Observable-Map Invariance in Transported Linear Growth Manifolds

This work investigates the dependence of rank-1 compression and geometric transport in linear cosmological growth observables on the choice of observable projection. Starting from the same underlying linear growth equation, we compare distinct observable maps, including the growth suppression ratio and the normalized growth-rate ratio. We show that although compression strength (eigenvalue hierarchy) varies moderately across projections, effective one-dimensional structure and background-locked principal-direction drift are preserved. Across representative examples, both observables exhibit global rank-1 dominance (PC1 ≳ 99.6%) and smooth principal-direction transport with monotonic correlation to background evolution. Differences appear only in the magnitude of the subdominant mode, demonstrating that compression intensity is projection-dependent while structural dimensionality is not. These results establish that geometric transport of low-dimensional growth manifolds is a property of the underlying dynamical system rather than of a particular observable representation. This paper complements companion works on structural compression and transport in linear growth dynamics.