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.

The rank map on a disconnected compact space

Example

Let $X$ be compact Hausdorff and let $X = X_{1} ⨿ X_{2}$ , where $X_{1}$ and $X_{2}$ are nonempty clopen subspaces. The bundle that is $ε^{1}$ on $X_{1}$ and $ε^{2}$ on $X_{2}$ has rank class

$(1_{X_{1}}, 2_{X_{2}}) \in H^{0} (X_{1}; Z) \times H^{0} (X_{2}; Z) ≅ H^{0} (X; Z) .$

Thus the rank map on a disconnected compact space is genuinely a locally constant function and cannot in general be replaced by one integer.

Facts & Assumptions

Given: the stated compact Hausdorff space and clopen decomposition with both pieces nonempty.

[F1]

The rank homomorphism sends a bundle to its integer-valued fiber-dimension function, viewed in $H^{0}$ , and respects virtual differences (Grothendieck ring structure and rank map).

[F2]

Bundles of locally constant finite rank, including rank zero, are admitted componentwise in the Whitney-sum monoid (The Whitney-sum monoid of complex vector bundles).

Verification

technique · direct construction and calculation

1.1

Form $E = (X_{1} \times C) ⨿ (X_{2} \times C^{2})$ with projection to $X$ . Since $X_{1}$ and $X_{2}$ are disjoint open subsets, these product charts make $E$ a complex vector bundle whose fiber dimension is $1$ on $X_{1}$ and $2$ on $X_{2}$ .

F2construct

2.1

Degree-zero cohomology of a disjoint union is the product of the degree-zero groups, and the fiber-dimension function in step 1.1 corresponds to $(1_{X_{1}}, 2_{X_{2}})$ . Therefore [F1] gives the displayed rank class.

F1step 1.1

3.1

If this class came from one integer $m$ , its restrictions to both nonempty pieces would be the same constant $m$ . Step 2.1 would force simultaneously $m = 1$ and $m = 2$ , a contradiction. Hence a single global integer cannot encode rank in general.

step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

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