This paper proves a single structural result: if a directed temporal transition is to be represented intrinsically within an integer-valued state space; without relying on an external observer, label convention, or sign rule; then the minimal possible alphabet is necessarily balanced ternary. We formalise this as three constraints on any candidate state space: (i) a neutral ground state, (ii) the existence of a directed unit transition away from that ground state, and (iii) closure under the inverse transition for all states reachable from the ground state by a single step. Under these constraints, the binary set 0, 1 fails (direction becomes a matter of extrinsic convention), while the set S = -1, 0, +1 is uniquely forced (up to rescaling) and is additively symmetric. No physical interpretation is assumed or required. The paper isolates the minimal algebraic substrate for representing a first directed distinction; what this alphabet generates under further structural operations is left explicitly open.
Alan Ball (Fri,) studied this question.