LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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 interval homotopy operator is coordinate independent

Statement

The interval operator $K : Ω^{k} (M \times [0, 1]) \to Ω^{k - 1} (M)$ is coordinate independent and maps smooth forms to smooth forms.

Facts & Assumptions

Given: A smooth form $ω$ up to the endpoints of $M \times [0, 1]$ .

[F1]

Integration along the unit interval for a differential form: Let $ω \in Ω^{k} (M \times [0, 1])$ be smooth up to the endpoints. For $k \geq 1$ , its interval integral is the $(k - 1)$ -form $K ω = \int_{0}^{1} β_{t} d t$ , where $ω = α_{t} + d t \land β_{t}$ and both families are tangential to $M$ . Set $K = 0$ on degree zero and on zero terms. Use the product structure of prop-products-of-smooth-manifolds-have-a-canonical-product-smooth-structure, restricted from $M \times R$ . The families are intrinsically $α_{t} = i_{t}^{*} ω$ and $β_{t} = i_{t}^{*} (ι_{\partial_{t}} ω)$ , using def-interior-product-of-a-form-by-a-vector-field; evaluation on tangential tuples and on $(\partial_{t}, v_{1}, \dots, v_{k - 1})$ proves existence and uniqueness of the decomposition. The integral is in the fixed finite-dimensional fibre $⋀^{k - 1} T_{x}^{*} M$ . Coefficients have smooth local extensions across endpoints. thm-differentiation-under-the-integral-sign-on-a-compact-rectangle supplies parameter differentiation; coordinate independence and full smoothness are proved in lem-the-interval-homotopy-operator-is-coordinate-independent.

[F2]

Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral: Let $a < b$ and $c < d$ . Suppose $g, h : [a, b] \times [c, d] \to R$ are continuous and, for every fixed $t \in [c, d]$ , the function $x \mapsto g (x, t)$ is differentiable on $(a, b)$ with derivative $h (x, t)$ . Define $G (x) : = \int_{c}^{d} g (x, t) d t .$ Then $G$ is differentiable on $[a, b]$ as a function on that interval and $G^{'} (x) = \int_{c}^{d} h (x, t) d t (x \in [a, b]) .$ At $a$ and $b$ the derivative is relative and one-sided. The derivative hypothesis is imposed only for interior parameter values; continuity of $h$ supplies its endpoint values.

Proof

technique · direct

1.1

The coefficient family $β_{t} = i_{t}^{*} (ι_{\partial_{t}} ω)$ is intrinsically a form in the fixed fibre at $x$ . A change of coordinates on $M$ multiplies its coefficient vector by the exterior-power transition matrix $A (x)$ , which does not depend on $t$ . Finite-dimensional integration gives $\int_{0}^{1} A (x) β_{t} (x) d t = A (x) \int_{0}^{1} β_{t} (x) d t$ . Thus the local integral expressions transform as a form.

F1given

2.1

Fix a smaller closed coordinate rectangle about a point of $M$ . Each coefficient $b (x, t)$ and all its derivatives are continuous on that rectangle times $[0, 1]$ , by local smoothness up to endpoints. Applying compact-parameter differentiation with the other coordinates fixed gives $\partial_{x_{j}} \int_{0}^{1} b (x, t) d t = \int_{0}^{1} \partial_{x_{j}} b (x, t) d t$ . The right side is jointly continuous, since uniform continuity on the compact rectangle bounds the difference of integrals by the supremum difference of integrands. Repetition for every multi-index proves all coordinate derivatives exist and are continuous. Hence $K ω$ is smooth. For $k = 0$ , $K = 0$ is smooth directly.

F1F2step 1.1∎

Source locator

Lee, Introduction to Smooth Manifolds, 2nd ed., Lemma 17.9 and Proposition 17.10, pp.444–445; the proof here computes the product differential directly.

Depends on

Used by

De rham homotopy formula on a product Theorem

Cited to discharge well-definedness by Integration along the unit interval for a differential form.

Dependency tree · two levels

11 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

Nigel Hitchin, Differentiable Manifolds (2014) (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)