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.

Cohomological indexing of the actual graded groups

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

Statement

Assume the Axiom of Choice. For the actual motivic tower, the additive tower spectral sequence indexed from page2 has $E_{2}^{a, b} = π_{- a - b} {gr}^{- b} F_{m o t} K (X) = H_{m o t}^{a - b, - b} (X, Z),$ with zero terms when $- b < 0$ . Its differentials are $d_{s} : E_{s}^{a, b} \to E_{s}^{a + s, b - s + 1}$ , and the corresponding target homotopy degree at $(a, b)$ is $- a - b$ . Convergence is a separate assertion requiring additional hypotheses, such as the degreewise bounds of the tower interface.

Facts & Assumptions

[A1]

The Axiom of Choice and the fixed size conventions of the cited constructions are inherited through Actual cofiber triangles and source grading of the motivic tower. They discharge that supplier’s local-realization premise at its first application below. The Axiom of Choice

Given: The Axiom of Choice and the fixed size conventions of the cited constructions, together with the actual cofiber triangles and source graded equivalences.

Goal: Check the stage, weight, shift and differential labels explicitly.

[F1]

Spectral sequence of a tower supplies the page2 indexing $π_{- a - b} {gr}^{- b}$ and differential degrees without convergence hypotheses. Its convergence clause additionally requires degreewise high-stage vanishing and low-stage isomorphisms, which are not assumed here.

[F2]

Actual cofiber triangles and source grading of the motivic tower supplies the actual graded complex; Complex shifts and spectral homotopy indices gives $π_{n} (C [2 r]) = H^{2 r - n} (C)$ . Motivic complexes and cohomology supplies the two-index notation.

Proof

1.1A1F1F2algebra

Use the supplied realization under [A1]. Substitute $r = - b$ and $n = - a - b$ into [F2]. Then $2 r - n = - 2 b + a + b = a - b$ , so the actual graded equivalence yields the displayed group identification. If $b > 0$ , the stage weight is negative and its group is zero by the zero extension in [F2].

2.1F1F2step 1.1algebra∎

The exact-couple source and target labels are those already calculated in [F1]; their differential sends $(a, b)$ to $(a + s, b - s + 1)$ and lowers total homotopy degree by one. Indeed $- (a + s) - (b - s + 1) = - a - b - 1$ . Thus the shift calculation is compatible with the differential and with the corresponding target homotopy degree. It establishes these indices; convergence requires the separate additional hypotheses in [F1].

Depends on

Used by · 3

Dependency tree · 3 levels

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