We study the optimal placement of k ordered points on the unit interval for the bounded pair potential K_q(d)=e^(-d^q), q>0. The family interpolates between strongly cusp-like kernels for 0<q<1, the threshold kernel e⁻ᵈ, and the flatter Gaussian-type regime q>1. Our emphasis is on the transition from collision-free minimizers to endpoint-collapsed minimizers. We reformulate the problem in gap variables, record convexity, symmetry, and the Karush-Kuhn-Tucker conditions, and give a short proof that collisions are impossible for 0<q<1. At the threshold q=1 we recover the endpoint-clustering law for e⁻ᵈ, while for q>1 we identify critical exponents qₖ beyond which interior points are no longer optimal. For odd k we derive the exact universal value q₂ₘ₊₁ = (log(1/(-log((1+e⁻¹)/2))))/(log 2) ≈ 1.396363475, and for even k=4,6,…,20 we compute the numerical transition values beginaligned q₄≈ 1.062682507, q₆≈ 1.155601329, q₈≈ 1.206132611, q₁₀≈ 1.238523533, q₁₂≈ 1.261308114, q₁₄≈ 1.278305167, q₁₆≈ 1.291510874, q₁₈≈ 1.302082885, q₂₀≈ 1.310744185. endaligned We also include comparison tables and diagrams for the kernels e^(-√d), e⁻ᵈ, and e^(-d²), briefly relate the bounded family to the singular Riesz kernel d⁻ˢ, and identify the q→ 0⁺ limit with the Fekete/Chebyshev–Lobatto configuration on [0,1].