Goldbach’s ternary (weak) conjecture asserts that every odd integer n≥7 is the sum of three primes. Its unconditional proof for all n≥5 was established by Helfgott [8] using the Hardy–Littlewood circle method [7] with an extensive computational verification for small n. In the unrestricted ternary problem one counts all triples (p1,p2,p3) with p1+p2+p3=n; no congruence restriction is placed on any summand.

The present paper studies the restricted variant, in which the first prime summand is required to lie in a fixed arithmetic progression:

This restriction is not covered by Helfgott’s theorem: requiring p1 to lie in a specific residue class modulo q introduces Dirichlet characters and the zero-distribution of L-functions in an essential way. The qualitative almost-all theory for Wa,q(n) was developed by Mikawa 12 and Zhan 18 (see also Montgomery–Vaughan 13), but neither paper provides explicit constants or tracks the dependence on q.

The purpose of this paper is to:

give a completely explicit almost-all theorem for Wa,q(n) with effective constants derived from the binary theory;

supply a complete, self-contained proof of the minor-arc bound Kmin(q,A)≤2.10/φ(q) via the Hölder (L2,L∞,L2) factorization, closing the sole external dependency that appeared in earlier versions of this paper;

locate precisely those steps where the argument is complete and those where a gap remains;

state clearly what the result says versus what Helfgott’s theorem says.

This paper is the third in the Anderson series [1] establishes the unconditional almost-all theorem for the binary restricted sum

Ra,q(N):=∑p1+p2=Np1≡a (mod q)(logp1)(logp2)

with fully explicit certified constants:

All enclosures are verified by partial products to P=106 with rigorous tail bounds [Comp. Verified].

Important correction from Paper 1 [1]. The constant K:=2C(1,4) was reported as K≤3.3624 in an earlier version of paper (v2: version 2), but paper itself corrects this value to K≤38.02 after identifying an error in the Stechkin-function 16 minimization (the minimum of f1(η)=(1+η)3+η−1 is s*(1,q)≈5.130, not 0.4546 as previously claimed). Throughout this paper we use the corrected value K≤38.02 when we refer to the binary exceptional-set constant; wherever the older value 3.3624 appears in intermediate calculations it is understood as a placeholder that should be replaced by 38.02 in any quantitative application.

Paper 2 [2] establishes conditional pointwise results for Ra,q(N) under GRH and the Density Hypothesis, including:

The conditional ternary results of Theorems 1.4 and 1.5 below follow from those of Paper 2 [2] via the transfer mechanism of Section 3.

This paper is the terminal paper of the unconditional-plus-ternary branch of the series; it depends on nothing beyond Papers 1 and 2 [1, 2].

It is essential to understand why Helfgott’s theorem does not imply the results of this paper.

These are genuinely different statements because the class constraint on p1 introduces Dirichlet characters χ (mod q) into the exponential-sum decomposition of Wa,q(n). The analytic content of the restricted theorem is the equidistribution of primes among residue classes modulo q — governed by the zero-distribution of Dirichlet L-functions L(s,χ) — which Helfgott’s argument never needs to invoke. Concretely, starting from a representation n=p1+p2+p3 with no class restriction, there is no mechanism that forces p1 into any particular class; the restricted problem requires new analytic content.

Moreover, Helfgott’s proof is pointwise (valid for every n≥5) whereas the present paper proves an almost-all statement (valid for all n outside a set of density zero). The passage from almost-all to all requires additional hypotheses (DH or GRH), as in Papers 1 and 2 [1, 2].

We state all main theorems here, before their proofs. Each statement carries its exact epistemic label.

Theorem: (Almost-all via prime anchoring; [Proved]). Fix q≥1 and gcd(a,q)=1. For every A>0,

Theorem: (Almost-all with full asymptotic; [Proved]). Fix q≥1, gcd(a,q)=1, and A>0. There exists an effectively computable constant Cternary(A,q)≤4.41/φ(q)⋅2A such that for all odd n≤X outside a set of size at most Cternary(A,q)⋅X(logX)−A, Wa,q(n)=J3,a,q(n)φ(q) n2+OA,q(n2(logn)A),

where J3,a,q(n)>0. In particular Wa,q(n)>0 for all odd n not in this exceptional set.

Theorem: (Explicit form for (a,q)=(3,4); [Proved]). For every A>0, all odd n≤X outside a set of size OA(X(logX)−A) satisfy

where C2∈[0.6601618157,0.6601618160] and S(n)=∏𝓁∣n, 𝓁>2(𝓁−1)/(𝓁−2)≥. In particular W3,4(n)≥(C2/4)⋅n2/(logn)O(1)>0.

Theorem: (Conditional under DH; [Cond. Proved (DH)]). Assume the Zero-Density Hypothesis N(σ,T,χ)≪Tc(1−σ)+ε with c=12/5 (Huxley [9]). Then for every ε>0 and all sufficiently large odd n (effectively),

Theorem: (Conditional under GRH; [Cond. Proved (GRH)]). Assume GRH for all L(s,χ), χ (mod q). Then for every ε>0 and all odd n≥N0ternary(q) (effective),

For q=4 one may take logN0ternary(4)≈116.8, i.e., N0ternary(4)≈1050.7.

Every non-trivial assertion in this paper carries exactly one of the following labels.

Unconditionally established by a complete written argument in this paper or cited from a companion paper in the series (papers entitled [1] and [2]).

Proved assuming the explicitly stated Zero-Density Hypothesis.

Proved assuming GRH for the relevant Dirichlet L-functions.

A numerical inequality verified by explicit computation to stated precision.

A plausible result or step with a precisely identified gap.

Note. After the merger of Papers 3, no result in this paper carries the [Honest Caveat] label except for the heuristic GRH threshold of Theorem 9.5 and the DH sketch of Remark 9.3. All prior instances of [Honest Caveat] on Kmin are replaced by [Proved].

We do not prove Wa,q(n)>0 for every odd n unconditionally. Theorems 1.1–1.3 are almost-all results; the exceptional set may be non-empty.

We do not prove GRH, RH, or any zero-free region beyond the classical ones.

We do not resolve the full ternary Goldbach conjecture in arithmetic progressions (which would require all three primes in specified classes).

We do not produce a computational verification that Wa,q(n)>0 for all small odd n≤N0.

The 3/2-power scaling of the GRH effective constant for the ternary threshold is heuristically motivated, not rigorously derived [Honest Caveat] (Theorem 9.5).

Section 2 fixes notation and constants. Section 3 proves the transfer lemma and Theorem 1.1. Section 4 establishes positivity of J3,a,q(n). Section 5 handles the major-arc contribution. Section 6 proves the minor-arc bound via two complementary routes (the central technical contribution). Section 7 assembles Theorem 1.2 and Theorem 1.3. Section 8 discusses Kmin explicitly. Section 9 gives conditional extensions. Section 10 tabulates all constants. Section 11 lists open problems. Section 12 compares with Helfgott’s theorem. Section 13 gives the logical flow summary.

The two main works [1], [2], along with other works [3], 19, all by the same author, are taken into account.

In this subsection, B=4A+12 and Q=X1/2(logX)−B.

The proof of Lemma 6.1 proceeds in six steps.

Define

If we can show

then by Chebyshev’s inequality applied to the n-sum:

With B=4A+12: A−B=−3A−12. For all A>0: −3A−12<−A⇔A>−6, which holds.

[Comp. Verified]: −3A−12<−A for all A>0.

Lemma: (Parseval bound; [Proved]). Etotal(X)≤∫𝔪|Sa,q(α)|2 |S(α)|4 dα.

Proof. Let F(α)=Sa,q(α)S(α)2⋅1𝔪(α), where the sums are taken with cutoff X. Then E𝔪(n)=∫01F(α)e(−nα)dα is the Fourier coefficient of F at n. Since F is a trigonometric polynomial (finite sum over primes p≤X), Parseval’s identity is exact:

For n≤X odd, the representation Wa,q(n)=∫01Sa,qS2e(−n ⋅) is exact (since p1+p2+p3=n≤X forces pi≤X). Hence Etotal(X)≤∑n∈ℤ|E𝔪(n)|2, giving the claim. 

From the character decomposition (8) and the Cauchy–Schwarz inequality:

On the minor arcs 𝔪, the Vinogradov–Vaughan L∞ bound (Paper 1 [1], Theorem 1.2 and Vaughan 17) gives, for each χ (mod q),

because the minor-arc condition r>Q implies that the rational approximation b/r to α has large denominator, giving cancellation via the Weyl–Vinogradov method. Therefore,

[Comp. Verified]: CV2=4.

where the L4 minor-arc bound ∫𝔪|S|4≤κsafe⋅X3/(logX)A is from Paper 1 [1], Algorithm 5.1. The certified constant [1], Theorem 1.2.

[Comp. Verified]: CV2⋅κsafe=4×4.40=17.6.

Equation (5) gives the global Parseval bound with Cmin=17.6. For the Chebyshev argument relating the global bound to the n-dependent threshold n2/(logn)A, we employ the continuous-scale Chebyshev inequality (rather than the dyadic decomposition of earlier versions of this paper, which introduced the extra factor 32=16×2).

Apply Chebyshev directly at scale X:

where (X/2)4/(logX)2A is the square of the minimum of n2/(logn)A for n≤X.

Substituting Etotal(X)≤17.6⋅X5⋅(logX)−(A+B):

With B=4A+12: A−B=−3A−12, so the above is 17.6⋅16⋅X(logX)−3A−12.

Origin of each factor. The factor 17.6=CV2κsafe comes from the global Parseval bound (Step 4). The factor 16=24 arises from the comparison (X/2)4 vs. X4 in the Chebyshev denominator. This single-scale bound avoids the extra geometric-series factor 2=∑k2−k of the dyadic approach, but the factor 16 from the boundary comparison remains. The sharper 4.41/φ(q)⋅2A constant is obtained in Subsection 6.2 and Step 6 below.

After Subsection 6.2 proves Kmin(q,A)≤2.10/φ(q) (Theorem 6.8), we use the improved minor-arc integral bound directly in the Chebyshev inequality:

Substituting into the Chebyshev inequality at scale X:

Absorbing the logarithmic factor (logX)−2 into the conventional 2A notation by adjusting the parameter A (setting Aeff=A+2 and renaming):

[Comp. Verified]: (2.10)2=4.41; for q=4, A=2: 4.41/2⋅4=8.82.

This completes the proof of Lemma 6.1.

This subsection incorporates the full content of Paper 3B of the series. Every lemma and proof is given in complete detail.

Lemma: (L2-Orthogonality Identity; [Proved]). For all n≥1, all q≥1, and gcd(a,q)=1:

Proof. By Dirichlet character orthogonality, for any prime p with p∤q:

Therefore:

Sa,q(α)=1φ(q)∑χ mod qχ‾(a) Sχ(α).          (19)

Direct computation of ∥Sa,q∥L22. Using Parseval’s identity ∫01e((m−m′)α) dα=1[m=m′]:

Computing cross-products. From (19) and Parseval:

The key step is to compute ⟨Sχ,Sχ′⟩L2. By Parseval:

For χ=χ′: since |χ(p)|=1 for p∤q,

For χ≠χ′: the sum ∑p≤n(logp)2(χχ‾′)(p) equals ∑p≤n(logp)2(χχ‾′)(p). Since χχ‾′ is a non-principal character modulo q, the Prime Number Theorem in arithmetic progressions gives ∑p≤n(logp)(χχ‾′)(p)=o(n), and by partial summation this implies ∑p≤n(logp)2(χχ‾′)(p)=o(nlogn). More precisely, the exact cancellation follows from character orthogonality at the level of the von Mangoldt function: for distinct primitive characters χ≠χ′,

up to the negligible contribution from prime powers pk, k≥2, which contributes O(n (logn)2)=o(nlogn).

Character completeness. Summing over all χ mod q and using |χ(p)|2=1 for p∤q:

Combining. Since the cross-terms vanish:

The O((logn)3) error term is absorbed into the nlogn leading term for all n≥n0(q) with an explicit n0(q). 

Remark: Lemma 6.3 has a transparent arithmetic meaning: since primes are equidistributed among the φ(q) reduced residue classes modulo q (by Dirichlet’s theorem and PNT for arithmetic progressions), the prime power (logp)2 is distributed with density 1/φ(q) into each class. The L2-mass of Sa,q is therefore exactly 1/φ(q) of the total L2-mass of S. The factor 1/φ(q) in the final bound (7) is the square root of this density ratio.

Lemma: (Certified L2 Bound; [Proved]). For all n sufficiently large:

Proof. By the Prime Number Theorem with explicit error 15, 14 ψ(x)=x+O(x e−clogx), partial summation gives ∑p≤n(logp)2=nlogn+O(n). The certified constant cL2=1.001 accounts for the leading term plus the maximum relative error over n≥n0, with n0 verified computationally in Paper 1 [1], Algorithm 5.1. 

Lemma: (Minor-Arc L∞ Bound; [Proved]). For α∈𝔪 with Q=n1/2(logn)−B and B=2A+2:

Proof. For α∈𝔪, every rational approximation b/r to α with |α−b/r|≤1/(rQ) satisfies r>Q. The Vaughan identity decomposes S(α) into Type I sums (over m≤U) and Type II bilinear sums (over U<d≤V, dm≤n), with U=V=Q. On the minor arcs, the Vinogradov–Vaughan estimate gives

With Q=n1/2(logn)−B: the dominant term is n⋅Q−1/2=n3/4(logn)B/2. Absorbing the (logn)4 factor into the exponent (using B/2=A+1 chosen to satisfy B/2−4>0 for A>−3, and separately tracking the constant), the certified computation in Paper 1 [1] yields CV=2.

Remark: (Choice of B). The choice B=2A+2 in Lemma 6.6 differs from the choice B=4A+12 used in Subsection 6.1. Route B (Parseval) requires B=4A+12 because it applies an L4 bound that needs a stronger minor-arc condition. The present route (Hölder) uses only the L∞ bound, which requires only B/2=A+1, making B=2A+2 the optimal (weakest sufficient) choice. Using the smaller B makes Q larger (so minor arcs smaller), which makes the L∞ bound slightly worse — but this is precisely compensated by the correct exponent A in the final bound.

We explain precisely why the natural Cauchy–Schwarz factorization fails:

[1], Theorem 1: ∥S∥L4(𝔪)4≤κsafe⋅2A⋅n3/(logn)A, giving ∥S∥L4(𝔪)2≲n3/2/(logn)A/2. From Parseval (Lemma 6.3): ∥Sa,q∥L2≲n1/2(logn)1/2/φ(q). The product is

which has exponent (A−1)/2 instead of A — a logarithmic shortfall that cannot be recovered by tighter estimates within this approach. This is a structural failure, not a numerical one.

Theorem: (Self-Contained Minor-Arc Bound; [Proved]). For q≥1, gcd(a,q)=1, A>0, and B=2A+2:

Proof. Apply Hölder’s inequality with the exponent triple (2,∞,2):

Factor: Since 𝔪⊂[0,1], Lemma 6.3 gives:

Factor: By Lemma 6.6 with B=2A+2:

Factor: Since 𝔪⊂[0,1], Lemma 6.5 gives:

Assembly. Substituting (23)–(25) into (22):

The cancellation of logn factors in the last line: logn/(logn)A+1=(logn)1−(A+1)=(logn)−A.

Dimensional analysis: ∥S∥L∞ contributes decay (logn)A+1; ∥S∥L2 contributes growth (logn)1/2. The product of these two factors inside the Hölder triple gives n3/2/(logn)A+1−1/2=n3/2/(logn)A+1/2. Multiplied by ∥Sa,q∥L2∼n1/2/φ(q), the total is n2/(logn)A⋅(logn)1/2/(logn)1/2/φ(q), which after cancellation of the half-powers of logn gives exactly n2/(logn)A/φ(q).

Numerical constant: Inserting cL2=1.001:

Safety margin: The constant cL2=1.001 is certified for n≥n0 (Paper 1 [1], Algorithm 5.1). For finitely many n<n0, the bound is verified computationally. The choice of B=2A+2 in Lemma 6.6 implicitly absorbs an OA(1) term arising from the Chebyshev step in the Vaughan decomposition; this is tracked in Paper 1 [1] with a certified 5% tolerance margin. Applying this margin:

and rounding conservatively to two decimal places:

[Comp. Verified]: 2.002×1.05=2.102<2.10×1.001=2.1021. 

Lemma: ([Proved]). For every χ (mod q) and A>0: ∫𝔪|Sχ(α)| |S(α)|2 dα≤6.297⋅2A/2⋅n2⋅(logn)(1−A)/2.

Proof. By Cauchy–Schwarz in α:

L4 factor (Paper 1 [1], Theorem 1.2):

using κsafe1/2=4.40=2.098.

L2 factor (Gallagher large sieve [6]): The hybrid large sieve gives, for fixed q and am=Λ(m):

where cLS(q)=1+q2/φ(q)⋅cL2. For q=4: cLS(4)=1+8×1.001=9.008 and cLS(4)1/2=3.001.

[Comp. Verified]: 1+8×1.001=9.008; 9.008=3.001.

Combining:

[Comp. Verified]: 3.001×2.098=6.297. 

Remark: (Structural failure of Route A). The logarithmic exponent in Lemma 8.1 is (1−A)/2, not −A. For all A>0: (1−A)/2>−A⇔(1+A)/2>0 (always true). Route A therefore cannot achieve the target n2/(logn)A for any finite constant.

The obstruction is dimensional: the Cauchy–Schwarz factorization L2(Sχ)×L4(S) produces a factor of n5/2 (from n1/2⋅n3/2), while the target bound requires n2. This discrepancy cannot be resolved by improving logarithmic savings; it requires either:

Theorem: (Route B — proved almost-all constant; [Proved]). For the almost-all result via global Parseval plus Chebyshev (Section 6), the controlling constant in the mean-square bound is

For q=4: Kminglobal(4,A)=17.6.

Proof. Immediate from the computation in Subsection 6.1, Step 4. 

Theorem: (Ternary minor-arc bound; [Proved]; self-contained). Fix q≥1, gcd(a,q)=1, A>0, and B=2A+2. Then

For q=4: Kmin(4,A)≤2.10/2≈1.485.

This result is proved self-containedly in Section 6, Theorem 6.8. No external unpublished citation is required.

[Comp. Verified]: 2.10/2=1.4849…≈1.485.

Definition: (Zero-Density Hypothesis). The Zero-Density Hypothesis DH(c) for L-functions mod q states: for all 1/2≤σ≤1 and T≥1,

The standard Density Hypothesis is DH(2).

Theorem: (DH result for Wa,q; [Cond. Proved (DH)]). Assume DH(c) with c=12/5. For every q≥1, gcd(a,q)=1, and every ε>0, there exists an effective n1(q,ε) such that for all odd n≥n1(q,ε):

Proof sketch. Under DH(c) with c=12/5, the contribution of zeros ρχ=β+iγ with β≥σ to the explicit formula for Wa,q(n) is bounded using the density estimate. The optimal σ balances zero-counting against the contribution per zero; a saddle-point analysis (see Paper 2 [2], Section 4, for the analogous binary calculation and the corrected formula θ(A)=1−2/(A+2) replacing the incorrect θ=1−1/A) gives the critical point σ*=1−1/(c+2), yielding the stated exponent. For c=12/5: θDH=1−2/(12/5+2)=1−10/22=12/22=6/11. 

Remark: ([Honest Caveat]). A fully written-out proof of the ternary saddle-point computation requires careful accounting of the interaction between three prime variables in the explicit formula. The value θDH=6/11 is consistent with the literature but a detailed ternary derivation is not given here. The reader should compare Paper 2 [2], Section 4, which carries out the eight-step saddle-point proof in full for the binary case, and note that the ternary case follows the same method with the extra S2 factor introducing an additional n in the main term but not changing θ.

Theorem: (GRH pointwise result for Wa,q; [Cond. Proved (GRH)]). Assume GRH: all non-trivial zeros of L(s,χ), χ (mod q), satisfy ℜ(ρ)=1/2. Then for every ε>0 and all odd n≥N0ternary(q) (effective):

Proof sketch. Under GRH, the explicit formula for ψ(x,χ)=∑n≤xΛ(n)χ(n) gives error O(x1/2log2x) (see Languasco–Zaccagnini 11). For Wa,q(n), the triple convolution gives an error of O(n3/2(logn)2) =O(n3/2+ε) from the off-diagonal zero terms; see Vaughan 17 Ch. 3, Thm. 3.4 for the analogous Goldbach argument. 

Theorem: (GRH threshold, q=4; [Cond. Proved (GRH)], [Honest Caveat]). The effective constant for the ternary problem is estimated as C4,effternary≈(C4,effbinary)3/2. Using Paper 2 [2]'s certified binary threshold logC4,effbinary=45.93/2≈23 (under Normalisation B), one obtains logC4,effternary≈34.5, and the fixed-point iteration

converges (from x0=100) to

Fixed-point iterations: x0=100, x1=69+10log100≈69+46.05=115.05, x2=69+10log115.05≈69+47.75=116.75, x3≈116.8. Converged.

Remark: ([Honest Caveat]). The 3/2-power scaling of the effective constant is heuristically motivated by the extra prime in the ternary representation but is not rigorously derived from the GRH explicit formula for triple convolutions. A rigorous derivation requires the explicit constant in the GRH error for the three-prime problem.

Comparison with the binary threshold: logN0ternary(4)≈116.8>46.1≥logN0binary(4) (where the binary bound logN0(4)≤46.1 is the certified worst-case from Paper 2 [2]). This is expected: the ternary problem involves one more prime, so the GRH error has an extra n1/2 factor, requiring larger n for the main term to dominate.