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

This paper systematically studies a class of high‑dimensional hyperrectangles with strict self‑similar properties, termed the RFW‑Orthotope. This construction generalizes the self‑bisecting property of the two‑dimensional $\sqrt{2}:1$ rectangle (ISO‑A paper format) to $N$‑dimensional Euclidean space, with side‑length ratios \[2^{(N-1) / N}: 2^{(N-2) / N}: \cdots:2^{1/ N}: 1 .\] In three dimensions this yields the distinctive ratio $\sqrt[3]{4}:\sqrt[3]{2}:1$.We prove that the self‑similar bisection condition uniquely enforces geometric‑progression side lengths within the family of hyperrectangles; the RFW‑Orthotope is the unique non‑trivial solution under this condition. When bisected through the mid‑point of its longest edge, each sub‑block can be brought into congruence with the original geometry via appropriate rotation, edge‑reordering and global rescaling. Within the $N$‑dimensional setting, we rigorously derive the volume formula $V_{N}=2^{(N-1) / 2}$ , together with the volume ratio $R_{N}=\pi^{N / 2}\big/\big[2^{N}\Gamma(N/2+1)\big]$ between the RFW‑Orthotope and its maximal axis‑aligned inscribed ellipsoid. We show that $R_{N}$ decays super‑exponentially for $N \ge 9$ , yielding a shell‑dominated, finely spiked geometric structure. We further investigate its combinatorial skeleton: vertex and edge counting, combinatorial explosion induced by recursive bisection, vertex‑associated pyramids and sector decomposition. It is proved that sector volumes increase strictly monotonically with the corresponding edge lengths, giving rise to strongly anisotropic geometric behaviour. In three dimensions, each vertex cone carries a traceless second‑moment tensor whose components are fully determined by its three edge lengths. We demonstrate that the three‑dimensional RFW‑Orthotope is the unique non‑trivial discrete scale‑invariant eigen‑mode for this tensor under recursive‑bisection while preserving principal‑axis directions. All results are derived within flat Euclidean space using volume measures, moment integrals and combinatorial counting; no physical assumptions or dynamical postulates are introduced. Supplementary heuristic discrete‑tensor constructions induced by recursive bisection, intended as potential motivation for future follow‑up investigations, are collected in Appendix C.