algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety algebra of motivated cycles; standard conjectures; tannakian category; motivic Galois group; Hodge conjecture; Tate conjecture; algebraicity of the Lefschetz involution; base pieces; motivated \(E\)-correspondences; algebraic gerb; motivic cohomology; Hodge conjecture for abelian varieties; motif with integer coefficients; motives in characteristic \(p\); numerical equivalence coincides with homological equivalence 