Movement is decodable from nearly every region of the mouse brain — but do distant regions share one low-dimensional global state, or do they develop region-specific geometries that merely carry the same scalar? DECOMP reframes this as geometry, not decodability, and asks how much of the shared subspace between mouse V1, M1, and cerebellum (CB) reflects a common population code versus parallel codes.
Across 15 dual-region IBL Brain-Wide Map recordings, we combine ridge-GLMs, SVCA, partial CCA with Frisch–Waugh–Lovell partialling against behavioral covariates, and lag plus peri-event analyses. All three pairs retain 0.69–0.85 of their shared geometry after partialling, but temporal structure dissociates them: V1↔CB is zero-lag with flat task responses (global state); V1↔M1 is zero-lag and task-locked (shared cognitive state); CB↔M1 shows a ~40ms cerebellar lead locked to motor events (cerebellothalamocortical communication). Scalar survival ratios are necessary but not sufficient — dynamics expose three mechanisms hidden behind nearly identical statistics.