What a Lattice Horizon Counts

Jacobson’s derivation of the Einstein equation needs one number: the entropy a local horizon carries per unit of its area, the same for every horizon through every point in every direction. If the horizon is a surface in a cubic lattice of recognition events, that number has to be a count. We ask which count. On a fixed lattice with unit spacing and unit light speed, the area of a horizon cross-section is a Lorentz invariant with no ambiguity, and exactly one of the lattice counts examined here matches it with coefficient one in every direction: the number of rest worldlines the null plane sweeps through per tick. Every other candidate examined fails in a specific way. Faces on a tilted plane carry the staircase factor |n1| + |n2| + |n3|. Lattice events lying on the plane have density exactly one at every rational null direction and 1/√2 at the irrational tilt we study, so that count is discontinuous in direction; so is the count by generators, whose classes are a column, a single site, or a chain according to the arithmetic of the direction. The proper shadow of a cell at an irrational tilt is irrational and is not a count at all. The one count that works is the record a cut exposes, the sites whose worldlines the plane crosses during the tick of the cut, one per unit area in the lattice frame, and it carries no dynamics: it is the same for every local clock rate, and on a fixed lattice the area it defines has zero second variation along the generators, so the horizon cannot focus. A lattice of recognition events therefore supplies the area coefficient as a limit law and supplies no time dilation by counting; the gravitational response has to live in a geometry that depends on what is posted.