Inverse Goldbach Conjecture: How OpenAI’s Proof Rules Out Primes as A + B

Two sets of integers, one addition sign, and every sufficiently large prime: that is the deceptively simple target of the Inverse Goldbach Conjecture. OpenAI’s September 24, 2026 manuscript, The Additive Indecomposability of the Primes, presents a proof that this construction is impossible when each set contains at least two nonnegative integers. Even allowing finitely many mistakes cannot rescue it. The argument combines sieve restrictions, character cancellation, Fourier analysis, and binary-tree comparisons. Its decisive move is to preserve two requirements simultaneously: generate every large prime, and stop generating large composites.

1. What Is the Inverse Goldbach Conjecture, and How Is It Different From Goldbach?

A sumset contains every sum obtained by choosing one element from each input set:

For example, $A=\{0,2\}$ and $B=\{3,5\}$ produce $A+B=\{3,5,7\}$. The repeated representation of five counts once. Crucially, one cannot keep selected pairs and discard the inconvenient sums. Every pair contributes to the same output set. This small example generates only primes, but says nothing about generating the entire prime sequence eventually.

Ostmann’s conjecture, traced to his 1956 treatise, rules out that eventual construction. Classical strong Goldbach instead asks whether every even integer greater than two is a sum of two primes. The outputs and quantifiers differ. This inverse Goldbach conjecture proof does not resolve classical Goldbach.

1.1. Inverse Goldbach Conjecture: Key Facts

The requirement is stronger than matching a density or finding infinitely many prime sums. As our Erdős–Turán explainer illustrates, the precise hypothesis on an integer set matters as much as its apparent abundance.

2. What OpenAI Proved: The Exact Statement of Ostmann’s Conjecture

Let $\mathbb P$ denote the positive primes and $\mathbb N_0$ the nonnegative integers. Theorem 1.1 says that whenever $A,B\subseteq\mathbb N_0$ and $|A|,|B|\ge2$,

The symmetric difference, marked by the triangle, contains numbers belonging to exactly one set. Here it records both missed primes and unwanted nonprimes. “Infinite” means mistakes persist arbitrarily far out, rather than disappearing after some threshold.

This also proves that any finite modification of the primes remains additively indecomposable. Removing two awkward primes or adding several composites cannot create a nontrivial decomposition.

One useful consequence sharpens the interpretation: if a candidate sumset contains every sufficiently large prime, it must contain infinitely many composites. Conversely, a sumset producing only finitely many composites must miss infinitely many primes. Individual representations, such as seven being three plus four, remain perfectly possible. The obstruction concerns two fixed sets and all their cross-sums together. Allowing one set to be a singleton would trivialize the question: adding zero to the primes reproduces them. Requiring two elements on each side removes exactly that escape.

3. Why Earlier Sieve Methods Could Not Solve the Inverse Goldbach Problem

Laffer and Mann excluded finite summands. Elsholtz placed hypothetical infinite summands near the square-root scale and excluded three nontrivial summands. Elsholtz and Harper later sharpened the binary counting estimates. These restrictions were substantial, but left room for two sets.

Write $A(x)=|A\cap[0,x]|$, with the same convention for $B(x)$. Earlier upper bounds and the coverage requirement could coexist:

Here $\ll$ allows a fixed multiplicative constant. A lower bound smaller by a logarithm does not contradict the upper bound. Nor does excluding three summands automatically exclude two.

The sharpened Elsholtz–Harper bounds, in their positive-integer convention, range from a constant times $\sqrt x/(\log x\log\log x)$ to a constant times $\sqrt x\log\log x$ for each counting function. Even setting both sizes near $\sqrt x$ leaves enough pairs to meet coverage. Size information alone cannot determine how many different prime outputs those pairs produce.

Green and Harper showed that a suitable inverse sieve conjecture would imply Ostmann’s conjecture through a quadratic structural description near the square-root boundary. Large intersections with quadratic images still fall short of that near-containment conclusion. A large structured fragment does not account for almost all of a set. The proof must constrain the remaining elements too, because they can participate in representations of primes that the fragment misses.

The new argument bypasses a general classification. Its “sieve barrier” is this specific insufficiency of earlier restrictions, rather than every limitation of sieve theory. Compare the finite sieve obstruction in the Gaussian moat problem: local restrictions become decisive only after an additional global argument.

4. The First Step: Finite Summands and Complementary Prime Residues

Lemma 2.1 supplies a fixed-shift sieve. If distinct fixed shifts $b_1,\ldots,b_j$ make every sufficiently large $c+b_i$ prime for $c\in C$, then

The subscript permits dependence on the shifts. If one summand were finite, two of its elements would force the other to have at most a constant times $Y/(\log Y)^2$ elements. Coverage needs order $Y/\log Y$ represented primes. That mismatch excludes a finite summand.

Now assume both sets are infinite and agreement holds above a threshold $N$. Put $D=-B$, so sums become differences. If long-tail elements $a\in A$ and $d\in D$ share a residue modulo a prime $p$, then $a-d$ is divisible by $p$. Taking both tails beyond $p+N_*$, with fixed $N_*>|N|+10$, makes this difference exceed both $p$ and $N$, contradicting primality.

Thus their tail supports are disjoint. Complete them to a partition $S_p,S_p^c$ of the residues modulo $p$. Unoccupied residues can initially go on either side.

5. Collision Stability: Why the Hypothetical Summands Must Be Nearly Balanced

A collision means two independently sampled elements agree modulo a prime. Restricting samples to fewer residue classes raises the probability of collision. Unequal integers cannot collide modulo too many primes, since those primes divide their difference. This is the counting mechanism behind the larger sieve.

Lemma 2.4 gives the bounds used throughout the paper:

Lemma 2.5 refines size control into stability. For diffuse probability measures on opposite long tails, with each point mass at most $(\log X)^b/\sqrt X$, it works up to $Q=\sqrt X/(\log X)^{b+1}$.

Its nonnegative error budget is $O_b(\log\log X)$. The budget combines squared distances from uniform distributions on the respective supports, weighted by $\log p$, and a density penalty weighted by $(\log p)/p$. Writing $\sigma_p=|S_p|/p$, that penalty is $\sigma_p^{-1}+(1-\sigma_p)^{-1}-4$, minimized at one half.

Consequently, substantial imbalance or nonuniformity cannot persist across too much weighted prime mass. This is an averaged conclusion, not perfect balance at every prime. The modular collisions in subset sum offer a useful computational comparison for the underlying divisibility count.

6. Quadratic Characters: How the Proof Eliminates Hidden Residue Bias

Balanced supports might still conceal structure. A quadratic character returns one on nonzero squares, minus one on nonsquares, and zero at zero. Translating its input tests whether a support favors squares around a center.

Proposition 3.1 rules out persistent translated quadratic bias on appropriate prime averages, even when every prime chooses its own center. That last freedom matters: assuming one integer center initially would leave an enormous gap.

An amplified even moment turns sustained bias into a large positive weighted sum. Poisson summation rewrites this sum through dual frequencies. Comparing the resulting congruences forces many initially unrelated centers to agree with a common rational number modulo their primes.

The quadratic large sieve then confines positive fractions of both tails to a small family of squarefree kernels, the signed squarefree factors left after removing square factors from the translated integers. Their populations force more collisions than stability permits.

Our Artin primitive-root analysis also discusses quadratic large-sieve estimates. Here their specific job is to eliminate hidden quadratic bias, not classify the summands completely.

7. Beyond Quadratic Characters: Higher-Order Decorrelation

Quadratic tests see only two multiplicative phases. Higher-order characters take other roots of unity and could detect biases those tests miss. Section 4 therefore handles every character order greater than two separately.

Repeated Cauchy–Schwarz transfers create copied prime variables. Smaller and larger anchor variables distinguish their positions, so changing one position exposes an interaction involving the square of a chosen character. That square remains nonprincipal precisely because the original order exceeds two. Quadratic characters required their own argument because their squares are principal.

The anchors also control diagonal contributions, where copied configurations coincide. Ignoring those coincidences would discard terms capable of overwhelming cancellation.

Corollary 4.2 combines the results: outside a set of harmonic prime mass $o(L)$ in fixed bands $\alpha L\le\log\log p\le\beta L$, the normalized transforms have uniformly vanishing correlations against mixed multiplicative characters and additive phases. Harmonic mass means summing $1/p$, and $o(L)/L$ tends to zero.

As in the two-point Chowla argument, cancellation concerns specified correlations under specified averages. It does not license treating all arithmetic quantities as independent random variables.

8. Fourier Analysis: Why Prime Coverage Forces Nonsparse Additive Transforms

Center the residue indicator and normalize its Fourier transform:

Here $\mathbb F_p$ is the field of residues and $\mathbb E$ means uniform averaging. The indicator is one inside the support and zero outside. These normalizations give $g_p(0)=0$ and $\mathbb E_v|g_p(v)|^2=1$.

Define $\gamma_p=\mathbb E_v|g_p(v)|$, its probability $L^1$ norm. Small $\gamma_p$, despite unit squared norm, signals concentration on relatively few frequencies. For intuition, a vector with constant absolute value one has both norms equal to one. Concentrating the same squared mass on fewer coordinates reduces its average absolute value. This is a norm comparison, rather than an assumption that every transform has a literal sparse support. Proposition 5.1 supplies an absolute $\delta_0>0$: balanced primes with $\gamma_p\ge\delta_0$ have harmonic mass at least $0.15L$ in $0.05L\le\log\log p\le0.9L$, for large $L$.

Coverage is essential here. If too much mass had sparse transforms, a nonnegative weight built from those spectra would contract on summand pairs. Its average over primes, combined with coverage, forces an incompatible lower bound.

8.1. Two Inputs That the Proof Keeps Separate

This proposition does not use character decorrelation. The proof develops the two inputs separately before combining them. Fourier analysis here measures structure, unlike the algorithmic cost studied in our DFT complexity explainer.

9. The Finite-Field Binary-Tree Comparison That Connects the Argument

Transfers generate full binary trees. Leaves carry multiplicative residue variables, while internal nodes reconstruct frequencies and arguments. Comparing two histories means measuring correlations between two such diagrams subject to shared product constraints.

Lemma 6.1 assumes a function vanishes at zero, has probability squared norm at most one, and has small mixed-character correlations. It does not assume a special algebraic formula or a uniform pointwise bound. That flexibility admits the actual residue-mask transforms, which may have large individual values.

For depth $\ell$, there are $r=2^\ell$ leaves. A single diagram has second moment at most $3^r$. Two diagrams have correlation tending to zero when their paired product constraints have at most $3r/4$ components, with $\ell\ge2$ fixed.

The proof groups leaves into quartets and finds a cycle in the constraint graph. Reciprocal rescaling around that cycle preserves products. Mellin decomposition, the expansion in multiplicative characters, exposes cancellation while second-moment bounds control the rest. Orthogonality links the character indices around the cycle, so one index is determined by the others. That provides a small correlation factor while bounded total coefficient mass pays for the remaining choices. Preserving this distinction between norm control and pointwise control is essential, as it is in Fourier arguments for Falconer distances.

10. Positive Statistics and Arithmetic Transfers: Preserving the Lower Bound

Section 7 turns the nonsparse supply into tests whose average over a long tail of $A$ is positive. Exact tests at smaller primes retain the residue restrictions. Squaring a real averaged test produces a nonnegative smoothed statistic:

Here $\psi$ is a nonnegative smooth weight, $T$ the assembled test, and $m$ a scale-dependent number of sampled positions. Positivity prevents cancellation from destroying the initial lower bound. Poisson summation converts it into a normalized frequency amplitude.

Further transfers double the history tree while retaining a lower bound. Section 8 justifies replacing arithmetic prime averages by finite-field comparisons. Coprimality and nonzero-frequency conditions stay attached to every history. Repeated internal primes are grouped by equality patterns, not silently declared impossible.

Prime progression estimates retain a possible exceptional real character. The proof handles its conductor through controlled deletions and bounds its surviving contribution where required. These qualifications connect the ideal finite-field argument to actual primes. Without them, the comparison would describe a cleaner model than the arithmetic statistic actually supplies. The spectator primes remain shared across leaves, whereas internal samples are copied according to the transfer rules. Keeping these roles distinct is what makes the eventual local cancellation applicable to the global statistic.

11. The Final Contradiction: Why Primes Cannot Be A + B

Section 9 averages over permutations of the bulk variables. An overlap graph records which leaves share variables between arrangements. Good pairs have few components and tiny correlation. Bad pairs can correlate more strongly, but their fraction has an exponential saving involving $-\tfrac34rm\log r$.

Fix a sufficiently large transfer depth $k$, put $r=2^k$, then let the scale tend to infinity. With a fixed positive constant $B$, the retained amplitude $\eta_k$ must satisfy both

The second bound is strictly smaller. The order of choices matters: constants and depth are fixed before taking the large-scale limit. Increasing depth during that limit would not justify the uniform estimates being used. This contradiction closes the inverse Goldbach problem.

The official Lean scope notes state coverage of the full theorem, its two-infinite-summands case, and finite modifications of the primes. This article does not report a fresh compilation or independent peer review. OpenAI’s release announcement attributes the results to an internal frontier model.

The Inverse Goldbach Conjecture establishes a structural impossibility, without providing a practical prime-generation algorithm or proving the general inverse-sieve conjecture. Follow Binary Verse AI for source-led explanations and verification updates that keep those distinctions visible.