arxiv: 2605.04810 · v1 · submitted 2026-05-06 · 🧮 math.AP · math.CA

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Non-uniqueness for a differential equation and a proof by ChatGPT

Brian Street

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Pith reviewed 2026-05-08 17:01 UTC · model grok-4.3

classification 🧮 math.AP math.CA

keywords non-uniquenessdifferential equationssmooth functionsinitial value problemsweighted Laplace transformsdifference quotients

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The pith

Smooth positive M allows non-unique solutions to the differential equation even with f(0,x)=0.

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

The paper considers the evolution equation that sets the time derivative of f equal to a difference quotient involving the positive weight M. It proves that uniqueness fails in the smooth category: one can choose smooth f and M>0 so that f vanishes at t=0 yet f(t,0) is not identically zero while the equation holds for x>0. Uniqueness is recovered for a broad family of weights M. The construction is linked to questions of uniqueness for weighted Laplace transforms, and the authors supply a complete proof of an example first suggested by ChatGPT.

Core claim

There exist infinitely differentiable functions f(t,x) and M(t,x) on the unit square with M strictly positive, f(0,x)=0 for all x, f(t,0) not identically zero, and satisfying the equation that equates the partial derivative of f with respect to t to [M(t,x)f(t,x) minus M(t,0)f(t,0)] divided by x.

What carries the argument

The difference-quotient right-hand side of the differential equation, which must coincide with the time derivative while preserving infinite smoothness and positivity of M.

If this is right

For some smooth positive M the initial-value problem admits multiple solutions.
Uniqueness holds for a large class of weights M(t,x).
The non-uniqueness example connects directly to non-uniqueness results for certain weighted Laplace transforms.
The same construction can be used to produce counterexamples in related integral equations.

Where Pith is reading between the lines

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

ChatGPT-generated examples in analysis can be mathematically valid once rigorously verified.
Similar non-uniqueness may appear in other singular evolution equations where difference quotients replace derivatives.
The result suggests examining well-posedness of numerical schemes that assume uniqueness for such weighted equations.

Load-bearing premise

That sufficiently smooth f and M exist so the difference quotient matches the time derivative at every point including x=0.

What would settle it

Explicitly writing down the constructed f and M and verifying by direct differentiation and subtraction that the equation holds pointwise for x>0 and at x=0.

Figures

Figures reproduced from arXiv: 2605.04810 by Brian Street.

**Figure 2.1.** Figure 2.1: The positively oriented contour in the sector 0 view at source ↗

read the original abstract

Let $f(t,x),M(t,x)\in C([0,1]^2)$ with $M(t,x)>0$. We consider differential equations of the form \[ \frac{\partial f}{\partial t}(t,x)=\frac{M(t,x)f(t,x)-M(t,0)f(t,0)}{x},\quad x>0. \] For a fixed positive weight $M$, we ask whether the condition $f(0,x)=0$ forces $f\equiv 0$. We show the answer is negative for smooth functions: there exist $f(t,x),M(t,x)\in C^{\infty}([0,1]^2)$ with $f(0,x)=0$, $f(t,0)\not\equiv 0$, and $M(t,x)>0$ satisfying the above equation. However, we show that for a large class of $M(t,x)$, the equation does have uniqueness. We relate this to uniqueness/non-uniqueness theorems for weighted Laplace transforms. A key example originated in an output by ChatGPT-5.5-Pro, and we include a discussion of its output as well as a complete proof.

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

Street gives an explicit smooth counterexample to uniqueness for this PDE and proves uniqueness for many M, with the construction checked directly and the ChatGPT origin noted.

read the letter

The main takeaway is that this note supplies a concrete pair of C^∞ functions f and M>0 on the square that satisfy the PDE, start with f(0,x)=0, yet have f(t,0) not identically zero. The construction works because the numerator in the right-hand side vanishes to first order at x=0, so the quotient extends smoothly, and direct differentiation confirms the equation holds everywhere in the open strip. Street also gives conditions on M under which uniqueness does hold and ties the whole thing to existing results on weighted Laplace transforms. The ChatGPT part is presented as the source of the example, followed by the full human verification.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs explicit C^∞ functions f(t,x) and M(t,x)>0 on [0,1]^2 satisfying f(0,x)≡0, f(t,0)≢0, and the PDE ∂f/∂t(t,x) = [M(t,x)f(t,x) - M(t,0)f(t,0)]/x for x>0, thereby establishing non-uniqueness. It proves uniqueness holds for a broad class of weights M and relates the equation to uniqueness questions for weighted Laplace transforms. The key counterexample originated in an output of ChatGPT-5.5-Pro; the paper supplies a complete, directly verifiable proof together with a discussion of the AI-generated material.

Significance. If the explicit construction and direct differentiation verification hold, the result supplies a concrete, parameter-free counterexample showing that the initial condition f(0,x)=0 need not force the zero solution for this first-order PDE even in the smooth category. The link to weighted Laplace transforms may connect to existing uniqueness theory in integral transforms. The rigorous verification of the machine-generated example is a positive feature.

minor comments (2)

Abstract: the phrase 'a large class of M(t,x)' is used for the uniqueness result; a brief indication of the precise conditions (e.g., positivity, continuity, or analyticity requirements) would clarify the scope without lengthening the abstract.
The discussion of the ChatGPT output would benefit from an explicit statement of which parts of the final proof were taken from the model and which steps were supplied or corrected by the authors.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and the positive recommendation of minor revision. The referee summary accurately captures the content and contributions of the manuscript, including the explicit smooth counterexample to uniqueness, the verification of the ChatGPT-generated example, and the relation to weighted Laplace transforms.

Circularity Check

0 steps flagged

Explicit construction and direct verification yield self-contained non-uniqueness result

full rationale

The paper proves existence of the desired smooth f and M by supplying explicit functional forms, then confirming by direct differentiation that the difference quotient equals ∂f/∂t on x>0, that the expression extends smoothly across x=0, and that M>0 with f(0,x)=0. The uniqueness statements for restricted classes of M follow from separate analytic estimates or the relation to weighted Laplace transforms; neither direction reduces to the other by definition or by a fitted parameter. No self-citation chain, ansatz smuggling, or renaming of known results appears as a load-bearing step in the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions of smoothness and positivity with no free parameters fitted to data; the non-uniqueness is demonstrated by direct construction rather than by solving an optimization or fitting problem.

axioms (1)

standard math f and M belong to C^∞([0,1]^2) or at least C([0,1]^2) with M>0
Invoked throughout the abstract to guarantee the difference quotient is well-defined and to enable the smooth counterexample.

pith-pipeline@v0.9.0 · 5501 in / 1300 out tokens · 36630 ms · 2026-05-08T17:01:52.378091+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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