DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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.

Degree-zero differential cohomology

Definition

For a smooth manifold $M$ , set ${\hat{H}}^{0} (M; Z) = H^{0} (M; Z) = Z (M)$ , the locally constant integer-valued functions. Define $I : {\hat{H}}^{0} \to H^{0} (M; Z)$ to be the identity and $R : {\hat{H}}^{0} \to Ω^{0} (M)$ to send each integer-valued function to the same real-valued function. Set ${\hat{H}}^{j} (M; Z) = 0$ for $j < 0$ .

Definition references

Singular cohomology with coefficients

De rham cochain complex

Remarks

This convention is separate from the degree- $k \geq 1$ smooth-singular character definition. It is required for fiber integration when the input degree equals the fiber dimension, including dimension zero.

Depends on

Dependency tree · 2 levels

Sources

Ulrich Bunke, Differential Cohomology, arXiv:1208.3961v6