Arithmetic Geometry and Birational Analysis of the G2 Fibre, and the Twin-Elliptic Perfect Gate: Each Coordinate Fibre Is a Genus-1 Curve and With the Perfect Gate It Is a Genus-5 Curve

P8: each G2 coordinate fibre is a smooth genus-1 quartic birational to E_r: Y² = (X + 706)(X² − 202500) and E_s: Y² = (X + 2657)(X² − 2458624); G2 maps to non-torsion points (torsion Z/2 × Z/4), so the rank is ≥ 1 and each fibre has infinitely many structural points. P9: imposing the open face as well gives a genus-5 curve for every rational ρ ≠ 1 (Jacobian ~ five elliptic factors), hence by Faltings (ineffective) only finitely many perfect points per fibre; 30,397,485 reduced rationals (denominator ≤ 10000) per fibre give only the 2 known structural points and 0 perfect points.