One Bit per Area, Not One Bit per Face: Around a Black Hole, No Lattice Count Is the Entropy of Local Horizons

A black hole's entropy is one bit for every 4 ln 2 Planck areas of its horizon. On a cubic lattice of spacing λ the simplest reading is one bit for each lattice face the horizon crosses, and through Jacobson's derivation of Einstein's equations from Clausius's relation δQ = T dS on local horizons that reading would set the strength of gravity, Għ = λ²c³/(4 ln 2). We prove that the thermodynamics of gravity rules it out, together with every count of lattice records of fixed range whose density is positive in every direction, and we identify what survives. Every such count depends on the horizon's direction. As the lattice moves, a light ray's direction measured against its axes turns unless sixteen linear conditions hold, and a lattice meets them throughout a region only where the Weyl tensor vanishes. On Jacobson's local horizons the turning changes a count at first order while the heat starts at second order, so Clausius's relation fails at every temperature whose ratio to the Unruh temperature is bounded below. Around every Kerr black hole, rotating or not, this holds whatever slicing carries the horizons' cuts and however the lattice moves. What survives is one bit per λ² of area, the same in every direction: a density, not a count, which is Għ = λ²c³/(4 ln 2) written in lattice units. The theorems hold in the continuum limit; the passage to finite spacing is stated as an open problem.