Proposition 8.4 with corrected constant: $L^2$ estimate off the strongly major arc ($7.4$)

$L^2$ estimate off the strongly major arc (Proposition 8.4, corrected constant). Let $8.7\times10^{36}\le x\le e^{3100}$ be an integer, $N_0=4\times10^{14}$, $T_0=3.29\times10^9$. Then

Why $7.4$ and not the paper's $7.09$. The proof subtracts a lower bound for the strongly-major-arc contribution from the full-circle large-sieve bound $8.001\,(N_0/3)^2x$ (Proposition 4.10). The paper asserts the major-arc lower bound $\int_{\|\alpha\|\le T_0/(3.6\pi x)}|S_{\eta_1,\sqrt x\sharp}|^2\ge 0.92x$ from Corollary 4.9. But Corollary 4.9 is normalised to $\|\eta\|_{L^2}=1$, while $\|\eta_1\|_{L^2}^2=2/3$ by (8.2); applied to $\eta_1/\|\eta_1\|_2$ it gives $\int_{\text{major}}|S_{\eta_1}|^2\ge 0.94\cdot\tfrac23 x\approx0.627x$, not $0.92x$. This is consistent with the Plancherel heuristic: on the major arc $S_{\eta_1}(x,\alpha)\approx x\hat\eta_1(\alpha x)$, so $\int_{\text{major}}|S_{\eta_1}|^2\approx x\|\hat\eta_1\|_{L^2}^2=x\|\eta_1\|_{L^2}^2=\tfrac23x$ — the major arc cannot carry $0.92x$ of $L^2$ mass. With the lower bound $0.627x\cdot(1-10^{-10})^2(N_0/3)^2$ from (8.13), the difference is $8.001-0.6266<7.4$.

The theorem is unaffected: the companion $L^\infty$ estimate minor_arc_Linfty_corrected holds with constant $0.05$ in place of the paper's $0.078$ (the paper's regime boundary $\alpha\ge20/N_0$ was not optimised), and $0.05\times7.4=0.37<\tfrac23-0.1$. This node replaces the retired minor_arc_L2 (constant $7.09$), which the analysis above indicates is not provable as stated.