Generative Structural Theory v1.1: Relational Neighborhoods, Transitive Closure, and Topological Admissibility

Generative Structural Theory (GST) v1.1 develops a relational-neighborhood layer over the selected stable sectors inherited from GST v1.0. Starting from a non-diagonal, generally asymmetric relation strength R and a fixed threshold ε, it constructs a directed threshold relation, reflexively anchored candidate neighbourhoods, an induced preorder through transitive closure, and an Alexandrov topology. The paper proves that three of the four neighbourhood-system conditions hold automatically, whereas the fourth condition, N4, is equivalent to transitivity of the reflexively anchored threshold relation. A topology is induced at every threshold without assuming N4. However, the original candidate family forms a genuine neighbourhood basis for that topology exactly when N4 holds. When N4 fails, transitive closure provides a repair that preserves the induced topology but does not preserve the original direct relation or candidate family. The paper distinguishes candidate neighbourhoods from minimal open neighbourhoods, establishes an exact non-recoverability result together with a positive characterization of what the topology does determine, introduces the closure-added relation fraction Lε, proves threshold filtration and separation results, and characterizes structural continuity as monotonicity for the specialization preorders. A finite five-sector computed model supplies explicit witnesses for non-transitivity, non-monotonic admissibility, separation–admissibility independence, and a homeomorphism that does not preserve candidate neighbourhoods. The standard correspondence among Alexandrov spaces, preorders, minimal open neighbourhoods, separation properties, and monotone maps is explicitly acknowledged as established mathematics. The contribution of GST v1.1 is the assembled construction and the exact admissibility, repair, and non-recoverability results proved within it—not a claim to have discovered those classical correspondences. The finite model is illustrative and non-empirical. The symbols R, ε, and the sectors are not assigned physical, temporal, causal, metric, probabilistic, or information-theoretic meanings. The computed model supports reproducible examples but does not replace the general proofs. This deposit contains the complete 50-page paper, LaTeX source, twelve publication figures with TikZ sources, the finite-model verification script, byte-exact expected output, reproducibility instructions, licensing information, and a SHA-256 manifest. The archive contains 31 hashed payload files plus the self-excluded manifest, for 32 regular files in total. Archive SHA-256: ca2717f3deac7b51a5860a0e272e3743d3139b2a56884c4c936c2b296076d512 Previous version: GST v1.0 — https://doi.org/10.5281/zenodo.22182415