In this work, we propose a novel continuous-variable source-independent quantum random number generator (CV-SI-QRNG) and establish its security through a semidefinite programming (SDP) approach. We also make a finite-size seurity analysis against general attacks by using concentration inequalities for correlated random variables. Unlike previous approaches that rely on the entropic uncertainty relation between two conjugate observables such as Pauli X and Z measurement, our protocol eliminates this requirement by using a single phase-insensitive detector, significantly simplifying the experimental setup. Furthermore, we address a key challenge in continuous-variable quantum protocols: the reliance on photon number cutoff assumptions in numerical methods for security analysis. These cutoff assumptions, while commonly used, are often non-rigorous and can introduce inaccuracies in the security proof, particularly when the system operates in high photon number regimes. To overcome this limitation, we introduce a dimension reduction technique enables us to perform a precise and reliable security analysis without the need for arbitrary cutoff assumptions. By leveraging SDP, our method transforms the security analysis into an optimization problem that can be solved efficiently, providing strong security guarantees for the CV-SI-QRNG protocol. These innovations enhance both the practicality and rigor of continuous-variable quantum random number generation, making it a promising approach for secure cryptographic applications.