OpenAI claim summary for Math-2645
mathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 100; 2026-09-27).
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
Finite cylinder approximation of ruled sets
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 100; 2026-09-27), the asserted result is that compact ruled segment families under the stated hyperbolic or square-zero differential conditions admit finite-cylinder approximations within any positive error of their integral projection cost.Mathematical statement of the claimed result
Manuscript 1.
For compact $D\subset\mathbb R^2$, a $C^1$ vector field $V$ on a neighborhood, and $L,h>0$, define \[ E(D,V,L)=\{(s,p+sV(p)):p\in D,\ |s|\le L\},\quad J_h(g)=(1+g_1^2+h^2g_2^2)^{-1/2}. \] Assume $\operatorname{tr}DV(p)=0$ and $-L^{-2}<\det DV(p)\le0$ for every $p\in D$. For every $\varepsilon>0$, the claimed finite compact nondegenerate square tiles $T_i$ and velocities $g_i$ satisfy \[ E(D,V,L)\subseteq\bigcup_i\{(s,\xi):\xi\in T_i+sg_i\},\qquad \sum_i|T_i|J_h(g_i)\le\int_DJ_h(V(p))\,dp+\varepsilon. \] After $F_h(s,\xi)=(s,\xi_1,h\xi_2)$, the perpendicular bases are compact parallelograms of total area $h\sum_i|T_i|J_h(g_i)$. No measure-zero-boundary assumption or smallness bound on $DV$ is imposed.Outline of the argument
For manuscript 1 (family 100; 2026-09-27), each coarse square freezes the field to an affine map, while a finer square lattice captures nonlinear target points with only a thin collar. Cost is summed with multiplicity; logarithmic product cells provide an alternative when the parameter boundary has zero area.Formal verification
For manuscript 1 (family 100; 2026-09-27), OpenAI lists Lean statements for this assertion: compact ruled segment families under the stated hyperbolic or square-zero differential conditions admit finite-cylinder approximations within any positive error of their integral projection cost.The central assertion is within the documented formal scope; the additional applications retain their own scope limits.
The linked family Comparator checks are CylinderCovering.lean, RuledCovering.lean, TriangularCovering.lean.
Their named declarations include OAI.CylinderCovering.fourPaperMain, OAI.RuledApproximation.fullMain, OAI.TriangularCovering.main.
This site has not executed the Lean code. Described statement coverage is not an independent proof check.
Status caveat
The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.Sources
- OpenAI, Finite cylinder approximation of ruled sets, 2026-09-27; citation key
OAI:Finite-cylinder-approximation-of-ruled-sets-September-27-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: b:approximation, b:main-condition, b:main-cover. - Family 100: Lean scope.
- Comparator CylinderCovering.json.
- Comparator CylinderCovering.lean.
- Comparator RuledCovering.json.
- Comparator RuledCovering.lean.
- Comparator TriangularCovering.json.
- Comparator TriangularCovering.lean.
- OpenAI catalogue entry.