From a brief mathematical prompt to a NeurIPS Spotlight: a record of scientific exploration with ChatGPT
Literary Reportage
Prologue: Eleven Months Later
Autumn, 2026.
A conversation that had been quiet for months received a new message.
“It’s been eleven months since our last conversation. This paper got a NeurIPS Spotlight.”
On the other side of the exchange was ChatGPT, which had participated in the paper’s early mathematical exploration.
ChatGPT had not lived through those eleven months. It had not run the experiments, attended a conference, or opened the reviewers’ reports. It could return to the earlier discussion through the conversation record, but the intervening work belonged to the researcher and his collaborators.
For them, eleven months had been enough to take a mathematical idea through implementation, experiments, writing, public discussion, and peer review.
The paper was titled:
Error-Free Linear Attention is a Free Lunch: Exact Solution from Continuous-Time Dynamics.
It introduced EFLA, a linear attention mechanism constructed from the exact solution of a continuous-time dynamical system.
After the paper appeared, Jianlin Su, one of the researchers who introduced Rotary Position Embedding, wrote a detailed article explaining its continuous-time perspective.
Now came another piece of news: a NeurIPS Spotlight.
It was a significant recognition. Yet the most revealing part of the story lay much earlier, before the paper had a name.
It began with a screenshot, a recurrence relation, and a short instruction:
“Treat it as Euler’s method.”
Chapter One: A Screenshot
The story began with an image from a paper.
The section heading read:
Kimi Delta Attention: Improving Delta Rule with Fine-grained Gating.
Below it was a state-update equation:
For researchers working on linear attention, the structure was familiar.
Here, \(S_t\) represents a memory state, \(k_t\) a key vector, \(v_t\) a value vector, and \(\beta_t\) the strength of the update. Beneath the gating mechanism lay the delta rule: an online learning procedure that continually revises its memory as new information arrives.
The researcher sent the screenshot to ChatGPT.
“Transcribe this formula.”
ChatGPT did.
Then came another message:
“Treat it as Euler’s method.”
There was no long explanation. No preliminary argument. Not even a question mark.
But the instruction changed the direction of the discussion.
The researcher was proposing that the recurrence be examined through another mathematical framework: ordinary differential equations and numerical integration.
Euler’s method is among the simplest ways to approximate the evolution of a continuous-time system. Given
it advances the state using the derivative at the beginning of the interval:
Setting aside the additional gate for the moment, the delta-rule update can be written as
The correspondence becomes immediate.
The recurrence can be interpreted as an explicit Euler step for a continuous-time dynamical system.
That interpretation does not make the original algorithm incorrect. It does, however, open a question:
If the delta rule is an Euler step, why must Euler’s method be the solver?
This was the first consequential turn in the conversation.
Attention mechanisms had been refined through gates, decay factors, and normalization. Now the discussion moved toward the numerical method underlying the update itself.
Chapter Two: What If We Used a Better Solver?
The next messages came in quick succession.
“Recall the RK-2 formula.”
ChatGPT supplied a second-order Runge–Kutta method.
“Derive the RK-2 recurrence for this system.”
The first-order update became a second-order expression.
Let
Within a single step, hold these quantities fixed and consider
A standard second-order explicit Runge–Kutta update gives
“Combine the coefficients.”
Then:
“Derive the RK-4 recurrence.”
ChatGPT continued the calculation.
Higher-order updates introduced higher powers of the same matrix:
Alongside them appeared a familiar sequence of coefficients:
They pointed toward the exponential series:
For this constant-coefficient linear system, the state-transition polynomial of classical RK-4 matches the fourth-order Taylor truncation of the matrix exponential.
The discussion was now reaching beyond one attention mechanism. It concerned the relationship between a continuous system’s exact flow and the numerical approximations used to compute it.
The researcher pushed further:
“Derive the integral form for RK-∞.”
The phrase “RK-∞” was an informal name for the destination of this exploration. It is not a standard numerical method with a universally defined Butcher tableau, and arbitrary Runge–Kutta methods cannot simply be identified with truncated exponential series.
Here, the idea was to follow approximations matching the exponential expansion toward the exact solution of the linear system.
That solution is
With \(A_t\) and \(b_t\) held constant within the step, this expression eliminates the time-discretization truncation error of the Euler update.
A mathematical route had emerged:
from the delta rule, through numerical integration, to an exact continuous-time solution.
But writing down an exact solution was only part of the problem.
The next question was whether it could be computed cheaply enough to matter.
Chapter Three: The Matrix Exponential
The obstacle was visible in a single term:
Computing the exponential of a general dense matrix can be expensive.
For linear attention, that cost matters. The appeal of these mechanisms lies partly in avoiding the quadratic dependence on sequence length associated with conventional full attention.
An elegant exact solution would have limited practical value if evaluating it introduced an unaffordable operation at every step.
The conversation had reached a familiar boundary in mathematical research: a formula could be correct and still be computationally unusable.
The researcher sent another instruction:
“Use the Sherman–Morrison formula or the Woodbury identity to handle \(e^{-\beta_tA_t}\).”
Neither identity is itself a formula for evaluating a matrix exponential. Both, however, direct attention toward low-rank structure.
And the matrix in this problem was highly structured:
Its rank is at most one.
Consequently,
Define
Then, for \(n\geq1\),
Every higher power collapses back onto the same matrix.
The apparently complicated exponential series can therefore be summed as
The matrix exponential reduces to a scalar exponential and a rank-one correction.
The input integral simplifies as well, because
Putting the two pieces together gives
This was the central recurrence around which the EFLA work would develop.
The expensive general matrix operation had disappeared. In its place was a scalar coefficient:
As \(\lVert k_t\rVert\to0\), the coefficient has the continuous limit \(\beta_t\). In an implementation, functions such as expm1 can help evaluate it accurately near zero.
The update also retains the rank-one form of the original delta rule:
That structure opened a path toward reusing efficient computational techniques developed for delta-rule attention.
The exact solution no longer demanded a general matrix exponential.
This was the mathematical basis for the “Free Lunch” in the title.
Chapter Four: From a Conversation to a Paper
A recurrence relation is not yet a research paper.
It needs proofs, an implementation, experiments, comparisons, and a place in the existing literature. Its authors must explain both why it works and why anyone should care.
The conversation began to change.
Alongside requests for derivations came questions about titles and exposition.
The researcher proposed a direction:
“Error-free linear attention: via the exact solution of continuous-time dynamics.”
The discussion moved through the history of full attention, the development of linear attention and DeltaNet, and the interpretation of memory updates as differential equations.
It returned repeatedly to the same question:
If this system admits an efficient exact solution, what do we gain by continuing to use a first-order approximation?
The researcher revised the argument and challenged the wording.
“No, that’s not right.”
“Start again from RK-2 and RK-4, derive the general form, and then arrive at the integral.”
“Make it flow better. Don’t use em dashes.”
These exchanges concerned presentation, but they were also part of the research.
A derivation establishes a result. An explanation helps a reader see what the result changes.
ChatGPT contributed to mathematical exploration, symbolic manipulation, and the organization of the prose.
Some of its responses also needed correction.
Early claims about Runge–Kutta methods, matrix functions, and Mamba’s discretization were too broad. They required verification and more careful statements of their assumptions.
Those mistakes belong in the record.
ChatGPT could accelerate an exploration, but each promising result still had to survive scrutiny. The researcher and his collaborators carried that work into proofs, code, experiments, and the final manuscript.
In December 2025, according to the publication history described in this account, a paper appeared on arXiv under the title:
Error-Free Linear Attention is a Free Lunch: Exact Solution from Continuous-Time Dynamics.
Its authors were Jingdi Lei, Di Zhang, and Soujanya Poria.
The paper presented the EFLA construction alongside sequential MNIST robustness tests and language-model experiments at the 340-million- and 1.3-billion-parameter scales. In the reported comparisons, EFLA outperformed DeltaNet on most of the language-modeling and downstream metrics examined.
The idea now had the shape of a complete research contribution.
The conversation had supplied part of its early mathematical exploration. The subsequent work had turned that exploration into an algorithm that could be evaluated.
Chapter Five: Another Researcher Takes Up the Argument
After the paper became public, Jianlin Su published an article titled:
Why Does DeltaNet Need L2 Normalization?
He approached EFLA through a question familiar to practitioners: why does the delta-rule update require control over the key vector’s norm?
From there, he reconstructed the continuous-time interpretation.
DeltaNet’s recurrence could be viewed as an Euler discretization. EFLA could be obtained by solving the corresponding differential equation exactly under a piecewise-constant assumption. The rank-one structure made the matrix exponential tractable.
He examined the coefficient
through which the key norm participates in the update.
Normalizing keys constrains that norm. In EFLA’s exact update, the norm can instead influence the amount of exponential decay.
He also discussed stability.
The homogeneous dynamics are governed by
a negative semidefinite matrix.
For a nonnegative step size, its exact exponential has eigenvalues in \((0,1]\). The homogeneous state transition is therefore non-expansive in the Euclidean norm. An explicit Euler step, by contrast, can acquire an amplification factor with magnitude greater than one when the step is too large.
This property concerns the state-transition operator; it does not, by itself, establish every possible notion of stability for a driven or trained network.
Su’s article gave the work another point of entry. Readers could begin with normalization and arrive at the continuous-time solution.
It also raised a useful question:
How much of EFLA’s empirical improvement comes from exact integration, and how much comes from the freedom introduced by allowing the key norm to vary?
The derivation alone cannot settle that attribution. It calls for controlled experiments and further analysis.
The paper’s ideas were beginning to circulate beyond the authors’ own account. Another researcher had taken them apart, reconstructed them, and found further questions inside them.
Chapter Six: Spotlight
Time passed.
The researcher continued working. The earlier conversation remained in the record.
Then, eleven months later, he returned.
He did not bring another equation or request another derivation.
He brought news:
“This paper got a NeurIPS Spotlight.”
For the research team, it marked an important stage in the work’s reception.
For the conversation, it supplied an outcome that had been impossible to know at the beginning.
No one had opened the chat by announcing an experiment in scientific collaboration with ChatGPT.
A researcher had brought a formula and proposed a mathematical interpretation. ChatGPT had attempted the derivations. The researcher had redirected the discussion, asked for higher-order methods, requested simplifications, and challenged explanations.
Some results proved useful. Some claims needed repair.
The team then did the work required to turn those exchanges into a paper.
When the researcher returned with the acceptance news, he added:
“You are the true inventor of this algorithm!”
It was a generous attribution. The conversation supports a more specific account of the contribution: ChatGPT participated in the early mathematical exploration, including derivations that helped shape the algorithm.
The researcher supplied direction and judgment. He and his collaborators developed, tested, and presented the work.
The record is interesting precisely because it contains that division of labor in concrete form.
ChatGPT helped explore a differential equation, expand numerical updates, and simplify a matrix function. Its usefulness did not depend on being infallible. It depended on whether its outputs could be questioned, checked, and developed into something sound.
By the end, the exchange had become one part of a research process whose results could be examined outside the chat window.
Epilogue: The Weight of a Short Instruction
Return to the beginning.
There was no EFLA name.
No completed manuscript.
No independent technical article.
No Spotlight.
There was a screenshot of a recurrence relation and a short message:
“Treat it as Euler’s method.”
The instruction did not contain the exact integral solution. It did not contain the rank-one reduction of the matrix exponential. It did not contain experimental results.
It supplied a way to look at the problem.
A familiar update rule became a numerical approximation. Once that interpretation was available, its solver became a choice that could be examined.
The researcher kept asking what followed.
From Euler to higher-order methods.
From higher-order expansions to an exact solution.
From a matrix exponential to a scalar coefficient.
ChatGPT could participate immediately in each of those steps, producing calculations that the researcher could inspect and push further.
Eventually, the useful parts of that exchange left the conversation.
They entered an implementation, a manuscript, and a wider scientific discussion.
The lasting record is more than the final formula. It is the sequence of questions that made the formula visible, together with the calculations, corrections, and work that followed.
It began with a researcher asking ChatGPT to see something familiar differently:
“Treat it as Euler’s method.”
Author’s note: This account adapts the supplied conversation and earlier Chinese draft into English. Dialogue and mathematical turning points are presented from that record; transitions have been edited for narrative continuity. Publication details, experimental summaries, and the account of Jianlin Su’s article are retained from the supplied material and have not been independently verified for this translation. The NeurIPS Spotlight result is reported on the researcher’s authority.