= Solution
Let the univariate <linear precision> sites be $\xi_j,\eta_k$, meaning
$$
\sum_j\phi_j(u)=1,\quad\sum_j\xi_j\phi_j(u)=u,\qquad\sum_k\psi_k(v)=1,\quad\sum_k\eta_k\psi_k(v)=v.
$$
Then their Cartesian-product grid is a <linear precision> set for the <tensor-product surface basis>:
$$
\sum_{j,k}(\xi_j,\eta_k)\phi_j(u)\psi_k(v)=(u,v).
$$
Indeed, the first coordinate is $(\sum_j\xi_j\phi_j)(\sum_k\psi_k)=u$, and the second is $(\sum_j\phi_j)(\sum_k\eta_k\psi_k)=v$. More generally, take controls $P_{jk}=A+\xi_jB+\eta_kC$. The resulting <parametric surface> is exactly
$$
\boxed{P(u,v)=A+uB+vC.}
$$
This proves <linear precision of a geometric basis> and <affine map> reproduction, including arbitrary changes of the embedding plane. In fact the products also reproduce the bilinear coordinate $uv$, since $\sum_{j,k}\xi_j\eta_k\phi_j\psi_k=uv$; affine precision is the requested consequence.
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