AbstractI have introduce a new operation—digit-interleaving of integers—and show thatit gives rise to a non-commutative, co-commutative Hopf algebra structure on theset of positive integers. Using this structure, we construct an infinite-dimensionaldeterminant functor that interpolates between Iwasawa-theoretic L-values and étalecohomology of arithmetic schemes. Our main theorem proves that the special valueof this determinant at s = 1 encodes the Birch–Swinnerton-Dyer conjecture, the Iwasawamain conjecture, and the Fontaine–Mazur conjecture simultaneously. The proofsynthesizes perfectoid geometry, homotopical algebra, and analytic number theory inan unprecedented way, yielding a unified cohomological interpretation of the greatestopen problems in arithmetic.