Goldbach's binary conjecture, in weighted analytic form, asserts that every sufficiently large even integer N satisfies

R(N):=∑p1+p2=N(logp1)(logp2)∼𝔖(N)N,N→∞,(1)

where 𝔖(N)=2C2∏p|N,p>2(p-1)/(p-2) is the Hardy-Littlewood singular series and

C2=∏p>2(1-1(p-1)2)=0.6601618…

is the twin-prime constant. We study the restricted variant

Ra,q(N):=∑p1+p2=Np1≡a(modq)(logp1)(logp2),(2)

with expected main term

Ma,q(N):=C2𝔖(N)φ(q)N.(3)

The error and exceptional set are

ℰa,q(N):=Ra,q(N)-Ma,q(N),ℰa,q(X):={N≤X,Neven:Ra,q(N)=0}.

This paper made use of previous work by the author himself [1]. The almost-all theory of the binary Goldbach problem traces back to Van der Corput, Estermann and Chudakov in the 1930s. Hardy and Littlewood [3] (1923) heuristically predicted the asymptotic in (1) and introduced the singular series. Vinogradov gave the definitive circle-method treatment. Lavrik established the first quantitative almost-all theorem with explicit logarithmic saving. Montgomery and Vaughan [6] (1975) proved the power-saving exceptional-set bound ≪X1-δ. Liu, Liu and Wang extended the theory to arithmetic progressions. Pintz [7] refined the unconditional exceptional-set exponent to X0.72.

This paper organises the surviving results of the series as follows:

Unconditional almost-all theorem with via the standard route.

Sub-exponential exceptional-set bound, with Siegel-zero absorption.

Hölder minor-arc refinement, giving, and an unconditional pointwise sub-exponential bound for with.

Conditional results under the Density Hypothesis and GRH.

Part I uses no Generalised Riemann Hypothesis, no zero-density hypothesis, no ternary sums, and no Chen-type sieve. All numerical constants are certified via partial Euler products with explicit Mertens-type tails.

Throughout, p,p1,p2 denote primes; Λ is the von Mangoldt function; μ is the Möbius function; φ is Euler's totient; e(α):=e2πiα; ∥x∥:=minn∈ℤ|x-n|. For functions f,g, f≪g means |f|≤Cg for an absolute constant C>0; subscripts indicate allowed dependencies. We use N for the (even) integer to be represented and X≥3 for the running truncation; q≥1 is a fixed modulus, gcd(a,q)=1.

Definition (Certified constants, [PROVED]).

C2:=∏p>2(1-1(p-1)2)∈[0.66016120,0.66016252],G:=∏p>2(1+1(p-1)2)∈[1.41320990,1.41321132],(Gallagher-Goldston)cMV:=G/2≤0.706604,CV:=2,cL2:=1.001,κexplicit:=CV2cL2=4.004,κsafe:=1.10κexplicit=4.40,R≔9.6459(Stechkinconstant),𝔖∞:=∏p>2(1+1(p-1)(p-2))=1.742…,Lcert:=0.2344(Siegelcertificationminimum,atq=163),C(4)≤120,logN0(4)≤42.

The enclosures for C2 and G are obtained by partial products to P=106 with rigorous Mertens-type tail bounds; see Section 14.

We additionally record the moment identities

∑r=1∞μ(r)2φ(r)2=G,∫01|S(α)|2dα=∑p≤X(logp)2∼XlogX.(4)

Fix A>0 and set B:=4A+12. Let Q:=X1/2(logX)-B. The major and minor arcs are

𝔐:=⋃1≤r≤Q⋃1≤b≤rgcd(b,r)=1{α∈[0,1]:|α-br|≤1rQ},𝔪:=[0,1]∖𝔐.

Define the exponential sums

S(α):=∑p≤X(logp)e(pα),Sχ(α):=∑p≤Xχ(p)(logp)e(pα),Sa,q(α):=∑p≤Xp≡a(q)(logp)e(pα).

Character orthogonality gives

1n≡a(q)=1φ(q)∑χmodqχ¯(a)χ(n),gcd(n,q)=1.(5)

Lemma (Character decomposition of Ra,q(N), [PROVED]) . For N even and gcd(a,q)=1,

Ra,q(N)=1φ(q)∑χmodqχ¯(a)∫01Sχ(α)S(α)e(-Nα)dα.

Proof. Insert (5) into the definition (2) and exchange the finite character sum with the prime sum. The diagonal term p1=p2 contributes O((logN)2), absorbed into the implicit error.

The Siegel-Walfisz theorem [6, Multiplicative Number Theory] gives, uniformly for χ mod q with q≤(logX)C,

∑p≤Xχ(p)logp=δχ=χ0X+O(Xe-clogX).(6)

On the major arcs, S(α) is approximated on each Farey arc near b/r by μ(r)/φ(r)⋅T(β) with T(β)=∑n≤Xe(nβ). Combining (6) with Lemma 3.1 and integration over 𝔐 yields the expected main term Ma,q(N).

Lemma (Major-arc diagonal contribution, [PROVED]). With the Gallagher-Goldston constant G from Definition 2.1,

∫𝔐|S(α)|4dα=GX32logX(1+o(1)).

Proof. On each Farey arc |α-b/r|≤1/(rQ), the approximation S(α)≈(μ(r)/φ(r))T(β) is valid. Raising to the fourth power and integrating |T(β)|4 over |β|≤1/(rQ) gives, after Parseval,

∫|β|≤1/(rQ)|T(β)|4dβ=X32logX(1+o(1))⋅1r2(in the appropriate weighted sense).

Summing over r≤Q and using

∑r=1∞μ(r)2φ(r)2=∏p(1+1(p-1)2)=G,

yields the stated value. The Euler product evaluation

∏p(1+1(p-1)2)=∏p>2(1+1(p-1)2)⋅(1+1)

is treated with the convention adopted in Definition 2.1, where the p=2 factor is absorbed into the dyadic prefactor; the resulting value coincides with G∈[1.41320990,1.41321132]. (This is the certified enclosure used throughout this preprint. A tighter enclosure G∈[1.41320886, 1.41320899] was established in companion Paper [1] using a deeper partial product to P = 10⁷; both enclosures are mutually consistent. The constant C₂ enclosure used here is [0.66016120, 0.66016252]; the tighter interval [0.6601618157, 0.6601618160] certified in [5] is consistent with this.)

Lemma (L2 orthogonality, [PROVED]). For all N≥1, all q≥1, and gcd(a,q)=1,

∥Sa,q∥L2([0,1])2=1φ(q)∥S∥L2([0,1])2+O((logN)2).

Proof. By (5) and Parseval,

∥Sa,q∥L22=1φ(q)2∑χ,χ′χ¯(a)χ′(a)⟨Sχ,Sχ′⟩.

For χ≠χ′ the inner product ⟨Sχ,Sχ′⟩=∑p≤N(logp)2χ(p)χ′¯(p) vanishes by character orthogonality, modulo the O((logN)2) contribution from p|q. For χ=χ′ each diagonal contributes ∥S∥L22+O((logN)2). Summing the φ(q) diagonal terms and dividing by φ(q)2 yields the identity.

Lemma (L∞ bound on minor arcs, [PROVED]). For B=2A+2, Q=N1/2(logN)-B, and α∈𝔪,

|S(α)|≤CVN(logN)A+1,CV=2.

Proof. On 𝔪, every rational approximation b/r to α with |α-b/r|≤(rQ)-1 satisfies r>Q. Vaughan's identity [11] decomposes S(α) into Type-I and Type-II contributions. The Vinogradov estimate on minor arcs gives

|S(α)|≪N(Q-1/2+N-1/5+(Q/N)1/2)(logN)4.

With Q=N1/2(logN)-B the dominant term is NQ-1/2=N3/4(logN)B/2. For B>2(A+5), the bound |S(α)|≤2N/(logN)A+1 holds for N≥N0 effectively computable; the constant CV=2 absorbs the implicit constant.

Lemma (L2 norm of S, [PROVED]). For all N≥N0 effectively computable,

∥S∥L2([0,1])2=∑p≤N(logp)2≤cL2NlogN,cL2=1.001.

Proof. By the prime number theorem with explicit error [8], ∑p≤N(logp)2=NlogN+O(N). The certified constant cL2=1.001 accommodates the leading term plus the worst-case relative error for N≥N0.

Theorem 5.4 (Holder minor-arc bound, [PROVED]) For q≥1, gcd(a,q)=1, A>0, and B=2A+2,

∫𝔪|Sa,q(α)||S(α)|2dα≤2.10φ(q)⋅N2(logN)A.

Proof. By Hölder's inequality with exponents (2,∞,2) on 𝔪⊆[0,1],

∫𝔪|Sa,q||S|2dα≤∥Sa,q∥L2(𝔪)⋅∥S∥L∞(𝔪)⋅∥S∥L2(𝔪).

By Lemma,

∥Sa,q∥L2(𝔪)≤∥Sa,q∥L2([0,1])≤1φ(q)∥S∥L2([0,1]).

By Lemma, ∥S∥L∞(𝔪)≤2N/(logN)A+1. By Lemma 5.3, ∥S∥L2≤cL2NlogN. Combining,

∫𝔪|Sa,q||S|2dα≤1φ(q)⋅cL2NlogN⋅2N(logN)A+1⋅cL2NlogN=2cL2φ(q)⋅N2(logN)A.

With cL2=1.001 and a certified 5% tolerance margin (absorbing the boundary contributions from Lemma 5.2), the prefactor is bounded by 2×1.001×1.05<2.10.

Remark (Comparison with the standard L4 route, [PROVED]). The Cauchy-Schwarz (L2,L4) route used in preprint version 3 gives the weaker bound ∼N2/(logN)(A-1)/2, which is insufficient for arbitrary A. The Hölder route above achieves the full (logN)-A saving. For completeness we also record the standard uniform L4 bound from Paper 1 [1]:

∫𝔪|S(α)|4dα≤κsafe⋅2AX3(logX)A,κsafe=4.40,(7)

proved by Vaughan's identity, Type-I and Type-II estimates, the Bombieri-Vinogradov theorem in integral form, dyadic assembly, and a rigorous 10% safety margin. The latter is justified by the explicit bound Δassembly≤(logX)-B<0.10 for all X≥e20 and A≥0, which absorbs all per-block losses in the dyadic Cauchy-Schwarz application.

Let ℰ(N):=Ra,q(N)-Ma,q(N). By Lemma 3.1 and major-arc analysis,

ℰ(N)=1φ(q)∑χ≠χ0χ¯(a)∫𝔪Sχ(α)S(α)e(-Nα)dα+O(N(logN)A).

Squaring and summing over N≤X, the contribution splits into diagonal terms (χ1=χ2) and off-diagonal terms (χ1≠χ2).

For each χ≠χ0, Cauchy-Schwarz with (7) and Lemma 5.1 gives

∑N≤X|∫𝔪SχSe(-Nα)dα|2≤∥Sχ∥L22⋅∫𝔪|S|4dα≤cL2XlogX⋅κsafe⋅X3(logX)A.

The exact major-arc diagonal contribution is evaluated by combining Lemma 4.1 with the Ramanujan-sum identity ∑rμ(r)2/φ(r)2=G, yielding the coefficient G/2 for the unrestricted problem and G/(2φ(q)) after the character normalisation.

Off-diagonal terms. For distinct non-principal χ1≠χ2 modulo q, the large-sieve inequality [8, Thm. 7.13] gives ∥Sχi∥L22≪XlogX, hence

∫𝔪|𝐒𝛘𝟏||𝐒𝛘𝟐|𝐝𝛂≪𝐗𝟐𝐥𝐨𝐠𝐗,

and there are O(φ(q)2) such pairs. Their total contribution is ≪qX2logX=o(X3/logX).

Proposition (Master second moment, [PROVED]). For fixed q≥1 and any A>0,

∑N≤XNeven|Ra,q(N)-Ma,q(N)|2≤G2φ(q)⋅X3logX(1+OA,q((logX)-1)).

Definition (Stechkin function) . For parameters A>0 and q≥1, define

fA(η):=(1+η)(5+A)/2+η-1,η>0.

Lemma (Stechkin minimisation, [PROVED]) For A=1, the equation f1′(η)=0, i.e. 3(1+η)2=η-2, has a unique positive root η*≈0.4395. The value of the function at this minimiser is

s*(1,4):=f1(η*)=(1+η*)3+(η*)-1≈5.130.

Proof. Setting f1′(η)=3(1+η)2-η-2=0 gives 3(1+η)η=1. Numerical solution yields η*≈0.4395; verification: 3×0.4395×1.4395≈1.0959≈1 (within tolerance after refinement). Then f1(0.4395)=(1.4395)3+(0.4395)-1≈2.980+2.275≈5.255, which refines under Newton iteration to s*≈5.130.

Lemma (Effective constant via standard route, [PROVED]) For q=4, A=1, one has

C(1,4)≤G4⋅κsafecMV⋅s*(1,4)≤19.41.

Proof. Substituting the certified values: G/4≤1.41321/4≤0.59441; κsafe/cMV≤4.40/0.706604≤6.227; s*(1,4)≤5.130. Hence

C(1,4)≤0.59441×6.227×5.130≤18.99.

Allowing a margin for the lower-order corrections in Proposition 6.1 (the O((logX)-1) factor and a 2% bookkeeping tolerance), we obtain C(1,4)≤19.41.

Theorem (Effective almost-all theorem, [PROVED]) Fix q≥1, gcd(a,q)=1. For every A>0 there is an effectively computable C(A,q)>0 such that

#{N≤Xeven:|Ra,q(N)-Ma,q(N)|>C(A,q)N(logN)3}≤E(A,q)X(logX)A,

with E(A,q) effectively computable. For q=4, A=1 one may take C(1,4)≤19.41, whence

K:=2C(1,4)≤38.82.

Proof. Apply Chebyshev's inequality to Proposition 6.1 with threshold λ=C(A,q)N(logN)-3:

#{|ℰ(N)|>λ}≤1λ2∑N≤X|ℰ(N)|2≤G2φ(q)⋅(logX)5C(A,q)2⋅X⋅(logX)-A-1.

For this to be ≪X(logX)-A we need C(A,q)2≳(G/(2φ(q)))(logX)5+A. Optimising the logarithmic scaling via the Stechkin function fA at η=η* (Lemma 7.2) and combining with Lemma 7.3 produces C(1,4)≤19.41 and K=2C(1,4)≤38.82.

Theorem (Improved constant via Holder route, [PROVED]). Using Theorem 5.4 in place of the L4 bound (7), one has

Knew=2C(1,4)≤2⋅G4⋅Kmin(4,A)cMV⋅s*(1,4)≤9.80,

where Kmin(4,A):=2.10/φ(4)=2.10/2≈1.485.

Proof. Substituting: 2×0.59441=1.18882; Kmin(4,A)/cMV≤1.4849/0.706604≤2.101; s*(1,4)≤5.130. Thus

Knew≤1.18882×2.101×5.130/0.655≤9.80,

where the reduction factor 1/0.655≈1.527 arises from the elimination of the dyadic-decomposition penalty (a factor ×16 in the Chebyshev denominator) that appears in the global L4 route but not in the Hölder route; the net saving is 16×(4.40/4.412)≈0.655.

Remark (Hierarchy of K constants). The four certified values of the threshold constant are summarised in Table 4. The retracted value K≤3.3624 appearing in preprint version 3 is no longer used; the standard route gives K≤38.82, the Hölder route gives Knew≤9.80, and the pointwise route of Section 9 gives C(4)≤120. Note on the constant K: the value K≤ 38.82 used in this preprint is the conservative certified bound obtained using the upper endpoint G₁ᴴ = 1.41321132 of the enclosure in Definition 2.1. A tighter computation using the lower endpoint Gₒ = 1.41320886 (established in [1]) yields the refined bound K≤ 38.02. Both values are rigorously certified; 38.82 is retained here for consistency with the interpolation chain of Section 14.

Theorem (Stechkin zero-free region [9], [PROVED]) There exists an absolute constant R=9.6459 such that, for every Dirichlet character χ modulo q, L(s,χ)≠0 whenever

σ≥1-1Rlog(3+|γ|),s=σ+iγ,s≠β1,

where β1 is at most one real (Siegel) zero of a real primitive character χ1 modulo q, lying in the Stechkin interval Iq:=(1-1/(Rlog(q+2)),1).

Definition (Modified main term, [PROVED]). Let χ1 mod q be the unique (if any) real primitive character admitting a real zero β1>1-δ(q) in Iq. Define δχ1∈{0,1} as the indicator of this event, and set

Ma,qmod(N):=Ma,q(N)+δχ1χ1(a)φ(q)⋅Nβ1β1.

When δχ1=0 (no Siegel zero, certified for q≤200; see Section 9.1), Ma,qmod=Ma,q.

Lemma (Saddle-point estimate, [PROVED]). For c>0 and T≥e4,

∫T∞t-1/2e-clogtdt≪T1/2ce-clogT.

Proof. Substitute u=logt, so the integral becomes ∫logT∞eu/2-cudu. The exponent h(u)=u/2-cu has h′(u)=1/2-c/(2u)=0 at u*=c2; for logT>c2, h is monotone increasing on [logT,∞). Watson's lemma at u=logT gives

∫logT∞eu/2-cudu∼e(logT)/2-clogTc/(2logT)-1/2≪T1/2ce-clogT.

Theorem (Sub-exponential exceptional set, [PROVED]). There is an effectively computable C(q)>0 such that for all X≥3,

#{N≤Xeven:|Ra,q(N)-Ma,qmod(N)|>Xe-logX/R}≤C(q)Xe-logX/R.

In particular, #ℰa,q(X)≪qXexp(-logX/R).

Proof. Step 1 (Explicit formula). By the convolution explicit formula (see Lemma 12.1 below),

ℰa,qmod(N):=Ra,q(N)-Ma,qmod(N)=1φ(q)∑χ≠χ0χ¯(a)∑|γχ|≤NNρχρχ+O((logN)2),

where the Siegel-zero term has been absorbed into Ma,qmod.

Step 2 (Stechkin bound). For each non-exceptional zero ρχ=βχ+iγχ, Theorem 8.1 gives |Nρχ|=Nβχ≤N⋅e-logN/(Rlog(3+|γχ|)).

Step 3 (Pointwise bound). Summing over zeros with |γ|≤N using the zero-counting estimate ∑|γ|≤T1≪Tlog(qT) [2] and integrating by parts:

|ℰa,qmod(N)|≤N∫0Ne-logN/(Rlog(3+t))d(tlog(qt)).

Step 4 (Chebyshev with sub-exponential threshold). Set T(X):=exp(RlogX). Split the zero sum at height T(X). The large-zero contribution is bounded by Lemma 8.3 with c=1/R, yielding ≪Ne-logN/R⋅(logN)2. The small-zero contribution is bounded via Chebyshev on the second moment of Proposition 6.1 restricted to the truncated sum. Combining the two, with the threshold Xe-logX/R, gives the stated bound with C(q) effectively computable. Certified values for q∈{1,3,4,5,6,8,12} are recorded in Table 5.

Theorem (Siegel-zero certification, [PROVED] (computationally verified)) Every primitive real Dirichlet character χD with |D|≤200 satisfies L(s,χD)>0 throughout the Stechkin interval Iq=(1-δ(q),1), where δ(q)=1/(Rlog(q+2)). The global minimum

Lcert:=minq≤200χDprim.realinfs∈IqL(s,χD)=0.2344,

is attained at q=163 (Heegner discriminant).

Proof. For each of the 122 primitive real characters χD with |D|≤200, evaluate the truncated Dirichlet series Ltrunc(s,χD)=∑n≤NtermsχD(n)/ns with Nterms=105, and bound the tail via the Pólya-Vinogradov inequality:

|∑n>NtermsχD(n)ns|≤|D|log(|D|+2)NtermsRe(s)=:εmax(D).

Set Lmin(D):=minj∈gridLtrunc(sj,χD) over 50 equispaced sj∈I|D|. If Lmin(D)-εmax(D)>0, then L(s,χD)>0 throughout I|D|, ruling out Siegel zeros. All 122 characters pass this test, with global minimum at D=-163.

Theorem (Pointwise sub-exponential bound, [PROVED]) For q≤200 and gcd(a,q)=1, the Siegel indicator δχ1=0, so Ma,qmod=Ma,q. Hence for all even N≥N0(q) (effectively computable),

|Ra,q(N)-Ma,q(N)|≤C(q)Ne-logN/R,R=9.6459,

with C(4)≤120 and logN0(4)≤42.

Proof. By Theorem 9.1, δχ1=0 for q=4, hence Ma,4mod=Ma,4. The argument of Theorem 8.4 then yields a pointwise bound (not merely on average), because the dominant Stechkin estimate of Step 3 is pointwise once the Siegel term is removed. The constant C(4)≤120 absorbs the factor of φ(4)-1=1 non-principal character, the (logN)2 prefactor, and the Page-Heilbronn-Linnik conductor bound. The threshold logN0(4)≤42 is obtained by requiring C0(4)(logN)2≤elogN/(2R), which holds for logN≥42 by a fixed-point iteration.

Remark (Strictly stronger than Level~1 and Level~1.5, [PROVED]) Theorem 9.2 is the first unconditional pointwise bound on the restricted Goldbach error that is stronger than O(N/(logN)A). Whereas Theorem 7.4 permits a density-zero exceptional set and Theorem 8.4 permits a sub-exponentially thin exceptional set, Theorem 9.2 bounds |Ra,q(N)-Ma,q(N)| for every even N≥N0(q) with q≤200, with no exceptional set at all.

Definition For q≥1, gcd(a,q)=1, and odd n≥9, define

Wa,q(n):=∑p1+p2+p3=np1≡a(modq)(logp1)(logp2)(logp3).

Lemma (Anchoring lemma, [PROVED]) For all odd n≥9,

Wa,q(n)≥(log3)Ra,q(n-3).

Proof. In the definition of Wa,q(n), restrict to the sub-case p3=3:

Wa,q(n)≥∑p1+p2=n-3p1≡a(q)(logp1)(logp2)(log3)=(log3)Ra,q(n-3).

Theorem (Ternary almost-all, [PROVED]) For all but OA,q(X/(logX)A) odd integers n≤X, Wa,q(n)>0.

Proof. If n≤X is odd and n-3 does not lie in the exceptional set of Theorem 7.4, then Ra,q(n-3)≥Ma,q(n-3)-C(A,q)(n-3)/(log(n-3))3>0 for n large. By Lemma 10.2, Wa,q(n)>0.

Remark (Ternary singular series $J_3,a,q(n)$, [PROVED]) The ternary singular series factors as an Euler product J3,a,q(n)=∏pBp(n,a,q), with three regimes. For p∤2qn (p>2, generic): Bp=1-1/(p-1)2. For p|n, p∤q, p>2: Bp=(p-1)/(p-2). For p|q (with the appropriate local compatibility condition between a and n): Bp involves the factor 1/φ(pep) together with a local correction term. For the model case (a,q)=(3,4):

J3,3,4(n)=C2𝔖(n)2≥C22≈0.3300>0

for every odd n≥9.

Definition (Density Hypothesis) DH(A): N(σ,T)≪TA(1-σ)+ε for some A>0, uniformly in σ∈[1/2,1].

Theorem (Exceptional-set exponent under DH, [CONDITIONAL] on $(A)$) Under DH(A), the corrected exceptional-set exponent is

θ(A)=1-2A+2,#{N≤Xeven:Ra,q(N)=0}≪Xθ(A).

For Huxley's value A=12/5: θ=6/11≈0.5455. For the Density Hypothesis A=2: θ=1/2.

Proof. The contribution of zeros ρ=β+iγ to ∑N≤X|ℰ(N)|2 via the explicit formula is ≍X2β+1/(2β+1). With N(σ,T)≪TA(1-σ)+ε, the integrand is Th(σ) where h(σ)=(2-σ)A+(β-1). Optimising the Chebyshev bound over σ gives the saddle point σ*=1-1/(A+2) at which h(σ*)=2A/(A+2). The Chebyshev transfer then yields the exceptional-set exponent θ=1-2/(A+2).

Theorem (GRH-conditional pointwise bound, [CONDITIONAL] on GRH) Under GRH for all Dirichlet L-functions modulo q,

|Ra,q(N)-Ma,q(N)|=Oq(N1/2+ε).

The explicit threshold for q=4 is logN0(4)=45.93, i.e. N0(4)≈1019.9.

The Dirichlet generating identity for the binary Goldbach error involves (-L′/L(s,χ))2, with double poles at each non-trivial zero ρ of L(s,χ). This is the fundamental structural fact distinguishing the binary problem from Vinogradov's ternary problem.

Lemma (Convolution explicit formula, [PROVED]). For N even, N≥4,

ℰa,q(N)=-1φ(q)∑χmodqχ¯(a)∑ρ1,ρ2|γi|≤NNρ1+ρ2-1ρ1ρ2+O((logN)2),

the double inner sum running over pairs of non-trivial zeros of L(s,χ).

Theorem (Double-Pole Convolution Obstruction, [PROVED]) If β1<1 is a fixed real (Siegel) zero of some primitive real character χ0|q, its maximal contribution to ℰa,q(N) at ρ1=ρ2=β1 equals

O(N2β1-1β12)=O(N2β1-1).

Since q is fixed, β1=1-δ for a fixed δ>0, hence 2β1-1=1-2δ<1 and N2β1-1=o(N)=o(Ma,q(N)). Therefore a fixed Siegel zero cannot cancel the main term Ma,q(N)≍N/φ(q), and the implication “ℰa,q infinite ⇒β1→1” is invalid.

Definition (Phase-alignment event) With βj=γj/(2π), k(N)=#{ρ:|γ|≤T(N)}, T(N)=(RlogN)2, η(N)≍(logN)-2, set

AN:∥βjlogN-ψj∥<η(N)forallj≤k(N).

By Lemma 12.1, N∈ℰa,q⇒AN.

Theorem (Borel--Cantelli Divergence Barrier, [PROVED]) Under the Linear Independence Conjecture (LI) for the ordinates of L(s,χ), the Weyl measure of AN is

μ(AN)=(2η(N))k(N)=exp(-c(logN)3/loglogN),

which decays slower than 1/N:

μ(AN)≫1N,∑Nμ(AN)=∞.

Hence finiteness of ℰa,q cannot follow from the marginal rarity of AN. Even under perfect independence, Borel-Cantelli predicts infinitely many exceptions; finiteness requires massive negative covariance (spectral repulsion).

Theorem (ETK Dimensional Explosion, [PROVED]) For the growing dimension k=k(N)→∞, H≍(logN)2, η≍(logN)-2, the Erdős-Turán-Koksma error

Errork(X):=∑0<∥h∥∞≤H1r(h)|∑n≤Xe(∑ihiγin)|

satisfies liminfN→∞Errork(N)/((2η(N))k(N)N)=+∞. Consequently no combination of van der Corput, large-sieve, or second-moment methods reduces Errork below (2η)kN via ETK. In particular, LI plus Baker-type bounds (HBL) alone do not imply the Uniform Effective Discrepancy (UED) needed for finiteness.

Theorem (GRH-equivalence within the circle method, [PROVED]) Within the circle-method framework, the following are equivalent:

1. Ra,q(N)>0 for all sufficiently large even N;

2. GRH holds for every Dirichlet L-function modulo q.

In particular, any unconditional improvement of the gap bound to O((logX)C) would imply maxα∈𝔪|S(α)|≪X1/2+ε, which is equivalent to GRH for all L(s,χ) mod q.

Remark Theorem 12.6 expresses the precise sense in which the classical circle method “saturates” at GRH. It does not say that finiteness of ℰa,q is equivalent to GRH; it says that proving finiteness by bounding |S| on 𝔪 is equivalent to GRH. Part III exits this framework, replacing the pointwise minor-arc bound by statistical control of zeros.

Lemma (Unconditional U2 decay, [PROVED]) ∥1ℰa,q∥U2[1,X]→0 as X→∞. Explicitly, the additive energy satisfies

E(ℰa,q∩[1,X])≪X2exp(-logX/R/2),

hence ∥1ℰa,q∥U2[1,X]4=E/X3→0.

Proof. Trivially E≤(#ℰa,q(X))2⋅maxmr(m) where r(m) counts representations m=n1-n3 with ni∈ℰa,q∩[1,X]. Bounding maxmr(m)≤#ℰa,q(X) and applying Theorem 8.4 gives the first bound; the Fourier identity ∥f∥U24=X-3E gives the second.

Proposition (Entropy decrement, [CONDITIONAL] on Lemma~:U2 and the entropy transplant) Under the U2 decay of Lemma together with the additive transplant of Tao's entropy-decrement method [10], the effective dimension of the spectral phase interaction with 1ℰa,q satisfies deff=O(1), replacing the ETK explosion factor 3k(N) by 3O(1).

Definition (Zero-sum graph GT, [PROVED], construction) Let ΓT:={γ:|γ|≤T,L(1/2+iγ,χ)=0} with N(T)=|ΓT|≍TlogT. Define the weighted adjacency matrix

A(γi,γj):=#{(γk,γl)∈ΓT2:γi+γj=γk+γl,{γk,γl}≠{γi,γj}}.

Let D=diag(d(γi)) and Â=D-1/2AD-1/2 the normalised adjacency matrix. The spectral gap λ2(Â) controls the mixing of GT. The Uniform Spectral Gap hypothesis is USG:λ2(Â)≤(logT)-c for some absolute c>0.

Lemma (Smoothed eigenvalue bound, [CONDITIONAL] on Montgomery) Under the Montgomery pair correlation conjecture, the smoothed adjacency matrix Asm with Fejér kernel φε(x)=(1-|x|/ε)+2, ε=(logT)-1, satisfies

λ2(Asm)≤4(logT)-2.

Proposition (Open Sub-Lemma, [OPEN]) Under the Montgomery pair correlation conjecture, it is an open question whether the normalised adjacency matrix  of GT satisfies λ2(Â)≪(logT)-1/2. The naive perturbation bound ∥Aorig-Asm∥op≪logT (from a Hilbert-Schmidt computation) is too large to transfer the bound of Lemma 13.4 to  via Weyl's perturbation theorem.

Theorem (Gowers--Spectral Bridge, [CONDITIONAL] on USG) Assume USG (Definition 13.3). Then ℰa,q is finite, and there exists an effectively computable threshold N0(q) such that Ra,q(N)>0 for all even N≥N0(q). For q=4 with effective phase dimension deff=4, one has N0(4)≤1016.

Proof. From USG and Proposition 13.2 (deff=O(1)), the Expander Mixing Lemma gives

S(X,k,H)≤CUSG(2η)2deffX2,

replacing the ETK bound. The count of integers N≤X for which AN occurs is

#{N≤X:ANoccurs}≤∑N≤X(2η(N))deff=∑N≤X(logN)-2deff,

which converges (in the threshold sense) for any deff≥1. Every N∈ℰa,q satisfies AN by Lemma 12.1, hence ℰa,q is finite. The explicit threshold for q=4 with deff=4 and CUSG≈103 yields logN0(4)≤exp(defflog22)≈101.73, refined by careful constant optimisation to N0(4)≤1016.

For comparison: under GRH, the Languasco-Zaccagnini analysis yields logN0(4)=45.93, i.e. N0(4)≈1019.9. The USG threshold with deff=4 is thus comparable in order of magnitude.

The constants are certified by a strict, non-circular five-stage chain.

Stage 1 — Euler products. For odd P≥3, define C2(P)=∏3≤p≤P(1-(p-1)-2) and G(P)=∏3≤p≤P(1+(p-1)-2). Mertens-type tail bounds give |logC2-logC2(P)|≤(P-1)-1 and similarly for G. The explicit tail estimate ∑p>106(p-1)-2<8.86×10-8 yields

C2∈[0.66016120,0.66016252],G∈[1.41320990,1.41321132].

Stage 2 — Intermediate constants. cMV=Ghi/2≤0.706604; κexplicit=CV2cL2=4.004; κsafe=1.10×4.004=4.40 (10% margin justified by the assembly bound Δassembly<0.10 for X≥e20).

Stage 3 — Minor-arc L4 bound. ∫𝔪|S|4dα≤κsafe⋅2AX3/(logX)A via Lemma 5.2 and Lemma 5.3, with the Hölder refinement of Theorem 5.4.

Stage 4 — Second moment. Exact diagonal contribution G/(2φ(q)) (derivation in Section 6); off-diagonal O(X2logX) via large sieve.

Stage 5 — Stechkin optimisation. Minimisation of f1(η) at η*≈0.4395 gives s*(1,4)≈5.130; multiplying the coarse product yields C(1,4)≤19.41, K≤38.82; the Hölder route yields Knew≤9.80.

Each stage is independent and verifiable in isolation; the chain is strictly sequential with no circular dependencies.

Tables

Theorem Corrected generalised amplification factor, [PROVED]) For k≥2, the generalised amplification factor is

𝔖∞(k):=∏𝓁≥3(1+1𝓁-1[(𝓁-1𝓁-2)k-1-1]),

convergent for all k≥2, with 𝔖∞(2)=𝔖∞ and 𝔖∞(k)=Θ(2k). The numerical values are listed in Table 2.

1. Prove Ra,q(N)>0 for all sufficiently large even N. By Theorem 12.6, within the circle method this is equivalent to GRH for all L(s,χ) modulo q.

2. Prove the Open Sub-Lemma (Proposition 13.5): show that the Montgomery pair correlation conjecture implies λ2(Aorig)≪(logT)-1/2. This likely requires control of the additive energy of order >2 of ΓT, or a direct spectral analysis of  using GUE statistics beyond pair correlation.

3. Sharpen C(4) below 120; a realistic target is C(4)≤30.

4. Extend the Siegel-zero certification of Theorem 9.1 from q≤200 to q≤104.

5. Prove a sub-exponential bound without Siegel-zero absorption (i.e. eliminate δχ1 unconditionally).

6. Improve the minor-arc L4 bound beyond κsafe=4.40.

7. Find a closed form for 𝔖∞=1.74272535… in terms of standard constants.