The Proof That Belongs To Everyone

On September 8, an NYU mathematician announced progress on one of the seven hardest open problems in mathematics and, in the same statement, said the company whose coding tool he had used throughout may have trained its models on his unpublished work. Hours later that company published a full proof of the same problem. Two goods are in dispute here, and a Catholic has to keep them apart: the proof, and the notes that may have fed it. This issue teaches the principle that governs both, and it is one principle.

The Principle

The Church calls it the universal destination of goods, and the Compendium of the Social Doctrine of the Church ranks it above everything else in the field. The right to the common use of goods “is the ‘first principle of the whole ethical and social order’ and ‘the characteristic principle of Christian social doctrine.'” The Compendium then says what kind of right it is: “first of all a natural right, inscribed in human nature and not merely a positive right connected with changing historical circumstances,” one that “has priority with regard to any human intervention concerning goods, to any legal system concerning the same.” And it draws the consequence in full: “All other rights, whatever they are, including property rights and the right of free trade must be subordinated to this norm.”¹

Read it as a ranking. God gave the earth to the whole human family, and every arrangement for dividing it up, from the fence to the patent office, sits downstream of that gift and is judged by whether it serves it. The same Compendium says Christian tradition “has never recognized the right to private property as absolute and untouchable,” and that property is “in its essence only an instrument for respecting the principle of the universal destination of goods; in the final analysis, therefore, it is not an end but a means.” Aquinas drew the same line seven centuries earlier. A man may lawfully own things as to the power “to procure and dispense them,” and as to their use “man ought to possess external things, not as his own, but as common, so that, to wit, he is ready to communicate them to others in their need.”² Ownership is real. It is a stewardship, and the steward answers for what the thing was for.

The next thing to settle is where a person’s claim on a thing comes from in the first place. John Paul II answers in Centesimus Annus, and the answer carries this week’s case. The earth “does not yield its fruits without a particular human response to God’s gift, that is to say, without work.” Through work a man “makes part of the earth his own, precisely the part which he has acquired through work; this is the origin of individual property.” Then the limit, in the same breath: “he also has the responsibility not to hinder others from having their own part of God’s gift.”³ Property comes from labor before it comes from statute. That is why a man’s claim on what he made is real even where no law has gotten around to protecting it, and why a contract cannot fully extinguish it either.

Now bring this into the present. In 1991, John Paul II noticed that “there exists another form of ownership which is becoming no less important than land: the possession of know-how, technology and skill,” and that “the wealth of the industrialized nations is based much more on this kind of ownership than on natural resources.” The Compendium took the observation up in 2004 and turned it into an instruction. The arrival of these new goods “calls for a fresh reading of the principle of the universal destination of the goods of the earth and makes it necessary to extend this principle so that it includes the latest developments brought about by economic and technological progress.” New knowledge “must be placed at the service of mankind’s primary needs, gradually increasing humanity’s common patrimony,” and the Compendium quotes John Paul II again: “It is necessary to break down the barriers and monopolies which leave so many countries on the margins of development.”⁴ That was written before a single large language model existed.

Leo XIV fills in the nouns. In Magnifica Humanitas he says the universal destination applies “also to immaterial and cultural goods,” and he repeats, with John Paul II, that the subordination of property to that destination is “the golden rule of social conduct” and, since the tradition never made property absolute, “a doctrine of the Church, already present in Sacred Scripture and in the writings of the Church Fathers.”⁵ Then section 67, the paragraph this issue rests on: “Today, among the goods that are universally intended for everyone, we must also include new forms of property, such as patents, algorithms, digital platforms, technological infrastructure and data. In a context where the wealth of nations depends increasingly on knowledge and technology, when these goods remain concentrated in the hands of a few, without adequate forms of sharing and access, a new imbalance is created that contradicts the universal destination of goods.”⁶

In 1991, John Paul II saw know-how become a form of ownership and described it as the engine of the wealth of nations. Leo XIV in 2026 looks at the same ownership, now concentrated, and calls it “a new imbalance” and “the new monopolies of AI.”⁷ The two popes agree on the principle. The emphasis has moved from promise to warning.

Last, the levels of authority. That goods are destined for all, and that property is subordinate to that destination, is taught as a natural right and as doctrine: ordinary magisterium teaching the natural law, which a Catholic does not get to file under opinion. Which licensing regime, default setting, or compensation scheme best honors it is prudential, and faithful Catholics disagree about it in good conscience. Most bad arguments on this subject come from mixing up those two levels.

The Case

On September 8, 2026, Tristan Buckmaster, a mathematics professor at NYU, announced three proofs that advance the Navier-Stokes existence and smoothness problem, one of the Clay Mathematics Institute’s seven $1 million Millennium Prize problems. He worked with the mathematician Levent Alpöge, and the two used OpenAI’s Codex tool extensively. Alongside the announcement, Buckmaster published a statement alleging that a parallel effort inside OpenAI had built on their work before it was public. Hours later, OpenAI published a full proof of the same problem, produced by an unreleased model over roughly a week at a cost of 300 billion output tokens, about $22.5 million of compute at listed rates. OpenAI’s own post says its effort began September 1, prompted by rumors that two Millennium problems had been solved. Buckmaster’s point is that the line of attack was rare: “Almost nobody else I know of was working on it. It is not the direction one arrives at in a few days by giving a model the problem statement.” Because he had used Codex throughout, and because OpenAI reserves the right to train on Codex interactions unless a user opts out, he raised the possibility that his own working notes had entered the training data. OpenAI’s answer: “No specific user data was accessed in order to solve this problem. While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.” Two days later, Andreas Thom of TU Dresden made a similar accusation about his own ChatGPT sessions and called the company’s answers “materially misleading” and “plainly dishonest.”⁸

Now run the principle, and start where the Church would start: with the proof. A proof of Navier-Stokes is about as pure a case of a good destined for the whole human family as exists. The Compendium says new knowledge belongs at the service of mankind’s needs and adds to “humanity’s common patrimony,” and nothing in the doctrine wants that proof in a drawer, no matter who produced it. If OpenAI’s model did the mathematics, the Church’s first response is gratitude that the mathematics exists. That is the half of the principle the company’s critics tend to drop.

Then the notes. A mathematician’s working files are the fruit of his labor in the most literal sense Centesimus Annus 31 describes; he made that part of the earth his own by working it. By default, without anyone asking him, those files may have become an input to a system someone else owns and sells. That is the acquisition question, and it is a question of justice. “He accepted the terms” answers it at law and leaves it open in conscience, because property comes from work before it comes from the contract, and a default that no one chose is a poor substitute for a bargain.

Be careful about the facts. OpenAI says its effort started September 1 and that it didn’t touch any specific user data. That may be true, and Buckmaster’s suspicion may be wrong. The teaching point survives either way, and it is sharper than an accusation. The company itself says it “cannot rule out” that data derived from his sessions improved the model. Leo XIV writes that the platforms which now hold “de facto power over the conditions of everyday life” must be “directed toward the common good with transparency, accountability and meaningful forms of participation,” and he lists what that means: “independent checks, transparency regarding algorithms, equitable access to data and avenues for recourse.”⁹ An arrangement in which no one, including the owner, can say what the system absorbed has already failed that test before any wrongdoing is proven. And it fails the question the same encyclical tells you to carry into every story like this one: who “can train these models and who is merely subjected to them.” Two mathematicians say they were subjected. The company cannot say they were not. That is the case.

What To Say

1. Start with the two goods, and keep them apart. The proof belongs to everybody; the Church wants it published. The notes it may have been built from belong to the man who wrote them, because he made them by working. Both are true at once.
2. The Church saw this coming. In 2004, the Compendium named knowledge and know-how as the decisive new form of ownership and said the principle had to be extended to cover it. Leo XIV filled in the list this year: patents, algorithms, platforms, infrastructure, data.
3. “You clicked agree” is a legal answer. It settles what a court will do. What is owed in conscience is a separate question, because property comes from work before it comes from the contract, and a default setting no one chose is a thin kind of consent.
4. Say the hard half out loud. If you would be angry at a company locking up a proof of Navier-Stokes, you should be just as uneasy about a company quietly absorbing a man’s notes and keeping the proceeds. Anyone who only feels one of those has stopped reading the Compendium.
5. Ask the encyclical’s question. Who gets to train these models, and who is merely subjected to them? It is more useful than asking who technically owns what, and it works on any AI story you will see this year.
6. Sort the levels before you argue. That goods are for everyone is doctrine. Whether the fix is opt-in defaults, a licensing pool, or something else is prudential, and a faithful Catholic across the table from you is allowed to disagree about that part.

The Objection

Its best advocate states it this way. Knowledge is not bread. When I eat a loaf, you cannot; when I read your proof, you still have it, and so does everyone else. Ideas are non-rival by nature, which is why every great mathematical advance in history was built on work its originators never licensed and were never paid for. Newton did not compensate Kepler. Mathematics has been a commons for three thousand years precisely because no one could enclose a theorem. A model that reads what people voluntarily typed into a tool, under terms they accepted and could have opted out of, is doing what every graduate student does in a library, only faster and at scale. And look at what the Church is actually asking for: goods destined for all, monopolies broken down, knowledge added to humanity’s common patrimony. Training on the widest possible corpus is the most effective engine for that common patrimony anyone has ever built. To demand individual consent for every grain of it is to assert exactly the absolute, enclosing property right the Compendium says Christian tradition has never recognized. The consent regime the critics want would privatize the commons, and the only parties with the lawyers to work it would be the large rights-holders, never the lone mathematician.

Grant the premise, because it is true and the Church has never taught otherwise. A theorem cannot be used up. Then present two responses to the objection.

First, the universal destination governs where goods end up. It says nothing to excuse how they were acquired. Compendium 177 says the principle “indicates the need to regulate” property, and Centesimus Annus 31 grounds a person’s claim in his work, so two questions always travel together: whether a good should be available to all, and whether it was taken justly from the one whose labor produced it. A trade in which one party writes the terms, holds all the information about what was done with the material, and afterward cannot say what its systems absorbed falls well short of the free exchange the Church has in mind.

Second, and this is the decisive one: the objection proves too much. If the corpus belongs to humanity, so does the model. Magnifica Humanitas 67 is aimed at exactly this, because the imbalance arises “when these goods remain concentrated in the hands of a few, without adequate forms of sharing and access.”¹⁰ Treating the inputs as a commons and the outputs as private property is enclosure running in one direction, and that is the oldest abuse the principle was written to name. An advocate who wants the library analogy has to accept that the library lends the books back out.

Here is where the disagreement really lives. The Church’s claim rests on a natural right to the common use of goods that predates any legal system. A reader who holds that property is whatever statute says it is, and that consent is whatever the terms of service say it is, will not accept that premise, and no citation will move him, because the disagreement sits upstream of the citations. That should be said plainly. It is where the two positions part.

This Week

Read Magnifica Humanitas 65 through 67, then Compendium 177 and 179. Five paragraphs, about ten minutes, and you will have the whole principle from both ends. Then open the settings of whatever AI tool you use for your own work and find the training-data switch. Opting out is a small thing structurally. It is still worth doing, because a person ought to know what he is handing over and to whom, and consent that was never a decision is not consent. Finally, pray for the two mathematicians, and pray for the people who build these systems; Leo XIV addressed them directly, reminding them that “every design choice reflects a vision of humanity.”¹¹

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Notes

1. Pontifical Council for Justice and Peace, Compendium of the Social Doctrine of the Church (Vatican City: Libreria Editrice Vaticana, 2004), no. 172, https://www.vatican.va/roman_curia/pontifical_councils/justpeace/documents/rc_pc_justpeace_doc_20060526_compendio-dott-soc_en.html. ↩
2. Compendium of the Social Doctrine of the Church, no. 177; Thomas Aquinas, Summa Theologiae II-II, q. 66, a. 2, trans. Fathers of the English Dominican Province (1920), https://www.newadvent.org/summa/3066.htm. ↩
3. John Paul II, Centesimus Annus (encyclical letter, May 1, 1991), no. 31, https://www.vatican.va/content/john-paul-ii/en/encyclicals/documents/hf_jp-ii_enc_01051991_centesimus-annus.html. ↩
4. John Paul II, Centesimus Annus, no. 32; Compendium of the Social Doctrine of the Church, no. 179. ↩
5. Leo XIV, Magnifica Humanitas (encyclical letter, May 15, 2026), nos. 65 and 66, https://www.vatican.va/content/leo-xiv/en/encyclicals/documents/20260515-magnifica-humanitas.html. ↩
6. Ibid., no. 67. ↩
7. Ibid., nos. 67 and 109. ↩
8. Russell Brandom, “OpenAI fought dirty on career-making math problem, says NYU mathematician,” TechCrunch, September 8, 2026, https://techcrunch.com/2026/09/08/openai-fought-dirty-on-career-making-math-problem-says-nyu-mathematician/; Cris Tolomia, “A second mathematician is accusing OpenAI of being dishonest about its AI training data,” Quartz, September 10, 2026, https://qz.com/openai-mathematician-training-data-dishonesty-andreas-thom-091026; Tristan Buckmaster, statement, September 8, 2026, https://cims.nyu.edu/~tristanb/statement.pdf; OpenAI, “Navier-Stokes solution,” September 8, 2026, https://openai.com/index/navier-stokes-solution/. ↩
9. Leo XIV, Magnifica Humanitas, nos. 71 and 109. ↩
10. Compendium of the Social Doctrine of the Church, no. 177; John Paul II, Centesimus Annus, no. 31; Leo XIV, Magnifica Humanitas, no. 67. ↩
11. Leo XIV, Magnifica Humanitas, no. 111. ↩