A Note You Cannot Explain Is Incomplete

Somewhere in your notes is an answer a model gave you that you could not now defend. Mathematics — the discipline that runs on proof — has written down what to do about that. Two rules: disclose which operations the tool performed, and be able to explain the result yourself. Neither is a flag you tick.

The rules come from two public documents, neither of them written about note-taking. The Leiden Declaration on Artificial Intelligence and Mathematics — dated 2 June 2026, endorsed by the International Mathematical Union — asks authors to "Transparently disclose the use of automated tools" and to include a "Tool and computational resource disclosure" section in their papers ¹.

The second rule comes from Terence Tao. His essay Mathematics in the age of AI, posted to arXiv on 17 August 2026 as the write-up of a public lecture given at the International Congress of Mathematicians in July 2026, states it as a personal standard rather than a policy: "A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified" ².

Neither document binds anyone. Both recommend. What follows is the version that fits inside a file you already keep.

What mathematics wrote down about AI assistance

Two rules, from two documents. The Leiden Declaration — twenty-three recommendations addressed to three audiences ³, endorsed by the International Mathematical Union — asks authors to disclose which tools they used. Tao adds the second: if you cannot give a clear, expert-level talk on a result, it should not be published.

The Declaration was not written quickly. It came out of a conference at the Lorentz Center in Leiden, Mechanization and Mathematical Research, held in September 2025 with "around 60 participants from 10 countries"; a smaller working group of sixteen then spent eight months drafting, and the finished text states that it reflects technology and practice "as of May 2026" ⁴. More than 3,500 people have signed it ⁵.

Tao's version of rule two is stricter than it first sounds:

"My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published." ²

The reason to write any of this down is not compliance. It is the failure Tao names one paragraph earlier: "The scenario to be avoided at all costs is one in which authors use AI tools covertly to aid their work, but conceal that usage in order to avoid criticism from their peers" ³. Concealment is the problem. A record nobody can read is the same thing by accident.

Peter Scholze, director of the Max Planck Institute for Mathematics, signed the Declaration, and one line of his endorsement carries the whole argument: "The goal of mathematical research is human understanding of mathematics, and so mathematics can only thrive in a community of human mathematicians" ⁶. Scale that down. The goal of your notes is your understanding of them.

The two blocks to add to your note today

Add two things to the note. A short disclosure listing which operations the tool performed, under a heading borrowed from the Declaration. Then a block of three to six sentences explaining the result in your own words — written with the note closed. If you cannot write the second block, the note is not finished.

---
title: Bond duration vs. convexity
created: 2026-08-20
generated-by: mixed              # who wrote it
model: <exact snapshot id>       # which model, exactly
---

## Tool and computational resource disclosure
- Used AI to: find the sources, draft the worked example, check the arithmetic.
- Did NOT use AI for: the conclusion under "What I think" — that part is mine.
- Checked by me, against the source: the two figures in the table.

## In my own words
<three to six sentences you could say out loud to a colleague, with this file closed>

Three lines of that block are borrowed — and it is worth knowing from whom. The heading is the Declaration's phrase, lifted verbatim so that a convention with a published definition does the naming instead of you ¹. The heading in your file is ours; the phrase in it is theirs.

The model: field is not new here either. Pinning the exact snapshot rather than the moving alias is the practice MNMNOTE wrote up separately — this block simply gives it a home next to everything else about the run.

The generated-by: field is likewise already spoken for. Marking which notes the AI wrote is a separate habit with a separate purpose, and it answers a different question.

That difference is the whole point of the second block. A flag says who wrote this. The disclosure and the explanation together say whether this record is complete. A note can be correctly marked generated-by: ai and still fail the test, because the test is not about authorship.

Write the second block last, and write it with the file closed.

The mechanism is old and better documented than the habit deserves. In two experiments published in Memory & Cognition in 2014, Nestojko, Bui, Kornell and Bjork had participants study text passages under one of two expectations — a later test, or teaching the passage to another student.

"In reality, all participants were tested, and no one actually engaged in teaching." The ones who expected to teach "produced more complete and better organized free recall of the passage" ⁷. Expecting to explain changed how they read.

Developers found the same lever at a smaller scale. Jamie Tanna, writing about why he still hand-writes his commit messages, puts it plainly: "The process of writing the commit message is also greatly helpful in determining whether I do actually understand the changes I've just made."

He adds the ownership half — the part a disclosure line cannot supply: "By having to explain the choices in the commit message, I'm not only working out why the decisions were made, but taking full ownership of the changes" ⁸. A commit message is a note about work done with a machine. So is yours.

It is the same argument that has developers writing by hand again, arriving from a different direction.

Why the second rule is the hard one

Because disclosure is a fact you can look up — and explanation is not. In the First Proof project's second batch, four AI systems attempted ten research-level problems, and seven problems drew at least one passing grade ⁹. What the referees kept finding was not error. It was polish laid over the hard step.

The denominator matters as much as the headline. Abouzaid, Srivastava, Ward and Williams tested four systems on ten problems and sent thirty-nine solutions to grading — each to at least two referees expert in its area, with the assistance of 30 expert mathematicians in total ¹⁰.

One problem defeated everything: "At the other extreme were problems which saw complete failure, most notably Problem 4 (metric geometry), where no system made substantial progress" ¹⁰. The systems' own cost accounting ran from $8 to $951 per problem ¹¹.

Then the finding that should change how you keep notes:

"A recurring theme in the referee reports was that AI solutions tended to handle routine parts of an argument in meticulous detail while glossing over the most difficult steps, sometimes asserting that a key claim follows from "standard arguments" without justification, or citing papers that do not actually contain the claimed results." ¹²

Read that as a description of your own notes. The output is longest and most careful exactly where the work was easy, and it thins to a phrase at the point you most needed it to hold. A disclosure line cannot catch that. It records that a tool was used, not where the argument went quiet.

The explanation block catches it, because the thin spot is the sentence you cannot finish.

What the disclosure should actually say

One sentence, four clauses, naming operations rather than intent. Tao's own paper discharges the convention he endorses in a single line: "AI assistance was used to perform literature search, to generate diagrams, to autocomplete text, and to convert the slides into a paper format." That is the whole disclosure, and it is enough ¹³.

Where that sentence sits is the more useful detail. It is not under a section called "Tool and computational resource disclosure" — the essay has no such section. Section 10 of the paper is titled Acknowledgments, and the disclosure is one sentence inside it ¹³. The convention belongs to the Declaration; Tao endorsed the Declaration separately, in his own signature on its site: "I wholeheartedly endorse the statements and recommendations in this declaration" ¹⁴.

So the person arguing for the standard discharged it in one line at the bottom of the page — because there is not yet an agreed place to put it. That is exactly the scale a person keeping notes can copy. It is also why "add a heading you will never fill in" is the wrong advice, and why the block above lists operations: found the sources, drafted the example, checked the arithmetic.

The Declaration sets no threshold for how much help counts. It asks for transparent disclosure and says outright that "the precise form of such a section will necessarily evolve" ¹. Do not invent a percentage. Name what the tool did.

What the disclosure cannot do is transfer responsibility. The Declaration is explicit: "When automated techniques are employed in published mathematical research, the responsibility for the correctness and adequacy of the arguments and results, as well as for the completeness and accuracy of citations to relevant prior work, remains exclusively with the human authors" ¹⁵.

The caveats

Nobody is required to do any of this. The Leiden Declaration recommends; the International Mathematical Union endorsed it; endorsement is not enforcement. Both arXiv papers cited here are version-one preprints, and the First Proof report is not peer-reviewed — its solutions were refereed by experts, the paper itself was not ¹⁶.

The 2014 study is smaller than its folk version. It ran on text passages, in a laboratory, with participants who never taught anybody; what it measured is the effect of expecting to explain, not of explaining ⁷. That is still the right mechanism for the second block, and it is a narrower claim than "teaching is the best way to learn."

Your notes are not a proof, and this is not mathematicians' business. The borrowed claim is not about proofs. It is about records of work done with a machine — which is why Tanna reaches the same rule for a commit message with no reference to any declaration ⁸.

And the first block has a boundary of its own. A disclosure line is a human convention written by you, for you. It is not authentication, it does not prove anything to anybody, and it makes no claim about what happened to your text elsewhere. If the question you actually have is who legally owns the output, that is a separate question with a separate answer.

Frequently asked questions

Do I need to disclose AI assistance? For your own notes, nobody requires anything. The convention exists because of a specific failure the Leiden Declaration and Terence Tao both name: work done with tools that were quietly left out of the record. Write the line because six months from now you will not remember which parts were yours ³.

What should an AI disclosure statement actually say? Which operations the tool performed, in one sentence. The shortest worked example in existence is Terence Tao's own, in the acknowledgments of his 2026 essay: "AI assistance was used to perform literature search, to generate diagrams, to autocomplete text, and to convert the slides into a paper format" ¹³. Name verbs, not feelings. Four clauses is a complete disclosure.

How do I write down that AI helped me with a note? Two blocks in the file itself. A short disclosure under the heading "Tool and computational resource disclosure" listing what the tool did and what it did not do, then a block titled "In my own words" holding three to six sentences you could say to a colleague. Both are plain text.

Is it still my work if AI wrote most of it? Responsibility does not move. The Leiden Declaration puts it exactly: correctness, adequacy and citation accuracy remain "exclusively with the human authors" ¹⁵. Jamie Tanna reaches the same place from the other side, calling the written explanation the act of "taking full ownership of the changes" ⁸. Ownership in the legal sense is a different question.

Do I have to disclose AI if I edited the output heavily? The Leiden Declaration draws no line at any percentage. It asks for transparent disclosure of tool use and states that the precise form of such a section "will necessarily evolve" ¹. So do not invent a threshold you will then argue with. List the operations the tool performed and let the list be short when the help was small.

How do I know if I actually understand something I did with AI? Close the file and write three to six sentences explaining the result. That is Terence Tao's talk test at the scale of one note ². If the sentences stall at the same place the model got vague, you have found the unverified step, which is precisely where referees found the failures in graded AI solutions ¹².

Can AI solve real research problems yet? Partly, expensively, and unevenly. In the First Proof project's second batch, four systems attempted ten research-level problems; seven drew at least one passing grade, one problem saw complete failure, and costs ran from $8 to $951 per problem ^{9 10 11}. The referees' recurring complaint was meticulous detail on easy steps and hand-waving on hard ones ¹².

The record is what survives the tool

A model can produce the answer. Only you can produce the account of the answer, and the account is the part that still works in five years, when the snapshot in your model: field has been retired and the chat log is gone. Two blocks of plain text are the whole practice.

Mathematics arrived here first because it had to. The rest of us get the rules for free — already argued, already signed by more than 3,500 people ⁵.

A note you cannot explain is incomplete. Not wrong, not useless — incomplete, in the specific sense that the work of understanding it has not been done yet, and the file is where that work goes.

Both blocks are just text in a file you keep, which is the argument for writing them in open Markdown on your own device, the way mnmnote.com does.