Unique Determination of the Lam\'e Parameters in Three Dimensions

We prove that the full Dirichlet-to-Neumann map for static isotropicelasticity uniquely determines both smooth Lam\'e parameters in abounded smooth domain of \( R^3\) under strong convexity.The proof uses an exact four-component Laplace-type reduction, but itsdecisive identities are obtained from physical elastic solutions.The characteristic quotient of the reduced system has rank three,whereas the leading physical adjoint lift supplies only two directions.We construct the missing polarization from the first nonvanishingcorrection of a reduced amplitude in the kernel of that lift.Paired with fixed primal and adjoint quotient frames, the resultingcomplex geometrical optics solutions yield a complete family ofholomorphically weighted \(3×3\) matrix moments through thephysical Betti identity. Cauchy--Plemelj factorization converts thesemoments into an invertible relative transport normalized in the commonexterior. Holomorphic dependence on the projective null direction andtransport rigidity then determine a material ratio and the shearmodulus, hence both Lam\'e parameters. No smallness, symmetry,separability, or analyticity hypothesis is imposed.