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A primary bottleneck in contact-rich manipulation is the difficulty of collecting real-world data. Sim-to-real reinforcement learning offers a scalable alternative, but the simulation-reality gap prevents information-dense modalities like touch from being effectively used. Existing sim-to-real methods often mitigate this gap by simplifying tactile data into coarse low-dimensional features -- sacrificing the richness required for complex manipulation. In this work, we introduce Center-of-Pressure (CoP), an effective tactile representation grounded in physical principles that preserves dense contact information while maintaining robustness for sim-to-real transfer. To support this representation, we propose a sensor calibration scheme based on differentiable dynamics, enabling the estimation of taxel orientations without requiring ground-truth force measurements. We evaluate CoP on two blind, challenging contact-rich manipulation tasks: peg-in-hole insertion and ball balancing. Across both tasks, policies conditioned on CoP achieve zero-shot sim-to-real transfer on a multi-fingered hand, and outperform both coarse binary-contact and raw-taxel baselines. Analysis of learned policy states further suggests that CoP-conditioned policies encode task-relevant physical properties, such as object mass, as an emergent byproduct of control.