whatTwo gases -- up-ticks left, cross-ticks right -- behind a wall, on a lattice. The dynamics are a lattice gas (HPP): integer state, exact collision rules, no rounding anywhere. That is the point: a floating-point gas is deterministic but not reversible in practice -- chaos doubles the rounding error at every collision. On the lattice, time reversal is exact, forever.

whyThe claim is Loschmidt's 1876 objection, staged: the microscopic rules are perfectly reversible, so REVERSE TIME -- every velocity flipped -- and the impossible happens: the gas retraces its history, the counter runs downhill, the species unmix into their halves. Boltzmann's reply is one click away: nudge a single particle by one part in a million before reversing, and chaos amplifies the flaw collision by collision -- the gas returns to a past that no longer exists, and the counter never comes down.

howHPP on an integer grid with species riding each particle; head-on pairs rotate by a fixed self-inverse rule; entropy is coarse-grained over an 8x8 grid of mixing fractions, normalized by ln 2, traced live. Left alone the piece stages the whole argument: mix, reverse, exact return; then turn one particle, reverse, no return. One integer seeds both gases.