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.

Stabilizing a map between spheres

Example

Let $f : S^{n + k} \to S^{n}$ be based, where $n \geq 0$ and $n + k \geq 1$ . Then $f$ determines a stable class

$[f]_{s t} \in π_{k}^{s}$

represented at every later stage $n + r$ by the iterated suspension $Σ^{r} f : S^{n + k + r} \to S^{n + r}$ .

Facts & Assumptions

[F1]

The sphere-stem bonding relation identifies a stage representative with its suspension (Stable stems of the sphere).

[F2]

Higher homotopy groups are invariant under based homotopy (Higher homotopy groups are functorial and based homotopy invariant).

Verification

Given: A based map $f : S^{n + k} \to S^{n}$ with $n + k \geq 1$ .

1.1

The homotopy class $[f] \in π_{n + k} (S^{n})$ is a legal representative in the colimit defining $π_{k}^{s}$ .

1.2

By definition of that colimit's bonding map, $(n, [f]) \sim (n + 1, [Σ f])$ . Iterating gives $(n, [f]) \sim (n + r, [Σ^{r} f])$ for every $r \geq 0$ .

2.1

A based homotopy $f ≃ f^{'}$ gives the same stage class by [F2], and its suspensions do likewise, so it gives the same stable class. This is consistent with strict-map functoriality for suspension prespectra. Nothing here says that the original unstable class can be recovered from its stable image. $□$

F1F2

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

J. P. May, A Concise Course in Algebraic Topology (standard reference, not scraped)