Additive indecomposability of the primes

Source and verification

This article explains family 013 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Sumsets and eventual equality

For $A,B\subseteq {\mathbb{N}}_0$, their sumset is

$$A+B=\{a+b:a\in A,b\in B\}.$$

Two sets are eventually equal if their symmetric difference is finite. Let $\mathcal{P}$ denote the positive primes. Ostmann’s inverse Goldbach question asks whether $\mathcal{P}$, up to finitely many changes, can be a sumset of two sets each containing at least two elements.

The restriction on cardinality removes translations such as $\{0\}+\mathcal{P}$. The question is called inverse Goldbach because it asks for additive building blocks of the primes, rather than representations of other integers as sums of primes.

Claimed additive indecomposability

For every $A,B\subseteq {\mathbb{N}}_0$ with $|A|,|B|\ge 2$, the manuscript asserts that $(A+B)△\mathcal{P}$ is infinite. (OpenAI, 2026)

Both parts of eventual equality matter: every sufficiently large prime must occur, and every sufficiently large sum must be prime. A density estimate on either side is weaker.

For example, if a nontrivial sumset contains all sufficiently large primes, the claimed theorem forces it to contain infinitely many composites. It does not prohibit infinite sumsets contained in the primes without the coverage requirement.

🏷️ The first counting obstruction

Write $A(x)=\#(A\cap [0,x])$, and similarly for $B$. Since nonnegative summands representing a prime at most $x$ are themselves at most $x$, coverage implies

$$A(x)B(x)\ge \pi (x)-O(1).$$

Thus $A(x)B(x)\gg x/\log x$. However, even an upper bound $A(x)B(x)\ll x$ would be compatible with this lower bound. Merely placing both sets near the square-root scale does not give a contradiction.

The source first reduces to the case of two infinite summands. Its main argument then uses local residue restrictions and global prime coverage simultaneously. Neither information is discarded after obtaining preliminary size bounds.

🏷️ Local avoidance and Fourier energy

Fix a prime $p$. If large $a\in A$ and $b\in B$ satisfy $a\equiv -b(\mathrm{mod}p)$, then $p\mid a+b$. Once $a+b>p$, that sum is composite.

After deleting finite initial segments, the residue images of $A$ and $D=-B$ must therefore be disjoint modulo $p$. They can be enlarged to a partition $S_p$ and $S_p^c$ of the residue classes. These sets may vary independently with $p$.

For $f:\mathbb{Z}/p\mathbb{Z}\to \mathbb{C}$, normalize

$$\hat{f}(r)=\frac{1}{p}\sum _{x\mathrm{mod}p}f(x)e^{-2\pi irx/p}.$$

Orthogonality of the exponential functions gives Parseval’s identity

$$\sum _{r\mathrm{mod}p}|\hat{f}(r){|}^2=\frac{1}{p}\sum _{x\mathrm{mod}p}|f(x){|}^2.$$

If $f=1_{S_p}-|S_p|/p$ and $\alpha =|S_p|/p$, the right-hand side is $\alpha (1-\alpha )$. This measures residue energy, but does not determine where that energy lies in frequency.

The source’s collision estimates show that, on suitable prime averages, the two parts are nearly balanced and the summands nearly uniform within them. A later prime-coverage argument forces enough additive Fourier mass to remain.

🏷️ Excluding structured correlations

An indicator supported on approximately half the residues can correlate with a quadratic character. Here a quadratic character is the sign distinguishing nonzero squares from nonsquares modulo an odd prime. Such correlations could defeat a naive claim that the residue indicators behave randomly.

The manuscript allows a different translating center at every prime. A moment argument and Poisson summation are used to force a common rational center if substantial quadratic correlation persists. A quadratic large sieve then concentrates positive fractions of the original summands in a small collection of quadratic patterns, contradicting their collision bounds.

Higher-order character correlations are treated by repeated Cauchy–Schwarz transfers. The resulting variables must retain enough separation to control diagonal terms. This yields the mixed-character decorrelation needed for the later finite-field comparison.

These arguments do not establish a general classification of all sets near equality in the large sieve. Their hypotheses come from the particular simultaneous prime-sum restrictions.

🏷️ Coverage and the final contradiction

Prime coverage re-enters through a nonnegative sumset statistic compared with its average on the primes. The comparison prevents the normalized additive transforms from having small $L^1$ norm on too much harmonic prime mass.

Repeated transfers create binary-tree expressions. The finite-field comparison is formulated with probability $L^2$ bounds rather than pointwise boundedness, because Fourier transforms of the residue indicators need not be uniformly small.

The arithmetic implementation keeps repeated prime labels, coprimality restrictions, and possible exceptional real characters in the progression estimates. The final averaging permutes a large collection of variables. Most pairs of arrangements overlap in few components and therefore have small correlation; the remaining diagonal terms are bounded separately. Their combined upper bound contradicts the positive statistic inherited from coverage.

The transfer depth is chosen large but fixed before the integer scale tends to infinity. This is the source’s proof architecture; the complete character and arithmetic estimates are not independently verified here.

🏷️ Key steps and proof outlines

Exclude finite summands before using the inverse sieve

Key lemma. An eventual decomposition of the primes with both summands nontrivial would have both summands infinite.

Proof outline. If $A$ were finite, two distinct elements $a_1,a_2\in A$ would force $b+a_1$ and $b+a_2$ both to be prime for every sufficiently large $b\in B$. The fixed-two-shift upper-bound sieve gives $B(Y)\ll Y/(\log Y{)}^2$. On the other hand, representing all large primes forces $Y/\log Y\ll |A|B(Y)$. These inequalities contradict each other as $Y\to \infty$. The argument for finite $B$ is identical. Thus the later analysis may use large finite portions of both summands. The two-shift sieve is an input; the contradiction from it is the elementary deduction. (OpenAI, 2026)

Near equality in collision counting forces balanced residue supports

Let $\mu ,\nu$ be probability measures on suitable large portions of $A$ and $-B$, with individual masses at most $(\log X{)}^b/\sqrt{X}$. Their reductions modulo $p$ are supported on complementary sets $S_p,S_p^c$. Write ${\sigma }_p=|S_p|/p$ and use counting ${\ell }^2$ norms for these probability mass functions.

Proof outline. Two distinct integers in $[-X,X]$ can be congruent modulo primes whose product divides their nonzero difference. Summing the logarithms of those primes costs at most $\log (2X)$. Separating equal pairs gives

$$\sum _{p\le Q}\log p\Vert {\mu }_p{\Vert }_2^2\le \log X+O(1),Q=\frac{\sqrt{X}}{(\log X{)}^{b+1}},$$

and the same bound for $\nu$. If $U_S$ is uniform probability on $S$, then for any probability supported there,

$$\Vert {\mu }_p-U_{S_p}{\Vert }_2^2=\Vert {\mu }_p{\Vert }_2^2-\frac{1}{|S_p|}.$$

Subtract these two uniform baselines. Mertens’ estimate gives $4{\sum }_{p\le Q}\log p/p=2\log X-O_b(\log \log X)$. The result is a bound $O_b(\log \log X)$ on the weighted sum of three nonnegative defects: the two deviations from uniformity and

$$\frac{1}{p}\left(\frac{1}{{\sigma }_p}+\frac{1}{1-{\sigma }_p}-4\right).$$

The last defect vanishes precisely at ${\sigma }_p=1/2$. Hence most of the relevant weighted prime mass has almost balanced supports and almost uniform summand distributions. This is a quantitative near-equality argument, stronger than merely knowing the supports are disjoint.

Prime coverage supplies Fourier mass that local avoidance cannot provide

Normalize the centered residue indicator so that its additive transform $g_p$ has probability $L^2$ norm one, and put ${\gamma }_p=p^{-1}{\sum }_v|g_p(v)|$. An $L^2$ normalization alone permits ${\gamma }_p$ to be small if the transform is concentrated on few frequencies.

Proof outline of the source’s supply proposition. Suppose too many primes have small ${\gamma }_p$. Use those spectra to construct a nonnegative weight on differences $a-d$, with $a\in A$ and $d\in -B$. The collision estimate controls the energy of products of centered residue coordinates. Tensoring these estimates bounds the sum of the weight over all such pairs. The sparse spectra produce a contraction in this upper bound. Separately evaluate the weight on large primes using prime-distribution estimates. Every sufficiently large prime is represented by at least one pair, so nonnegativity makes that prime average a lower bound on the pair sum. The resulting bounds are incompatible. The proposition therefore supplies a positive harmonic mass of primes for which the supports are balanced and ${\gamma }_p$ has a fixed positive lower bound. The two uses of the decomposition remain visible: avoidance gave collision stability, while coverage gives this new Fourier supply.

Character obstructions must be removed before tree comparison

A half-density set can be organized by quadratic residues, so balance does not imply the character decorrelation needed later. The source first treats quadratic characters with translating centers that can vary with $p$.

Proof outline. A high moment of persistent translated quadratic correlation expands into tuples of summand elements. Poisson summation separates the terms whose phases can remain coherent; substantial coherence forces a common rational translating center. A quadratic large-sieve estimate then puts positive fractions of the summands into a small number of the resulting patterns, violating collision stability. Repeated Cauchy–Schwarz transfers handle higher-order mixed character expressions. Each transfer retains separation conditions on the variables, so terms on the diagonals can be counted rather than mistaken for cancellation. These are specialized inverse estimates under the simultaneous sumset hypotheses, not a universal classification of half-density residue sets.

Symmetrization creates a quantitative excess of independent arrangements

Repeated transfers yield a binary-tree amplitude with $r=2^k$ leaves and $m$ bulk variables per leaf. The source averages this amplitude over all $(rm)!$ arrangements of the bulk variables. The averaging is valid because their sampling laws and the common transform are permutation invariant; the individual history restrictions stay inside their respective amplitudes.

Proof outline. Expand the squared norm of the averaged amplitude into pairs of arrangements. Their overlap graph records which leaf groups share variables. For a bad pair with more than $3r/4$ connected components, let $a_i$ be the number of leaves on either side of its $i$th component. Counting edges shows the two sides have equal sizes, and

$$\sum _i{a}_i=r,\sum _i(a_i-1)<r/4.$$

There are at most ${\prod }_i(a_{i}m)!$ arrangements with fixed component partitions. Since $(am)!\le (m!{)}^a{a}^{am}$, their fraction among all arrangements is at most

$$\exp \left[\left(-\frac{3}{4}\log r+C\right)rm+O_k(1)\right].$$

Good pairs instead satisfy the finite-field tree decorrelation estimate after the arithmetic comparison, including its repeated-prime and exceptional-character terms. The bad-pair saving must pay for all previous transfer costs. The source bounds those costs by a leading $(2/5)rm\log r$, leaving the strict coefficient $-3/4+2/5=-7/20$. Other depth-dependent terms grow only like $rm\log k$. Choose a large but fixed depth $k$, then send the integer scale to infinity. The resulting upper bound is smaller than the positive amplitude propagated from coverage. This final comparison explains why controlling the diagonals and the order of parameter choices is part of the proof, rather than an afterthought.

🏷️ Connections

Jacobsthal’s function also turns prescribed residue classes into a global obstruction, but asks for surviving integers in every interval. Large prime-gap frequency uses positive sieve weights to ensure a prime next to an empty interval.

🏷️ Manuscripts and verification scope

• The additive indecomposability of the primes — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The page selects the eventual-equality obstruction and its two-infinite-summands case. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. The additive indecomposability of the primes.