The Generic Three-Family Elliptic Decomposition, the Möbius Gate-Swap Automorphism and the Hyperelliptic E_0 Bridge

R5: via binary-quartic invariants the five factors collapse to three families, Jac(C_n) ~ E_f² × E_0² × E_−, which at n = 7/32 match the previously certified rank-one curves (each rank 1 is cited from earlier bundles, not derived here), so rank = genus = 5 and ordinary Chabauty does not apply. R6: finds the Möbius gate-swap involution α on the fibre, an explicit order-32 symmetry subgroup, a canonical genus-2 quotient G_n with Jac ~ E_f × E_0 and the family E_0′(N), which at n = 7/32 is the historical curve Y² = X(X − 448²)(X + 1073²). R7: puts G_n in even-sextic form H² = 2(t⁴ − (N−1)⁴)((N² + 6N + 1)t² − (N−1)⁴), identifies both elliptic quotients explicitly, and at n = 7/32 obtains Y² = 2(T⁴ − 1)(1352033T² − 950625) mapping to the historical E_0′.