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Statement

Theorem 2.3 (named "Two infinite summands" in the source). There are no two infinite sets $A,B\subseteq\mathbb N_0$ such that

$$(A+B)\mathbin{\triangle}\mathcal P$$

is finite, where $\mathcal P$ is the set of positive primes and $\mathbb N_0$ the nonnegative integers. Equivalently, for infinite $A,B\subseteq\mathbb N_0$ there is no threshold $N$ with $\mathcal P\cap(N,\infty)\subseteq A+B$ and $(A+B)\cap(N,\infty)\subseteq\mathcal P$ (the source's display (2.3), the form the proof assumes for contradiction). Sets of positive integers are the special case $0\notin A\cup B$. The source's Lemma 2.2 reduces Theorem 1.1 (summands of any size at least two) to this statement.

Source. OpenAI, The additive indecomposability of the primes, release folder the-additive-indecomposability-of-the-primes-September-24-2026; TeX sections/02-preliminaries.tex lines 113--116 (label thm:main), PDF p. 5; the proof occupies the rest of Section 2 and Sections 3--9 (sections/02-preliminaries.tex line 123 to sections/09-conclusion.tex line 386, PDF pp. 5--79), ending with the source's closing sentence, "This proves Theorem 2.3", on PDF p. 79.

Read depth. Claims checked: the statement and the statements of the intermediate results listed on the card (Lemmas 2.4, 2.5, 5.2, 5.3, 6.1 and 8.1, Propositions 3.1, 4.1, 5.1 and 7.1, Corollaries 3.2 and 4.2, Remark 5.4) were read clause by clause in the TeX source. The 75-page proof was read for its structure only, as summarized below; no step was checked, no constant was recomputed and the order of parameter choices was not audited. Nothing here is independently reviewed.

Proof pointer

The proof is by contradiction from display (2.3) with a fixed threshold $N$, an integer $N_*>|N|+10$ and $D=-B$. Its stages, in the source's order:

1. Residue partitions and sizes (Section 2, pp. 5--7). For each prime $p$, the residues of elements $a\in A$ with $a>p+N_*$ and of $d\in D$ with $-d>p+N_*$ are disjoint (a common residue would make a sum in $A+B$ a proper multiple of $p$ beyond $N$), giving a partition $S_p,S_p^c$ of $\mathbb F_p$ with density $\sigma_p$. Lemma 2.4 gives $\sqrt Y/(\log Y)^3\ll A(Y),B(Y)\ll\sqrt Y(\log Y)^2$ by the large sieve, a collision count and prime coverage. Lemma 2.5 (collision stability) bounds, with weight $\log p$ over $p\le\sqrt X/(\log X)^{b+1}$, the squared distances of the projected measures on the two tails (probability measures with every point mass at most $(\log X)^b/\sqrt X$) from uniform on $S_p$ and $S_p^c$ together with the imbalance $(\sigma_p^{-1}+(1-\sigma_p)^{-1}-4)/p$ by $O_b(\log\log X)$.
2. Quadratic decorrelation (Section 3, Proposition 3.1 and Corollary 3.2). On $\log p/p$ averages, the maximal translated quadratic-character bias of $S_p$ is $o(1)$, with translating residues allowed to vary with $p$. A biased prime block is amplified by a moment; Poisson summation produces a common rational center for many primes; Heath-Brown's quadratic large sieve confines positive fractions of the tails to few quadratic kernels, and their populations contradict Lemma 2.5.
3. Higher-order decorrelation (Section 4, Proposition 4.1 and Corollary 4.2). Repeated Cauchy--Schwarz transfers with anchor primes force the same conclusion for every character of order greater than two; combining with stage 2 and the imbalance term gives, outside harmonic mass $o(L)$ in a band $\alpha L\le\log\log p\le\beta L$, $\sigma_p=\tfrac12+o(1)$ and $o(1)$ for every multiplicative-twisted additive correlation of the normalized transform $g_p$ of $\mathbf 1_{S_p}$.
4. Fourier supply from coverage (Section 5, Proposition 5.1). Primes with balanced $\sigma_p$ and probability $L^1$ norm $\gamma_p\ge\delta_0$ of $g_p$ carry harmonic mass at least $.15L$ in $.05L\le\log\log p\le.9L$. Otherwise a nonnegative tensor weight built from the sparse spectra has a sum over $A\times D$ that prime coverage bounds below, and a sum over primes evaluated by Lemma 5.3 (prime sums twisted by primitive characters, with the Landau--Page exceptional character omitted) bounds above; a local contraction (Lemma 5.2) makes the two incompatible. Remark 5.4 notes a variant through the release's Quasi-Riemann Hypothesis preprint and states that the proof does not use it.
5. Finite-field tree comparison (Section 6, Lemma 6.1). For a function $g$ on $\mathbb F_q$ with vanishing twisted correlations, diagram values on binary trees of depth $l$ have second moment at most $3^r$, and two diagrams of depth $l\ge2$ whose leaves are paired into at most $3r/4$ product-equality sets correlate at most $C_l(\varepsilon_q+q^{-1/4})$.
6. Positive statistic and transfers (Section 7). With a fixed depth $k$ and $m\asymp k^4L$, half-lists of giant, bulk, spectator and compensation primes with harmonic priors on disjoint bands (spectators from stage 3) give a nonnegative statistic bounded below by $\sqrt Xe^{-Cm}$ (display (7.5), from Lemma 2.4 and stage 4); Poisson summation turns it into an amplitude $\eta_0$, and exact transfers produce $\eta_1,\dots,\eta_k$. Proposition 7.1 states the one-assignment norm bound and the paired correlation bound $\exp(-\omega_k(m))$, at level $l\ge2$, for arrangements whose overlap graph has at most $3r/4$ components.
7. Arithmetic comparison (Section 8). Proves Proposition 7.1: Lemma 8.1 (prime sums in progressions on short logarithmic intervals, with a retained possible exceptional zero) idealizes the giants and bulk primes to real coordinates and independent unit residues; coincidences in the line conditions are removed; the signed comparison uses the spectator primes through Lemma 6.1.
8. Conclusion (Section 9). Bad arrangements are counted (display (9.1)), the diagonals are bounded with the parameter order $B_s$, then $B_z$, then $B_D$, then $k$, then $L$, and induction through the transfers gives $|\eta_k|\ge\exp(-B2^km)$; averaging $A_k$ over permutations of the $rm$ bulk values, Cauchy--Schwarz with the norm of the common regular transform, and Proposition 7.1 give $|\eta_k|^2\le\exp(-(2B+1)rm)$ for large $L$, the contradiction.

Dependencies

At statement level: the additive large sieve (Montgomery and Vaughan 1973); Gallagher's larger sieve in the quantitative form of Green and Harper (2014); Elsholtz (2006, Theorem 1.9) for the square-root bounds, which the source reproves; Heath-Brown's quadratic large sieve (1995); Bonami's hypercontractive inequality (1970); Kneser's addition theorem (1953, with DeVos's proof); Montgomery's zero-density theorem (1969) in the form stated by Inoue (2021); the classical zero-free region and Page's theorem (Montgomery and Vaughan 2007, Theorem 11.3 and Corollary 11.10), stated for a whole family of conductors in the form of Ford, Green, Konyagin, Maynard and Tao (2018, Lemma 7.1), whose one-prime deletion of an exceptional conductor is also followed; Siegel's theorem (Montgomery and Vaughan 2007, Corollary 11.15); the smoothed explicit formula and a zero-count bound (Montgomery and Vaughan 2007, Chapter 10; Helfgott's manuscript); Rota's Möbius inversion on set partitions (1964); mixing bounds for quasirandom groups (Gowers 2008; Babai, Nikolov and Pyber 2008; Gill 2016); and the Schwartz--Zippel--DeMillo--Lipton lemma. The release's Quasi-Riemann Hypothesis preprint is cited only in Remark 5.4, which the source says the proof does not use. None was checked here.

Bears on

• Problem 431: claimed resolution, negative. This statement is the problem's question with the answer no for subsets of $\mathbb N_0$: no two infinite sets of nonnegative integers have a sumset agreeing with the primes up to finitely many exceptions. The problem page leaves the ambient set unstated. The claim is unverified here; the page's status rests on acceptance evidence, which this record does not supply.