A Double Swiss-Cheese Calabi-Yau with a Three-Valley Kähler Sector: Construction, Moduli Roles, and an Integrality Bound on the Flux Window

We work out a specific Calabi–Yau threefold from the Kreuzer–Skarke database, (h¹¹, h²¹) = (3, 165), χ = −324, whose double Swiss-cheese geometry supports a Large Volume Scenario with a three-valley Kähler sector. For flux data in the window |W₀| ~ 10⁻² and integer gauge ranks (14, 10), the construction yields fibre inflation with nₛ ≈ 0.967 and unobservably small tensors (r ~ 5×10⁻¹¹), a three-valley racetrack with a volume-independent energy-splitting ratio, and right-handed neutrino masses compatible with thermal or resonant leptogenesis.