Tail-projection certificates for cyclotomic minimum-degree polynomials beyond the reciprocal condition

For an odd squarefree integer n=pq_1...q_k with k at least 3, this paper gives an explicit divisor-residue sufficient condition under which the minimum degree of a nonzero nonnegative real multiple of the n-th cyclotomic polynomial is (p-1)q_1...q_k, with equality only for positive scalar multiples of the regular p-gon polynomial. For every fixed k, the condition yields infinitely many clustered prime tuples outside Steinberger's reciprocal sufficient condition. Status: Public Beta; internally verified candidate proof; external review pending.