How to Learn

1 · The objects, before the argument

Six things run through this chapter. None is difficult; several are usually left undefined by the people who use them. Here is what each one is, in language you can picture, before anything is argued from it.

2 · The technique, and the step that carries it

The technique attributed to Feynman is genuinely good advice, and this chapter is not a debunking. Explaining something aloud, without notes, to someone who will not politely nod — that is a real and underused thing to do, and most people never do it.

But look at where the work is. Steps 1, 2, 4 and 5 are procedures: pick, explain, look up, simplify. You can follow them mechanically. Step 3 is not a procedure. Notice where you get stuck is a request for a capability, and the technique supplies no way to acquire it.

That would be a quibble if the capability were common. It is not. The characteristic property of a gap is that it does not feel like one. Every error corrected in this corpus — and there are a great many, logged in detail — was written by someone who could have explained it fluently at the time, because that is what an error you have not yet found is. Fluency is not evidence of soundness; producing fluent wrong sentences is the easiest thing a trained mind does.

That is an empirical claim, so it wants evidence, and this project happens to keep exactly the record that tests it.

3 · The census: what actually caught each defect

This repository carries docs/audit-log.md — 4,362 lines recording every defect the author and his sessions have found in their own work: what was wrong, what was true instead, and, usually, how it surfaced. It was not built to answer this question. It is simply the only honest data anyone here has.

The companion script parses it into dated entries and classifies each by what caught it. An explicit sentence of attribution decides an entry; counting marker words is only the fallback. Both lists are in the script's source, and --entries prints every verdict, because a classification you cannot inspect is not evidence.

Two things in that table matter and they point in different directions.

The first is the obvious one. The instruments that can mechanically say no — a kernel and a recomputation — account for fourteen of the twenty entries that name a route. Reading it back, which is precisely Feynman's step 3, accounts for three. If gap detection were a skill that improves with care and experience, this record is by someone who was being careful, in writing, on purpose, for months. It caught three.

The second is not obvious, and it is the more important one. A person is the smallest column. Three entries. And one of those three is the only entry in the entire log that says, in so many words, that nothing in this repository could have caught the thing at all.

4 · Four instruments, four blind spots

Each of these is genuinely good at something, and each is structurally — not accidentally — unable to see something else. The blind spots below are taken from the log's own reasoning about its own failures, not invented for the argument.

Instrument 3 deserves its caveat spelled out, because it is the one this corpus leans on hardest. A kernel is exhaustive within a formal statement. It will return zero admitted steps on a theorem nobody needed, stated on a space it does not apply to, in service of a claim that means nothing. It certifies the proof; it has no opinion about the sentence around it. The corpus states this elsewhere as a market fact — closure prices, significance does not — and it is the same fact here in pedagogical clothes.

5 · The case no instrument caught

On 7 September 2026 a tool in this repository measured something correctly and the measurement was reported as a defect list. The tool was backlinks.py. It found 121 one-way citations across 48 of 80 papers: places where a later paper cites an earlier one and the earlier one does not cite back. The heading called them a worklist. Eleven files were queued for repair.

Nothing was miscounted. There really are 121 such edges, and there really are 48 papers involved. What was wrong was the word defect. A first edition cannot cite a paper that does not exist yet. The asymmetry is not damage; it is what publishing in time looks like.

What caught it was a person saying eight words: this happens naturally — one can usually only look back in time.

Read that sentence for what it does not contain. It has no data. It cites nothing. It corrects no arithmetic. Every number in the report survived it unchanged. What it supplied was a category — a kind of thing the measurement was, which decided what the measurement meant. And no amount of re-running a measurement produces a category, because a category is not in the data.

This also explains the shape of the census. A person appears three times and a kernel ten, and that ordering is real — but the three include the one case where the alternative was not a slower catch, it was never. Instrument 4 is rare and irreplaceable at once, and if you rank by frequency you will build a practice that has no room for it.

6 · IMPA, and instrument 2 at the scale of a country

Instrument 2 — a problem you cannot solve — is the one most learners actually have access to, and it is bottlenecked on something that does not scale: somebody has to write the problem. A good problem is a device that fails precisely where a specific understanding is missing, and building one takes a person who understands both the subject and the learner.

The Instituto de Matemática Pura e Aplicada, in Rio de Janeiro, runs the engineering answer to that bottleneck. OBMEP — the Brazilian Mathematical Olympiad for Public Schools — reached its 21st edition with a first phase on 9 June 2026.

The medals are not the mechanism. Fewer than one student in three hundred receives one, which means 99.7% of whatever OBMEP does, it does to people who win nothing. What happened on 9 June is that eighteen million children each met twenty problems they had not seen before, and found out — in the only way anyone finds out — which of them they could not do.

And the design decision that makes it possible is worth stating exactly, because it is the general answer to instrument 2's cost. The first-phase papers are administered and marked by the schools themselves. You do not scale the problem-setter. You scale the problem. One scarce team writes twenty questions; fifty-eight thousand schools run the check.

One caution, and it belongs in this chapter rather than in a footnote. Everything above is about a door being open. Whether a door is open is not the same question as whether a particular child can reach it, and this series treats that as a separate matter with its own evidence, in Book 7, Chapter Pr · What a Child Can Enter. Nothing here should be read as saying that a method is enough.

7 · The protocol

Five steps, each naming which instrument it hands you. It is the Feynman technique with the missing equipment attached.

1. Say it out loud, to a person, in one paragraph, without notes. instrument 1
   You will hear the sentences you cannot finish. This is free and you should do it first, precisely because it costs nothing — but stop expecting it to find the fluent errors. It will not.
2. Ask them what kind of thing you just described. instrument 4
   Not "does that make sense" — they will say yes. Ask what sort of object it is, what it would be an instance of, what it is not. This is the step the popular version omits, and it is the only one in the list that catches a right measurement under a wrong sentence.
3. Write down a question your explanation should be able to answer, then answer it before looking anything up. instrument 2
   Set the problem before you know whether you can do it. A gap that blocks an inference becomes visible immediately; a gap that blocks nothing is not one you need today.
4. Make the claim checkable by something that can say no. instrument 3
   A script that recomputes the number. A proof a kernel will read. A prediction with a figure and a tolerance. If nothing in your setup is capable of returning no, you have not learned the thing, you have memorised a description of it.
5. Write down what you did not establish — in the same place, at the same time. all four
   This is the habit that made this chapter possible. The census exists because the errors were written down as they were found, by someone with every incentive not to. A log of your own mistakes is the only instrument on this list that gets better the longer you own it.

Notice what the protocol does not say. It does not tell you to explain things simply, and it does not ask you to find an analogy. Those are steps 4 and 5 of the original and they are fine; they are also what you do after the hard part, and putting them in a list of five makes the hard part look like one item among equals. It is not. It is the item.

8 · Honest inventory

9 · Where this sits

• Book 7 · Ch Fy · Richard Feynman — the physics. Feynman's actual contribution here is the path integral, and that chapter is where the man's work is read; this one is about a study technique that carries his name.
• Book 7 · Ch Pr · What a Child Can Enter — the access question, which this chapter deliberately does not answer.
• Study Mathematics at IMPA — the practical route: programmes, deadlines, documents.
• Ch 6 · Pedagogy — the fourteen-week structure this book teaches on, and how cognitive demand is kept below linguistic capacity.
• docs/audit-log.md — the record the census reads. It is long, unflattering, and the reason any of this is checkable.