The Collapse of the Theorem Economy

“The product of mathematics is clarity and understanding. Not theorems, by themselves.” — Bill Thurston

The Art of the Unwritten

My most significant mathematical achievement is a theorem I never actually bothered to write down.

It came to me during a vivid morning in Lausanne, Switzerland, while I was prepping for my final invited talk. The logic felt so transparent and the conclusion so undeniable that I took a gamble: I edited my presentation at the very last second. Because I was out of time, I couldn't present it as a formal theorem; instead, I tucked it away as an informal observation at the bottom of my final slide.

At that point, my life had shifted. I had exited the ivory tower of academia to launch a machine-learning venture. I recognized that the grueling process of polishing a proof for publication was a luxury I no longer possessed. I left that remark there like a message in a bottle, hoping a talented young mathematician would eventually discover it and integrate it into a larger theoretical framework. I entertained the fantasy that, through the chaotic nature of academic credit, it might one day be known as the Bessis cellular decomposition theorem.

However, by simply stating the result, I effectively destroyed the motivation for anyone else to do the hard work of formalizing it.

The "Find-and-Replace" Proof

If I were to rank my second-best contribution, it would be Theorem 0.5 from an old preprint regarding Garside categories. Despite my initial enthusiasm, I never submitted the paper. The sheer mental exhaustion of the creative phase left me without the stamina to refine the introductory sections.

The irony is that once the preliminaries are established, the actual proof is relatively mundane group theory. The process is essentially a conceptual bulk edit:

# Conceptual logic of Theorem 0.5
for paper in classical_garside_theory_papers:
    original_axioms = paper.get_axioms()
    generalized_axioms = replace(original_axioms, "standard", "general")
    paper.apply(generalized_axioms)
    # Result: Theorem 0.5 emerges organically

If you possess the correct intuition, the execution is almost mechanical. This highlights a common misconception about mathematics: the "hard part" isn't the proof itself. The true difficulty lies in the intuition—the ability to sense that "something like Theorem 0.5" should exist and the capacity to build the conceptual scaffolding that makes the proof trivial.

The "Miracle" of Flow

There are rare days in research where you simply "click" into a state of acceleration. Jean-Pierre Serre once remarked that writing his landmark paper on coherent sheaves required almost no conscious effort; the typewriter seemed to produce the 100 pages automatically, as if the text already existed in the ether.

Unfortunately, I am not Serre, and I didn't have his magical typewriter.

Status vs. Innovation

My preprint remains on the arXiv, cited by prestigious papers, yet the real breakthrough wasn't the theorem itself. It was the language—specifically Definition 2.4 and Definition 9.3. This linguistic framework eventually ended up in a 700-page tome on Garside theory.

I admit to a certain level of strategic selfishness. I sacrificed the prestige of that innovative preprint to focus on a more tedious one. By using Definition 9.3 as a catalyst, I solved a classical problem: the $K(\pi, 1)$ conjecture for finite complex reflection groups.

This achievement solidified my professional standing. In a sense, the version of me that is "famous" for the $K(\pi, 1)$ conjecture is merely a social parasite living off the brilliance of the version of me that conceived of the original definitions.

The AI Crisis and the Two Mathematics

As AI evolves, I find myself in a state of emotional contradiction:

• Excited: I see a genuine revolution unfolding.
• Vindicated: My scientific and epistemological preparations feel justified.
• Puzzled: The sheer speed and frenzy of the transition are jarring.
• Sad: I feel a nostalgia for a value system that is rapidly evaporating.

I believe the existential threat to mathematics stems from a fundamental misunderstanding of what the discipline actually is. In my book, Mathematica: a Secret World of Intuition and Curiosity, I distinguish between two realms:

Official vs. Secret Mathematics

Official math is a sterile paradise for nerds—a world where "bullshit" is mathematically impossible. Secret math is the messy, human engine that actually drives discovery. Because secret math lacks the rigidity of the official version, it is ignored in curricula and dismissed as peripheral.