Multiplicity Thresholds for Minimal Vanishing Sums of Roots of Unity

How many terms must a minimal vanishing sum of roots of unity contain before one root can occur with a prescribed multiplicity? Principal results Analytical proofs give the first thresholds for multiplicities two through five: 21, 33, 44, and 56. The weight-21 equality case has relative order 210 and two possible fiber-weight patterns. The weight-to-multiplicity ratio converges to a constant strictly greater than 64/7 and less than 15.165. Explicit elevenfold circuits give the upper bound. Finite rational lower optimizations and explicit circuit upper bounds characterize the optimal coefficient and prove that it can be approximated to any prescribed accuracy. Deviations below the limiting linear trend are unbounded. Methods and scope The proofs combine prime decomposition, integer layers, cycle structure, triangle contraction, rational circuits, and the Conway-Jones conductor bound. Efficient circuits extracted from large minimal sums can be amplified using fresh primes at negligible relative cost. All principal results have analytical proofs; exact programs provide independent checks. A sharp numerical evaluation or closed form for the limiting coefficient, its arithmetic nature, finer fluctuations, and exact thresholds beyond multiplicity five remain undetermined.