Casimir effect for a gapped scalar field: Exact crossover at the Compton scale

We study the Casimir effect for a scalar field with a mass gap Δ between two parallel plates with Dirichlet boundary conditions. Starting from the exact formula of Toms (1980), we derive the exact pressure P(d) and analyze its asymptotic behavior. We identify the crossover separation d* as the position of the minimum of the ratio R(d) = P(d)/P_massless(d). This definition is unambiguous and requires no arbitrary threshold. We prove that d* is proportional to the reduced Compton length λc = ℏc/Δ, with a pure-number coefficient c₀ = 0.834330. At this point, the minimum ratio is R(d*) = 0.656442. These numbers are independent of the gap value and of any material parameter. We explicitly derive the Toms formula from the divergent zero-point energy using zeta-function regularization and the Abel-Plana formula. We compare the result with the massless Casimir effect, the exponential suppression at large distances, and the Ambjørn-Wolfram derivation. We state three strong falsifiable predictions and propose a concrete experimental realization in a Bose-Einstein condensate. We provide a fully reproducible Python script that confirms the numerical values to machine precision.