PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-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.

Linearity and additivity of the form integral

Statement

For compactly supported smooth top forms $ω, η$ on an oriented $M^{n}$ and $a, b \in R$ , $\int_{M} (a ω + b η) = a \int_{M} ω + b \int_{M} η .$ Also $\int_{M} ω = \sum_{C} \int_{C} ω ∣_{C}$ , where $C$ ranges over connected components with their restricted orientations; only finitely many meet $supp ω$ .

Facts & Assumptions

[F1]

Finite chart localization of compactly supported top forms, including boundary defines the compact-support integral by a finite nonnegative chart localization, proves independence, linearity and locality, and identifies it with the global partition integral whenever that construction is available. It includes manifolds with genuine boundary and dimension zero.

[F2]

Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ : Let $Q = \prod_{j < m} [a_{j}, b_{j}]$ be nondegenerate. For integrable $f, g : Q \to R$ and scalars $α, β$ , the function $α f + β g$ is integrable and its integral is $α \int_{Q} f + β \int_{Q} g$ . If $f \leq g$ , then $\int_{Q} f \leq \int_{Q} g$ . Also $∣ f ∣$ is integrable and $∣ \int_{Q} f ∣ \leq \int_{Q} ∣ f ∣$ . If $a_{r} < c < b_{r}$ , cutting $Q$ at the coordinate hyperplane $x_{r} = c$ gives two nondegenerate subrectangles; integrability on $Q$ is equivalent to integrability on both restrictions, and their integral values add to the integral over $Q$ .

Proof

Given: The objects and hypotheses in the statement above.

1.1F1F2

Choose a common finite chart localization near the compact union of the two supports by [F1]. In positive dimension, each chart coefficient for $a ω + b η$ is the corresponding linear combination. Riemann linearity, followed by summation of finitely many terms, gives the first formula. Independence from the chosen localization, empty supports and zero scalars are covered by [F1].

2.1F1step 1.1

Manifolds have connected small ball or half-ball neighborhoods. Every connected component is therefore open. Its components form an open cover, so a compact support meets only finitely many of them. Choose the finite chart localization inside those components, group its terms accordingly, and use the locality of [F1].

3.1step 1.1algebra∎

In dimension zero use the finite sums $\sum_{p} ε (p) ω (p)$ ; both distributivity and grouping are identities of finite sums. A singleton contributes its one signed value.

Depends on

Dependency tree · 2 levels

Sources

Lee Proposition 16.6(a), pp.407–408