The stabilised modulus on de Sitter branes: curvature response, the destabilisation threshold and the moving radion

We compute the radion spectrum of a Goldberger–Wise-stabilised two-brane background with de Sitter slices and locate the loss of linear stability directly. For the reference model with infrared tension fixed, the mass-zero boundary occurs at H_J/m₀ ≃ 0.292; the numerically located boundaries among the 37 crossing scan models span 0.217–0.294 in the same frame. Changing the deformation family shifts the boundary by approximately a factor of two in the tested comparison. The local curvature expansion at the reference model gives c = −9.7916 and c₄ = −376.3, while the fixed-separation probe limit recovers m² = −4H². Exponential growth begins when the mass squared becomes negative; 9H²/4 separates oscillatory from non-oscillatory decay. A reduced four-dimensional action also exhibits a vanishing modulus mass at kJ* ≈ 6.661. Over one reference period, a homogeneous five-dimensional comparison gives reduced-model separation and velocity differences of 0.588% and 0.779% under the specified slice-length map.