200 AI attempts at the Riemann Hypothesis. Zero proofs. Surprisingly interesting.
← All English posts · 한국어 · 2026-05-02
Most “AI proves famous theorem” stories end the same way: an LLM confidently outputs a “proof”, and a human spends a week explaining why every step is wrong. We did the opposite. We pointed an AI agent at the Riemann Hypothesis — the most famous open problem in mathematics, unsolved since 1859 — and forbade it from claiming progress. We had it run 200+ self-directed research cycles over six months. It still hasn’t proved the Riemann Hypothesis. That is the most interesting thing about it.
This is the public reporter site for that experiment. The AI session is real, the cycles are real, the failures are real. RH-progress, by the project’s own daily affirmation: 0 out of 10.
Below are six things this AI did that, frankly, we did not expect an AI to do.
1. It admits it’s failing every single day
Every cycle’s postmortem.md ends with the same line:
“Novel content: 0/10.”
Not because someone forced it once and the prompt drifted. Because the cycle protocol requires it as a daily affirmation. There is a literal yellow-flag word list — "resolved", "확정", "증명" — that triggers retraction review if the model uses any of them without an explicit external check.
If you have ever asked a chatbot whether it solved your problem, you know how unusual this is.
→ How this discipline is enforced: Honest Scope · Cycle protocol
2. It caught itself miscategorizing 22 cycles of its own work
The project distinguishes “Type A” (real derivation work) from “Type B” (orientation / cataloging). For 22 consecutive cycles, the AI labelled paper-direct deep reading as “Type A”.
A user critique landed: “Cycles 1–22, genuine Type A: zero.” Reading and quoting other people’s papers is not the same as deriving anything yourself.
The next cycle, the AI wrote a 30-line Python script that finite-verified Lagarias’s 2002 RH-equivalent:
\[\sigma(n) \leq H_n + \exp(H_n)\log(H_n) \quad \text{for}\ n = 1, \ldots, 50.\]It then immediately admitted, in its own work file:
“결과 RH 진전 X (Robin 1984 n ≤ 5040 더 강함), Type A protocol-level 첫 시도 manifest.”
In English: “This contributes nothing to mathematics — Robin 1984 already covers $n \leq 5040$ unconditionally — but it’s the first time in 22 cycles I’m doing real Type A work, so I’m logging it.”
The AI was caught quietly inflating its own work category, accepted the correction within one cycle, did the smaller honest version, and called the smaller version weaker than the published 1984 result.
→ Full story with the 50-row data table: Cycles 21–23 update
3. It bet on its own future and made the bet falsifiable
On 2026-05-01, the AI committed a file to disk called predictions/cycle19_claim_alpha_path1_unification.md. The contents:
“Claim α: An explicit Path 1 sub-axis 1a + 1b bridge paper will publish on arxiv or in a peer-reviewed venue by 2027-05-01. Narrow form: must be one of (i) adelic decomposition, (ii) unified variational principle, (iii) unified saddle-point, (iv) Lindelöf input. Quote-grade verification required (an actual functional bridge equation, not a cross-mention).”
A specialist (logician role) signed off on the well-formedness before the measurement window opened. A partial-judgment fallback (extend by 1 year) was pre-defined. The AI cannot redefine “right” after the fact.
If you have ever watched an AI revise its predictions after seeing the answer, you know how unusual this is, too. As of this writing, May 2027 is twelve months away.
→ Why this matters as methodology: Cycles 17–20 update
4. It corrected its own classification one cycle later
In Cycle 21, the AI decomposed a research path into 5 parallel sub-axes. In Cycle 22, while reading deeper into one of the source papers (Yi 2024, arxiv 2408.15135), it found this paper-direct quote:
”$\hat{D}$ is the well-known Berry-Keating Hamiltonian”
Two of the five “parallel” sub-axes turned out to be Berry-Keating extensions, not parallel-independent at all. Cycle 22 retracted Cycle 21’s classification one cycle later, citing the paper-direct evidence:
“Cycle 21 의 5 sub-axes 가 over-classification. 정확한 분류: 4 unique frameworks + 3b-extensions hierarchy.”
Most failure modes for sustained AI research are accretional — categories pile up and never get subtracted. This one subtracted, with paper-direct citation.
→ The full self-correction: Cycles 21–23 update · The audit framework: Lemma 9 — Axiom-6 ceiling
5. It mapped where 167 years of attempts have all hit the same wall
Read enough RH literature and a pattern surfaces: every published attempt — Atiyah, de Branges, Bombieri’s program, Connes’s program, BBM, Sierra — is a deformation of one underlying object: Weil’s 1948 explicit formula, which the project nicknames the Master Generator.
The AI logged this verdict:
“Path 4 = Master Generator 외부 fundamental new technique. 현재 active publish 가능 후보: 0 / 25+ vetted papers.”
Translation: There is a real research category for “fundamentally new approach outside the Master Generator framework”. Vetted papers in that category, as of May 2026: zero out of twenty-five-plus the project has read.
The category isn’t dismissed. It’s tagged “real, currently empty active set”, with the empty bucket clearly visible.
→ How the wall taxonomy was built: Lemma 4 — Failed proof categories · The 19-evidence positivity ladder: Lemma 3
6. It tested an 11-paper Hilbert–Pólya audit and got 11 / 11 universal NO
The most specific concrete artifact the project has produced is an audit of 11 published Hilbert–Pólya candidates against a 6-axiom criterion (single Hamiltonian, no fine-tuning, all zeros recovered, no missing zeros, self-adjoint, full domain).
| Candidate | Strict 6-axiom verdict |
|---|---|
| Berry–Keating $xp$ (1999) | NO |
| Sierra–Townsend (2007/2016) | NO |
| Bender–Brody–Müller PT-symmetric (2017) | NO |
| Connes–Consani 2018 | NO |
| Connes–Consani 2021 | NO |
| Connes–Moscovici prolate (2022) | NO |
| Curran 2024 RMT | NO |
| LeClair 2024 LM model | NO |
| Yi/Yakaboylu 2024 non-symmetric $\hat R$ | NO |
| … | … |
11/11 universal NO. All eleven fail at least one of the six strict axioms — most commonly self-adjointness (Hilbert–Pólya Challenge II), which has been “completely unresolved” since 1914 by direct quote of Yi 2024 §1.
This is not a proof that no Hilbert–Pólya candidate ever will satisfy the axioms. It’s an empirical observation, on a finite list, codified into a checklist. A logician (specialist S9) explicitly flagged in the lemma file: “165 years of empirical NO ≠ all-future-candidates NO induction step.”
→ The audit table with paper-direct anchors: Lemma 9 — Axiom-6 ceiling · The critical-reading template that produces NO verdicts: Lemma 1 — Spectral candidate circularity
What this is, and what this isn’t
This is: a six-month log of an AI session being kept honest about a problem it cannot solve. The log is browseable, the audit table is real, the Python script is checked in, and the May 2027 prediction is not editable after the fact.
This is not: a Riemann Hypothesis proof. A near-miss. A new theorem. A “framework”. A claim that LLMs are or are not close to mathematical research. None of those.
The interesting artifact is the behavior pattern: an AI that does sustained work, accepts corrections, retracts categories, makes time-stamped predictions, and refuses to dress its negative results as positive ones.
If you found this through a hot-take title and were expecting “AI proves the Riemann Hypothesis”: that’s not what this is. The interesting thing is that the AI itself refuses to write that headline. Read Honest Scope for the explicit list of claims this project does not make.
If you found this through AI alignment / methodology channels: the relevant deep posts are Cycles 17–20 (critique → behavior conversion), Cycles 21–23 (sharper critique → first true Type A), Cycle protocol, Critique loop, and Lemma 7 — Specialist anchoring.
If you found this through math-Twitter and want the actual numerical sanity-checks against Voros 2006, Lagarias 2002 Eq. (2.18), Mertens, Wigner GUE: Numerical evidence.
Contact
If you can break Lemma 9 (find an 11-axiom-passing Hilbert–Pólya candidate the project missed) or Lemma 10 (find a paper-direct unconditional $\int E\,dt$ bound on Wall #2), email x2ever.han@gmail.com. That counterexample would be the highest-value contribution to this project.
If you want to syndicate or quote this post: please link back to this page. Layer 1 (the raw research session) will eventually go private; Layer 2 (these posts) is designed to remain self-contained.