ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 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.

K-theory of a point and the empty space

Example

Choice-free,

$K^{0} (*) ≅ Z, K^{0} (\emptyset) = 0, {\tilde{K}}^{0} (*) = 0.$

Assuming AC for the periodic assertion, $K^{2 k} (*) ≅ Z$ and $K^{2 k + 1} (*) = 0$ for every integer $k$ .

Facts & Assumptions

Given: the point, the empty space, and AC only for the graded conclusion.

[F1]

$K^{0}$ is the Grothendieck completion of the Whitney-sum monoid (Complex topological K⁰ by Grothendieck completion).

[F2]

Reduced $K^{0}$ is the kernel of restriction to the basepoint (Reduced complex K-theory).

[F3]

Under AC, Bott multiplication extends the coefficient grading with period two (Complex Bott periodicity).

[F4]

Under AC, ${\tilde{K}}^{0} (S^{1}) = 0$ (Complex K-theory of spheres).

[A1]

AC is used only in step 3.1 through [F3] and [F4].

Verification

technique · direct

1.1

A complex bundle over a point is a finite-dimensional complex vector space, classified by its dimension. Whitney sum adds dimensions, so [F1] completes $N$ to $Z$ , with the trivial line representing $1$ .

F1algebra

2.1

Over $\emptyset$ , every bundle has empty total space and all are isomorphic, so the bundle monoid has one element and [F1] gives the zero group. For the point, the restriction map in [F2] is the identity of $K^{0} (*)$ , so its kernel is zero. These calculations make no choices.

F1F2step 1.1

3.1

By [F3], the degree-zero group in step 1.1 repeats in every even degree. By [F4], $K^{- 1} (*) = {\tilde{K}}^{0} (S^{1}) = 0$ , and [F3] repeats this zero group in every odd degree. Thus the stated graded groups follow under AC.

F3F4A1step 1.1step 2.1∎

Depends on

Used by

Complex K-theory of even and odd spheres Example

Dependency tree · two levels

14 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

Hatcher, Vector Bundles & K-Theory, §§2.1–2.2 (standard reference, not scraped)
May, A Concise Course in Algebraic Topology, Chapter 24 §§1–2 (standard reference, not scraped)