Rigidity of integral Gram representations in quadratic orders, with an application to Pythagoras numbers of biquadratic fields

We prove thatP( OQ(√3,√ s))≥6for every squarefree integer $s>14$ with 3 s. The proof develops a rigidity principle for Gram representations in quadratic orders. Let R be a Dedekind domain of characteristic different from $2$ and let S be a finite locally free R-algebra of rank two. Multiplication induces μS:ΓR²(S)→ S. Writing tS=TrS/R(S), we proveμS(R²S)⊗2⊗R tS⁻¹.At split real places, positive semidefiniteness confines each fiber parameter to a compact interval; uniqueness among the integral parameters then preserves Gram length. Over Z[√3], we use a fixed symmetric binary form of Gram length six and give a self-contained proof of its length. Suitable generators in the three classes s≡1,2,34 reduce the application to one-dimensional fiber bounds. Explicit test vectors isolate the zero integral parameter in the odd cases; in the even case, a parity restriction modulo $2$ is combined with the real-place bounds. Dombek's finite verifications cover the remaining small values of s.