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The upper-bounded (lower-bounded) interaction acting upon Fock subspaces ranging from $\\left\\vert 0\\right\\rangle $ to $\\left\\vert M\\right\\rangle $ ($\\left\\vert N\\right\\rangle $ to$\\ \\infty$), and the sliced one confined to Fock subspace ranging from $\\left\\vert M\\right\\rangle $ to $\\left\\vert N\\right\\rangle $, whatever $M<N$. 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