Constructible Ordering from Distinguishable Steps

We present a self-contained characterization of when a well-dened ordering can be constructed from a primitive step relation on a state space, without presupposing time, dynamics, probability, or physical structure. Starting from a nonempty state space S equippedwith a binary relation ⇒ (the step relation), we dene an ordering value t(s) as the minimal number of steps required to reach state s from a xed reference. We prove that thisconstruction yields a well-dened function t : S → N ∪ {∞} and an induced preorder.Our main contribution is a characterization theorem: we establish both suciency(explicit construction) and necessity (showing that each condition is logically required).While the mathematical constructionminimum path length in a directed graphis wellknown in combinatorics, our contribution lies in the conceptual reframing: we isolate theminimal pre-theoretic conditions for ordering, independent of physical interpretation, andposition this within the philosophical tradition from Robb (1914) through the causal theoryof spacetime.This work makes no claim about time, causality, or physics. The value lies in conceptualclarication: showing exactly what ordering requires and what it does not require.Keywords: ordering construction, step relation, characterization theorem, causal structure,foundations, graph distance, computational complexity