  I show that the basic structure of symplectic integrators is governed by a
theorem which states {\it precisely}, how symplectic integrators with positive
coefficients cannot be corrected beyond second order. All previous known
results can now be derived quantitatively from this theorem. The theorem
provided sharp bounds on second-order error coefficients explicitly in terms of
factorization coefficients. By saturating these bounds, one can derive
fourth-order algorithms analytically with arbitrary numbers of operators.
