The Modular Character as a Relative Entropy: A Lattice Bridge to the Type-II Crossed Product

Every region of a quantum system carries a hidden mathematical mirror — the modular conjugation of Tomita–Takesaki theory. This paper studies the number obtained by composing two regions’ mirrors and taking a trace: the modular character. The main theorem is that, for separated regions, this number is exactly the exponential of a relative entropy — a measure of how much the two regions know about each other. Why that matters: the absolute entropy of a region diverges in continuum physics (the algebras of quantum field theory are type III — they have no trace to define it with), but relative comparisons stay finite there. So the character is a bridge: a quantity computable exactly on a finite lattice that remains meaningful, unchanged in kind, in the continuum — and its n → 1 limit is precisely the quantity that the gravitational crossed product turns into black-hole (generalized) entropy. One computable object thus connects lattice simulations to continuum field theory and to gravity’s entropy bookkeeping. Along the way the character measures pure-state entanglement completely — turning its dial reads the entire entanglement spectrum, including the entanglement entropy itself, while provably ignoring everything else — reads topological ground-state degeneracy, distinguishes phases of matter by its decay law, and — when the two regions overlap — develops an imaginary part that detects chirality (broken time-reversal symmetry).