Sun's conjecture on a certain ternary quadratic form

We establish an asymptotic form of Sun's conjecture concerning representations by the ternary quadratic polynomial \[ P(x,y,z)=x(5x+1)+\dfrac{y(5y+1)}{2}+\dfrac{z(5z+1)}{2}. \] More precisely, we prove that every sufficiently large integer is represented by $P$, with an effectively computable threshold. The proof begins by reducing the representation problem to that of a positive definite ternary quadratic form. We then study its genus and associated theta series, using explicit representation identities to determine the Shimura lift of the relevant weight-$3/2$ cusp form. Through Waldspurger's formula, this allows us to relate its Fourier coefficients to central values of quadratic twists of a weight-two modular form. Finally, effective lower bounds for class numbers, together with subconvexity estimates, show that the cuspidal contribution is dominated by the genus average for all sufficiently large integers, yielding the desired result.