From Primitive Global Non-Distinction to Relational Causality: A Conditional Construction of 1+1-Dimensional Lorentzian Pregeometry

Abstract

This preprint investigates whether an effectively differentiated causal geometry can be constructed under a primitive-description constraint containing no intrinsic differentiation. Primitive Global Non-Distinction (PGND) is represented by a single ground that, under an explicit articulation principle, occupies two non-equivalent relational positions without becoming two primitive occupants. Conditional on independently stated principles of persistent consequence, recentering, two-orientation articulation, and repeatable-extension closure, finite relational compositions form a binary history monoid. Count-based endpoint recombination produces an endpoint space isomorphic to the nonnegative integer lattice ℕ², while extension-induced reachability defines a locally finite product order. After additional causal and metric specialization, this order is identified with the causal order induced on a regular null lattice in (1+1)-dimensional Minkowski spacetime. At fixed lattice scale, the construction provides causal classification, causal diamonds, a proper-time function, and quadratic interval growth. Under refinement, rescaled interval counting converges to the corresponding Minkowski Alexandrov-volume law. Independently, the large-interval ordering fraction approaches the (1+1)-dimensional Myrheim–Meyer value. A finite-generator obstruction shows why a fixed finite alphabet of independent endpoint directions cannot exactly generate the non-polyhedral Lorentz cone in spacetime dimension three or greater. The claimed contribution is not the independent discovery of the product order, its familiar null-coordinate realization, or the non-polyhedral character of higher-dimensional Lorentz cones. Rather, it is the explicit same-occupant relational route leading to these structures, together with a dependency and countermodel audit identifying the additional principle required at each stage. PGND functions as a restrictive primitive-description constraint rather than a generative axiom. The construction therefore establishes a conditional admissibility witness: it shows that Lorentzian causal organization is compatible with primitive global non-distinction under sufficient additional conditions. It does not derive physical realization, uniqueness, higher-dimensional spacetime, microscopic Lorentz symmetry, dynamics, quantum structure, matter, or empirical predictions from PGND alone.