Openai 2026 finite angular cylinder covers below half area bound

../

corollary_1_2: Transfers Theorem 1.1 by affine invariance of the directionwise ratios |B_i| / |π_{u_i^⊥} T|; the manuscript presents it as a counterexample to the Bezdek–Khan 1-Codimensional Cylinder Covering Conjecture in R³. Formally verified here only for the regular tetrahedron of Theorem 1.1, which already gives the counterexample; the extension to every nondegenerate tetrahedron is not.

theorem_1_1: The manuscript's main claim: for every tilt 0 < ε ≤ 1/2000, the edge-two regular tetrahedron is covered by 2⌈2/ε²⌉ cylinders with compact triangular perpendicular bases of normalized total area 1/2 − (13/6000)ε² + O(ε⁴), strictly below 1/2, a negative answer to the half-area question; formally verified here in full, the prose proof unreviewed.

OpenAI, Finite angular cylinder covers below the half-area bound, OpenAI Math Release preprint, September 27, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026; the held PDF, main.pdf in the release, is retained as openai_2026_finite_angular_cylinder_covers_below_half_area_bound.pdf, and the release's TeX bundle sits beside main.pdf in that folder.

@misc{OAI:Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026,
  author = {{OpenAI}},
  title = {{Finite angular cylinder covers below the half-area bound}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf}{OAI:Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026}},
  year = {2026}
}

Attestation as the release states it, recorded here as the source's own account and not as this corpus's review: the release README says 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". The manuscript's own README adds nothing beyond the title, the author line "OpenAI", the date and the bibtex entry above; it carries no statement about human assistance. The manuscript text names no author other than OpenAI and carries no arXiv identifier of its own. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

Formalization, as the release lists it: the release's Lean catalogue lean/formalization.yaml has no entry for this manuscript, while the release's contents page marks the whole family, not the manuscript, with a link to the family page lean/docs/100.md. That page's first scope paragraph says the formalization "constructs finite covers of a regular tetrahedron by cylinders with compact triangular bases whose total area is strictly below that bound", with normalized area $1/2-(13/6000)\varepsilon^2+O(\varepsilon^4)$ for the explicit small parameter, and "also gives a counterexample to the directionwise normalized half-bound" (its later sentences on covers with parallelogram bases describe the slope-field companion, not this manuscript). The page names the comparator statement files lean/ComparatorChallenges/TriangularCovering.lean (the explicit triangular cover: one theorem main stating the conjunction of $A_{\min}=\sqrt2$ for the manuscript's tetrahedron, an asymptotic cover statement with existential constants in place of the manuscript's explicit $1/2000$ and $2$, and a cover of relative cost below $1/2$; its proof body in the challenge file is sorry, with a solution module named in the paired JSON file) and lean/ComparatorChallenges/CylinderCovering.lean (an aggregate for all four manuscripts of the family). The family page says the affine extension to every nondegenerate tetrahedron, Corollary 1.2's second half, is outside the selected statements. This listing is read statically from the release's catalogue; the build of the two statements here is recorded below. No Lean file is a proof of an Erdős problem here; the manuscript names none.

Formal verification here: this corpus's verification built OAI.CylinderCovering.fourPaperMain and OAI.TriangularCovering.main at the release's revision adc7f1241b42e322a6451854ab7e4b4c146bf78a (2026-10-06) with toolchain leanprover/lean4:v4.34.1 on 2026-10-08. The axioms of each are exactly propext, Classical.choice and Quot.sound, no sorry appears, and each declaration's fingerprint is identical to its comparator challenge, CylinderCovering.lean and TriangularCovering.lean. Checked clause by clause against the manuscript, fourPaperMain certifies Theorem 1.1 in full, with its explicit constants: $A_{\min}(K)=\sqrt2$ for the tetrahedron (1.4) and, for every $0<\varepsilon\le1/2000$, a cover of $K$ by $2\lceil2/\varepsilon^2\rceil$ cylinders with compact nondegenerate triangular perpendicular bases, normalized total area within $2\varepsilon^4$ of $1/2-(13/6000)\varepsilon^2$, and total area strictly below $A_{\min}(K)/2$, area being two-dimensional Hausdorff measure, which agrees with Lebesgue area on planes. TriangularCovering.main certifies the same theorem in qualitative form, with an unspecified range $0<\varepsilon<\delta$ and an unspecified remainder constant in place of $1/2000$ and $2$. Corollary 1.2 is certified only for the regular tetrahedron $K$: a finite cover with compact triangular bases and directionwise relative area below $1/2$, a formal counterexample to the Bezdek--Khan conjecture. Its extension to every nondegenerate tetrahedron by affine invariance is not certified; the aggregate reaches every regular tetrahedron only with parallelogram or open bases. The aggregate's remaining conjuncts, and the release's OAI.RuledApproximation.fullMain, belong to companion manuscripts the library does not hold and are credited to no result here. The prose proofs remain unreviewed, and no refereed or independently reviewed version of the manuscript is known.

Companions: the release groups this manuscript with three others under the family "Cylinder coverings below the half-area bound": Slope-field perturbations of the two-cylinder covering (an alternate construction for the same two claims, with parallelogram rather than triangular bases according to the family page), Finite cylinder approximation of ruled sets and Finite triangular approximation of radial sweeps (approximation theorems whose abstracts present the tetrahedron cover as an application). None of the three is held in this library, so no card is linked.

Read status: claims checked for Theorem 1.1, Corollary 1.2, Lemma 2.1, Propositions 3.1, 4.1 and 6.1, Corollaries 3.2 and 4.2 and Theorem 5.1, read clause by clause in the TeX source (main.tex, sections/introduction.tex, sections/history.tex, sections/construction.tex, sections/coverage.tex, sections/area.tex, sections/angular-extensions.tex, sections/cubic-buffer.tex; the appendices sections/alternative-estimates.tex and sections/scaling-specializations.tex read for their statements and table) on 2026-10-07; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

• Section 1, Introduction (sections/introduction.tex with sections/history.tex; pp. 1--3). Defines a cylinder $C=B+\mathbb Ru$ with measurable base $B\subset u^\perp$ of finite area, the minimum projection area $A_{\min}(K)$ over two-dimensional subspaces, and the half-area question (1.1): must every finite cylinder cover of a convex body $K\subset\mathbb R^3$ have $\sum|B_i|\ge A_{\min}(K)/2$. The history paragraphs recall Bang's plank theorem and Ball's directionwise refinement for symmetric bodies, attribute the two-cylinder equality example for a regular tetrahedron and the half-area question to Bang's 1951 paper through Bezdek (2009, Problem 3.1) and Bezdek and Litvak (2009), define the directionwise relative area $\mathcal R(\mathcal C;K)=\sum|B_i|/|\pi_{u_i^\perp}K|$ (1.2), recall the Bezdek--Litvak bounds $\mathcal R\ge1/3$ in general and $\mathcal R\ge1$ for ellipsoids, the Bezdek--Khan "1-Codimensional Cylinder Covering Conjecture" $\mathcal R\ge1/2$ (2016, Conjecture 4.13), and Verreault's 2026 survey listing the question as open (Question 4.14). The tetrahedron $K=\{(x,y,\sqrt2\,t):0\le t\le1,\ |x|\le1-t,\ |y|\le t\}$ (1.4), a regular tetrahedron of edge $2$, is fixed. Theorem 1.1 (p. 2): $A_{\min}(K)=\sqrt2$ and for $0<\varepsilon\le1/2000$ a cover by $2\lceil2/\varepsilon^2\rceil$ cylinders with compact triangular perpendicular bases has normalized total area $1/2-(13/6000)\varepsilon^2+O(\varepsilon^4)$, remainder at most $2\varepsilon^4$, strictly below $1/2$. Corollary 1.2 (p. 2, proved on p. 3): every nondegenerate tetrahedron has a finite triangular-base cylinder cover with $\mathcal R<1/2$, by affine invariance of the directionwise ratios (cited to Bezdek--Litvak, Section 3, and proved directly). A paragraph "Idea of the proof" (p. 3) explains the mechanism: subdivide the two base triangles into narrow angular sectors, tilt each sector's axis, match adjacent sector sides in common planes, and enlarge each sector radially by order $\varepsilon^2$ so the two families still meet near $t=1/2$.
• Section 2, Geometry and the finite construction (sections/construction.tex; pp. 4--6). Lemma 2.1 (p. 4): $A_{\min}(K)=\sqrt2$, by a face-by-face form of Cauchy's projection formula (cited to Martini 1991 for the projection-body version), giving the shadow area $A(u)=\max(\sqrt2|u_x|,|u_z|)+\max(\sqrt2|u_y|,|u_z|)\ge\sqrt2$. Then the explicit parameters: $\eta=1/1000$, $n=\lceil2/\varepsilon^2\rceil$, $\Delta=2/n$, nodes $q_j=-1+j\Delta$, the boundary displacement $\phi(q)=(1-q^2)/4$, secant coefficients $\alpha_j=(1+q_jq_{j+1})/4$ and $\beta_j=(q_j+q_{j+1})/4$ (2.2) with the exact shared-boundary identity (2.3), the radial enlargement $d(q)=q^2(1+q^2)/16$, $M_j=\eta+\max_{[q_j,q_{j+1}]}d$, cutoff $T_j=1/2+\varepsilon^2M_j$ (2.4), and the $2n$ cylinders $P_j+\mathbb Rv_j$, $Q_j+\mathbb Rw_j$ (2.6)--(2.7) with intercept triangles in the planes $x=0$ and $y=0$ and axes $v_j=(1,\varepsilon\alpha_j,\sqrt2\varepsilon\beta_j)$, $w_j=(-\varepsilon\alpha_j,1,\sqrt2\varepsilon\beta_j)$. Figure 1 shows the zero-tilt cover and the matching of sides after the tilt.
• Section 3, Coverage of the entire tetrahedron (sections/coverage.tex; pp. 6--8). Proposition 3.1 (p. 6): for $0<\varepsilon\le\eta/2$ the $2n$ cylinders cover the closed $K$. The proof chooses a sector in each family by a first-crossing argument on a finite sequence that need not be monotone, then rules out a point that lies beyond both selected cutoffs: a radial budget identity (3.5) whose mixed first-order terms cancel, the bound $x^2\alpha_j+y^2\alpha_k\le d(p)+d(q)+(3/2)\varepsilon+\Delta/4$, and the margin $2\eta-(2+2\eta)\varepsilon-\varepsilon^2/4>0$. Corollary 3.2 (p. 8): the same coverage criterion with a variable margin $\eta\ge0$ and tilt $0<\tau<1$, sufficient when $2\eta-(2+2\eta)\tau-\tau^2/4\ge0$.
• Section 4, The strict area decrease (sections/area.tex; pp. 8--10). The exact finite area formula (4.1), $S(\varepsilon)/\sqrt2=\sum_j\Delta T_j^2[1+\varepsilon^2(\alpha_j^2+2\beta_j^2)]^{-1/2}$. Proposition 4.1 (p. 9): for $0<\varepsilon\le1$, $|S(\varepsilon)/\sqrt2-1/2-(2\eta-1/240)\varepsilon^2|\le2\varepsilon^4$, by a second-derivative bound $59/64$ for each sector's integrand, Taylor's theorem, and replacement of the Riemann sum by the integrals $\int_{-1}^1d=1/15$ and $\int_{-1}^1A=17/30$. Proof of Theorem 1.1 (p. 10): Lemma 2.1, Proposition 3.1 and Proposition 4.1 with $\eta=1/1000$ give $1/2-(13/6000)\varepsilon^2+2\varepsilon^4<1/2$ for $0<\varepsilon\le1/2000$. Corollary 4.2 (p. 10): the area estimate with a variable margin $0\le\eta\le1/8$, uniform and not assuming coverage.
• Section 5, Other caps and angular partitions (sections/angular-extensions.tex; pp. 10--13). Theorem 5.1 (p. 11), a general criterion: for partitions and coefficients satisfying $\beta_j=q/2+O(\tau^2)$, $\alpha_j=(1+q^2)/4+O(\tau^2)$, continuous compatible boundaries, and radial caps $R_j(q)=1/2+\tau^2(d(q)+\eta)+O(\tau^4)$ with fixed $0<\eta<1/480$, the $2m$ cylinders cover $K$ for all sufficiently small $\tau$ and have normalized area $1/2+(2\eta-1/240)\tau^2+O(\tau^4)$, with threshold and constants independent of $m$; constant caps give triangles. Section 5.2 lists caps satisfying the hypotheses (midpoint, maximum, continuous radial, squared-radius, and a bounded Cartesian aperture). Section 5.3 gives node-centered sectors with $\tau=1/k$ and exactly $2(k^2+1)$ triangular bases at normalized cost $1/2-(13/6000)\tau^2+O(\tau^4)$ for large $k$.
• Section 6, A vanishing radial margin (sections/cubic-buffer.tex; pp. 13--15). Proposition 6.1 (p. 14): for every $0<\tau\le1$, a cover by $2\lceil8/\tau^2\rceil$ triangular-base cylinders with cutoff $T_j=1/2+\tau^2d_j+(25/4)\tau^3$; its normalized area is $1/2-\tau^2/240+(25/2)\tau^3$ up to $2\tau^4$ for $0<\tau\le1/50$, strictly below $1/2$ for $0<\tau\le1/4000$, and equals $1/2-\tau^2/240+O(\tau^3)$ as $\tau\to0$. The manuscript distinguishes the coverage interval $0<\tau\le1$ from the interval $0<\tau\le1/4000$ on which it proves a saving.
• Appendix A, Alternative coverage and area calculations (sections/alternative-estimates.tex; pp. 15--17): direct arguments for squared-radius caps, a Cartesian discriminant, and node-centered sectors, presented as alternative explanations of cases already covered by Theorem 5.1, not as inputs to the main construction.
• Appendix B, Counts and formulas under changes of scale (sections/scaling-specializations.tex; pp. 17--19): a table of cylinder counts and normalized costs for eight cap and scale choices, obtained by substitution into Theorem 5.1; the edge-$\sqrt2$ and edge-one dictionaries; Remark B.1 (p. 19), the edge-four form of Proposition 6.1 with $2\lceil2/\varepsilon^2\rceil$ cylinders covering for $0<\varepsilon\le1/2$ and total area strictly below $A_{\min}/2$ for $0<\varepsilon\le1/8000$.
• References (pp. 19--20): Bang 1951; Bezdek 2009 (arXiv:0903.4637v1); Bezdek and Litvak 2009 (J. Geom. Anal.); Bezdek and Khan 2016 (arXiv:1602.06040v2); Verreault 2026 (Bull. London Math. Soc.); Ball 1991; Martini 1991.

External inputs the proofs rest on: Cauchy's projection formula in its face-area-vector form (Martini 1991, cited; the manuscript proves the needed case for this tetrahedron directly) and the affine invariance of the directionwise ratios (Bezdek and Litvak 2009, cited; proved directly in Corollary 1.2). Everything else is elementary calculus and inequalities written out in the text. The manuscript flags nothing as numerical, computer-assisted or conditional; its constants ($1/1000$, $1/2000$, $13/6000$, $59/64$, $1/15$, $17/30$) are stated as exact. The release holds no verification/ folder for this manuscript.

Bears on

The manuscript names no Erdős problem; no problem page is linked. Its target is the half-area cylinder-covering question (Bezdek 2009, Problem 3.1; Verreault 2026, Question 4.14) and the Bezdek--Khan directionwise conjecture, neither of which is an Erdős problem in this corpus.

• Bezdek and Litvak 2016, packing convex bodies by cylinders: that card frames its paper by Bang's question on the base areas of cylinders covering a three-dimensional convex body and records Theorem 3.1 as the $r$-fold covering lower bound; that card's source (Bezdek and Litvak 2016, equation (1)) states the constant, which for 1-codimensional cylinders in $\mathbb R^3$ reads $\sum\operatorname{crv}_K(C_i)\ge1/3$ with $\operatorname{crv}_K$ taken relative to the projection area, as the manuscript's $\mathcal R$ is; this manuscript gives a cover of a regular tetrahedron with directionwise relative area $\mathcal R<1/2$, which answers Bang's half-area question negatively and shows the $1/3$ bound cannot be raised to $1/2$ in dimension three, while contradicting nothing the card states. That cover is formally verified here, as recorded above; the prose proof is unreviewed. The one problem page that card links, Problem 1121 (a circle-covering statement), is not touched by this manuscript, and its status rests on its own acceptance evidence.