Can Metadata Fix the Gauge? Calibration Turns Sparse Multi-Source Learning from Sparse PCA into Sparse Mean Recovery

Source labels tell us where a sample came from, but not how that source oriented its coordinates. In a sparse Gaussian multi-source model with source-specific sign gauges, that distinction changes the learning problem. Without calibration, pooling across sources erases the mean signal, and the source-labelled experiment reduces to sparse PCA. With task-independent relative-gauge metadata, synchronization aligns sources and the same observations become a sparse mean problem. We formalize the value of metadata through aligned mass, which quantifies how much first-order signal a metadata-induced alignment restores. Source labels create only limited aligned mass, whereas valid calibration can create aligned mass proportional to the full sample size. This yields a statistical separation in micro-source regimes and, under standard sparse-PCA hardness assumptions, a computational separation as well. Experiments confirm the predicted phase transition, the failure of invalid metadata, reuse of one calibration graph across many tasks, and controlled real-feature settings with imposed source gauges.