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Deep watermarking methods often share similar encoder-decoder architectures, yet differ substantially in their functional behaviors. We propose DiM, a new multi-dimensional watermarking framework that formulates watermarking as a dimension-aware mapping problem, thereby unifying existing watermarking methods at the functional level. Under DiM, watermark information is modeled as payloads of different dimensionalities, including one-dimensional binary messages, two-dimensional spatial masks, and three-dimensional spatiotemporal structures. We find that the dimensional configuration of embedding and extraction largely determines the resulting watermarking behavior. Same-dimensional mappings preserve payload structure and support fine-grained control, while cross-dimensional mappings enable spatial or spatiotemporal localization. We instantiate DiM in the video domain, where spatiotemporal representations enable a broader set of dimension mappings. Experiments demonstrate that varying only the embedding and extraction dimensions, without architectural changes, leads to different watermarking capabilities, including spatiotemporal tamper localization, local embedding control, and recovery of temporal order under frame disruptions.