Methods based on implicit neural representations have demonstrated superior performance in Screen Content Image Super-Resolution (SCISR) . However, they overlooked the inherent frequency characteristics, leading to suboptimal performance. We propose a frequency decoupled framework (FDF) that rethinks SCISR from a phasor perspective by capturing structured energy in amplitude and relational continuity in phase, and jointly exploiting them with bespoke implicit representations to faithfully recover the regular textures and global configuration of Screen Content Image (SCI). Amplitude-Phase Factorization Network (APFN) first separates images into amplitude and phase streams, where Amplitude Clustering Module (ACM) organizes sparse yet high-energy amplitude responses into representative prototypes for periodic pattern extraction, while Phase Consistency Self-Attention (PCSA) progressively reinforces configuration through continuous consistency propagation. And Oscillation-Anharmonic Implicit Fitting Network (OAIF-Net) integrates periodic and coherent implicit representations for efficient exploitation of the periodic patterns and coherent context embedded in SCI. Experimental results show FDF achieves state-of-the-art SCISR performance at multiple scales across four public SCI datasets. Ablation experiments further demonstrate the effectiveness of each component in extracting and exploiting periodic patterns and coherent context.