BPB/MBE Paper VIII v1.16 — Rational HR Branch, Square Closure, 2-adic/3-adic Descent, and Critical-Completion Ledger

Paper VIII of the BPB/MBE series presents the full theory/dictionary closure of the rational Hassan–Rosen anchored BPB/MBE branch. The manuscript preserves the exact rational branch structure, square-closure identities, integer-tower elliptic-curve filter, rational-tower height minimality, the 2-adic/3-adic descent theorem, and the guarded critical-completion ledger. The 20/9 value is recorded as a spectral plus conditional Schur critical-completion value, while the fully constrained action coefficient coef_total(Q_D^2) remains explicitly open. Paper IX and later full-nuisance likelihood work are intentionally excluded from this bundle.