Spatial Organization Emerging from Atemporal Relational Information Structures

Prior work established that intrinsic order and a unit-cover clock-like grading can be constructed from a static relational information structure without introducing an external time variable. Motivated by that result, the present study applies the same methodology independently to the problem of space, without reusing the temporal structure as an input. We treat the information repository itself, in which data items and their relations accumulate, as a pre-spatial information substrate and ask what spatial organization is forced, without auxiliary choices, by relational incidence alone, with no coordinate system, continuous background space, physical distance, causal order, probability, or preassigned object semantics. In a finite relational incidence dual object with two marked terminal classes, separator inclusion defines a pair of mutually reversed intrinsic orders {P,Pop} and a unit-cover grading {r,−r}; rank-2 Boolean diamonds provide same-grade local adjacency, while higher-order minimal nonfaces provide local constraints that cannot be reduced to a pairwise graph. A reaction grammar is then constructed from the parity-dependent unary rule for complete carriers Km and a proper-overlap strict-majority binary product. By binary source recoverability, the one-step reaction factors exactly through DF=EF,BF, and each D-fiber coincides with an autonomous future-equivalence class. The canonical core CF obtained by removing predictive redundancy is the unique least exact representative of each equivalence class, and facet rigidity implies that every reaction output is automatically a core. Hence CR=R=RC, and the core reaction is an injective self-map. From this injectivity, each connected component of the core reaction graph is restricted to a rooted ray N0, a bi-infinite line Z, or a finite cycle; relative displacement, chain order, an intrinsic integer metric, and the topology of the reaction 1-complex are then recovered. Because finite reaction-compatible quotients separate exact core states, residual finiteness follows. The componentwise inverse-limit completions are computed exactly as Cp, Z, and the rooted compactification N0⊔Z. Moreover, whenever a nonprincipal ultrafilter fiber concentrating on finite-cycle components exists, that fiber is classified as a procyclic torsor of supernatural order determined by the common divisibility spectrum. In the directly computed generated recurrent subsystem, 54,180 primitive rooted rays are realized. Whether a nontrivial exact cycle or a genuine Z component is actually realized in the full arbitrary core remains an open realization boundary of this study.