The Hidden Structure of the Euro System

This paper identifies a hidden mathematical structure in the euro system. It shows that the fifteen denominations across five layers satisfy a linear Diophantine system N = pA + qB + rC + sD + tE with parameters (19, 9, 3, 3, 1), and that the conditions p ≡ 1 (mod q) and t | s | r | q guarantee a single law: A₀ = dr(N). The digital root is the compass of every valid denomination construction, computable in one step for any amount.