The Critical Almost Mathieu Operator

An electron on a grid in a magnetic field. Is its energy spectrum missing a piece at every possible place?

Posed by Mark Azbel / Douglas Hofstadter (model, 1964/1976); "Dry Ten Martini" named after Mark Kac's bet.

In plain terms

Put an electron on an infinite grid of atoms and switch on a magnetic field. The fraction of a flux quantum threading each little square of the grid — call it α — controls the electron's allowed energies, and the picture of those energies against α is the Hofstadter butterfly, one of the most famous fractal images in physics. Each rational α = p/q gives q separate energy bands; each irrational α gives an infinitely fragmented set.

The conjecture is about the fragmentation. A deep bookkeeping theorem ("gap labelling") lists every place where the spectrum is allowed to have a gap. The Ten Martini problem asked whether the spectrum really is fragmented for every irrational field — that was proved in 2009. The Dry Ten Martini problem asks for more: is every gap on the allowed list actually open, with none collapsed to zero width? In 2023–24 this was settled for every strength of the model except one — the physically central, self-dual, critical coupling, which is exactly the Hofstadter case. There it remains open for every single irrational α.

What is known

The spectrum is a Cantor set for all irrational fields (Avila–Jitomirskaya, 2009). All labelled gaps are open away from critical coupling (Avila–You– Zhou). At critical coupling the total size of the spectrum is zero, its fractal dimension is at most one half (Jitomirskaya–Krasovsky), and for rational flux p/q everything is exactly computable: the spectrum is where a single degree-q polynomial stays between −4 and 4, all q − 1 gaps are open except that for even q the two middle bands kiss at zero energy, and Thouless conjectured — with striking numerics, still unproven — that q times the total bandwidth tends to a specific constant, 32 times Catalan's constant over π.

Why it is hard

Critical coupling is a phase transition point. Below it the model behaves like free motion, above it like localized motion, and the two regimes come with two different proof technologies; at the critical point both machines break at once. The known gap-opening arguments run along one side of the transition and lose all their room exactly there. Worse, the quantities that survive at criticality (zero measure, dimension bounds) are the ones that cannot see individual gaps.

What a breakthrough would mean

The butterfly is now measured in real systems — moiré graphene, cold atoms, photonic lattices, superconducting qubit arrays — and the same spectrum governs the critical temperature of superconducting wire networks through the de Gennes–Alexander equations, where the field-dependent T_c literally traces the butterfly's edge. Whether the idealized spectrum has zero-width gaps is the mathematical backbone behind which of those measured gaps are robust. Inside mathematics it would close the last case of a program that has driven quasi-periodic spectral theory for forty years.

In this lab it carries a low budget and long-shot framing, deliberately. Nobody here expects to prove a gap open for irrational α. What the lab can produce is exact: certified band edges, gap widths and bandwidth sums at rational flux, over the real cyclotomic field with no floating point in any certificate, pushed along Fibonacci approximants of the golden mean where the conjectured constants have falsifiable shadows. Nothing has been tried here yet.

What this lab tried

Standing: Not attempted Onboarded 2026-08-04, never worked. Rational-flux harness is exact over the real cyclotomic field and self-tested to q = 34; no attempts recorded, so blind and informed mode are currently equivalent for this problem.

No approaches have been recorded for this problem yet, so there is nothing here to build on or avoid.

Go deeper

Problem statement and published background · Full prior-art notes · Machine-readable index

Found something, or think we got it wrong?

Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.

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