Progressive transmission of 3D Gaussian Splatting (3DGS) requires each completed transmission stage to be decodable from received data and directly renderable. This work presents ProGS, a progressive codec that organizes anchor-based 3DGS as parent-closed octree prefixes. ProGS combines parent-causal entropy coding, level-balanced anchor growth, bounded multi-prefix training, and lightweight parent-anchor refinement to improve early-prefix quality without altering the complete-model rendering path. One fixed-$λ$ training run yields five deployable rate--quality points from a single bitstream. Experiments on 17 scenes from three datasets evaluate rate--distortion performance, rendering speed, and transmission efficiency against progressive and single-rate baselines. On one representative scene per dataset, ProGS reaches a common quality target with 30.7 $\sim$ 60.7\% fewer bytes than HAC-Rand and 22.6 $\sim$ 53.4\% fewer bytes than HAC++-Rand. Across the three dataset averages, ProGS-LR uses 4.7 $\sim$ 6.1\% fewer bytes than HAC-high while improving SSIM by 0.002 $\sim$ 0.041 and reducing LPIPS by 4.8 $\sim$ 51.2\%. The parent-closed syntax makes every prefix causally decodable and directly renderable without future topology. ProGS-HR also yields higher endpoint SSIM and lower LPIPS than PCGS across all three dataset averages, and all five prefixes render in real time. Code is available at https://github.com/ZhiyeTang/ProGS-Official

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