On the minor arcs, should behave like a 'random' sum and hence be small, but proving this rigorously with strong enough exponents was the main obstruction to lowering the threshold below . Helfgott developed sharper, 'log-free' bounds — meaning the bound on depends only on (the denominator of the nearby rational) without extra factors of creeping in from the usual decomposition of via Vaughan's identity, combined with large-sieve-type inequalities and a technique adapted from the Barban–Vehov–Graham smoothing sieve. These sharper bounds are precisely what let the minor-arc error stay small enough for as low as , rather than requiring astronomically larger .
Because major and minor arcs behave differently at the boundary between them, Helfgott further introduces 'intermediate' arcs — tails where the standard major/minor treatment loses efficiency — with their own tailored estimates. Putting the major-arc main term (Step 3) together with these minor- and intermediate-arc error bounds (all fully explicit, with every implied constant tracked numerically) yields the unconditional statement: , and hence is a sum of three primes, for every odd .