Elliptic Orbit Generation, the Local Mordell–Weil Sieve and the Five-Elliptic Decomposition of the Fixed-k Genus-5 Closure Curve

The first gate at k = 7/32 is an elliptic curve with torsion Z/4 × Z/2 and a seed point P of infinite order; all 200 positive gate-1 states from nP + T (n ≤ 100) fail gate 2, and a local sieve forces n to be divisible by 19,070,251,200 (resp. 12,700,800) in the two orbit families; the V4 symmetry of the genus-5 curve gives Jac(C_k) ~ E_f² × E_0 × E_+ × E_− over Q, through explicit genus-1 quotients and the genus-3 quotient H_k.