AI exposition

Working specification

Where AI-drafted explanatory writing fails a human reader, what the writing guide in this repository already does about it, what Tao’s account of AI proof exposition adds, and the instructions to test next.

Started 2026-09-15, with no settled scope. Section 1 names two questions, one on exposition and one on how a field changes under abundance. Sections 2 and 3 record what two sources say. Section 4 maps the failures onto a modeling report. Section 5 lists instructions to test and Section 6 says how a test would be scored. Nothing in Section 5 has been tested.

1. Scope

The standard the project measures against is Thurston’s (reference 2), as Tao quotes it:

We are not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about math.

Substitute models, reports and analyses for definitions, theorems and proofs, and the sentence holds for pharmacometrics without other change. A correct report that leaves nobody’s thinking clearer has not met the measure.

The project carries two questions, and Tao’s essay is the source for both.

The first is how to instruct an AI so that the explanatory documents it drafts serve a human reader: a modeling report, a vignette, a literature review, a project specification. The documents are correct on arrival and fluent on arrival, and still go back for another round, because the emphasis is in the wrong place. Sections 2, 5 and 6 work on this one.

The second is what happens to a field when the first stage of its pipeline becomes cheap: which stages the queue moves to, which of its institutions were built for scarcity, and what its members owe each other once generation is no longer the work. Tao asks it for mathematics, and the pharmacometric version runs in parallel stage for stage. Section 3 sets both fields side by side. This second question is the wider one, and the project will split if it grows sources of its own beyond Tao.

Out of scope. Correctness of the content. Tone, grammar and formatting, which are already close to flawless. Choosing between models or tools. Prompting technique in general. The question is what to ask for in a document.

2. What the writing guide already answers

The writing guide is a set of rules derived from one reader’s hand edits of AI drafts, kept with the evidence for each rule. It is a working answer to the question above for one reader, and it is the baseline any new instruction is tested against.

It covers the surface failures completely. Its machine-prose tic list names the vocabulary, the contrastive antithesis, the rule of three, the significance announcement and the closing summary, and its checklist item 11 turns the list into a search run over every file before handover.

It covers the emphasis failures partly, by observation rather than by rule:

• O8 says the reader is an expert on one side of a field boundary and a novice on the other, and that uniform explanation depth is wrong in both directions at once.
• O3 says locating prior art is a deliverable, and that being told an idea already has a name is welcome.
• O9 says “where do I start” is a literal request, and a survey ends with a ranked entry path.
• O14 says essay-shaped sections do not earn their length, and a numbered subsection per item, each opening with what it asks, is what survives review.
• O16 says the design tradeoff is to be argued in the reply, with the reason, before the code exists.

What it does not have is a rule that says where in a document the difficulty lies, or an instruction that makes a draft show it. That is the gap Section 3 names.

3. What Tao adds

Terence Tao’s essay on mathematics after AI (reference 1) carries an account of AI-drafted proofs that names four failures. The first three are in one paragraph:

Current AI tools have a decidedly mixed record with proof exposition. On the one hand, the spelling, the grammar, and the formatting are close to flawless. On the other hand, the writing very often dwells at length on trivialities while passing briefly through, or even actively obscuring, the most interesting and novel portions of the argument. AI-generated mathematical texts also frequently fail to situate the result in the prior literature, or to offer the high-level overview that lets a reader decide whether the argument is worth their time.

The fourth is the one he calls under-appreciated, and it is the one that applies to the document even after the first three are fixed:

A proof can be too slickly written, with the routine steps and the genuinely difficult steps presented as being equally easy to digest.

In a human-written proof, the parts of the argument that the author found difficult typically retain some natural friction: an apologetic remark, an unusually careful lemma, a change of notation, a paragraph that has clearly been rewritten several times. This friction is informative. It signals to the reader where to slow down and pay attention, and it is one of the main channels by which the tacit knowledge of a field is transmitted. An excessively AI-polished proof may sand away both the “artificial” friction (typos, awkward phrasing, disorganization) and the “natural” friction, leaving a text that is easy to read and hard to learn from.

His footnote on the flawless formatting reads, in full, “One can argue that they are too flawless.”

The four, named for use below:

1. Misplaced emphasis. Length on the routine steps, brevity on the novel one.
2. No prior literature. The result is not placed against what was already known.
3. No overview. Nothing at the top lets the reader decide whether to continue.
4. Uniform friction. The hard step and the easy step read as equally settled, so the reader cannot tell where to slow down.

Failure 4 is a claim about what a reader uses a document for. A report that is read once for its conclusion is served by a smooth text. A report that is read to learn how the conclusion was reached, or to check it, or to reuse the method, needs the difficulty visible. The guide’s O2 is the same observation from the other side: implementing a method is not understanding it, and a smooth description of an implemented method leaves the reader where they started.

The pipeline the failures sit in

Tao arrives at the four failures by asking what problem solving is for, and answering in five attempts. The last reads:

Goal 6.5 (final attempt?). Solve unsolved problems, verify them to be correct, ensure they are clearly communicated, and have them digested, accepted, and incorporated into the definitive theory of the field.

His Figure 4 draws the goal as a chain of five stages, with digestion as a dotted arrow that bypasses publication. Redrawn here with his labels:

flowchart LR
  A[Open problems] -- proof generation --> B[Unverified solutions]
  B -- proof verification --> C[Verified solutions]
  C -- proof exposition --> D[Well-written solutions]
  D -- proof publication --> E[Accepted solutions]
  E -- proof canonicalization --> F[Definitive solutions]
  C -. proof digestion .-> F

Tao’s Figure 4. Five stages, of which he says only the first was ever an explicit goal of the community.

The pharmacometric pipeline has the same five stages, and the same imbalance in which of them is treated as the job:

1. Generation. The question and the data go in; a fitted model comes out. The explicit goal, and the stage the modeler is judged on.
2. Verification. Diagnostics, a second pair of eyes on the code, a check of the parameters against what was published. Required by procedure, and done.
3. Exposition. The report. Written because the procedure requires one, after the answer is known, and this is where all four failures in the list above sit.
4. Acceptance. A dose is chosen, a design is agreed, a regulator accepts the argument. Slow, human, and outside the modeler’s control, as Tao says of publication.
5. Canonicalization. The model becomes the class model that the next program starts from. Few reports reach this stage, and the ones that do are the ones whose exposition let someone else reuse the reasoning.

Digestion, the dotted arrow, is the team understanding the model well enough to think with it rather than cite it. It is the stage Thurston’s sentence in Section 1 is about, and failure 4 is the one that blocks it: a report can be verified, written and accepted and still not be digested, if nothing in it says where the reasoning was hard.

Model abundance

Tao’s Section 7 says what happens to the pipeline when the first stage stops being the slow one. If AI tools solve problems at the rate he conditions on, proofs pile up at every later stage: generated faster than they can be verified, verified faster than they can be written up, written up faster than volunteer referees can read them, and published faster than the field can work them into definitive form. He calls the transition one from proof scarcity to proof abundance, and says the institutions of mathematics, journals, priority, hiring, prizes, were designed under scarcity and should be expected to behave poorly under abundance. The strain predates AI, in the growth of the literature and the load on refereeing; AI makes it worse.

The pharmacometric version is model abundance, or analysis abundance. An agent that fits fifty candidate models overnight, each with its diagnostics, moves the queue to the stages that still run at human speed:

1. Fits accumulate faster than a modeler can look at the diagnostics.
2. Checked fits accumulate faster than reports can be written for them.
3. Reports, even correct and well written, accumulate faster than the reviewer, the clinical team and the regulator can read them.
4. Accepted analyses accumulate faster than anyone works them into the model the next program starts from.

The institutions here were also designed under scarcity: one analysis plan, one report, one reviewer, one submission. The early signs Tao names have counterparts that predate agents: sensitivity analyses run because they were cheap rather than because anyone asked, and model libraries nobody opens.

For this project the consequence is that abundance moves the four failures from a nuisance to the constraint. When fitting is cheap, the reviewer’s hour is the scarce resource, and a report is judged by how little of that hour it wastes. The overview (failure 3) is what lets the reviewer triage fifty reports; the marked hard steps (failure 4) are what tell the reviewer where in one report to spend the hour. Tao’s own suggestion for the abundance regime is a filter that flags submissions for “incoherent exposition” before a human reads them, which is the writing guide’s checklist item 11, applied by the journal rather than the author.

The recommendations

Tao’s Section 8 quotes four recommendations to individual mathematicians from the Leiden Declaration on Artificial Intelligence and Mathematics (reference 9), and adds a rule of thumb of his own. The four, in his order, with their form for an analysis:

1. Disclose tool use. A “tool and computational resource disclosure” section in the paper. For a report, a section saying which parts an agent drafted, fitted or checked. Tao’s worst case is covert use concealed to avoid criticism.
2. Support the needs of reviewing. Disclose tool use, give “precise and complete references to previous results”, and provide formal proofs where feasible. For a report, the prior-art paragraph of instruction 3, and code that reruns.
3. Affirm the humanity of authorship. Credit and responsibility stay with people. A report is signed by the modeler who can defend it, whichever tool fitted it.
4. Put effort into proper attribution. Tools attribute badly, so the author owes proactive effort to find and credit sources, and states it explicitly where attribution is not possible. The ❌ marker on a references page here is that explicit statement.

Between the second and third he states the change in emphasis the recommendations serve: decrease the weight the culture places on proof generation and on being first, and increase the weight on digestion, which he lists as exposition, refereeing, publication and canonicalization.

His own rule of thumb closes the section:

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. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

The analysis version: a model that nobody on the team can explain in an expert-level talk is incomplete, whatever its diagnostics show and however carefully the code was reviewed. The talk is a test the instructions in Section 5 can be checked against. A talk has to open with the overview (instruction 4), has to say which choice decided the answer (instruction 1), and has to say where the speaker is unsure (instruction 2), because an expert audience asks. A report from which that talk cannot be given has failed on one of the four, and the question in the room usually says which.

4. The modeling report

Tao writes about proofs. A pharmacometric modeling report is a document of the same shape, an argument from stated assumptions to a conclusion that someone else has to accept, and each of the four failures has a form in it.

The four failures, their form in a modeling report, and where the writing guide touches each

Failure	In a proof	In a modeling report	Guide entry
1. Misplaced emphasis	Pages on routine lemmas, a paragraph on the novel step	Pages on the base model and the covariate search, a paragraph on the mechanism that decided the dose	O8, O14
2. No prior literature	The result is not placed against known results	No comparison with published models of the same drug or class, so the reader cannot tell what is new and what was inherited	O3
3. No overview	Nothing at the top says whether to read on	No one-screen statement of the question, the answer and what the answer rests on	O9, rule 10
4. Uniform friction	Hard and easy steps read as equally settled	The two-compartment choice and the choice of a mechanism the data barely identifies read as equally settled	O16, rule 10 in part

Failure 4 in a modeling report has a specific consequence. The reader of a report is usually deciding something: a dose, a design, whether to believe the model. The choices the modeler agonized over are the ones that decide the answer, and they are the ones a reviewer should question. A report in which every choice reads as routine invites no question anywhere, so the reviewer either questions everything or nothing.

One example of a document in this repository that carries the friction is the TCE-IPDE specification. Its In brief section answers failure 3, its Section 1 carries an explicit fork between two approaches, and its references page marks every source with whether the claim drawn from it was checked. The markers are friction made visible: a ⚠️ on a transcribed parameter says to slow down there.

5. Instructions to test

Each is an instruction that could be added to a project’s agent instructions or to the writing guide. None has been tested. Each is paired with the failure it addresses and the shape of output that would show it worked.

1. Name the decisive step. Before drafting, state in one sentence which choice the conclusion depends on most, and give that choice the longest section. Addresses failure 1. Worked if the section lengths track a reader’s ranking of the choices by consequence.
2. Rank the choices by how much the data supported them. In the report, mark each modeling choice with one of three markers: decided by the data, decided by precedent, decided by judgment. Addresses failure 4. Worked if a reviewer’s questions land on the judgment entries. The ✅ ⚠️ ❌ markers on the references pages here are the existing precedent for a marker inside a body of text.
3. Write the prior-art paragraph first. Before the model is described, state what has been published for the same drug or class, and what in this analysis is inherited from it and what is new. Addresses failure 2. Worked if the reader can say from that paragraph alone what the analysis adds.
4. Lead with one screen. The document opens with the question, the answer, and the one thing the answer rests on, in under a screen. Addresses failure 3. Already the practice in this repository’s working specifications; the test is whether it transfers to a report.
5. Leave the rewritten paragraph rough. Where a passage was drafted more than once because the first version was wrong, keep a sentence saying what the first version got wrong. Addresses failure 4 directly, by Tao’s mechanism. This is the instruction most likely to be rejected on review, since it asks for prose the reader would normally delete, and it is the one the writing guide’s open convention already proposes: record what was tried and rejected, two lines per rejection. Tao’s own version, in the sentence after the Thurston quotation, is that authors assist digestion “by describing the insights, the false starts, and the stories from the period when they were working on the problem”, and that current tools are opaque about their own process.

Instruction 5 is where the guide and Tao are in tension. The guide’s rule 9 cuts characterization in favour of measurement, and rule 12 cuts the entry point thin. Tao’s friction is a characterization, and it lengthens the document. The resolution may be the guide’s own contract: the friction belongs in the reference-tier document, where length is permitted, and not in the entry point.

6. How a test is scored

The writing guide’s evidence base is the reader’s own edits, and that is the measure to reuse. An instruction has worked if a draft written under it comes back with fewer edits of the kind the instruction targets, over enough documents to tell.

The procedure for one instruction:

1. Pick a document to be drafted anyway.
2. Draft it under the current guide, and record the edits it comes back with, by rule number.
3. Add the instruction and draft the next comparable document. Record its edits the same way.
4. The instruction stays if the edits it targets fell and no other kind rose.

Two documents per instruction is thin evidence. The guide’s rules were each derived from one or two edits, so the standard is the same one it already applies, and a rule adopted this way carries its evidence with it in the same form.

7. Open questions

1. Whether failure 4 applies to the reader here at all. The guide’s O10 says the reader accepts a longer first draft to iterate less; whether he also wants the draft to show where it struggled is not in the evidence, and instruction 5 is the test.
2. Whether a model can locate its own difficulty. Tao’s friction is a by-product of effort, and a model’s effort is not visible to it in the same way. Instruction 2 asks the model to rank choices by data support, which is a question about the data rather than about effort, and may be answerable where instruction 5 is not.
3. Whether the modeling-report guidance already published (references 6 and 7) asks for any of this. Those documents say what a report must contain; if they already ask for the decisive choice to be named, the instruction has a precedent outside this repository.
4. Whether the reviewer’s hour is the right measure once analyses are cheap. Section 6 scores an instruction by the reader’s edits on the draft, which measures the author’s rounds. Under abundance the cost that grows is on the reading side, and the measure may need to be the time a reviewer spends before reaching a verdict, or the number of reports a reviewer can triage from their overviews alone.