OAI.CylinderCovering.fourPaperMain

The theorem states that seven formally defined propositions about covering sets in R^3 by cylinders all hold simultaneously. A cylinder has a unit axis u and a base lying in the plane orthogonal to u, and it covers the points of the base translated by all real multiples of u; areas are two-dimensional measures, and the shadow area of a set along u is the area of its orthogonal projection onto that plane. (1) AngularExplicitStatement: for the solid K of points (x,y,√2 t) with 0≤t≤1, |x|≤1−t, |y|≤t, the minimal shadow area over unit directions equals √2; for every 0<ε≤1 there is a family of 2⌈2/ε²⌉ triangular cylinders with prescribed tilted axes and projected-triangle bases whose total area divided by √2 equals an explicit Riemann-type sum and lies within 2ε⁴ of 1/2−(13/6000)ε²; and for every 0<ε≤1/2000 such an explicit family covers K, has the same estimate, total base area below half the minimal shadow area, and relative cost (the sum of base area over the shadow area of K along each axis) below 1/2. (2) HyperbolicStatement: if D⊂R² is compact, v is C¹ near D, L>0, and at each p∈D the derivative of v has eigenvalues λ and −λ with 0<Lλ<1, then for every e>0 the ruled set {(p+r v(p), r): p∈D, |r|≤L} is covered by finitely many parallelogram-based cylinders of total base area at most ∫_D 1/√(1+|v|²)+e. (3) NilpotentC1Statement: if D is compact, G is C¹ near D and its derivative squares to zero at every point of D, then for all h,eps>0 the corresponding physically scaled ruled set is covered by finitely many auxiliary cylinders over square tiles T_i (similar images of the unit square) with directions g_i, with Σ area(T_i)·weight(g_i) at most ∫_D weight(G)+eps, where weight(w)=1/√(1+w₀²+h²w₁²); moreover, for every real a there are two families of parallelogram cylinders whose carriers are the images of these auxiliary cylinders under two explicit coordinate maps and whose areas equal h·area(T_i)·weight(g_i). (4) RegularParallelogramCounterexampleStatement: for every regular tetrahedron K there is a finite family of parallelogram-based cylinders covering K with total base area below half the minimal shadow area of K and relative cost below 1/2. (5) RegularOpenCounterexampleStatement: the same conclusion holds with bases that are open, bounded and measurable rather than parallelograms. (6) RadialSweep.AlignedStatement: for sweep data consisting of a unit vector e, independent vectors a,b orthogonal to e, a base point p₀ orthogonal to e, an interval α<β, a length L>0, a continuous nonnegative radius R, and a continuous velocity w orthogonal to e whose derivative is λ(t)(a+tb) with λ continuous, the swept set {p₀+ρ(a+tb)+s(e+w(t)): α≤t≤β, 0≤ρ≤R(t), |s|≤L} and any tags in the N-cell subdivisions of [α,β], there exist triangular cylinders for each N>0 with axes along e+w(tag), prescribed projected frozen-triangle bases and prescribed areas, covering the sweep, whose total area tends to the weighted area κ∫R²/(2√(1+|w|²))dt, where κ is the area of the parallelogram spanned by a and b. (7) RadialSweep.ActualSweepStatement: for the same data and tags, for each N>0 the cylinders over the projections of the local sweeps of the cells have unit axes and compact, measurable bases of finite area orthogonal to the axes and contained in the frozen bases, they cover the sweep, the total base area is eventually at most the weighted area plus any ε>0, and its limsup is at most the weighted area. The proof is admitted, not verified.