The Square-Lift Tower and Quadratic-Chabauty Admissibility: From the Fixed-k Genus-5 Curve to a Rational Conic, and the Genus 0 → 1 → 3 → 5 Rank–Genus Locking

The symmetric quotient of the fixed-slice curve is a rational conic; C_{7/32} has exactly 16 known boundary points, only 16 points mod p = 11, 19, 31, and satisfies the quadratic-Chabauty rank inequality r = 5 < g + ρ − 1 (≥ 11) — but no Chabauty computation is performed; the curve is a tower of quadratic lifts C0 (conic) ← C1 (E_0′) ← C2 (genus 3) ← C3 (genus 5, unramified) with Jacobian ranks 0, 1, 3, 5 equal to the genera, and the genus-3 layer already meets the quadratic-Chabauty criterion.