Explicit odd-order sign matrices with condition numbers tending to one

For each odd prime $p\ge13$, an explicit real sign matrix $A_p$ of order $p^2$ is built as a convolution matrix over the additive group $\mathbb F_p^2$. Its kernel is the quadratic character of the anisotropic form $x^2-dy^2$, with $d$ a nonsquare, with the origin filled and with the signs flipped on the quadratic-residue points of a parabola on which the character is constant. An exact Gauss-sum computation shows that the unmodified kernel has all nontrivial Fourier coefficients of modulus $p$, the classical mixed Weil bound controls the flips, and the zero frequency is corrected exactly. Hence $\kappa_2(A_p)\le(p+2+3\sqrt p)/(p-2-3\sqrt p)=1+O(n^{-1/4})$ with $n=p^2$, and $\kappa_2(A_p)\le210/151<\sqrt2$ for every prime $p\ge361$. This gives an explicit infinite family of odd-order approximate Hadamard matrices; no search or randomness is used.