Emergent Pregeometry: Cellular Complexes of Fork/Join Histories and Reachability Obstructions in PL Filling

Without assuming external coordinates, continuous space, or physical time a priori, this paper formulates the process by which geometric structures are constructed from "finite directed history graphs" from the perspective of pure algebraic topology. By doing so, it mathematically proves the inevitable "topological reachability obstructions" that arise in local three-dimensional extensions based on causal rules, presenting the geometric limits faced by models of spacetime emergence. [Key Highlights] Construction of the base cell complex $K_B$ and determination of its homotopy type: We prove that the base cell complex $K_B$, obtained by applying the $E_B$ face attachment rule to a finite directed history graph generated by Fork/Join operations, is homotopy equivalent to a wedge sum of 2-spheres, and its 1-dimensional integral homology is trivial ($H_1 = 0$). Formulation of conditional PL filling (PHEN filling): We construct a framework for three-dimensional extension that imposes a link test for a PL 3-manifold with boundary (local geometric consistency) during the process of regularly attaching 3-cells to the configured sphere candidates. Mathematical proof of reachability obstructions under all admissible orders: We present an 8-operation history counterexample where two $\theta$-graphs are joined at a single point. We rigorously prove that even if filling proceeds while satisfying local PL conditions, a residual integral homology ($H_2 \neq 0$) inevitably emerges at the termination of all maximal admissible attachment sequences, making global topological obstructions unavoidable regardless of the sequence of steps. [Abstract] In this paper, we formulate the process by which geometric structures are constructed from causal networks based on finite-step generation rules, and mathematically prove the inevitable topological obstructions that arise in local three-dimensional extensions.In this study, we apply the $E_B$ face attachment rule to a fixed finite directed history graph $H_{hist}$ to construct a base cell complex $K_B$ of dimension at most two. We then determine the homotopy type of $K_B$ and formulate PHEN filling, restricted to the regular attachment of 3-cells on a fixed history. Ultimately, using an 8-operation single-point junction counterexample, we derive that although two sphere candidates can each be attached individually, every maximal admissible attachment sequence terminates with $H_2(K;\mathbb Z)\cong\mathbb Z$ . Thus, in this example, no admissible attachment order eliminates the residual second homology.