ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck pass

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.

The zero stable stem is the integers

Example

Degree gives a canonical isomorphism

$π_{0}^{s} ≅ Z,$

under which the stable class of the identity sphere map is $1$ .

Facts & Assumptions

[F1]

Degree is an isomorphism $π_{n} (S^{n}) ≅ Z$ and sends the identity to $1$ (Based sphere maps are classified by degree); suspension preserves degree (Suspension preserves sphere map degree).

[F2]

The zero stem is the colimit of the suspension system $π_{n} (S^{n})$ (Stable stems of the sphere).

Verification

Given: The sphere prespectrum and its zero-graded suspension system.

1.1F1

For every $n \geq 1$ , [F1] gives $\deg : π_{n} (S^{n}) \overset{≅}{\to} Z$ , with the identity sent to $1$ .

1.2F1

Suspension preserves degree by [F1]. Consequently the square [F1]

\begin{array}{ccc} π_{n} (S^{n}) & \to_{}^{E} & π_{n + 1} (S^{n + 1}) \\ \deg ↓ & ↓ \deg \\ Z & \overset{}{=} & Z \end{array}

commutes. Thus the degree identifications turn the entire system defining $π_{0}^{s}$ into the constant identity system on $Z$ .

2.1F1F2step 1.2

By [F2], its colimit is the zero stem and hence is $Z$ . The identity at any stage represents the compatible element $1$ , so its stable class maps to $1$ . $□$

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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)