ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Suspension prespectra of spheres are shifts

Example

For every $r \geq 0$ there is a canonical isomorphism of sequential prespectra

$Σ^{\infty} S^{r} ≅ {sh}^{r} S .$

With the grading convention of this page, it follows that

$π_{k} (Σ^{\infty} S^{r}) ≅ π_{k - r}^{s} .$

Facts & Assumptions

[F1]

Canonical smash associators are coherent with the leading $S^{1}$ structure coordinate (Canonical associativity, symmetry, and unit maps for smash products).

[F2]

The shift formula is $π_{k} (sh E) ≅ π_{k - 1} (E)$ (Shift and suspension of sequential prespectra).

Verification

Given: An integer $r \geq 0$ .

1.1F1

At level $n$ , the canonical smash associator gives [F1] $(Σ^{\infty} S^{r})_{n} = S^{n} \land S^{r} ≅ S^{n + r} = S_{n + r} = ({sh}^{r} S)_{n} .$

1.2F1

Both structure maps add a leading $S^{1}$ coordinate. Smash coherence says the level homeomorphisms commute with these structure maps, so they form a strict prespectrum isomorphism.

2.1F2step 1.2

Applying [F2] $r$ times gives $π_{k} ({sh}^{r} S) ≅ π_{k - r} (S) = π_{k - r}^{s}$ . The opposite sign in the Step-1 scaffold is incompatible with the direct reindexing and is not used. $□$

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Allen Hatcher, Spectral Sequences in Algebraic Topology, Chapter 2 (standard reference, not scraped)