Openai 2026 power saving polynomial differences prime arguments

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theorem_1_1: The manuscript's main claim: for a fixed prime-intersective integer polynomial h of degree at least 2 with positive leading coefficient, sets in {1,...,N} whose differences avoid every nonzero h(p), p prime, have at most C_h N^(1-c_h) elements; the proof rests on the release's claimed 7/8 zero-free half-plane. Claims checked only.

OpenAI, A Power Saving for Polynomial Differences at Prime Arguments, OpenAI Math Release preprint, October 5, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision fd4aeeb2e), folder preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026; the held PDF, prime-argument-polynomial-differences.pdf in the release, is retained as openai_2026_power_saving_polynomial_differences_prime_arguments.pdf, and the release's TeX bundle in that folder is the TeX source cited on this card. The license term was read in the release's root LICENSE file at that revision on 2026-10-12.

@misc{OAI:A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026,
  author = {{OpenAI}},
  title = {{A Power Saving for Polynomial Differences at Prime Arguments}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf}{OAI:A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026}},
  year = {2026}
}

The held PDF has 42 pages with a text layer; its title page is dated October 5, 2026, and its metadata records no creation date (the TeX source sets the pdfTeX switch that omits the date fields). The TeX source is build/main.tex with seven section files under build/sections/ and one TikZ figure file, 3,171 lines in all, and the bibliography build/references.bib (316 lines, 27 entries); it is the text read here.

Attestation. The release README states that the manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that "Not all have accompanying Lean formalizations" and that "Some of the unformalized results could have issues". Its account of the method says that "The vast majority of results were obtained with the same procedure", and names as an exception "work on a zero-free region for the Riemann zeta function", the subject of the manuscript that supplies this one's analytic input; it does not say which manuscripts the exception covers. The manuscript's own README carries only the title, the author line "OpenAI", the date and the citation block, and adds no sentence about authorship, assistance or verification. The title page names OpenAI as the sole author, and the text has no acknowledgment. These are the source's own statements, recorded as attestations and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here, and nothing on this card is independently reviewed.

Formalization. The release's Lean page for this family, lean/docs/182.md ("Power savings for intersective polynomial differences and prime arguments"), names only A power saving for square-difference-free sets as its accompanying paper. Its one comparator statement, lean/ComparatorChallenges/SquareDifference.lean (theorem OAI.SquareDifference.power_saving; SquareDifference.json names the solution module OAI.Combinatorics.SquareDifference.Main and permits the axioms propext, Quot.sound and Classical.choice), reads in words: there are real numbers $c>0$ and $C$ such that, for every $N\ge1$, every finite set $A$ of integers in $[1,N]$ in which no difference $a-b$ equals $m^2$ for an integer $m\ge1$ has $|A|\le CN^{1-c}$. Read here, that is the consequence this manuscript draws from its theorem with $h(x)=(x-1)^2$ (TeX build/sections/introduction.tex, lines 94--96, p. 2), but the formalization accompanies the square-difference companion's proof, not this one; nothing in the release formalizes Theorem 1.1 here or any statement with prime arguments. At this revision the release's catalog lean/formalization.yaml lists no manuscript of this family among its sources and no comparator of the family under status.main_results, and its review field reads status: unchecked. The release does list the zero-free input, Theorem 5.1, as formalized under another family: the same catalog lists under status.main_results the comparator ComparatorChallenges/DirichletSevenEighths.json (declaration OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re, file OAI/NumberTheory/DirichletL/Nonvanishing.lean), whose statement file DirichletSevenEighths.lean reads, in words: for every modulus $q\ge1$, every Dirichlet character $\chi$ modulo $q$ and every $s$ with $\operatorname{Re}s>7/8$, other than $s=1$ for the principal character, $L(s,\chi)\ne0$. This card reads that as the statement of Theorem 5.1; the release's page for it is lean/docs/003.md, recorded on the 7/8 card. All of this is read statically from the release; nothing was built, replayed or audited in this repository.

Companions. The release groups this manuscript, in its family, with A power saving for intersective polynomial differences with an exponent depending only on the degree (October 5, 2026), filed at the intersective-differences card, which the release describes as a bound $O_h(N^{1-c_k})$ for sets whose differences avoid the nonzero values $h(1),h(2),\ldots$, for every intersective $h$ of degree $k\ge2$ with positive leading coefficient, with $c_k>0$ depending only on $k$; this manuscript calls it its "direct methodological predecessor", adapts its tuple laws, Fourier lifts, signed kernels and contracting induction to prime arguments, and reproduces its Fourier-lift moment estimate with proof (Proposition 4.4). The family's third manuscript is A power saving for square-difference-free sets (September 24, 2026), filed at the square-difference card, whose theorem as the release states it, absolute constants $c,C$ with $|A|\le CN^{1-c}$ for sets with no nonzero square difference, Theorem 1.1 here would also give at $h(x)=(x-1)^2$, though here only given the zero-free input; Section 4 cites it for the square-difference counterparts of its signed-form and lift estimates. The release holds no earlier dated version of this manuscript. Outside the family, the analytic input (Theorem 5.1) is the Dirichlet part of Theorem 1.1 of The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8 (September 30, 2026), filed at the 7/8 card, whose read status there is claims checked only; the manuscript says only that assertion is needed, so the Hecke part over $\mathbb Q(\sqrt{-3})$ is not used, and Theorem 5.1 spells out that imprimitive characters and $\zeta$ are covered. The theorem here is proved only given that claim, whose Dirichlet statement the release lists as formalized (see Formalization). The companions' arguments were not read for this card.

Read status: claims checked for Theorem 1.1, together with the definition of a prime-intersective polynomial and the remark after the theorem, and for the statements of Theorem 5.1 (the zero-free input), Proposition 5.2 (primes in residue classes) and Proposition 7.5 (uniform power decay), read clause by clause in the release's TeX source (build/sections/introduction.tex, label thm:main at lines 19--29, the definition at lines 10--17 and the remark at lines 31--34; build/sections/primes.tex, labels thm:zero-free-input at lines 27--34 and prop:prime-progressions at lines 36--47; build/sections/descent.tex, label prop:uniform-descent at lines 414--420); the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

Results are numbered within sections in one sequence shared by theorems, propositions and lemmas (Lemma 2.1, Proposition 3.1, Theorem 5.1); the PDF and the TeX agree.

• Title page and Section 1, Introduction (pp. 1--3). The abstract, then the definition: $h\in\mathbb Z[x]$ is prime-intersective if every modulus $q$ has a root $r$ of $h$ with $(r,q)=1$. States Theorem 1.1 (p. 1) and notes that a prime root ($h(p)=0$) restricts nothing. The history runs from Furstenberg and Sárközy for squares, Kamae and Mendès France for intersective polynomials and Lucier's quantitative form, through Pintz--Steiger--Szemerédi, Bloom--Maynard, Green--Sawhney ($N\exp(-c\sqrt{\log N})$ for squares), Rice and Arala, to Adajar and coauthors ($N\exp(-c(\log N)^\mu)$, $\mu<1/2$). For prime arguments it cites Sárközy for $p\pm1$; Lucier, Ruzsa--Sanders and Wang; Green's power saving for $p-1$ and Thorner--Zaman's explicit exponent; Li--Pan for nonlinear $h$ with $h(1)=0$; and Rice's bound $|A|\ll N(\log N)^{-c}$, $c<1/(2k-2)$, for every prime-intersective $h$ of degree $k\ge2$, with Wierdl's qualitative characterization by the unit-root condition as Rice records it. It names the intersective companion as the method's source, observes that $h(x)=(x-1)^2$ yields the square-difference case, outlines the mechanism, and calls the 7/8 manuscript's Dirichlet half-plane, cited as Theorem 1.1 there, "a theorem input".
• Section 2, auxiliary polynomials and local admissibility (pp. 3--8). Fixes a unit $p$-adic root of $h$ at every prime, a completely multiplicative $\lambda$ recording its multiplicity, and for each $l$ the affine map $T_l(x)=r_l+lx$ with $(r_l,l)=1$ and the integral polynomial $h_l=h\circ T_l/\lambda(l)$ (Lemma 2.1, p. 4). A set avoids at $l$ when no nonzero $h_l(m)$ with $m\ge1$ and $T_l(m)$ prime is a difference; restriction to a progression of step $\lambda(D)$ carries avoidance at $l$ to avoidance at $lD$. Lemma 2.2 bounds local exponential sums at large primes over arguments $x$ with $p\nmid T_l(x)$, with savings $p^{-8\delta j}$ and $p^{-4\delta}$, $\delta=1/(100k2^k)$; Lemma 2.3 gives uniform lifting classes at the finitely many small primes; Lemma 2.4 fixes one cycle length $L$ for which $L$ admissible local values sum to zero at every prime.
• Section 3, reflection-positive laws on residue tuples (pp. 8--15). Proposition 3.1 (p. 8): for all large primes $p$, uniformly in $l$, a probability law on labels in $\mathbb F_p$ indexed by a fixed finite slot set, reflection positive across every cut, translation invariant, covariant under $l\mapsto lD$, conditionally mixing with rate $p^{-\gamma}$ ($\gamma=\delta/10$), and, off an exceptional part of mass at most $p^{-4}$, with admissible increments along a designated $L$-cycle of slots. It is a mixture of components in which marked blocks are forgotten, with index interactions supplying positivity (Lemma 3.2 gives the graph estimates; Figure 1, p. 13, illustrates the list concatenation).
• Section 4, multilinear estimates and Fourier lifts (pp. 15--22). The signed Hölder inequality, by repeated folding in the chessboard manner of Fröhlich, Israel, Lieb and Simon (Proposition 4.1); the resulting seminorm and its transport under $l\mapsto lD$ (Lemma 4.2); decay at large Fourier denominators from conditional mixing (Lemma 4.3); lifts of functions on an integer interval to products of the fields $\mathbb F_p$, keeping frequencies with squarefree denominator at most $Q$, with each $L^{2r}$ norm at most $C_{r,\epsilon}\|f\|_\infty Q^\epsilon$ when the interval has length at least $10Q^6$ (Proposition 4.4); and specialization of coordinates (Lemma 4.5).
• Section 5, prime estimates (pp. 22--29). Theorem 5.1 (p. 23) quotes the 7/8 manuscript's half-plane for Dirichlet $L$-functions: no such function, for any modulus and character, imprimitive ones and $\zeta$ included, vanishes in $\operatorname{Re}s>7/8$. Proposition 5.2 (p. 23) derives, uniformly in the modulus $b$, the count of primes up to $x\le Y$ in a reduced class weighted by $\log p$ as $x/\varphi(b)$ plus $O(Y^{31/32}\log^2(2bY))$, through a bound for $L'/L$ on $\operatorname{Re}s=15/16$, a smoothed Mellin integral and differencing of a nonnegative sum; as stated, the error is smaller than the main term only for $\varphi(b)$ below about $x^{1/32}$ (an observation made here, not in the manuscript). Proposition 5.3 (p. 24) gives a power saving for prime-weighted polynomial exponential sums when $(k!)^2$ times the leading coefficient has a rational approximation with denominator between $Y^\rho$ and $Y^{k-\rho}$, by Vaughan's identity and Weyl differencing (Lemmas 5.4 and 5.5), with the product cutoff $mn\in I$ kept inside the differenced sums.
• Section 6, a kernel on differences at prime arguments (pp. 29--35). Fixes the exponents ($\eta=1/(10000k)$ down to $\tau$, $\sigma$) and the cutoffs $Q=N^\beta$, $H=N^{\beta/2}$. Proposition 6.1 (p. 29): for $l\le N^\zeta$ and two functions bounded by $1$ on residue classes modulo $D\le N^\beta$ with admissible difference, whose ordered differences avoid the forbidden values at $l$, $D$ times the pairing of their Fourier coefficients weighted by a cycle edge's multiplier is $O(H^{-\gamma/2})$. The proof compares a model kernel with a signed kernel supported on values $h_l(m)$ with $T_l(m)$ prime (Lemma 6.2, p. 31), using Proposition 5.2 on major arcs and Proposition 5.3 on minor arcs.
• Section 7, descent to shorter intervals (pp. 35--41). Defines $\mathcal Y_l(N)$, the largest diagonal tuple integral of the lift of $\mathbf 1_A$ over sets avoiding at $l$, which bounds $(|A|/N)^d$ from above. Residue specialization (Lemma 7.1), comparison with shorter intervals at parameter $lD$ (Proposition 7.2), cancellation for residue assignments with admissible cycle increments (Proposition 7.3), the recurrence (Proposition 7.4), and Proposition 7.5 (p. 40): $\mathcal Y_l(N)\le C_0l^{b_0}N^{-a_0}$ for all $l,N\ge1$. The proof of Theorem 1.1 (p. 41) takes $l=1$ and $c_h=a_0/d$.
• References (pp. 41--42), 27 items, including the three release manuscripts named above.

The manuscript flags nothing as numerical, computer-assisted or conditional; it treats the quoted zero-free half-plane as "a theorem input" (p. 3). No constant is evaluated, and $c_h$ is given only as $a_0/d$ with $a_0$ chosen small after all other parameters. On dependence the manuscript says only that $c_h$ and $C_h$ "depend only on $h$" (abstract, p. 1, and in other words the remark after the theorem); it does not claim, as the integer-argument companion does, an exponent depending only on the degree, and the release's family description says the exponent is "allowed to depend on h". Read here, the proof as written gives no degree-only exponent: $d$ and $a_0$ are fixed through the cycle length $L$ and the modulus $M_0$, which depend on $h$ itself through its lifting exponents at small primes (Lemmas 2.3 and 2.4) and the prime thresholds. The release folder for this manuscript holds only the PDF, the TeX build directory and the README; there is no verification/ folder.

Bears on

The manuscript names no Erdős problem, and no problem page is identified as using it. The leads of its analytic input, the zero-free half-plane, are recorded on the 7/8 card.