Nested Sieve Structure and Equilibrium Zero Locations on the 6k ±1 Candidate Axis

We study cumulative equilibrium zero locations between the prime numbers and those composite numbers on the 6k ±1 candidate axis that survive sieving below a prescribed prime-factor threshold. Adding the sieve prime p removes precisely the composites whose least prime factor is p. Their number is exactly the scaled rough-number counting function Φ(x/p, p)−1, where prime factors below p are excluded. This finite nesting yields an exact compensation identity between consecutive equilibrium zero locations. The classical rough-number theorem and the prime number theorem then yield the power scale xp = pu∗ +o(1) for the first cumulative zero location, where u∗ω(u∗) = 2 and u∗ = 3.56584662. . .. The removed least-prime-factor mass satisfies pBp(xp)/P (xp) →C∗ = u∗ω(u∗−1) = 2.01292908. . .. These deductions connect discrete sieve transitions with the derivative of the continuous Buchstab balance function. The underlying sieve identities and asymptotic estimates are classical. No new prime-number error bound or controlled formula for gaps between consecutive zero locations is claimed; the contribution is a unified, exactly verifiable presentation of zero-location transitions and their asymptotic scales.