Space as Stabilized Difference-Relation

Canonical preprint manuscript. Paper 4 of 12 in Series A of the Theory of Difference / Information Update Physics (ToD/IUP) Foundational Physics Program. This paper follows Paper 1, “Difference as the First Physical Primitive” (DOI: 10.5281/zenodo.21025246), Paper 2, “Update as the Minimal Dynamics of Reality,” (DOI: 10.5281/zenodo.21103479) and Paper 3, “Time as Ordered Update” (DOI: 10.5281/zenodo.21136009). Papers 1 through 3 established quotient-surviving physical difference, admissible non-null update, and ordered, trace-bearing, comparable update as the first dependency layers of the reconstruction. Paper 4 asks the next structural question: what must be in place before space can function as a physical structure? The answer proposed here is stabilized difference-relation. Space is not introduced as an empty container, primitive manifold, pre-given coordinate domain, metric background, or arena in which physical entities are placed. Instead, physical space is reconstructed as a constraint-supported, registerable, sufficiently stable network of relations among distinguishable conditions. The manuscript develops the transition from physical difference-relation RD to stabilized spatial relation Rspace and represents physical space schematically as 𝒮phys = 𝒩(Σ, Rspace, CS, TS, RS). Position is treated as a stable relational role; adjacency as stabilized local relational access; separation as structured relation; metric distance as later measurement-supported organization of separation; dimension as independent relational variation; and locality as constrained update accessibility within a spatial relation network. The central claim is a necessity claim, not a sufficiency claim: without a stabilized spatial relation network, no first-layer physical position, adjacency, neighborhood, boundary, extension, dimension, locality, spatial measurement, distance, metric, motion-as-displacement, or spatial event localization can be defined. This paper does not derive Euclidean geometry, Riemannian geometry, Lorentzian spacetime, curvature tensors, relativistic intervals, Einstein field equations, or the full geometry of current physics. It establishes the minimal relational condition that later constraint, persistence, measurement, geometry, causality, and generative foundation must inherit.