A Geometric Determinant–Dispersion Proof of the Friedlander–Iwaniec Theorem on Primes of the Form \(x^2+y^4\)

We give a direct geometric proof that the polynomial $x^2+y^4$ represents infinitely many primes. The basic embedding\[z=x+i y^2,\qquad \mathrm{N} z=x^2+y^4,\]places the problem in the Gaussian lattice. If $z=wv$ with $w=a+ib$ and $v=c+id$, then fixing the square height $\operatorname{Im}(wv)=y^2$ cuts the cofactor plane by the affine line $bc+ad=y^2$. Pairing two Gaussian divisor rows therefore means intersecting two such line families. Their oriented area\[D(w_0,w_1)=\operatorname{Im}(w_0\overline{w_1})\]is simultaneously the crossing determinant, the Jacobian of the cofactor reconstruction, and the modulus governing the oscillatory congruence. The degenerate locus $D=0$ is the parallel ridge; the transverse locus $D\ne0$ is treated by determinant-frequency dispersion. A second $TT^*$ step resolves common and relative prime-power depth, and an asymmetric Vaughan decomposition places the bilinear range away from the parallel ridge. For fixed non-negative weights $W_1,W_2\in C_c^\infty((0,\infty))$ we prove, for every $A>0$,\[\sum_{x,y\in\mathbb Z}W_1(x/X^2)W_2(y/X)\Lambda(x^2+y^4)=\frac{4}{\pi}I_WX^3+O_{A,W}\!\left(X^3(\log X)^{-A}\right),\]where\[I_W=\left(\int W_1\right)\left(\int W_2\right).\]The proof does not invoke the asymptotic-sieve theorem used in the original Friedlander--Iwaniec proof. The only non-elementary input specific to Gaussian integers is a Siegel--Walfisz estimate for smooth Gaussian M"obius sums with polylogarithmic local conductor.