Goldbach's Conjecture

Goldbach's Conjecture

Definitions and mathematical statement

With $\mathcal P$ denoting the positive primes, the statement is \[ \forall n\in2{\mathbb{Z}},\quad n>2\quad\Longrightarrow\quad \exists p,q\in\mathcal P\text{ with }n=p+q. \] The primes are not required to be distinct, so $4=2+2$ is allowed. Every even integer is quantified, not merely all sufficiently large integers or a density-one subset.

Formulation and concept references: [S1].

Short English statement

Can every even integer larger than two be written as the sum of two primes?

Research attempt

Scope and source check.

The target includes every even integer greater than two, allows equal primes, and therefore includes $4=2+2$. An asymptotic argument for large integers needs an explicit finite-range completion before it proves this formulation. Oliveira e Silva--Herzog--Pardi [E1] report verification through $4\cdot10^{18}$; that published computation is not rerun here. It is a finite theorem, not a replacement for control of the unbounded tail.

The recent catalogue source [S2], Bhowmik--Grimmelt's August 2026 revision, is largely a survey of exceptional-set methods, with additional explicit formula work. It does not assert that the exceptional set is empty. Li--Liu [S3], version 2, state a representation $N=p+rq$, with $p,q$ prime, $r=1$ or prime, and $r\le q^{0.9}$ for sufficiently large even $N$. This still permits $r>1$, so its stated conclusion is weaker than binary Goldbach. The manuscript statements were inspected; their full proofs were not independently audited. Neither claim is used as a premise of the elementary arguments below.

The selected route asks exactly what the circle method would have to prove for one prescribed even integer, and compares that with what an average-square error bound actually guarantees. The goal is to identify the first missing uniform estimate, not to infer pointwise positivity from an exceptional set of density zero.

An exact Fourier representation without prime powers.

For an integer $X\ge4$, put $e(t)=\exp(2\pi i t)$ and define \[ S_X(\alpha)=\sum_{p\le X}(\log p)e(p\alpha),\qquad R(N)=\sum_{\substack{p+q=N\\p,q\text{ prime}}}\log p\log q. \] The sum counts ordered pairs. For $4\le N\le X$, orthogonality gives \begin{equation}\label{wsref-0-72-61-3a-32-35-3a-66-6f-75-72-69-65-72} R(N)=\int_0^1 S_X(\alpha)^2e(-N\alpha)\,d\alpha. \end{equation} This follows by expanding a finite double sum and using $\int_0^1e(m\alpha)\,d\alpha=1$ for $m=0$, zero otherwise. All weights are positive, so $R(N)>0$ is exactly the desired existence statement. Proper prime powers are absent by definition, and no subtraction estimate for them is hidden in this equivalence.

The square in (ra:25:fourier) is not an absolute square. Replacing it by $|S_X|^2$ counts prime differences $p-q=N$. For example with $X=N=4$, the difference count is zero, whereas $R(4)=(\log2)^2>0$. Positivity of $|S_X|^2$ therefore cannot be used to prove positivity of the required Fourier coefficient. The actual coefficient is real, but substantial complex cancellation occurs inside the integral.

Local factors give a positive predicted main term.

For an odd prime $r$, count residue pairs $(a,b)$ with $a+b=N$ modulo $r$ and both coordinates nonzero. The count is $r-1$ when $r\mid N$ and $r-2$ otherwise: choose $a$, excluding zero and, in the second case, the distinct residue $N$. Relative to independent unit densities, the local multiplier is \[ \frac{r\,\#\{a\bmod r:a\ne0,\ N-a\ne0\}}{(r-1)^2}. \] For even $N$, the factor at two is two. Consequently the formal singular series can be written \begin{equation}\label{wsref-0-72-61-3a-32-35-3a-73-65-72-69-65-73} \mathfrak S_G(N)=2C_2 \prod_{\substack{r\mid N\\r>2}}\frac{r-1}{r-2}, \qquad C_2=\prod_{r>2}\left(1-\frac1{(r-1)^2}\right). \end{equation} The infinite product is strictly positive: the factors lie in $(0,1)$, and the sum of their deficits converges by comparison with $\sum_{n\ge2}n^{-2}$. For instance $-\log(1-u)\le2u$ for $0\le u\le1/2$ bounds the logarithmic tail. Thus $\mathfrak S_G(N)\ge c_0:=2C_2>0$ for every even $N$.

This derivation is an exact local calculation, not a proof that $R(N)\sim\mathfrak S_G(N)N$. CRT similarly multiplies the residue counts over a fixed squarefree modulus, but actual primality is a global restriction. The positive lower bound for the proposed main term is useful only if an accompanying analytic error can be controlled.

The small prime two requires a separate boundary check. If $N>4$ is even and one summand were two, the other would be the even integer $N-2>2$ and could not be prime. Thus every representation for even $N>4$ uses two odd primes, consistent with the local unit calculation at two. The exceptional representation of four is handled by the exact Fourier identity and the finite check, not by an asymptotic local model. Similarly, ordered counting never doubles a diagonal pair: when $N/2$ is prime it contributes one term of weight $(\log(N/2))^2$. These conventions ensure that the positive coefficient being targeted matches the precise existence statement at both the boundary and large values of $N$.

A precise pointwise sufficient estimate.

Choose $Q=(\log X)^B$ with fixed $B>0$ and define major arcs as the union, modulo one, of \[ |\alpha-a/q|\le Q/(qX),\qquad 1\le q\le Q,\quad 0\le a<q,\quad(a,q)=1. \] Let this union be $\mathfrak M_X$, with complement $\mathfrak m_X$, and set \[ A_X(N)=\int_{\mathfrak M_X}S_X(\alpha)^2e(-N\alpha)\,d\alpha, \quad I_X(N)=\int_{\mathfrak m_X}S_X(\alpha)^2e(-N\alpha)\,d\alpha. \] The sets are invariant under $\alpha\mapsto-\alpha$, making both integrals real. They sum to $R(N)$ exactly. Their definition is valid even when arcs overlap; for large $X$ the usual disjointness also follows from $X>2Q^3$ and separation of reduced fractions.

Suppose, for some explicit $X_0$ and all $X\ge X_0$, one could establish uniformly for even $X/2\le N\le X$ that \begin{equation}\label{wsref-0-72-61-3a-32-35-3a-74-61-72-67-65-74-73} |A_X(N)-\mathfrak S_G(N)N|\le c_0X/8, \qquad |I_X(N)|<c_0X/8. \end{equation} Then $R(N)>c_0X/4>0$. Dyadic intervals would cover every sufficiently large even integer. An explicit check of the remaining interval $4\le N<X_0$ would finish the exact catalogue statement. This implication is proved by the displayed estimates; the estimates themselves are not proved here. Classical major-arc analysis and the role of exceptional sets are discussed in [S2]. In this notebook no theorem with different uniformity or an implicit exceptional set is substituted for (ra:25:targets).

What a mean-square estimate really provides.

The elementary Bessel inequality already quantifies the loss when one passes from a single integer to most integers. Fix $X$ and suppose a major-arc lower bound $A_X(N)\ge aX$ is available on its even half-interval, for a fixed $a>0$. If such an $N$ has no representation, then $I_X(N)=-A_X(N)$, so $|I_X(N)|\ge aX$. Let $U=\sup_{\alpha\in\mathfrak m_X}|S_X(\alpha)|$. Orthogonality and Bessel give \begin{align*} \sum_{N\in\mathbb Z}|I_X(N)|^2 &=\int_{\mathfrak m_X}|S_X(\alpha)|^4\,d\alpha\\ &\le U^2\int_0^1|S_X(\alpha)|^2\,d\alpha =U^2\sum_{p\le X}(\log p)^2 \le U^2 X\log^2 X. \end{align*} The first equality is Parseval for the square-integrable function $1_{\mathfrak m_X}S_X^2$; restricting the sum to even $N$ would give an inequality, which is all we need. Therefore the number $E_X$ of exceptional even integers in that interval satisfies \begin{equation}\label{wsref-0-72-61-3a-32-35-3a-65-78-63-65-70-74-69-6f-6e-61-6c} E_X\le\frac{U^2\log^2X}{a^2X}. \end{equation} This derivation uses only the elementary bound on the number of primes; sharper familiar second-moment estimates are unnecessary for the point.

For example, even a bound $U\le X/(\log X)^A$ with fixed $A>1$ would give only $E_X\ll_a X/(\log X)^{2A-2}$. Its density tends to zero, but the bound grows and does not force the integer $E_X$ to vanish. Applying Cauchy--Schwarz directly to one coefficient instead gives $|I_X(N)|\le U\sqrt X\log X$, which is useful for positivity only with a far stronger supremum bound. Neither estimate captures the specific oscillatory cancellation of $e(-N\alpha)$ needed pointwise.

A sharp stress test of the average-to-pointwise step.

Consider an auxiliary array with predicted main value $aX$ at every even index in $[X/2,X]$. Let its actual value equal that prediction except at one index, where it is zero. All values remain nonnegative, and the sum of squared errors is exactly $a^2X^2=o(X^3)$. Thus even a mean-square error small relative to the aggregate natural scale $X^3$ is compatible with a missing representation. Choosing one such exceptional index in each dyadic interval makes infinitely many exceptions of density zero. These arrays are not prime counts and do not disprove Goldbach; they refute the logical inference from the stated norm bound to universal positivity.

An average theorem supplemented by finitely many direct checks has the same issue: the hypothetical sparse exceptional indices may all lie beyond the checked range. To use integrality, the final upper bound on $E_X$ must actually be strictly below one for every remaining interval, or a separate argument must remove each possible exceptional integer. Calling a positive real upper bound "small" cannot replace this step.

Exact finite checks and counting conventions.

The local script builds the prime indicator by an integer sieve and computes its ordinary polynomial convolution. If $g(N)$ is the number of unordered pairs with $p\le q$, the ordered count satisfies \[ r(N)=2g(N)-1_{N/2\text{ prime}}. \] It checks this identity and verifies every even $4\le N\le10000$ has a representation. Selected values are \[ \begin{array}{r|r|r} N&r(N)&g(N)\\\hline 4&1&1\\10&3&2\\100&12&6\\1000&56&28\\10000&254&127 \end{array} \] The diagonal correction is essential at four and ten. The same script enumerates all residues modulo $210$ and checks the product of the local counts $r-1$ or $r-2$ in (ra:25:series); for example there are twenty admissible ordered residue pairs for $N=100$ modulo $210$. These exact local and finite computations do not estimate either arc integral. Reproduction files are checks.py and results.json in _Local/top_problems/continuation_15_25_2026_09_27/analytic/.

Remaining gap and outcome.

The work proves the exact prime-weighted Fourier identity, the positive local-series bound, and the exceptional-set inequality with all normalizations visible. The first unresolved analytic step is a bound for the individual minor-arc coefficient strong enough to preserve positivity, paired with a major-arc estimate of the stated uniformity. Even proving eventual positivity would leave an effective threshold and the finite interval below it to address the full target.

Outcome: UNRESOLVED. No pointwise estimate of the required strength and no counterexample even integer were obtained. The explicit norm obstruction explains why an almost-all conclusion cannot be promoted to the requested theorem. The process used direct derivation, primary source inspection, and exact finite checks; it does not claim that the open problem has been solved or independently certified.

Sources

• S1 C. K. Caldwell, "Prime Conjectures and Open Questions," The PrimePages, https://t5k.org/notes/conjectures/. https://t5k.org/notes/conjectures/. Formulation source.
• S2 The exceptional set of the Goldbach problem. https://arxiv.org/html/2607.27282v2. Further reference; not a proof endorsement.
• S3 Theorem $1+1.9$ on the Goldbach Conjecture. https://arxiv.org/abs/2606.05224. Further reference; not a proof endorsement.
• S4 Restricted Goldbach Sums in Arithmetic Progressions: Local Obstructions, an Explicit Almost-All Framework, and a General-Modulus Extension. https://www.preprints.org/manuscript/202607.2077. Further reference; not a proof endorsement.

• E1 Tomás Oliveira e Silva, Siegfried Herzog, and Silvio Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot10^{18}$, Mathematics of Computation 83 $2014$, no. 288, 2033--2060, DOI: 10.1090/S0025-5718-2013-02787-1. https://www.ams.org/mcom/2014-83-288/S0025-5718-2013-02787-1/S0025-5718-2013-02787-1.pdf. Primary computational reference; that large computation was not rerun in this notebook. Consulted 27 September 2026.
• E2 Gautami Bhowmik and Lasse Grimmelt, The exceptional set of the Goldbach problem, arXiv:2607.27282v2 (13 August 2026), especially Sections 1 and 4--7. https://arxiv.org/html/2607.27282v2. Version-specific identification of [S2]: survey and new explicit-formula work, not an assertion that every even integer is represented. Consulted 27 September 2026.
• E3 Jiamin Li and Jianya Liu, Theorem $(1+1.9)$ on the Goldbach Conjecture, arXiv:2606.05224v2 (8 August 2026). https://arxiv.org/abs/2606.05224v2. Version-specific identification of [S3]: stated representation permits a semiprime summand. Statement inspected; complete proof not independently audited or used here. Consulted 27 September 2026.

Goldbach's Conjecture

Definitions and mathematical statement

With $\mathcal P$ denoting the positive primes, the statement is \[ \forall n\in2{\mathbb{Z}},\quad n>2\quad\Longrightarrow\quad \exists p,q\in\mathcal P\text{ with }n=p+q. \] The primes are not required to be distinct, so $4=2+2$ is allowed. Every even integer is quantified, not merely all sufficiently large integers or a density-one subset.

Formulation and concept references: [S1].

Short English statement

Can every even integer larger than two be written as the sum of two primes?

Research attempt

Current frontier.

The target is every even integer greater than two, not eventual validity or a density-one result. The published verification through $4\cdot10^{18}$ is a useful finite benchmark, not an argument about unbounded integers. [E1]. Helfgott's theorem for odd integers greater than five is a three-prime theorem, not the binary assertion. [E2]. Bhowmik--Grimmelt's August 2026 version surveys exceptional-set estimates and develops an explicit major-arc formula; its additional Siegel-zero consequence assumes a sparse Hardy--Littlewood estimate. [S2]. Li--Liu's August 2026 revision states that every sufficiently large even $N$ has $N=p+rq$ with $p,q$ prime, $r=1$ or prime, and $r\le q^{0.9}$. That recent manuscript theorem, whose complete proof was not independently audited here, still permits $r>1$ and a finite initial range. [S3]. None of these statements is treated as the exact target.

Attack route.

I pursue a pointwise convolution lower bound, explicitly accounting for proper prime powers and the finite initial segment. The missing estimate is individual control of the minor-arc coefficient, not a mean-square bound allowing exceptions. A competing idea is that sufficiently small nonzero Fourier coefficients, together with the correct average density, might force positive pair counts. An exact quadratic-residue model below falsifies that proposed inference even with square-root Fourier bounds. Thus the calculation tests what more arithmetic information the selected route needs, rather than merely repeating a heuristic independence model.

Attempt.

Let $\Lambda(m)=\log p$ if $m=p^a$ for a prime $p$ and integer $a\ge1$, and zero otherwise. All logarithms in this entry are natural, except $\log_2$. Define \[ R(N)=\sum_{m=1}^{N-1}\Lambda(m)\Lambda(N-m),\qquad R_{\rm prime}(N)=\sum_{p+q=N}(\log p)(\log q), \] where the latter sum is ordered and includes $p=q$. Thus $R_{\rm prime}(N)>0$ is exactly the desired representation for $N$.

Lemma 25.1 (a fully explicit prime-power error). For every integer $N\ge4$, \begin{equation}\label{wsref-0-72-32-35-3a-70-6f-77-65-72-73} 0\le R(N)-R_{\rm prime}(N) \le B(N):=2\sqrt N\lfloor\log_2N\rfloor(\log N)^2=o(N). \end{equation} Proof. For each exponent $a\ge2$ with $2^a\le N$, the number of powers $p^a\le N$ is at most $N^{1/a}\le\sqrt N$. Summing over at most $\lfloor\log_2N\rfloor$ exponents bounds the number of proper prime powers, even with overcounting. In every term of the difference, one of $m,N-m$ is a proper prime power. There are at most twice that many ordered terms, and each weight is at most $(\log N)^2$. Nonnegativity is immediate, and $N^{-1/2}(\log N)^3\to0$ gives the last statement.$\square$

Let $e(t)=e^{2\pi it}$ and $S_N(\alpha)=\sum_{m=1}^{N}\Lambda(m)e(m\alpha)$. Orthogonality of finite exponential sums gives the exact identity \begin{equation}\label{wsref-0-72-32-35-3a-63-69-72-63-6c-65} R(N)=\int_0^1S_N(\alpha)^2e(-N\alpha)\,d\alpha. \end{equation} There is no infinite-series interchange here. For a chosen $P\ge1$, let $\mathfrak M_N$ be the union on ${\mathbb{R}}/{\mathbb{Z}}$ of intervals of radius $P/N$ centered at reduced fractions $a/q$ with $1\le q\le P$; write $\mathfrak m_N$ for the complement. Define $I_{\mathfrak M}$ and $I_{\mathfrak m}$ by restricting the integral in (r25:circle). Both are real: each domain is invariant under $\alpha\mapsto-\alpha$, which conjugates the integrand.

The expected major-arc coefficient is \begin{equation}\label{wsref-0-72-32-35-3a-73-69-6e-67-75-6c-61-72} \mathfrak S(N)=2C_2\prod_{\substack{p\mid N\\p>2}}\frac{p-1}{p-2}, \qquad C_2=\prod_{p>2}\left(1-\frac1{(p-1)^2}\right)>0 \quad(N\text{ even}). \end{equation} The positivity and convergence of $C_2$ follow because every factor is positive and $\sum_{p>2}(p-1)^{-2}<\infty$; for $0\le u\le1/4$, $|\log(1-u)|\le(4/3)u$. In particular $\mathfrak S(N)\ge2C_2$. Its role in the major-arc analysis and the limitations of the error terms are detailed in [S2], Sections 2--5. Positivity of this formal main term alone does not prove positivity of the actual coefficient.

Lemma 25.2 (a criterion retaining every quantified integer). Suppose an explicit $N_0\ge4$ and a constant $c>0$ are supplied such that for every even $N>N_0$ one has \begin{equation}\label{wsref-0-72-32-35-3a-6d-69-73-73-69-6e-67} I_{\mathfrak M}(N)\ge cN,\qquad |I_{\mathfrak m}(N)|\le cN/2,\qquad B(N)<cN/2. \end{equation} If every even integer $4\le N\le N_0$ is certified as a sum of two primes, then the full binary Goldbach statement follows.

Proof. For $N>N_0$, (r25:circle) gives $R(N)\ge cN/2$. Subtracting (r25:powers) yields $R_{\rm prime}(N)>0$. The finitely many remaining even integers are covered by the certificates, including $4=2+2$.$\square$

The implication is complete, but its analytic hypotheses have not been proved here. Standard major-arc asymptotics motivate the first condition; the unresolved universal minor-arc condition is much stronger than an estimate holding outside a small exceptional set. Nor does the bare existence of an ineffective threshold eliminate the need to handle its finite predecessors. An eventual result cannot be presented as a proof that no small exceptional integer exists.

Why average control does not force pointwise positivity. Suppose an analysis on even $N\in[X,2X]$ gives $R(N)=\mathfrak S(N)N+E_N$. If $N$ in this range is a counterexample, then $R(N)\le B(N)$, so for sufficiently large $X$ it satisfies $|E_N|\ge C_2X$. Consequently \begin{equation}\label{wsref-0-72-32-35-3a-65-78-63-65-70-74-69-6f-6e-61-6c} \#\{N\in[X,2X]:N\text{ even and exceptional}\} \le\frac1{C_2^2X^2}\sum_{\substack{X\le N\le2X\\N\text{ even}}}|E_N|^2. \end{equation} An upper bound $O(X^{3-\delta})$ with $0<\delta<1$ on the sum gives $O(X^{1-\delta})$ exceptions, which still allows one exception or an infinite sparse sequence. The inference is an exact counting inequality, not a criticism of the genuine strength of exceptional-set results. The distinction is central to [S2].

Also, discarding the oscillating factor in the minor-arc integral gives only \[ |I_{\mathfrak m}(N)|\le\int_0^1|S_N(\alpha)|^2d\alpha =\sum_{m\le N}\Lambda(m)^2. \] Even the elementary bound $N(\log N)^2$ is too large for (r25:missing); sharper average estimates do not automatically supply the required individual cancellation. The following finite-group model makes that logical obstruction particularly explicit.

Lemma 25.3 (square-root Fourier cancellation with a missing pair sum). Let $p\equiv3\pmod4$ be prime and let $\chi$ be its quadratic character, with $\chi(0)=0$. On the additive group ${\mathbb{F}}_p$ define $g(0)=0$ and $g(x)=1+\chi(x)$ for $x\ne0$. Then \begin{equation}\label{wsref-0-72-32-35-3a-70-61-6c-65-79} g\ge0,\quad \sum_xg(x)=p-1,\quad (g*g)(0)=0, \quad |\widehat g(a)|=\sqrt{p+1}\quad(a\ne0), \end{equation} where $\widehat g(a)=\sum_xg(x)e(ax/p)$ and $(g*g)(z)=\sum_xg(x)g(z-x)$.

Proof. Exactly half the nonzero elements are squares, so $g$ is two on them and zero elsewhere. Since $-1$ is not a square, $\chi(-x)=-\chi(x)$ for $x\ne0$, which gives $g(x)g(-x)=0$, including at zero. This proves the first three assertions. For $a\ne0$ put $\tau_a=\sum_x\chi(x)e(ax/p)$. Then $\widehat g(a)=-1+\tau_a$. Complex conjugation and $x\mapsto-x$ give $\overline{\tau_a}=-\tau_a$. To compute its modulus, substitute $x=ty$ for $x,y\ne0$: \[ |\tau_a|^2 =\sum_{t\ne0}\chi(t)\sum_{y\ne0}e(ay(t-1)/p) =(p-1)-\sum_{t\ne0,1}\chi(t)=p. \] The inner sum is $p-1$ for $t=1$ and $-1$ otherwise, while $\sum_{t\ne0}\chi(t)=0$. Thus $\tau_a$ is purely imaginary of modulus $\sqrt p$, and $|-1+\tau_a|=\sqrt{p+1}$, proving (r25:paley).$\square$

The average of $g$ tends to one and every nontrivial Fourier coefficient is of square-root order, but one convolution coefficient vanishes exactly. Such primes are arbitrarily large: if only finitely many primes $3\pmod4$ existed, $4$ times their product minus one would have a new prime divisor $3\pmod4$. Hence this is not merely a fixed-size anomaly. In Fourier inversion the nonzero frequencies can collectively cancel the positive zero-frequency contribution. The calculation is an exact stress test, not a claim about the distribution of actual primes, and not a Goldbach counterexample. It shows why a route based only on average density and generic Fourier cancellation requires further arithmetic structure in the two-summand case.

Stress test.

The threshold in Lemma 25.2 is part of its hypothesis. The published finite verification can close a gap only when an independently justified analytic threshold lies inside that verified range, not merely because both statements involve large numbers. Proper prime powers were subtracted explicitly; positivity of $R(N)$ alone need not yield two primes. The repeated-prime case is never excluded. No GRH, no Landau--Siegel-zero exclusion, and no unproved distribution conjecture has been silently used to establish (r25:missing).

Li--Liu's parameter $1.9$ restricts the smaller factor $r$ of the almost prime $rq$; it does not mean a nonintegral count of prime factors or force $r=1$. [S3]. The quadratic-residue model is not proposed as an arithmetic substitute for primes: it only falsifies the stated general Fourier implication. Better structured major/minor-arc arguments may avoid that obstruction, but must prove their extra estimates.

The companion program uses an exact sieve to produce a prime pair for every even integer from four through $20{,}000$. It also checks the proper-prime-power counting bound and the quadratic-character identities for small primes $3\pmod4$, without numerical Fourier approximations. These are reproducible finite checks, far below the published range; they add no evidence of a new analytic threshold and no proof for unbounded $N$.

\Needspace{8\baselineskip}

Outcome.

Outcome: Conditional reduction.

The attempt gives an explicit all-integer positivity criterion, with prime-power contamination and finite-range obligations separated, and proves a square-root Fourier countermodel to a tempting generic shortcut. The central uniform minor-arc estimate has not been established, and no Goldbach counterexample has been found. The classical circle-method framework and finite-field calculation are not claimed as historically new. The remaining step is a pointwise arithmetic estimate strong enough to exclude every exceptional even integer, together with a certified initial range, not just a smaller exceptional-set density.

Sources

• S1 C. K. Caldwell, "Prime Conjectures and Open Questions," The PrimePages, https://t5k.org/notes/conjectures/. https://t5k.org/notes/conjectures/. Catalogue formulation source.
• S2 The exceptional set of the Goldbach problem. https://arxiv.org/html/2607.27282v2. Catalogue research/status reference; not a proof endorsement.
• S3 Theorem $1+1.9$ on the Goldbach Conjecture. https://arxiv.org/abs/2606.05224. Catalogue research/status reference; not a proof endorsement.
• S4 Restricted Goldbach Sums in Arithmetic Progressions: Local Obstructions, an Explicit Almost-All Framework, and a General-Modulus Extension. https://www.preprints.org/manuscript/202607.2077. Catalogue research/status reference; not a proof endorsement.
• E1 Tomás Oliveira e Silva, Siegfried Herzog and Silvio Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot10^{18}$, Mathematics of Computation 83 $2014$, 2033--2060. https://doi.org/10.1090/S0025-5718-2013-02787-1. Author's project page: https://sweet.ua.pt/tos/goldbach.html. Published finite benchmark and author's account consulted 27 September 2026; not asserted to settle the unbounded problem.
• E2 Harald A. Helfgott, The ternary Goldbach conjecture is true, arXiv:1312.7748. https://arxiv.org/abs/1312.7748. Three-prime theorem and its distinct quantifiers; abstract consulted 27 September 2026.