For \(u\in[-1,1]\), set \(v(u)=(1,u,u^2)^{\mathsf T}\). Its convex hull is
Read the diagram in words
The normalized domain lies above the parabola and below the horizontal upper boundary. The three-dimensional wireframe shows its homogeneous geometry. Within the supplied parameter family, uncertainty changes a boundary posterior’s probability without moving its ray; arbitrary coordinate perturbations do not share that protection.
Choose a reference probability measure on \([-1,1]\) with moments \(\mathbb E U=\mathbb E U^3=0\), \(\mathbb E U^2=1/3\) and \(\mathbb E U^4=1/5\). With \(L_2(u)=(3u^2-1)/2\), define
Its normalized-state density is \(p_{a,b}(u\mid x,y)=1+axu+byL_2(u)\). If \(|a|+|b|\leq1-\rho\) for \(\rho\in[0,1]\), it is at least \(\rho\), and integrated mass is exact.
The mathematical point
Every positive-density posterior is the same boundary point \(v(u)\) throughout this parameter family. Uncertainty changes its probability without moving its posterior ray.
Scope
This protection depends on the supplied algebraic family. Free perturbations of output coordinates can leave the cone. The three-dimensional wireframe shows the homogeneous geometry, not sampled evidence.
Formal route
CurvedMomentGeometry.lean; CurvedMomentInstrument.lean; exact P83 map in COVERAGE.json.