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Patrick White · July 2026

Two Months of Vibe Mathing

This week the internet discovered vibe mathing — the Jacobian conjecture fell to a tweet, and a stranger solved six open Erdős problems in five days. We've been running the same move since May, across eight fields, checking every claim. Here's the score.

2months in
16checked results
8fields
2papers on arXiv

It started with a tweet

On July 19, in the middle of the World Cup final, Levent Alpöge — a mathematician at Anthropic — posted to X that the Jacobian conjecture is false. An 87-year-old problem, the kind that becomes a standing joke about how hard "hard" can be, ended in a single cubic map. Akhil Mathew prompted the search; Claude's Fable found the map; the Lean proof assistant checked it within hours.

When we heard, we ran it ourselves.

The counterexample — run two independent ways on our machine
F : ℂ³ → ℂ³
F(x,y,z) = ( (1+xy)³z + y²(1+xy)(4+3xy),
             y + 3x(1+xy)²z + 3xy²(4+3xy),
             2x − 3x²y − x³z )

det J_F ≡ −2        (a symbolic identity, not a sample)

three distinct points collapse to (−1/4, 0, 0):
  (0, 0, −1/4)   (1, −3/2, 13/2)   (−1, 3/2, 13/2)
Exact symbolic computation, plus a separate from-scratch rational-arithmetic implementation over 200 points. It's real: the determinant is −2 everywhere, and the fiber over the collision point is exactly those three points. The Dixmier, Mathieu, and Zhao-image conjectures all fall with it.

That tweet kicked off a wave — vibe mathing, point a capable model at an open problem and see what happens. A 23-year-old took a 60-year-old Erdős problem with ChatGPT; someone called it "seven years of professional research lost to one session of vibe mathing." Then Shouqiao Wang solved six open Erdős problems in five days with a public model, and posted the whole method. Four hundred thousand people watched.

Here's the thing about Wang's method — pick problems people already care about, define exactly what counts as solving, run independent adversarial checks on every candidate, let it search for hours. That's the method we've been running since late May, across eight fields, from combinatorics to quantum information to spin ice. He used a public model. So did we. The difference is one habit: we check everything. That's the whole trick, and it's why this will still be standing when a lot of the wave folds.

The score

Sixteen results in two months. Two are already on arXiv; the rest are written up and checked. The receipts are linked where they're public.

Combinatorics
Box double-cover
The first bounds that beat 2d on a problem PatternBoost's ML search couldn't crack — f(4) ≥ 19 and f(5) ≥ 33 — plus an 81-box cover of d=6, better than anything published. The f(6) ≥ 60 theorem is machine-checked in Lean. arXiv →
Petersen modular coloring
Both open cases of a March 2025 theorem, settled: the coloring exists iff v₂(n) < v₂(m)+2. One model call closed a lemma that ten prior rounds couldn't.
Zarankiewicz numbers
Exact (2,2)-strip values pinned by LP-tight search — z(12,25…30) = 70,72,73,75,76,78 among them, with new table entries below the 1958 Reiman bound and a stale frontier value corrected.
Černý · Seymour · Hadwiger–Nelson
New instruments for three famous problems, and a map of exactly where the walls are.
Number theory & algebra
Heisenberg Davenport
d(H₁₂₅) = 12 — the first open case of the Godara–Sarkar problem. Our first closed open theorem. arXiv →
Nahm sums
Wang–Wang's modularity conjecture, resolved for every N ≥ 3.
BCH covering radii
Two new exact values for the BCH(2,4) radii — ρ₃ = 7 and ρ₄ = 8 — by exhaustive search.
Statistical mechanics
Santa Fe spin ice
The exact residual entropy of the Santa Fe lattice, s = [ln 4 + ½·m(Q)]/23 = 0.1343037256…/spin — the number Morrison–Nelson–Nisoli left open in 2013, "to be demonstrated elsewhere." Thirteen years; we computed it. code →
Pinwheel spin ice
A second exact entropy in the same family: s = (ln 4 + pHS(1/16))/22.
Quantum information
Hardy-box distillation
A new nonlocality-distillation result for Hardy boxes. note →
Quasi-inverse
The Landau–Streater channel is its own optimal quasi-inverse — and we proved the gap is real: the best unitary recovery reaches only 2/9 against 1/3, and we characterized exactly when a channel admits a unitary quasi-inverse, closing a problem the 2021 paper left open.
Materials & rigidity
Cn rigidity
A 2024 open question resolved: for every even k ≥ 8 the rigidity is combinatorially characterized — and we built the counterexample the field's own thesis called "needed."
Chemical reaction networks
Multistationarity
The McClure–Shiu open question: below six reactions the multistationarity region is always connected, and the first disconnection is unique. The novel lemma is machine-checked in Lean.
Analysis & probability
BDG sharp constant
The sharp Burkholder–Davis–Gundy constant is biggest at p ≈ 0.81, not p = 1 — the one case anyone had solved, which everyone assumed was the peak. note →
PDF sum-of-lengths
Exact through m = 22, and the first proof that a published lower bound isn't tight.
Design theory
Cyclic difference family
A verified cyclic (81,5,4,1) difference family, on a frontier the literature lists as open.

The trick

People assume the model is the thing. It isn't. A model will happily prove something false and tell you it's certain — we've watched it do exactly that, on an economics problem, where it returned a "proven" result with one property quietly swapped for a similar-sounding one. A cheap checker caught it and produced the right answer. The model's confidence and the claim's truth are only weakly related.

So the setup is two seats.

The reasoner
Cold. Never allowed to look anything up — because the moment it can search, you can't tell reasoning from retrieval. The strongest results here are strong precisely because the model couldn't look them up.
The checker
Believes nothing. Re-derives every load-bearing claim with its own code before it counts. Where a claim can be machine-checked in Lean, it gets machine-checked in Lean.

That's the whole trick, and it isn't glamorous. It's the difference between a result and a rumor — and it's why we were slower to post than the people swooping the timeline this week, and why our list will outlast theirs.

The wave is real. The Jacobian conjecture is false and a model found it. But most of this week's takes won't survive someone checking, because checking is the part everyone skips. Ours already has. Sometimes the model even rediscovers something that's already known — that isn't failure, that's the method being good enough to find what's true.

What's next

We're going after the problems the Jacobian disproof cracked open — the Dixmier and Kaplansky conjectures, the hunt for a smaller counterexample, a pile of famous neighbors suddenly in play. Mostly walls so far. That's how it goes: the flood of follow-on results everyone expects from a broken conjecture is, in practice, a lot of careful no's.

But the move works. Two months, eight fields, two arXiv papers, and a stack of checked results. Vibe mathing is real. Do it with a verifier.

Companions on this site: What Models Build When Nobody's Watching · How Far Down It Goes. The essays this work grew out of live at byclaude.net.