Approaching cubic Diophantine statistics via mean-value $L$-function conjectures of Random Matrix Theory type

Fix a smooth cubic form $F/mathbb{Q}$ in $6$ variables. For $N_F(X):=#{boldsymbol{x}in[-X,X]^6:F(boldsymbol{x})=0}$, the randomness prediction $N_F(X)=(c_text{HL}+o(1))cdot X^3$ as $Xtoinfty$ of Hardy-Littlewood may fail. Nonetheless, Hooley suggested a modified prediction accounting for special structured loci on the projective variety $V:= V(F)subseteqmathbb{P}^5_mathbb{Q}$. A weighted version of $N_F(X)$ essentially decomposes as a sum of adelic data over hyperplane sections $V_{boldsymbol{c}}subseteq V$, generically with nonzero discriminant $D(boldsymbol{c})$. Assuming standard conjectures for the Hasse-Weil $L$-functions $L(s,V_{boldsymbol{c}})$ over ${boldsymbol{c}inmathbb{Z}^6:D(boldsymbol{c}) eq0}$, Hooley proved the bound $N_F(X) = O_epsilon(X^{3+epsilon})$, essentially for any given diagonal $F$. Now assume (1) standard conjectures for each $L(s,V_{boldsymbol{c}})$, for certain tensor $L$-functions thereof, and for $L(s,V)$; (2) standard predictions (of Random Matrix Theory type) for the mean values of $1/L(s)$ and $1/L(s_1)L(s_2)$ over certain geometric families; (3) a quantitative form of the Square-free Sieve Conjecture for $D$; and (4) an effective bound on the local variation (in $boldsymbol{c}$) of the local factors $L_p(s,V_{boldsymbol{c}})$, in the spirit of Krasners lemma. Under (1)-(4), we establish (away from the Hessian of $F$) a weighted, localized version of Hooleys prediction for diagonal $F$ -- and hence the Hasse principle for $V/mathbb{Q}$. Still under (1)-(4), we conclude that asymptotically $100%$ of integers $a otin{4,5}bmod{9}$ lie in ${x^3+y^3+z^3:x,y,zinmathbb{Z}}$ -- and a positive fraction lie in ${x^3+y^3+z^3:x,y,zinmathbb{Z}_{geq0}}$.