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Math-2645 · mathematics · Geometry

Finite cylinder approximation of ruled sets

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let $D\subset\mathbb{R}^2$ be compact, let $V$ be $C^1$ near $D$, and let $L,h>0$. If $\operatorname{tr}DV=0$ and $-L^{-2}<\det DV\leq0$ on $D$, can \[ E=\{(s,p+sV(p)):p\in D,\ |s|\leq L\} \] be covered, for each $\varepsilon>0$, by finitely many cylinders \[ \operatorname{Cyl}(T_i,g_i)=\{(s,\xi):\xi\in T_i+sg_i\}, \] with compact nondegenerate square tiles $T_i$, such that \[ \sum_i\frac{|T_i|}{\sqrt{1+g_{i,1}^2+h^2g_{i,2}^2}} \leq\int_D(1+V_1^2+h^2V_2^2)^{-1/2}+\varepsilon? \] Under $(s,\xi_1,\xi_2)\mapsto(s,\xi_1,h\xi_2)$, this is physical perpendicular-base cost divided by $h$.

Short English statement

For every compact $D\subset\mathbb{R}^2$, every $C^1$ field $V$ near $D$, and $L,h>0$ satisfying $\operatorname{tr}DV=0$ and $-L^{-2}<\det DV\leq0$ on $D$, can $\{(s,p+sV(p)):p\in D,|s|\leq L\}$, for each $\varepsilon>0$, be covered by finitely many $\operatorname{Cyl}(T_i,g_i)=\{(s,\xi):\xi\in T_i+sg_i\}$ with compact nondegenerate square tiles $T_i$ and cost $\sum_i|T_i|/\sqrt{1+g_{i,1}^2+h^2g_{i,2}^2}\leq\int_D(1+V_1^2+h^2V_2^2)^{-1/2}+\varepsilon$? This cost is the physical perpendicular-base cost divided by $h$ under $(s,\xi_1,\xi_2)\mapsto(s,\xi_1,h\xi_2)$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2645-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2645

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).

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

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