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We present the first detection of mass-dependent galactic spin alignments with local cosmic filaments with >2σ confidence using IFS kinematics. The 3D network of cosmic filaments is reconstructed on Mpc scales across GAlaxy and Mass Assembly fields using the cosmic web extractor DISPERSE. We assign field galaxies from the SAMI survey to their nearest filament segment in 3D and estimate the degree of alignment between SAMI galaxies' kinematic spin axis and their nearest filament in projection. Low-mass galaxies align their spin with their nearest filament while higher mass counterparts are more likely to display an orthogonal orientation. The stellar transition mass from the first trend to the second is bracketed between 1010.4, 1010.9 M⊙, with hints of an increase with filament scale. Consistent signals are found in the Horizon-AGN cosmological hydrodynamic simulation. This supports a scenario of early angular momentum build-up in vorticity rich quadrants around filaments at low stellar mass followed by progressive flip of spins orthogonal to the cosmic filaments through mergers at high stellar mass. Conversely, we show that dark matter only simulations post-processed with a semi-analytical model treatment of galaxy formation struggles to reproduce this alignment signal. This suggests that gas physics is key in enhancing the galaxy-filament alignment.