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.

Homotopy groups of integer spectral shifts

PDF p. 4: Shift [2j] in equation (2.1)

Statement

For a spectrum $E$ in the stable spectra interface and integers $i, k$ , there is a natural isomorphism $π_{i} (E [k]) ≅ π_{i - k} (E)$ , where $[1]$ is suspension and $[- 1]$ its inverse.

Facts & Assumptions

Given: $E$ and integers $i, k$ .

Goal: Compute the homotopy index after shifting.

[F1]

Stable spectra and exact triangles defines suspension and desuspension as inverse shifts and supplies $π_{i} (E [1]) ≅ π_{i - 1} (E)$ .

Proof

1.1F1given

For $k \geq 0$ , iterate the isomorphism in [F1] $k$ times, including the identity when $k = 0$ . This gives $π_{i} (E [k]) ≅ π_{i - k} (E)$ .

2.1F1step 1.1∎

For $k < 0$ , apply the nonnegative case to $E [k]$ and $- k$ : $π_{i - k} (E) ≅ π_{i} (E [k])$ . Inverting this isomorphism gives the stated formula. Every step is natural in $E$ .

Depends on

Stable spectra and exact triangles

Used by · 1

Complex shifts and spectral homotopy indices

Dependency tree · 3 levels

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