4 Major arcs
The live Chapter-14 major route lower-bounds the mixed major contribution by
It combines the primitive-character error envelope (14.2), the two sides of the (14.7) \(L^2\) estimate, Proposition 10.4.1, the smoothing main-term lower bound (14.10)–(14.14), and the odd singular-series lower bound (14.9).
Trust tier: EF-U. Above the Helfgott–Platt threshold, the primitive-character error term satisfies the explicit envelope used in equation (14.2).
Source: helfgott-ternary-2015, equation 14.2, p. 260 [adapted]
Specialize the proved explicit-formula and zero-sum bounds at the Chapter-14 scale. The external zero computations and finite interval certificates are exposed separately in the trust-boundary appendix.
Trust tier: EF-U. The quadratic major-arc error has the explicit upper bound required by equation (14.7) at the honest threshold.
Source: helfgott-ternary-2015, equation 14.7, p. 261 [adapted]
Insert the primitive error envelope into the character and denominator decomposition, then sum the certified per-modulus bounds.
Trust tier: EF-U. The main major-arc block has the complementary explicit lower estimate used in equation (14.7).
Source: helfgott-ternary-2015, equation 14.7, p. 261 [adapted]
Combine the per-denominator lower blocks, the disjoint-arc integral splitting, and the certified odd Mertens and smoothing-mass bounds.
Trust tier: EF-U. The \(L^2\) estimates imply the explicit Proposition 10.4.1 error budget for the mixed major integral.
Source: helfgott-ternary-2015, Proposition 10.4.1, pp. 213–215 [adapted]
Apply Cauchy–Schwarz to the main/error decomposition and substitute the upper and lower \(L^2\) envelopes.
Trust tier: BT-U. The smoothing main term in equations (14.10)–(14.14) has the required explicit positive lower bound.
Source: helfgott-ternary-2015, equations 14.10–14.14, pp. 261–263 [derived]
Evaluate the windowed smoothing integral and certify the remaining compact numerical inequalities by exact rational enclosures.
Trust tier: EF-U. For odd \(n\), the singular-series factor has the explicit lower bound used in equation (14.9).
Source: helfgott-ternary-2015, equations 14.8–14.9, p. 261 [adapted]
Isolate the factors at \(2\) and \(3\), check the finite prime deficit product, and bound the remaining Euler tail.
Trust tier: EF-U. For odd \(n\) above the finite threshold, the real part of the mixed major integral is at least \((1.058259/49)X^2\).
Source: helfgott-ternary-2015, equation 14.27, p. 267 [weakened]
Combine the (14.7) error envelope, Proposition 10.4.1, the positive main-term bound from (14.10)–(14.14), and the odd singular-series lower bound (14.9).