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Math-2628 · mathematics · Geometry

Triangular-cylinder approximation of aligned radial sweeps

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Let $P\subset\mathbb R^3$ be a two-dimensional vector plane with unit normal $e$, and take linearly independent $a,b\in P$. Let $J=[\alpha,\beta]$ with $\alpha<\beta$, let $R:J\to[0,\infty)$ be continuous, let $w:J\to P$ be continuously differentiable, and fix $p_0\in P$ and $L>0$. Suppose $w'(t)=\lambda(t)(a+tb)$ for a continuous real function $\lambda$. Define \[ S=\{p_0+\rho(a+tb)+s(e+w(t)):t\in J,\ 0\le\rho\le R(t),\ |s|\le L\}. \] For every finite tagged partition of $J$, can one cover the part of $S$ over each partition interval by a compact cylinder having a nondegenerate triangular base and axis $e+w(\text{tag})$, with the sum of the base areas perpendicular to their respective axes converging to \[ |\det_P(a,b)|\int_\alpha^\beta \frac{R(t)^2}{2\sqrt{1+\lVert w(t)\rVert^2}}\,dt \] uniformly over all partitions and tags as the mesh tends to zero?

Short English statement

Do aligned radial sweeps admit these finite triangular-cylinder covers, with the prescribed uniform limit of total perpendicular base area?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2628-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2628

The source uses the same planar data, continuous radius, continuously differentiable tilt and alignment equation as the question. It permits every finite partition and every selection of tags, supplies one compact nondegenerate triangular-base cylinder for each interval, and asserts uniform convergence of the sum of perpendicular base areas to the displayed integral as the mesh tends to zero. These are all the requested assertions, so the match is full.

Triangular-cylinder approximation of aligned radial sweeps

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 radial segment sweeps have finite triangular-base cylinder realizations whose area converges to the weighted parameter cost, including a tetrahedral cover below the half-area bound.

Mathematical statement of the claimed result

Manuscript 1.
Choose a Euclidean plane $P$ with unit normal $e$, independent $a,b\in P$, $J=[\alpha,\beta]$ with $\alpha<\beta$, continuous $R\ge0$, a $C^1$ map $w:J\to P$, $p_0\in P$ and $L>0$. Set \[ \mathcal S=\{p_0+\rho(a+tb)+s(e+w(t)):t\in J,0\le\rho\le R(t),|s|\le L\}, \quad\mathcal A=|\det_P(a,b)|\int_{\alpha}^{\beta}\frac{R(t)^2}{2\sqrt{1+\|w(t)\|^2}}\,dt. \] If $w'(t)=\lambda(t)(a+tb)$ for continuous real $\lambda$, every tagged finite partition gives one covering cylinder per interval, with frozen axis $e+w(c_i)$ and nondegenerate compact triangular perpendicular base, whose total base area tends uniformly to $\mathcal A$ as the mesh tends to zero. Frozen projections of the actual interval sweeps give the same limit. The tetrahedral consequence claims, for each regular tetrahedron $K$ of positive edge length, a finite triangular-base cover with \[ \sum_i|B_i|<\tfrac12 A_{\min}(K),\qquad \sum_i\frac{|B_i|}{|\pi_{u_i^{\perp}}K|}<\tfrac12. \]

Outline of the argument

For manuscript 1 (family 100; 2026-09-27), one triangular enclosure is assigned to each interval of a tagged partition. Complementary shortened beams leave a strip covered by horizontal cylinders; exact overlap and strip calculations cancel the first-order cost and leave a strict second-order saving.

Formal verification

For manuscript 1 (family 100; 2026-09-27), OpenAI lists Lean statements for this assertion: radial segment sweeps have finite triangular-base cylinder realizations whose area converges to the weighted parameter cost, including a tetrahedral cover below the half-area bound.

This coverage does not establish a formal proof of the complete problem as stated here.

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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