arXiv:2607.26797v2 Announce Type: replace Abstract: We establish sharp strong error estimates for a fully discrete approximation of semilinear stochastic convection--diffusion equations driven by additive noise subject to nonsmooth initial data. The problem is discretized in space using a standard Galerkin finite element method and in time via the linearly implicit Euler scheme. By decoupling the noise regularity index $\beta \in (0, 2]$ from the initial data regularity index $\mu \in (0, 2]$, we derive time-weighted spatial and temporal regularity bounds that isolate the initial singularity at $t = 0$ into an integrable temporal weight. By allowing the drift operator $F$ to map into negative-order fractional spaces to incorporate the convection term, we prove sharp strong convergence rates of order $O(h^\beta + k^{\beta/2})$ for $\beta \in (0, 2)$ and $O\bigl(h^2 + k(1+\ln(t_n/k))^{1/2}\bigr)$ at the critical endpoint $\beta = 2$ for any fixed $t_n > 0$, without requiring the regularity of the initial data to match that of the noise. Furthermore, these convergence rates are achieved under a relaxed differentiability condition requiring only a Lipschitz continuous first G\^ateaux derivative of $F$. Numerical experiments validate the theoretical results across both trace-class and space-time white noise regimes.