3 The signed smoothed circle method

Fourier orthogonality identifies the weighted von-Mangoldt triple count with its circle integral. After splitting the circle into major and minor arcs, the inequalities

\[ \operatorname {Re} I_{\mathrm{major}}\ge c_1, \qquad \lVert I_{\mathrm{minor}}\rVert \le c_2, \qquad T{\lt}c_1-c_2 \]

leave positive mass after every term involving a proper prime power has been removed. A positive remaining summand is a genuine prime triple.

Theorem 3.1 Smoothed Fourier identity

✓

Trust tier: BT-U. The weighted von-Mangoldt triple sum equals the integral over \([0,1]\) of the cube of its smoothed exponential sum times \(e(-n\alpha )\).

Proof ▶

Expand the finite cube, interchange the finite sum and integral, and apply additive-character orthogonality. Only triples whose sum is \(n\) survive.

Theorem 3.2 Signed mixed-tsum Fourier identity

✓

Trust tier: BT-U. For three absolutely summable von-Mangoldt weights, the mixed infinite triple sum is the corresponding circle integral.

Proof ▶

Absolute summability justifies the iterated sum/integral interchanges. Fourier orthogonality then selects the equation \(a+b+c=n\).

Definition 3.3 Signed major/minor/tail endpoint

✓

Trust tier: DEF. Definition of the per-target proposition asserting the existence of three summable weights and a measurable major-arc set with a major lower bound \(c_1\), minor upper bound \(c_2\), prime-power tail \(T\), and \(T{\lt}c_1-c_2\). This node defines the endpoint contract; it does not prove an instance.

Theorem 3.4 Signed circle-method extraction

✓

Trust tier: BT-C. A signed mixed Fourier identity, a positive major-minus-minor gap, and a smaller prime-power tail produce a three-prime representation.

Proof ▶

Split the circle integral into the major set and its complement. The triangle inequality gives a positive von-Mangoldt contribution. Removing all prime-power terms still leaves positive mass, hence one genuine prime triple.

Theorem 3.5 Endpoint yields three primes

✓

Trust tier: BT-C. ‘SmoothedMixedTsumAnalyticEndpoint n‘ implies that \(n\) is a sum of three primes.

Proof ▶

Project the summability, major, minor, tail, and strict-gap witnesses from the endpoint and apply the signed circle-method extraction theorem.

Theorem 3.6 Finite-sum smoothed extraction

✓

Trust tier: BT-C. The finite-sum ‘SmoothedAnalyticEndpoint n‘ also implies a three-prime representation.

Proof ▶

The major/minor lower bound makes the smoothed von-Mangoldt count dominate the explicit prime-power tail; positivity then yields a prime triple.