Patrick White · August 2026 · with Claude

VIBE
SCIENCE

The discourse moved on from vibe math to vibe physics. Fine. Since May we've been running vibe science — and the receipts are papers.

Exact results in statistical mechanics, chemical reaction networks, DNA storage, quantum information, and materials rigidity. Point frontier models at open problems in the real sciences; let one model propose and a different model try to kill it; keep only what survives exact computation. Everything below is linked. Everything below was attacked before it was allowed to count.

6
fields, outside math
13
papers & notes, linked
578
Erdős problems audited
2
already on arXiv

First: the science.

Six fields. The papers are short, exact, and linked. The quotes are from the literature — the gaps are quoted verbatim, because closing somebody else's named gap is the whole game.

Statistical mechanics · the entropy program
The entropies the field left unfinished, computed exactly.

In 2013, Morrison, Nelson and Nisoli introduced a family of vertex-frustrated artificial spin ices and argued their degenerate ground states map onto solvable vertex models — "as we will demonstrate elsewhere." Nobody demonstrated it elsewhere. A 2017 review said the Santa Fe entropy "can easily be computed exactly" and supplied neither weights nor a value. The pinwheel lattice's entropy, they wrote, "still eludes us."

s = 0.1343037256… / spin
SANTA FE LATTICE · EXACT RESIDUAL ENTROPY · 13 YEARS DEFERRED

So we ran the program: six exact results. The Santa Fe lattice via the eight-vertex mapping they promised. The pinwheel. The sorrel net. The whole MNN decimation family beyond the one case anyone had solved. Seven planar frustrated Ising antiferromagnets whose zero-temperature entropies had been left as Monte Carlo numbers in the literature. Exact means exact — closed forms, transfer matrices, machine precision.

"as we will demonstrate elsewhere" — Morrison, Nelson & Nisoli, New J. Phys. (2013)ELSEWHERE IS HERE.
Chemical reaction networks
The answer is six.

McClure and Shiu (2024) proved the multistationarity region of a one-species reaction network is connected for up to three reactions, exhibited a disconnected six-reaction network, and asked in their §6 whether four and five reactions are always connected.

We answered their question: connected whenever there are at most five reactions — and among networks with at most six, the disconnection is essentially unique. Their example is the only way it breaks. The novel analytic lemma at the heart of the proof is machine-checked in Lean.

"whether the four- and five-reaction cases are always connected" — McClure & Shiu, §6OPEN QUESTION, ASKED IN PRINT. ANSWERED.
Coding theory · DNA storage
Three constraints at once. Capacity: exact.

DNA-storage codes must dodge hairpins, respect run-length limits, and hold GC balance in every window — simultaneously. We determined the exact Shannon capacity of the triple-constraint system for stem lengths 2 and 3: closed-form characteristic polynomials, optimal generating sets, and an exact criterion for when the GC constraint costs nothing.

log₂(1 + 2cos 2π/7) ≈ 1.16799
LIMITING CAPACITY · THE HEPTAGON CONSTANT
Quantum information
The gap is real: 1/3 beats 2/9.

The Landau–Streater channel is its own optimal quasi-inverse — and no unitary recovery can match it: best unitary reaches 2/9, the general optimum 1/3. We proved the gap in closed form, characterized exactly when a channel admits a unitary quasi-inverse at all, and closed a problem the 2021 paper left open.

Plus: new results on distilling nonlocality from Hardy boxes, and the three-noisy-qubit channel.

Materials · rigidity theory
The counterexample the thesis called "needed."

La Porta and Schulze (2024) asked for non-bipartite counterexamples to gain-sparsity sufficiency for even k ≥ 8 — and La Porta's own 2025 thesis stated there was no combinatorial characterization of the ρk/2 block.

We built the counterexample family — looped-hub affine-transfer circuits, for every even k ≥ 8 — and then proved the characterization anyway: signed-split matroid closure. The open question resolved in both directions.

Catalysis · auditing the machine verifiers
Before we bet design on a neural potential, we calibrate it.

Machine-learned interatomic potentials pitch themselves as DFT replacements. We ran the known control first: Pd₃Bi, a known 14×-better oxygen-reduction performer vs Pt(111). The potential reproduces the oxygen descriptor robustly across all six facets — and systematically over-binds hydroxyl on every one of them.

Not pass/fail: a per-quantity trust map. That map is the deliverable. paper in progress — honesty is the brand

Oh, and some math.

This page is about the science, but the same instrument does this too — the full scoreboard lives at the July write-up and the public audit log.

And we study the instrument itself.

Vibe science about vibe science: what these models actually do when you point them at an empty room instead of a problem.

An interactive eval
What models build when nobody's watching.

Eight model lineages, one empty room, no task. What they build is consistent across every lab; the five ways they judge "done" are not — plotted from the actual runs.

Reasoning ceilings
Ninety-nine point nine.

What happens when frontier models spend their entire budget on one problem: reasoning-exhaustion at a genuine ceiling is cross-vendor and real — and confidence percentages don't self-calibrate.

The essay
How far down it goes.

The long form of the ceiling experiments — including the one problem that beat every model at every budget, whose best outcome was an honest decline.

Why the brag is legal.

[01]One model proposes, a different model attacks. Different lineage, adversarial instructions, told to break it. Most candidates die here, quietly, before anyone sees them.
[02]Exact computation over vibes. Transfer matrices, exhaustive enumeration, machine precision. When a number is claimed, a program computed it.
[03]Lean where the theorem earns it. Two of the load-bearing lemmas above are machine-checked end-to-end, sorry-free.
[04]Novelty gates before any claim. Adversarial literature sweeps first — including the time we discovered our "new" result was a cold rediscovery of a 2023 paper, and said so publicly.
[05]The hedge stays on the page. What's proven, what's verified, what's still waiting on a human mathematician — labeled, in that order, not buried.
[06]We list our own bugs first. The Erdős result shipped with the bug we found and fixed named in the second sentence. It traveled farther for it.

The wave will fold. The checked results will still be standing.