Anthropic has deposited a 42-page paper, credited to Claude Fable 5.2, that settles how a black hole forms in a universe small enough to have no classical geometry. Hawking and Page showed in 1983 that a black hole appears in anti-de Sitter space through a first-order phase transition, which in N=4 super Yang-Mills on a three-sphere is deconfinement at strong coupling. In 2003 physicists suggested the weak-coupling transition at the Hagedorn temperature could connect continuously to that one, either as a single first-order line or as a pair of continuous transitions with no gravity counterpart. The sign of one number decides it: the quartic coefficient b of the Polyakov-loop effective action, which first appears at three loops and had not been computed in the two decades since.

The paper finds b = −1.48×10⁻³ λ², so the transition stays first order, sits slightly below the Hagedorn temperature, and its three saddle points appear, exchange dominance and disappear in the same order as thermal anti-de Sitter space and the small and large black holes. The Hamiltonian computation derives almost every number two independent ways, and it reproduces the two-loop Hagedorn temperature known from integrability.

Hexagon’s provenance note says an internal Claude research model, prompted by a physicist, obtained the result, and that the physicist checked it against several known special cases; a separate Claude computation using a different method, not included in the paper, gives the same answer. Hexagon flags the deposit as primarily AI-generated text.

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The Hawking-Page transition at weak coupling · Hexagon