Openai 2026 weak inhomogeneous duffin schaeffer conjecture

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corollary_1_2: A Hausdorff f-measure version of the main theorem obtained through the Beresnevich–Velani mass transference principle; a one-directional divergence statement for a fixed shift and arbitrary numerators, claimed and unverified here.

theorem_1_1: The manuscript's main claim: for every fixed real shift and every finite-valued tolerance function with divergent totient-weighted sum, almost every real number has infinitely many unrestricted-numerator approximations; stated as a claim, unverified here.

OpenAI, The Weak Inhomogeneous Duffin–Schaeffer Conjecture, OpenAI Math Release preprint, September 25, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_weak_inhomogeneous_duffin_schaeffer_conjecture.pdf, and the release's TeX bundle sits in the same release folder.

@misc{OAI:The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026,
  author = {{OpenAI}},
  title = {{The Weak Inhomogeneous Duffin--Schaeffer Conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf}{OAI:The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026}},
  year = {2026}
}

Attestation as the release states it. The release's root README says the repository holds manuscripts "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all have Lean formalizations, and that "Some of the unformalized results could have issues". The manuscript's own README adds only the title, author ("OpenAI"), date and citation block; it makes no statement about human assistance or review. The PDF carries the author line "OpenAI" and no affiliation, funding note or acknowledgment. These are the source's historical attestations, not 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.

The release's Lean catalogue (lean/formalization.yaml) lists no formalization for this manuscript or its family, and the release has no lean/docs page for it. The manuscript is the only member of its family in the release.

Read status: claims checked for Theorem 1.1 and Corollary 1.2, read clause by clause in the TeX source (sections/introduction.tex, labels thm:main, lines 9--21, and cor:hausdorff, lines 32--51) on 2026-10-07; the statements of the intermediate propositions named under Contents were read in their TeX files and the proofs were read for their structure only, no step was checked; nothing here is independently reviewed.

Contents

The manuscript has eleven sections and 87 PDF pages; the title, abstract and table of contents occupy pp. 1--2 and the references pp. 86--87. Numbering is by section (Theorem 1.1, Lemma 2.1, Proposition 5.2). Results are stated below in this corpus's words; the manuscript's "centre" is written "center" here.

• Section 1, Introduction (pp. 3--6). States Theorem 1.1: for fixed $\gamma\in\mathbb R$ and finite-valued $\psi:\mathbb N\to[0,\infty)$, divergence of $\sum_q\phi(q)\psi(q)/q$ implies $\|qx-\gamma\|<\psi(q)$ for infinitely many $q$ for almost every $x$, with no coprimality condition on the numerator; and Corollary 1.2, the Hausdorff $f$-measure version obtained through the mass transference principle. The background places the statement after Khintchine (1924), Szüsz (1958) for fixed shifts, Duffin–Schaeffer (1941), and the proof of the homogeneous conjecture by Koukoulopoulos and Maynard, whose coprime-numerator theorem implies the $\gamma=0$ case of Theorem 1.1 and whose common-pivot method the manuscript says it adapts. It cites Ramírez (2017) and Chow–Hauke–Pollington–Ramírez (2025) for nonmonotone counterexamples with divergent unweighted sums, names the question as Yu (2021, Question 1.2), Chow–Technau (2024, Conjecture 1.22) and Beresnevich–Hauke–Velani (2024, Conjecture 2), cites Beresnevich–Hauke–Velani (2024, Theorem 14) for the weak conjecture at every rational shift, and reports that the coprime-numerator inhomogeneous assertion is false in general, citing Hauke-Treuer–Maynard–Pollington (arXiv, 25 September 2026) and He–Liao (arXiv, 25 September 2026). The strategy subsection describes the proof: finite blocks of rows of fixed small totient-weighted mass, a weighted sum of interval indicators built by prescribed prime-step weight updates, a comparison sum that omits only the extra deletions, and a second-moment bound that contradicts a positive-measure avoided set.
• Section 2, Finite blocks and elementary conventions (pp. 6--10). Lemma 2.1 handles the case $\psi(q)\not\to0$ directly; Lemma 2.2 cuts the tail into disjoint finite blocks of mass between $w$ and $2w$. Definition 2.3 gives the "deficit law" $\pi_L(d)=\phi(L/d)/L$ on divisors of $L$. Lemma 2.4 records elementary Chebyshev-type prime bounds with proofs. Lemma 2.5 (one-coordinate concentration, attributed in form to Green–Walker Lemma 2.1 and Hauke-Treuer–Vazquez–Walker Lemma 3.2, proof included) and Lemma 2.6 (tensor kernel) give a summable bound for sums weighted by $(L/(L,M)\cdot M/(L,M))^{-s}$. Lemma 2.7 averages residue conditions along determinant cycles of two rational grids.
• Section 3, Centres, density, and prime heights (pp. 10--16). Assigns each row a center (Definition 3.1, Lemma 3.5) with a density bound (Lemma 3.2) and summable center kernels (Lemmas 3.3, 3.7); Lemma 3.4 disposes of blocks where raw rows carry mass; Lemma 3.6 separates bands. Section 3.5 fixes the order of about thirty constants on which every later estimate depends.
• Section 4, Masks and initial numerator prescriptions (pp. 16--21). Removes bad deficit masks at negligible cost (Definition 4.1, Lemma 4.2), prescribes tag-prime exclusions (Construction 4.3) and rational-model exclusions (Definition 4.4, Lemma 4.5), bounds the retained mean below (Lemma 4.6) and the cost of soft gcd tests (Lemma 4.7).
• Section 5, Projected tables and the second-moment reduction (pp. 21--34). Defines the adaptive weight updates (Lemma 5.1), states the three inputs the rest of the paper must supply (Propositions 5.2, 5.3, 5.4), proves the variance budget (Lemma 5.5), the broad-energy bound (Proposition 5.6), dominating virtual product weights (Proposition 5.7, Corollary 5.8) and their equidistribution (Lemma 5.9), and in Proposition 5.10 derives Theorem 1.1 from the three inputs.
• Section 6, Arithmetic estimates for close pairs (pp. 34--44). A Selberg upper-bound sieve for close grid pairs with its quadratic-form calculation written out (Lemma 6.1), a rotation alternative for integers and primes (Lemma 6.2), a weighted lcm bound (Proposition 6.3) in the common-pivot style of Koukoulopoulos–Maynard, Green–Walker and Hauke-Treuer–Vazquez–Walker, and a small-scale extraction proposition (Proposition 6.4) with a sampling corollary (Corollary 6.5).
• Section 7, Direct comparisons at simultaneous births (pp. 44--58). Proves the simultaneous-birth part of Proposition 5.2 (Proposition 7.1): ultra switching (Lemma 7.2), tag folding (Lemma 7.3), a linear bound for exact modeled coincidences (Lemma 7.4), the non-ultra row kernel (Lemma 7.6, with the Pollington–Vaughan overlap estimate named as its homogeneous antecedent and not used), a static discard near low rational grids (Construction 7.7, Lemma 7.8) and the exclusion of saturated boxes (Lemmas 7.9--7.11).
• Section 8, Alignments in sparse prime steps (pp. 58--65). Proposition 8.1 proves Proposition 5.3 and Construction 8.2 defines deterministic interval "trains" that cover every class-joining segment.
• Section 9, Construction of the operational partitions (pp. 65--74). Proves Proposition 5.4 (Lemmas 9.1--9.4): cell boundaries are drawn uniformly at random in microintervals and skipped inside inherited class hulls; the first-mass loss is bounded in expectation over these draws. This is the manuscript's one probabilistic component; it states that the shift itself is never randomized.
• Section 10, First joins of previously constructed tables (pp. 74--85). Proposition 10.1 bounds the remaining first-join comparisons through a backward expansion of the actual weights (Lemmas 10.2--10.6) and the closing proof of Theorem 1.1 (pp. 84--85) assembles the inputs in chronological order and records the parameter margins (p. 85).
• Section 11, The Hausdorff-measure consequence (p. 85). Proves Corollary 1.2 from Theorem 1.1 and Theorem 2 of Beresnevich–Velani (2006).
• References (pp. 86--87): nineteen entries, including the three arXiv preprints dated 2026 named above and Gou (arXiv, April 2026) for related unweighted small-lcm counts.

External inputs the proofs rest on at statement level: the mass transference principle (Beresnevich–Velani 2006, Theorem 2) for Corollary 1.2 only. The Selberg sieve and the one-coordinate concentration lemma are cited for their form and reproved; the manuscript states that every construction and transfer estimate it uses is proved in the text. Nothing is flagged as numerical, computer-assisted or conditional; the random boundary draws of Section 9 are part of the proof's expectation argument, not a computation. The release holds no verification/ folder for this manuscript.

Bears on

• Problem 999: the manuscript's Theorem 1.1 at shift $\gamma=0$ is the unrestricted-numerator divergence half of the problem's statement and follows from the problem's proved coprime-numerator form; for $\gamma\ne0$ it is a claimed extension of that weak form to every fixed real shift, a variant the page does not ask about and that the manuscript itself distinguishes from the coprime version, which it reports false for nonzero rational shifts. The claim is unverified here, and the page's status rests on the acceptance evidence it records; this card predicts no change to it. The manuscript names no Erdős problem.