A black hole swallows a star, and its horizon expands. This is not just a growth in area: the geometry of the horizon records information about what fell in. Physicists have long suspected that the horizon area is tied to entropy, but ordinary entropy did not fit the bill: if an object in a pure state with zero entropy falls in, the area still grows. The new work shows that the horizon behaves like an accounting ledger, where positive and negative entries cancel each other out, leaving only the net energy flux.

To understand this balance, the author splits the black hole into two subsystems. The positive subsystem consists of ordinary matter that has fallen in. The negative subsystem includes the partners of Hawking pairs with negative energy. When an object falls into the hole, the positive subsystem undergoes scrambling: a rapid mixing of quantum information that conserves energy but destroys order. After scrambling, the positive subsystem settles into a microcanonical ensemble, that is, an equilibrium state with fixed energy. The entropy increase is computed by expanding the density of states around the initial energy, and the result takes a canonical form with the Hawking temperature. The temperature of the positive subsystem is calibrated using a perfectly reflecting cavity, where the outgoing Hawking channel and its time-reversed ingoing channel form a steady-state exchange.

The key result: the entropy increase of the positive subsystem does not depend on the initial entropy of the object. It equals the energy of the infalling object divided by the Hawking temperature, plus the logarithm of the statistical sum, that is, the sum of the probabilities of all states of the system. For paired modes of Hawking radiation and ingoing modes of the same frequency, the statistical sum contributions cancel. As a result, the change in horizon area is determined only by the change in energy, in agreement with the first law of black hole thermodynamics. The result generalizes to rotating and charged black holes, with the canonical ensemble replaced by the grand canonical one.

This work provides a microscopic justification for the coherent-information interpretation of the horizon area. Coherent information is a measure of quantum entanglement between systems, and it can be negative. Negative conditional entropy means that the horizon stores not thermal disorder but a quantum link between the interior and the radiation. This explains why thermal fluctuations of Hawking radiation make no contribution to the horizon area. Previously it was unclear how to interpret the horizon entropy in the picture of Hawking pair creation, especially when an object in a pure state falls in. Now it has become clear that the horizon area is tied to coherent information, and the entropy increase upon absorption has a canonical form at the Hawking temperature. This result can be used in theoretical studies of quantum gravity and in quantum simulations of black holes to test information protocols.

The horizon's ledger balances: positive entries from infalling matter and negative entries from Hawking partners cancel each other out. What remains is only the net energy flux, which is what expands the horizon. So the black hole keeps track not of thermal chaos but of quantum information.