RFW-Orthotope: Self‑Similar Geometry and Vertex‑Tensor Structure

This paper investigates a family of high‑dimensional orthotopes obeying a newly formulated condition termed bisective self‑similarity, which generalizes the well‑known self‑bisecting property of the two‑dimensional $\sqrt{2}:1$ ISO‑A rectangle to $N$‑dimensional Euclidean space. We prove a uniqueness classification theorem: up to translation, orthogonal transformation and global isotropic scaling, the RFW orthotope with edge‑length geometric‑progression $2^{(N-1)/N}:2^{(N-2)/N}:\cdots:1$ is the only non‑trivial orthotope satisfying this bisective self‑similarity.For the three‑dimensional special case the edge‑length proportions read $\sqrt[3]{4}:\sqrt[3]{2}:1$. After bisection at the midpoint of its longest edge, each resulting sub‑block becomes congruent to the original body upon edge‑order permutation and global rescaling. We derive closed‑form expressions for its volume $V_{N}=2^{(N-1)/2}$ and analyse the volume ratio $R_{N}=\pi^{N / 2}/\big[2^{N}\Gamma(N/2+1)\big]$ of the maximal axis‑aligned inscribed ellipsoid relative to the orthotope. This ratio, universal for all orthotopes, exhibits super‑exponential decay in high dimensions; nevertheless, the inscribed ellipsoid occupies a non‑negligible volume fraction for moderate $N$, and we avoid the unwarranted inference of shell‑dominated volume concentration. We further study combinatorial skeleton properties and introduce a sector decomposition for vertex‑corner domains. Sector volumes are proven to increase strictly monotonically with corresponding edge lengths, giving rise to a dimensionless bisection‑invariant anisotropy index that quantifies corner anisotropy. In three dimensions we construct a traceless second‑moment tensor defined over each vertex‑corner domain, which serves as a local shape invariant complementary to scalar angle defects from Regge calculus. All derivations are performed within flat Euclidean space using volume integration, moment calculation and combinatorial counting, without imposing physical assumptions or dynamical postulates. The RFW orthotope provides a fully analytically tractable one‑parameter test case for high‑dimensional convex geometry.