$ for every integer stage $r$ under the source’s constant extension to $r \leq 0$ . Equivalently its cohomological support, with $H^{m} (E) = π_{- m} E$ , is in degrees $\leq B (X) - r$ . The same actual tower is nonnegative, augmented by $F^{0} = K$ , multiplicative, and induces the source’s bounded multiplicative spectral sequence. The value of $B$ need not equal a Krull or valuative dimension.

Facts & Assumptions

Given: The Axiom of Choice and the actual source filtration of the completed filtration input and the source valuative convention.

Goal: Import the precise stagewise, rather than merely graded or total-K, boundedness statement.

[F1]

EM Theorems4.12(1) and4.27(1) explicitly assert boundedness. Section2.2 defines boundedness exactly by a common cohomological upper bound $B - r$ on the individual stages.

[F2]

Motivic filtration and image groups specifies the actual source filtration, its augmentation and integer extension. The assumed The Axiom of Choice discharges that definition’s Choice premise when its common-family perfect-complex realization of $K$ and essentially small nonconnective-functor domain are used. Elmanto–Morrow filtration input supplies the source existence and multiplicativity; Valuative dimension and degreewise bounded towers specifies the dimension convention. This is an inherited assumption of the selected local realization, not an additional geometric restriction on $X$ or a claim of logical necessity for the external theorem.

Proof

1.1F1F2given

Under the assumed Choice premise, use [F2] for the selected actual filtration and its prime-field realization. The hypotheses on $X$ are precisely the qcqs and finite-valuative-dimension hypotheses of the appropriate characteristic case in [F1]. Thus the imported boundedness assertion applies to this tower.

2.1F1F2step 1.1algebra∎

By the definition of boundedness in [F1], one integer $B (X) \geq 0$ bounds every stage in cohomological degrees $\leq B (X) - r$ . With $H^{m} (E) = π_{- m} E$ , this is exactly the displayed homotopical bound. The source convention already includes all integer stages, with the nonpositive stages equivalent to stage zero as in [F2]. The augmentation, multiplicativity and bounded spectral sequence are those of [F1] and [F2]; none uses the erroneous printed numerical estimates.

External input

Import [F1] with the full source records and reading limits. This actual bound is additional input beyond the completed existence-only supplier. The printed proof paragraphs contain false homological sign/type inequalities recorded in the assistance findings; those estimates are not used. Only the explicit theorem assertion interpreted by its definition, with some finite $B (X)$ , is imported. No deduction of stagewise vanishing from total K-theory vanishing is made.

Depends on

Used by · 1

The precise source stage bound and its homotopy interpretation

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §2.2 pp.11–12; Theorem4.12(1) p.22 and Theorem4.27(1) p.29; Theorem1.1(1) p.3

For a qcqs scheme over Q or F_p of finite valuative dimension, the actual multiplicative motivic filtration is bounded in the explicit §2.2 sense: there exists an integer B≥0 such that stage r is supported in cohomological degrees≤B−r, equivalently homotopical degrees≥r−B. It is N-indexed and stage0 is K. Boundedness yields completeness and a bounded multiplicative spectral sequence.

Hypotheses: Qcqs over the selected prime field; finite valuative dimension in the Jaffard/EHIK convention. Noetherianity is not required for this statement. No particular numerical value B=dim_vX+c is imported.

Reading limits: The prior carrier records reading §2.2, Theorems4.12 and4.27 with their printed proofs, and Theorem1.1(1). This adjudication reread those passages in the retrieved arXiv:2309.08463v1 PDF text (header 15 September2023; internal date 18 September2023), including §2.2 in full and both theorem statements and proofs. Fresh page-image retrieval failed; no fresh visual verification or byte-pinned download is claimed. The printed homological estimates fail the stage-zero test and are not imported. Trace-method/descent foundations, cited prerequisite proofs and optimal numerical constants were not reconstructed.

Sources

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §2.2 pp.11–12; Theorem4.12(1) p.22 and Theorem4.27(1) p.29; Theorem1.1(1) p.3