The Stellated Tetrahedron 𝒫11/20 is not Rupert: Cylinder Theorems and a Local-to-Global Method for Non-Rupertness

Jeff Pickhardt

Loading document…

The Stellated Tetrahedron 𝒫11/20 is not Rupert: Cylinder Theorems and a Local-to-Global Method for Non-Rupertness

Abstract

A convex polyhedron 𝑄 ⊂ ℝ3 is Rupert if a straight hole can be cut through it large enough to pass a second copy of 𝑄, or equivalently, if one shadow 𝜋(𝑅1𝑄) fits strictly inside another 𝜋(𝑅_2𝑄) under some rotations 𝑅_1, 𝑅_2 and a translation 𝑡 ∈ ℝ_2. Steininger and Yurkevich exhibited the first provably non-Rupert convex polyhedron, the 90-vertex Noperthedron, but their local theorem near diagonal configurations relies essentially on central symmetry, which eliminates the translation. Zeng conjectured that the 8-vertex stellated tetrahedron 𝒫_11/20, which is not centrally symmetric, is not Rupert, and identified the missing ingredient exactly: a local theorem that accounts for the translation term. We develop that theory and establish its central certificates. On the analytic side we prove cylinder theorems that exclude containment for a neighborhood of inner rotations and every translation at once; a Marginal Reduction Theorem exhibiting twelve diagonal copies and twelve reflection sheets of the marginal variety of 𝒫𝑎, glued by an exact involution (these organize the local theory; exhaustiveness of the equal-shadow locus is not claimed and not needed — §10, contribution 2); the exact first-order structure at the six critical directions; and a Global Assembly Theorem reducing non-Rupertness of 𝒫11/20 to four finite, precisely specified certified inputs. The assembly reduction is established here: the exact order-24 outer symmetry and the unconditional translate- exclusion of the whole marginal orbit are proved in the text (§16.1–16.3); the completeness-critical transport (Lemma GA-3), boundary handoff (Lemma HO), and straddle-termination lemmas are proved in Appendix A. The linear-order box certificate (SB-box) at the core of the local theorem required two new soundness lemmas that circumvent the provable 1/𝛿 ill-conditioning of the flag matrix at the marginal set, which defeats the standard inversion-based interval certificate. All four certified inputs are verified in outward-rounded interval arithmetic — the (SB-box) box certificate over its full 𝛿-continuum by a direct interval-𝛿 sweep — the (SF) order-4 Taylor-tail constant 𝐶4 ≤ 53 < 60 is proved, and the interface inequalities that glue the inputs are gap-free. We conclude that 𝒫_11/20 is not Rupert. As is standard for a proof of this kind, the finite certificate computations are carried out by public, independently replayable scripts that re-check every cell in outward-rounded interval arithmetic.

File Details

v2 paper.pdf