Coalescent Haplotype Dynamics of GBA1/GBAP1 Recombination, Hierarchical Wahlund Amplification, and Topological Orthogonality of Mating Networks in Regional Japanese Demes: A Mathematical Synthesis of Eurasian Gene Flow, Edo–Meiji Koseki DAG Partitioning, Ashkenazi–Japanese Structural Isomorphism, and QFS-zkVM Provenance

Building upon our companion monograph on the stochastic population bottlenecks of Gaucher disease (GBA1 deficiency at chromosome 1q21) and the deterministic Human-in-Command (HIC) legislative elimination of the Age-18 Pediatric Subsidy Cliff for Japan's historical N=90 cohort (Zenodo DOI: 10.5281/zenodo.23581228), this study establishes a unified mathematical and molecular-evolutionary framework resolving two foundational questions in human population genetics. First, we formalize the molecular discrimination between trans-Eurasian Silk Road gene flow (Identity by Descent: IBD) and recurrent local mutation (Identity by State: IBS) driven by Non-Allelic Homologous Recombination (NAHR) and inter-locus gene conversion between functional GBA1 and its 96%-homologous pseudogene GBAP1 located ~16 kb downstream. Using coalescent haplotype decay theory, where shared linkage disequilibrium (LD) block length L (in Morgans) across g generations follows E[L] ≈ 2/g, we prove that ancient continental migration (g ≈ 50–60 generations, ~1,500 years BP) yields compressed conserved haplotypes of 3.3–4.0 cM around private non-recombinant point mutations, whereas early-modern Edo-period village bottlenecks (g ≈ 10–15 generations) preserve extended 13.3–20.0 cM identity blocks. Second, we derive the exact hierarchical Wahlund decomposition (P(aa) = q̄^2 + Var(q_i) + ∑_i w_i q_i(1-q_i)FIS,i) and Depopulation Amplification Factor (Γ(q, F_t) ≈ 1 + F_t/q ≈ 101x) that elevated Japan's empirical patient incidence (P(aa) ≈ 7.1 × 10^-7 at N=90) two orders of magnitude above the panmictic expectation (q̄^2 ≈ 7.0 × 10^-9). Modeling historical Japanese civil registries (Koseki from Kumage District Ogo-mura and Kuga District Shinjo-mura / Yanai in Yamaguchi Prefecture) as partitioned Directed Acyclic Graphs (DAGs), we prove that social-stratum and marital-sphere boundaries enforced a Markov absorbing state (q_lineage = 0) in genealogically orthogonal samurai-administrative and mercantile lineages (such as the Hasebe and Nishimoto clans), while adjacent isolated demes experienced severe drift (q_k >> q̄) and consanguinity (FIS,k > 0). Finally, we demonstrate the mathematical isomorphism and structural contrast between Ashkenazi Jewish macro-endogamy (N370S) and Japanese micro-deme bottlenecks (L444P, F213I, and private alleles), sealing all genomic, genealogical, and anonymized civic-legislative provenance within the QFS-zkVM architecture over the 64-bit Goldilocks prime field (F_p, p = 2^64 - 2^32 + 1).