Intrinsic Order and Clock-Like Grading Derived from Static Unoriented Two-Ended Finite Relational Structures

We investigate whether a static, unoriented, two-ended relational structure with no external time variable or future/past orientation can define intrinsic order and a clock-like step coordinate on a separator-state space derived from the structure rather than on its original vertices. Here, information denotes incidence and adjacency data, not a probabilistic information measure. For a finite 2-vertex-connected terminal-marked graph with a nonempty family of terminal-separating connected sides, we prove that every Hasse cover in the inclusion poset adds exactly one vertex and that exchanging terminal roles induces an order anti-isomorphism between the two side posets. The reference complex K0 is the 10-vertex, 20-simplex standard staircase triangulation of ∂Δ⁴ × I. For all 15 legal 2→4 direct targets K1,…,K15 generated from K0, the core audit verifies dual 2-vertex-connectivity, nonempty separator families, connected Hasse graphs, and Δr=1 for r(U)=|U|−N/2 on every cover. Detailed Boolean/type/path calculations are performed on D14={K1,…,K5,K7,…,K15}, with exact separator-poset transport and q preservation verified for 350 double-shelling targets. Thus the core order/grading scope comprises 365 structures and the detailed transition/cubical-grammar scope 364. Exchanging terminal roles satisfies r1(c(U))=−r0(U), yielding a signed grading pair without selecting an absolute endpoint role. These results establish constructive pre-temporal orderability and a unit-cover step structure on a derived state space, not physical time emerging from a featureless substrate.