Constructive Method for Goldbach Representations Yielding Generated-Family Bounds and a Prime-Selector Heuristic

Abstract

Goldbach’s binary problem asks whether every even integer greater than two is a sum of two primes. This paper studies a direct two-stage interval construction that, from a sufficiently large even input 2n, selects a prime p′ just below 2n−5, forms the residual M=2n−p′, and then selects a prime q′ above M. Writing q′=M+2c yields a Goldbach representation p′+q′=2n+2c. The earlier shift identity p′=p+2k, q′=q−2k+2c is retained only as an equivalent parametrization when an input representation p+q=2n is available. Using the Baker-Harman-Pintz exponent θ=0.525, together with a finite treatment below the asymptotic threshold, the iterated construction generates Ω(x^0.724375) Goldbach even integers up to x. Under an explicit unproved Cramér-type O((log x)^2) gap assumption, the two-scale argument gives c=O((log log n)^2) and Ω(x/(log log x)^2) generated values. A separate selector heuristic uses the Hardy-Littlewood singular series but does not assume that candidate events are independent as a theorem. The revised numerical experiment reports the complete distribution of S_G(n) and the minimal shift c for 900,001 inputs. Neither full Goldbach nor generated density one is proved.

Keywords: Goldbach conjecture; primes in short intervals; prime gaps; Cramér model; singular series; constructive number theory; probabilistic heuristic.

Introduction and Literature Review

Exceptional sets in the binary Goldbach problem

Hardy and Littlewood supplied the classical circle-method heuristic for binary Goldbach, including the arithmetic correction now expressed through the singular series¹. On the unconditional side, Montgomery and Vaughan proved a power-saving estimate for the exceptional set: if E(x) counts even integers up to x that are not sums of two primes, then E(x)≪x^(1−δ) for some δ>0². Li obtained a later refinement of the exceptional-set estimate³. Matomäki studied the exceptional set in short intervals⁴, Lu obtained a further exceptional-set bound⁵, and Grimmelt studied Goldbach numbers in short intervals⁶. A 2026 paper of Bhowmik and Grimmelt reviews this line and adds an explicit major-arc formula and a conditional observation concerning exceptional zeros⁷.

What is known from this literature is therefore much stronger than mere infinitude: Goldbach-representable even integers have density one. What remains open is the elimination of the exceptional set altogether. The present paper does not improve E(x). Its unconditional result concerns a particular iteratively generated subfamily, so its counting theorem must be compared with exceptional-set theory only to clarify scope, not as a competing density-one result.

Primes and Goldbach numbers in short intervals

The unconditional interval input used in this paper is Baker-Harman-Pintz: for all sufficiently large x, the interval [x−x^0.525,x] contains a prime⁸. Explicit interval estimates have continued to be developed; for example, Cully-Hugill and Lee give explicit prime-location estimates⁹. Recent work has also obtained conditional short-interval results directly for Goldbach numbers. Assuming the Riemann Hypothesis, Cully-Hugill and Dudek proved that every interval (x,x+9696(log x)^2] contains a Goldbach number¹⁰, and Chirre and Hagen improved the constant 9696 to 123¹¹. In a different direction, Guth and Maynard proved new zero-density estimates that imply asymptotics for primes in intervals of length x^(17/30+o(1))¹².

These statements answer different questions. Baker-Harman-Pintz gives the one-sided existence theorem used here; Guth-Maynard concerns asymptotic prime counts on a longer power scale; the 2024 and 2026 Goldbach-interval papers are conditional on RH and concern the presence of at least one Goldbach number in a nearby interval. The construction below instead applies a prime-in-interval input twice, at two different scales, to control the jump it generates.

Prime gaps and local structure

Small-gap results show that bounded gaps occur infinitely often, but they do not furnish a uniform bound for the next prime after every integer. The Goldston-Yıldırım and Goldston-Pintz-Yıldırım correlation methods initiated the modern small-gap framework^13,14,15; Zhang, the Polymath project, and Maynard established and refined bounded-gap results^16,17,18. Large-gap results of Maynard, Ford-Green-Konyagin-Tao, and Ford-Green-Konyagin-Maynard-Tao demonstrate that unusually long prime-free intervals also occur^19,20,21. Fourier-optimization work provides another perspective on quantitative gap questions²².

Thus a single worst-case gap function suppresses information about the range of available primes inside a search window. That distinction matters here: the deterministic part of the paper uses a worst-case interval guarantee, whereas the unit-step question is a selector problem that asks whether at least one first-stage prime has a favorable complementary residual.

Cramér-type gap heuristics

Cramér introduced a random model for primality and used it to motivate a maximal-gap scale of order (log x)^2²³. Granville explained important arithmetic limitations of the naive independence model and refined the heuristic picture²⁴. Granville and Lumley later compared short-interval heuristics with computations²⁵, while Banks, Ford, and Tao introduced a sieve-aware probabilistic model that more faithfully incorporates residue-class obstructions and yields heuristic information on large gaps²⁶.

No uniform O((log x)^2) gap theorem is known for the primes. Accordingly, every use of that scale below is labeled an assumption or heuristic. The paper’s conditional result is not presented as a consequence of the prime number theorem, RH, or the Baker-Harman-Pintz theorem.

Singular series and probabilistic selector models

For binary Goldbach, Hardy-Littlewood’s heuristic predicts a representation count with a multiplicative local correction depending on the prime divisors of the target even integer¹. Modern work continues to analyze these local factors: Goldston and Suriajaya study a Cesàro mean of the prime-pair singular series²⁷; Leung proves conditional Poisson point-process limits under a uniform prime-k-tuple hypothesis and obtains mean-value results for singular series²⁸; and Chou, Haag, Huryn, and Ledoan relate the error term in the Hardy-Littlewood prime-pair formula to an exponential sum over primes²⁹.

The missing step for the selector model in this paper is not the local factor itself but a theorem controlling dependence among the different candidate complements N−r. The singular series corrects congruence biases; it does not by itself justify independence. This is why the Poisson calculation below is used only as a qualitative heuristic and is tested numerically rather than promoted to a theorem.

Position of the present work

Languasco and Zaccagnini study the number of Goldbach representations³⁰. Bauer and Wang treat binary Goldbach with primes restricted to arithmetic progressions with large modulus³¹, while Bauer applies a sparse-moduli large-sieve inequality to the Goldbach problem³². Oliveira e Silva, Herzog, and Pardi give large-scale computational verification of even Goldbach and prime gaps³³. Platt and Trudgian study Linnik’s approximation to Goldbach and related questions³⁴. Friedlander, Goldston, Iwaniec, and Suriajaya specifically examine exceptional zeros in the Goldbach problem³⁵. Distributional questions for primes in short and logarithmic intervals have also been developed by Montgomery-Soundararajan, Bazzanella-Languasco-Zaccagnini, and Mastrostefano^36,37,38, while Languasco studies a conditional short-interval exceptional-set problem³⁹.

Against this background, the research question here is deliberately narrower: how far can a simple two-stage residual construction be pushed from a rigorous prime-in-interval input, and what extra selector information would be required to force the unit transition 2n→2n+2? The proved contribution is a generated-family lower bound. The Cramér-scale improvement is conditional, and the density-one unit-step statement is only a model prediction.

Known Results and Heuristic Background

Baker-Harman-Pintz short-interval theorem

Baker, Harman, and Pintz prove that there exists x0 such that every x≥x0 has a prime in [x−x^0.525,x]⁸. The strict one-sided statements needed by the construction follow as corollaries. For a prime strictly below y, apply the theorem at x=y−1; after enlarging the constant and threshold, this gives y−B y^0.525<p<y. For a prime strictly above y, fix A>1 and put z=y+A y^0.525. For sufficiently large y, z^0.525<A y^0.525, hence z−z^0.525>y. Applying Baker-Harman-Pintz at z gives y<p≤y+A y^0.525. After enlarging constants and thresholds, one may therefore use a common B for the strict-below and strict-above searches.

Cramér gap conjecture and model

Cramér’s 1936 probabilistic model treats integers near x as independently prime with probability about 1/log x and motivates maximal gaps on the (log x)^2 scale²³. Arithmetic correlations invalidate literal independence, and Granville’s analysis explains why local congruence structure changes refined predictions²⁴. In this paper the statement ‘a prime occurs within A(log y)^2 on either side of y’ is therefore introduced explicitly as an unproved assumption, not as Cramér’s theorem.

Hardy-Littlewood Goldbach heuristic and singular series

For a large even integer N, the Hardy-Littlewood heuristic predicts a Goldbach representation count with leading factor 2C2 S(N), where C2 is the twin-prime constant and S(N) is the local correction¹. In the normalization used below,

C2 := ∏_{p>2}(1 − 1/(p−1)^2),

S(N) := ∏_{p|N, p>2}(p−1)/(p−2).

The factor S(N) is deterministic arithmetic data. A Poisson law for the selector count would require substantially more than inserting S(N) into a first-moment estimate. Conditional results such as Leung’s show how Poisson behavior can arise when a suitably uniform Hardy-Littlewood prime-tuple hypothesis is assumed²⁸; no such theorem is assumed here.

Construction Method and Prime Distribution Bounds

Let 2n be a sufficiently large even integer. Set X=2n−5. Choose a prime p′ in the first-stage interval and define the compressed residual

M := 2n − p′.

A second prime q′ is selected above M. Since M and q′ are odd, q′−M is even, so there is a positive integer c with

q′ = M + 2c,

and consequently

p′ + q′ = p′ + (2n − p′) + 2c = 2n + 2c.

This is the substantive two-stage construction.

Relation to the original shift parametrization

If an input Goldbach representation p+q=2n is available, the same first-stage prime can be written p′=p+2k with k=(p′−p)/2∈Z. Since p=2n−q,

q − 2k = q − (p′−p) = p+q−p′ = 2n−p′ = M.

Hence q′=q−2k+2c. The identities in p,q,k are therefore an algebraic parametrization of the direct construction. After p′ is selected, the residual M and the second prime do not depend on the particular starting representation.

Unconditional short-interval bound

By the Baker-Harman-Pintz theorem and the strict one-sided corollaries stated above, there are constants B>0 and y0 such that for every y≥y0 the required prime searches can be completed within B y^θ, with θ=0.525⁸. For sufficiently large n choose

p′ ∈ (X − B X^θ, X),    X=2n−5.

Then

5 < M = 2n−p′ < 5 + B(2n−5)^θ.

Let M0 be a threshold beyond which the upper-side corollary is valid. If M≥M0, choose a prime q′ with M<q′≤M+B M^θ. Writing q′=M+2c gives

0 < c ≤ (B/2)M^θ ≤ (B/2)[5+B(2n−5)^θ]^θ.

If 5<M<M0, only finitely many odd residuals occur. Let P+(M) be the least prime greater than M and define

C0 := max_{5<M<M0, M odd} (P+(M)−M)/2.

Then c≤C0 in the bounded range. Since the displayed power scale tends to infinity, C0 is absorbed for sufficiently large n, yielding

c = O(n^(θ^2)).

With θ=0.525, θ^2=0.275625.

Worked Numerical Example of the Construction

The following example illustrates the mechanics of the construction and the difference between an arbitrary admissible first-stage choice and a unit-step selector. It uses the normalized heuristic window G(x)=(log x)^2 only for illustration; it is not a Baker-Harman-Pintz certificate. Take 2n=200000. Then X=199995 and G(X)≈148.99. The prime p′=199967 lies in (X−G(X),X). It gives M=200000−199967=33. The least prime greater than 33 is q′=37, so q′=M+4=M+2c with c=2, and

199967 + 37 = 200004 = 200000 + 2(2).

A different admissible first-stage prime, p′=199961, gives M=39. The least prime above 39 is 41=M+2, so c=1 and

199961 + 41 = 200002.

Thus the interval construction itself guarantees a later Goldbach even, while the selector question asks whether one can choose p′ so that the later even is the immediate successor 2n+2.

Fundamental Counting and Generated-Family Bound

Starting from any fixed sufficiently large even input, iterate the two-stage construction and denote the successive outputs by 2n1<2n2<⋯. Each output is Goldbach by construction, and the next application depends only on the new even input. Let a(x) count generated values not exceeding x.

From c=O(n^(θ^2)), there is a constant C such that while 2nj≤x,

2(n_{j+1}−n_j)=2c_j ≤ C x^(θ^2).

A deterministic covering count therefore gives

a(x) ≫ x^(1−θ^2).

Equivalently, with d_gen(x):=a(x)/(x/2), one has d_gen(x)≫x^(−θ^2). For θ=0.525,

a(x) = Ω(x^0.724375).

This lower bound concerns only the family generated by the construction. It is not a density-one theorem for Goldbach numbers and is not a proof of binary Goldbach.

Corrected Two-Scale Cramér-Type Analysis

The fixed-exponent argument evaluates the second prime search at the compressed residual M. The same scale separation is essential under a Cramér-type hypothesis. Motivated by Cramér’s model and later refinements^23,24,25, assume explicitly that there exist A>0 and Y0 such that every y≥Y0 has a prime in (y−A(log y)^2,y) and a prime in (y,y+A(log y)^2].

Choose p′ in (X−A(log X)^2,X). Then

5<M<5+A(log(2n−5))^2,

so M=O((log n)^2). If M≥Y0, choose q′ with M<q′≤M+A(log M)^2; then

c ≤ (A/2)(log M)^2 = O((log log n)^2).

If 5<M<Y0, the finite set of possible residuals is bounded by a constant C1 exactly as in the unconditional argument. Hence

c = O((log log n)^2).

The same covering count yields the conditional bounds

a(x)=Ω(x/(log log x)^2),    d_gen(x)≫1/(log log x)^2.

These statements are conditional generated-family bounds; they do not imply full Goldbach or generated density one.

Limits of the Nested Max-Gap Certificate for Unit Steps

Let G(x) be a nondecreasing gap function that certifies the two prime searches at the relevant scales. The first stage gives 5<M<5+G(2n−5), and the second gives

2c ≤ G(M) ≤ G(5+G(2n−5)).

Therefore

G(5+G(2n−5)) < 4

is sufficient for this particular worst-case bound to force c=1, because 2c is a positive even integer. If G is nondecreasing and unbounded, the displayed sufficient condition eventually fails. This only shows that this nested worst-case certificate stops forcing c=1; it does not rule out a selector argument using the multiplicity of first-stage primes.

A Heuristic Prime-Selector Model for the Unit-Step Transition

Selector count

Let G(x) be the first-stage window length and put N=2n+2. A unit step requires q′=M+2 and p′+q′=N. Writing q′=r, both r and N−r must be prime, and the first-stage window implies 7<r≤7+G(2n−5). Define

S_G(n) := Σ_{7<r≤7+G(2n−5), r prime} 1_{N−r is prime}.

Then S_G(n)>0 is exactly the event that the allowed window contains a unit-step selector.

First-moment heuristic and local correction

A window of length G contains heuristically about G/log G prime candidates. For each candidate r, the complementary integer N−r has size about N. Using the Hardy-Littlewood local factor¹, the heuristic mean is

λ(n) := 2 C2 S(N) G(2n−5) / [log N · log G(2n−5)].

The singular series accounts for congruence obstructions. It does not establish independence among the events {N−r is prime}; modern singular-series and conditional Poisson results likewise distinguish local corrections from a full independence theorem^27,28,29. Consequently the approximation P(S_G(n)=0)≈e^(−λ(n)) is used only as a model.

For the normalized heuristic G(x)=(log x)^2, λ(n) is of order C2 S(2n+2) log(2n)/log log(2n), so the model predicts a decreasing zero probability. This is not a proved density-one statement.

Corrected large-n singular-series illustration

At an odd prime p, averaging the factor (p−1)/(p−2) over even N gives 1+1/[p(p−2)]. Hence the product of mean local factors is 1/C2≈1.5147801281, rather than 1. For n=10^69, direct factorization gives S(2n+2)=1.4983622183666705. With natural logarithms and G(2n−5)=(log(2n−5))^2:

Quantity	Value
log(2n)	159.5715185971491
log log(2n)	5.072492214185656
S(2n+2)	1.4983622183666705
G(2n−5)	25463.0695474003
λ(n)	31.11724960392055
e^(−λ)	3.061612377665808 × 10^−14

The numerical value is illustrative only because the independence approximation is unproved.

Numerical Experiment

Method

The finite experiment evaluates every integer n from 100,000 through 1,000,000 inclusive, for 900,001 inputs, with G(x)=(log x)^2 and natural logarithms. A sieve of Eratosthenes is constructed through 2,000,100, which covers every primality query in the experiment. For each n, set N=2n+2 and enumerate every prime r with 7<r≤7+G(2n−5); S_G(n) is the number for which N−r is also prime. The singular factor S(N), the model mean λ(n), and e^(−λ(n)) are computed from the formulas above.

For the minimal shift, enumerate every odd residual M satisfying 5<M<5+G(2n−5) for which p′=2n−M is prime. Let P+(M) be the least prime greater than M and define c(M)=(P+(M)−M)/2. The recorded quantity is c_min(n)=min_M c(M). By construction, c_min(n)=1 if and only if S_G(n)>0, providing an internal consistency check between the two computations.

Across the full range, the empirical mean of S_G(n) is 5.8146990948 and the empirical variance is 8.5190218486. There are 2,985 zero-selector inputs, a frequency of 0.3316663%. The average model mean is 5.2489892324, while the average independent-Poisson zero probability is 1.51714135%. Thus the model has the correct rough scale for the mean but substantially overpredicts zero selectors on this finite range.

Full distribution of S_G(n)

S_G(n)	Count	Percent
0	2,985	0.331666%
1	20,091	2.232331%
2	61,813	6.868103%
3	113,914	12.657097%
4	144,838	16.093093%
5	137,101	15.233428%
6	109,656	12.183986%
7	82,507	9.167434%
8	65,170	7.241103%
9	52,095	5.788327%
10	39,837	4.426328%
11	28,148	3.127552%
12	18,253	2.028109%
13	11,224	1.247110%
14	6,248	0.694221%
15	3,263	0.362555%
16	1,580	0.175555%
17	757	0.084111%
18	315	0.035000%
19	125	0.013889%
20	53	0.005889%
21	19	0.002111%
22	5	0.000556%
23	3	0.000333%
24	1	0.000111%

The distribution is concentrated around 4–7 selectors but has a visible right tail through 24 in this range. Its variance exceeds its mean, so a single Poisson law with one fixed parameter would not describe the mixture across n even before candidate dependence is considered.

Distribution of the minimal shift c

c_min	Count	Percent
1	897,016	99.668334%
2	2,976	0.330666%
3	9	0.001000%

The value c_min=1 occurs for 897,016 of the 900,001 inputs, or 99.668334%. The remaining 2,985 inputs are exactly the S_G(n)=0 cases: 2,976 have c_min=2 and 9 have c_min=3. No tested input requires c_min≥4. This strong finite-range concentration is empirical evidence about the chosen window; it is not an asymptotic theorem.

Comparison with Known Exceptional-Set Results

Let E(x) be the number of even m≤x that are not sums of two primes. Montgomery and Vaughan’s power-saving estimate already implies that Goldbach-representable even integers have density one². Li and Lu obtained later exceptional-set refinements^3,5; Matomäki studied exceptional sets in short intervals⁴; and Grimmelt studied Goldbach numbers in short intervals⁶. The 2026 treatment of Bhowmik and Grimmelt gives a current overview together with new major-arc information⁷.

The theorem proved here is of a different type. The unconditional bound Ω(x^0.724375) counts only values reached by one deterministic construction. It neither improves E(x) nor approaches the strongest density statements about all Goldbach numbers. Its purpose is structural: the two-stage argument makes explicit what follows from a prime-in-interval input alone and isolates the additional selector property needed for unit steps.

Limitations

The paper has five principal limitations. First, the generated-family lower bounds concern only outputs of the construction and do not prove binary Goldbach or generated density one. Second, Baker-Harman-Pintz is asymptotic, so finite residuals must be handled separately and no explicit numerical threshold is claimed here. Third, the O((log x)^2) prime-gap input is an unproved Cramér-type assumption rather than a theorem. Fourth, the selector calculation uses a first-moment Hardy-Littlewood correction together with a Poisson independence approximation that is not justified for the candidate events in this problem. Fifth, the computation covers only 100,000≤n≤1,000,000 with the specific normalization G(x)=(log x)^2; the observed 99.668334% unit-selector frequency and the empirical distributions do not establish any limiting density.

Open Problems and Future Directions

Residual prime-selector conjecture

The selector question can be stated directly. For a chosen window G, ask whether every sufficiently large n admits a prime

p′ ∈ (2n−5−G(2n−5), 2n−5)

such that 2n−p′+2 is prime. This remains an unproved conjectural condition. The Poisson model and the finite experiment do not prove it.

Correlation-sensitive analysis

A natural next step is a second-moment or sieve analysis of S_G(n) that tracks dependence among the complements N−r. The correlation and small-gap methods of Goldston-Yıldırım and Goldston-Pintz-Yıldırım provide relevant precedent^13,14,15, while the later bounded-gap literature supplies complementary sieve techniques^16,17,18.

Minimal-shift statistics

The computed distribution of c_min suggests studying its moments, tails, dependence on S(2n+2), and sensitivity to the first-stage window. Such questions are numerical or conjectural at present. Large-scale Goldbach and prime-gap computation provides a useful benchmark for designing those experiments³³.

Conclusion

The construction is best viewed as a direct two-stage residual procedure. From a sufficiently large even input 2n, it selects p′ near 2n−5, forms M=2n−p′, and selects the next controlled prime above M, thereby producing a later Goldbach even 2n+2c. The p,q,k notation is an equivalent parametrization when an initial Goldbach representation is available, but it is not the mechanism of the construction.

Using the Baker-Harman-Pintz exponent θ=0.525, with the strict one-sided searches derived as corollaries and the bounded residual range treated separately, the construction gives c=O(n^0.275625) and Ω(x^0.724375) generated values. Under an explicitly unproved Cramér-type gap hypothesis it gives c=O((log log n)^2) and Ω(x/(log log x)^2). The nested max-gap inequality is only a sufficient unit-step certificate.

The selector remains heuristic. The full computation on 900,001 inputs finds S_G(n)=0 only 2,985 times and c_min=1 in 99.668334% of cases, but the empirical zero rate is materially below the independent-Poisson prediction. That discrepancy reinforces the central unresolved point: local singular-series corrections are not a substitute for a correlation theorem. The proved contribution is therefore the generated-family analysis and the separation of worst-case interval control from the stronger selector behavior required for c=1.

Acknowledgment

The author thanks the mathematical community and the NHSJS reviewers for detailed comments that improved the precision, literature framing, and numerical presentation of the manuscript.

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