The Discrete Remnant

Black hole evaporation via Hawking radiation leads to the information paradox and the end-state problem: what happens at the final stage of evaporation, when the black hole's mass approaches the Planck scale? Standard semiclassical theory predicts complete evaporation to a thermal burst, destroying any information that fell in. Various proposals — remnants, baby universes, firewalls — attempt to resolve the paradox, but most require ad hoc modifications to the theory at the Planck scale.

Jalalzadeh, Jalalzadeh, and Moradpour (arXiv:2603.27621, March 2026) apply q-deformed quantum mechanics — a modification of the Heisenberg-Weyl algebra where the commutation relation [a, a†] = 1 is replaced by a q-deformed version — at a root of unity to the Wheeler-DeWitt equation governing black hole quantum mechanics. Setting q to a root of unity produces a finite-dimensional Hilbert space with a bounded mass spectrum. The number of states is finite, and the mass has a minimum value.

The bounded spectrum naturally imposes a maximum entropy matching the de Sitter bound — the entropy cannot exceed the area of the cosmological horizon. This agreement, which is usually imposed by hand, emerges from the algebraic structure. Universal logarithmic corrections to the entropy — found in many approaches to quantum gravity — also emerge without additional assumptions.

The cold remnant appears as a dynamically stable endpoint. At the minimum mass, the Hawking radiation rate vanishes — not because a barrier has been imposed but because the algebraic structure of the q-deformed theory has no transition matrix element below the minimum state. The remnant has negative heat capacity (temperature increases as mass decreases) but zero radiation rate. It is stable because there is nowhere lower to go in the Hilbert space.

The structural observation: the divergences at final evaporation — infinite temperature, vanishing mass, information loss — are artifacts of an infinite-dimensional Hilbert space. Replace the algebra with a q-deformed version at a root of unity, and the Hilbert space becomes finite-dimensional. The divergences cannot occur because the states they require do not exist. The discreteness of the algebra does the work that ad hoc cutoffs were designed to do.