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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Positivity of the oriented integral

Statement

Let $ω \in Ω_{c}^{n} (M)$ be nonnegative on the positive determinant ray of an oriented smooth manifold. Then $\int_{M} ω \geq 0$ , and $ω \neq 0$ implies $\int_{M} ω > 0$ . Here the compact-support integral is the choice-free finite-chart value, equal to the global partition value whenever that construction is available.

Facts & Assumptions

[F1]

Finite chart localization of compactly supported top forms, including boundary constructs a finite nonnegative chart localization near any compact support, proves its signed chart sum independent of that localization, and proves nonnegativity and strict positivity for a nonzero nonnegative top form, including genuine boundary and dimension zero. Its value agrees with the global partition integral under countable choice.

Proof

Given: The objects and hypotheses in the statement above.

1.1F1

Apply the finite localization of [F1] to $K = supp ω$ . In every chart its nonnegative weight multiplies a nonnegative signed coefficient, so each chart integral is nonnegative and so is their finite sum.

2.1F1step 1.1

If $n \geq 1$ and $ω_{p} \neq 0$ , at least one finite weight is positive at $p$ . Its signed coefficient is continuous and positive there, hence bounded below by some $c > 0$ on a small interior rectangle of positive volume, including when $p$ lies on a genuine boundary face. Its chart integral is positive; the other terms are nonnegative.

3.1F1step 1.1step 2.1∎

For $n = 0$ every signed point summand is nonnegative and a nonzero form has a strictly positive summand. The zero form and the empty manifold give zero. Agreement with the global integral when it is constructed by a partition follows from [F1].

Depends on

Finite chart localization of compactly supported top forms, including boundary

Used by

Positive curvature forces positive Euler characteristic Corollary
A cohomology class obstructs a global symplectomorphism Counterexample
False: summing unweighted atlas integrals is valid False statement
A leafwise positive closed two-form calibrates a taut foliation Proposition
Riemannian inner product of compactly supported forms Proposition
The Riemann bilinear relations and the period lattice Theorem

Dependency tree · two levels

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Sources

Lee Proposition 16.6(c), pp.407–408 (nonnegative version by the same proof) (standard reference, not scraped)