Base, found, ground, bottom, stay, rest are comparable when they mean to supply or to serve as a basis.
Base now rarely suggests a material support upon which a material superstructure is built, but rather something material or immaterial that underlies a nonmaterial superstructure (as a belief, a system, a judgment, a hope or an action).
Found comes so close to base as often to be indistinguishable from it and, therefore, to be interchangeable with it.
Often, however, it suggests not what merely underlies but what is consciously advanced as support (as for an opinion, a principle, a judgment, a belief, or an affection).
Ground denotes an implanting (as into the earth) that gives solidity and firmness; it may apply to something (as personal virtue, education, or an institution) which can grow and thrive only when it is firmly based (as if by deep roots).
But ground may be used, less strictly, in a sense approaching that of base and found.
Bottom implies a broad or strong base.
Stay implies a support that keeps upright or prevents from falling and may suggest adding a supplementary support to correct an observed or anticipated tendency.
Rest stresses reliance or dependence on something as a base or fundamental support.