Complete Integrability for Piecewise-Smooth Distributions

Jack McKee

arxiv: 2607.01569 · v1 · pith:K5NAKZJLnew · submitted 2026-07-02 · 🧮 math.DG · math.OC

Complete Integrability for Piecewise-Smooth Distributions

Jack McKee This is my paper

Pith reviewed 2026-07-03 06:31 UTC · model grok-4.3

classification 🧮 math.DG math.OC

keywords Frobenius theorempiecewise-C1 distributionscomplete integrabilitybi-Lipschitz coordinatesfoliationsdifferential distributionsdifferential geometry

0 comments

The pith

Piecewise-C1 distributions that may be discontinuous satisfy two generalized Frobenius criteria for complete integrability in bi-Lipschitz coordinates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves two extensions of the classical Frobenius integrability theorem to distributions that are piecewise continuously differentiable yet may fail to be continuous across interfaces. These criteria are shown to be sufficient for the distribution to arise as the tangent spaces to a foliation when the foliation charts are permitted to be bi-Lipschitz. A sympathetic reader would care because many geometric and applied settings produce distributions that jump but remain piecewise smooth, and the result guarantees local foliations under this relaxed regularity. The criteria are explicitly not claimed to be necessary.

Core claim

Two generalizations of the Frobenius integrability theorem are proved concerning distributions which are piecewise-C1 but may fail to be continuous. The criteria presented are sufficient, but not necessary, for complete integrability of such distributions with bi-Lipschitz coordinates.

What carries the argument

Generalized Frobenius-type integrability conditions adapted to piecewise-C1 regularity that replace the classical continuous involutivity requirement.

If this is right

Distributions with discontinuities across smooth interfaces can still be completely integrable when the stated criteria hold.
The integrability conclusion produces local foliations whose transition maps are bi-Lipschitz.
The result applies directly to any distribution that is C1 on each piece of a finite partition of the domain.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The bi-Lipschitz allowance may allow the result to apply to certain hybrid or switched dynamical systems where vector fields jump across switching surfaces.
Similar extensions could be tested for distributions that are merely piecewise Lipschitz rather than piecewise C1.
The approach might connect to questions of rectifiability or almost-everywhere integrability in geometric measure theory.

Load-bearing premise

The distributions are assumed piecewise-C1 with possible discontinuities, and integrability is required only in bi-Lipschitz coordinates rather than smoother ones.

What would settle it

A concrete piecewise-C1 distribution that meets one of the two generalized criteria yet fails to integrate to a foliation in any bi-Lipschitz coordinates would disprove the claim.

Figures

Figures reproduced from arXiv: 2607.01569 by Jack McKee.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The abstract states two sufficient conditions for integrability of piecewise-C1 discontinuous distributions in bi-Lipschitz coordinates, but the full paper is needed to check whether the proofs actually deliver anything beyond known results.

read the letter

The main claim is that the classical Frobenius theorem can be extended to distributions that are only piecewise C1 and allowed to jump, with two sets of sufficient conditions that guarantee integral manifolds exist in bi-Lipschitz charts. The work is positioned as a direct generalization rather than a restatement.

What stands out is the choice to work with bi-Lipschitz coordinates instead of smoother ones; that choice fits the discontinuous setting and avoids forcing artificial smoothness across interfaces. The paper also correctly notes that the conditions are sufficient but not necessary, which keeps the claim modest.

The soft spot is that everything rests on the abstract. No definitions of piecewise-C1, no description of how the jump sets are handled, and no sketch of the argument appear here, so it is impossible to tell whether the proofs close the gaps that usually appear at discontinuities or whether they simply assume away the hard parts. Without seeing the actual steps, the soundness rating stays low.

This is narrow material aimed at people already working on non-smooth distributions in differential geometry or control theory. A reader who needs a reference for piecewise-smooth integrability might find it useful if the details hold; otherwise it will not move the broader literature.

I would send it to referees. The topic is legitimate and the stated goal is clear, but the paper needs external verification of the arguments before it can be trusted.

Referee Report

0 major / 1 minor

Summary. The manuscript proves two generalizations of the Frobenius integrability theorem for distributions that are piecewise-C¹ but may fail to be continuous. It supplies sufficient (but not necessary) criteria ensuring complete integrability, with the resulting integral manifolds realized via bi-Lipschitz coordinate charts.

Significance. If the stated criteria and proofs hold, the work extends the classical Frobenius theorem to a strictly larger class of distributions admitting jump discontinuities. This could be relevant for geometric questions involving piecewise-smooth structures, such as singular foliations or hybrid dynamical systems, where the relaxation from C¹ to bi-Lipschitz charts is a natural and practical weakening.

minor comments (1)

The abstract asserts that proofs exist, yet the provided text supplies no explicit definitions of 'piecewise-C¹', no description of the discontinuity interfaces, and no outline of the bi-Lipschitz chart construction, preventing verification of the central claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for recognizing the potential relevance to singular foliations and hybrid systems. The report lists no specific major comments under the 'MAJOR COMMENTS' heading, so we have no individual points to rebut or revise at this stage. We remain available to supply additional details on the proofs or examples if the uncertainty concerns the validity of the stated criteria.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper is a pure existence proof establishing two sufficient (not necessary) criteria that generalize the classical Frobenius theorem to piecewise-C1 distributions, possibly discontinuous, with integral manifolds realized in bi-Lipschitz coordinates. No equations, definitions, or arguments are shown to reduce by construction to fitted inputs, self-citations, or renamed known results; the work relies on standard external mathematical definitions and theorems. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted or audited from the manuscript.

pith-pipeline@v0.9.1-grok · 5543 in / 998 out tokens · 24418 ms · 2026-07-03T06:31:00.468088+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 4 canonical work pages

[1]

, year =

Warner, Frank W. , year =. Foundations of Differentiable Manifolds and Lie Groups , ISBN =. doi:10.1007/978-1-4757-1799-0 , journal =

work page doi:10.1007/978-1-4757-1799-0
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A Generalization of Chow’s Theorem and the Bang-Bang Theorem to Nonlinear Control Problems , author=. Siam Journal on Control , year=
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Rampazzo, Franco , year =. Frobenius-type theorems for Lipschitz distributions , volume =. Journal of Differential Equations , publisher =. doi:10.1016/j.jde.2007.05.040 , number =

work page doi:10.1016/j.jde.2007.05.040 2007
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and Sussmann, H.J

Rampazzo, F. and Sussmann, H.J. , booktitle=. Set-valued differentials and a nonsmooth version of Chow's theorem , year=
[5]

Lipschitz Distributions and Anosov Flows , urldate =

Slobodan Simić , journal =. Lipschitz Distributions and Anosov Flows , urldate =
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Clarke , journal =

Frank H. Clarke , journal =. Generalized Gradients and Applications , urldate =
[7]

Integrability of normal distributions Part 2: Neat foliations by manifolds with boundary , publisher =

Perrella, David and Pfefferlé, David and Stoyanov, Luchezar , keywords =. Integrability of normal distributions Part 2: Neat foliations by manifolds with boundary , publisher =. 2021 , copyright =. doi:10.48550/ARXIV.2111.12970 , url =

work page doi:10.48550/arxiv.2111.12970 2021
[8]

On the inverse function theorem , volume =

Clarke, Francis , year =. On the inverse function theorem , volume =. Pacific Journal of Mathematics , publisher =. doi:10.2140/pjm.1976.64.97 , number =

work page doi:10.2140/pjm.1976.64.97 1976
[9]

2000 , publisher=

Differential Geometry: Cartan's Generalization of Klein's Erlangen Program , author=. 2000 , publisher=

2000

[1] [1]

, year =

Warner, Frank W. , year =. Foundations of Differentiable Manifolds and Lie Groups , ISBN =. doi:10.1007/978-1-4757-1799-0 , journal =

work page doi:10.1007/978-1-4757-1799-0

[2] [2]

Siam Journal on Control , year=

A Generalization of Chow’s Theorem and the Bang-Bang Theorem to Nonlinear Control Problems , author=. Siam Journal on Control , year=

[3] [3]

Frobenius-type theorems for Lipschitz distributions , volume =

Rampazzo, Franco , year =. Frobenius-type theorems for Lipschitz distributions , volume =. Journal of Differential Equations , publisher =. doi:10.1016/j.jde.2007.05.040 , number =

work page doi:10.1016/j.jde.2007.05.040 2007

[4] [4]

and Sussmann, H.J

Rampazzo, F. and Sussmann, H.J. , booktitle=. Set-valued differentials and a nonsmooth version of Chow's theorem , year=

[5] [5]

Lipschitz Distributions and Anosov Flows , urldate =

Slobodan Simić , journal =. Lipschitz Distributions and Anosov Flows , urldate =

[6] [6]

Clarke , journal =

Frank H. Clarke , journal =. Generalized Gradients and Applications , urldate =

[7] [7]

Integrability of normal distributions Part 2: Neat foliations by manifolds with boundary , publisher =

Perrella, David and Pfefferlé, David and Stoyanov, Luchezar , keywords =. Integrability of normal distributions Part 2: Neat foliations by manifolds with boundary , publisher =. 2021 , copyright =. doi:10.48550/ARXIV.2111.12970 , url =

work page doi:10.48550/arxiv.2111.12970 2021

[8] [8]

On the inverse function theorem , volume =

Clarke, Francis , year =. On the inverse function theorem , volume =. Pacific Journal of Mathematics , publisher =. doi:10.2140/pjm.1976.64.97 , number =

work page doi:10.2140/pjm.1976.64.97 1976

[9] [9]

2000 , publisher=

Differential Geometry: Cartan's Generalization of Klein's Erlangen Program , author=. 2000 , publisher=

2000