Layer-dependent spin-resolved electronic structure of ferromagnetic triple-layered ruthenate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Sr</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msub><mml:mi>Ru</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi mathv

High-resolution angle- and spin-resolved photoemission spectroscopy (ARPES) of the triple-layered ruthenate Sr$_4$Ru$_3$O$_{10}$ reveals features of the electronic structure that extend our understanding of the layered strontium ruthenates. The spectra near the Fermi energy are very different from the non-magnetic analogues Sr$_2$RuO$_4$ and Sr$_3$Ru$_2$O$_7$ with distinct Fermi surfaces for wide electron-like minority spin bands around the zone center and narrow hole-like majority spin Fermi surface contours around the zone corners. The most dramatic results are two narrow spectral peaks $\sim$30 meV below the Fermi-level, a spin-minority hole-like band at the Brillouin zone center, and a spin-majority saddle-band van Hove singularity at the zone edge, which exhibits almost 100\% spin-polarization at low temperature, and a strong temperature dependent coherence-incoherence crossover attributed to Hund metal correlations. Quantitative comparison of the ARPES to spin-polarized density functional theory (DFT) calculations identify the specific antibonding and nonbonding orbital origins of the narrow bands, with a prediction of different spatial localization in the central and outer layers. This is shown to be consistent with experimental ARPES multi-zone matrix element intensity variations, and implicates outer-layer-specific control of the in-plane metamagnetism. The renormalization of the bands relative to the mean-field DFT, the demonstration of spin-polarized oxygen bands, and of spin-minority and spin-majority band-crossing hybridization, provide a more complete picture of the magnetism which displays aspects of both delocalized and local moment behavior.